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Home > Accenture Preparation Dashboard > Accenture Critical Reasoning Questions and Answers 2025

Accenture Critical Reasoning Questions and Answers 2025

Accenture critical reasoning and problem solving questions 2025.

This page will provide you with a variety of questions to help you prepare for the Accenture 2025 and Accenture Critical Reasoning in particular. The questions on this page are also from the most recent pattern, which will help you understand the newly introduced pattern of Accenture 2025.

• This page has Accenture Critical Reasoning and Problem Solving Questions with Solutions as well as practice questions based on Accenture Questions 2025.

Note: Critical Reasoning and Problem-Solving Questions in Accenture falls under the banner of the Written Round.

Accenture Critical Reasoning and Problem Solving Test 2025

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Accenture Critical Reasoning and Problem Solving Curriculum 2025

Accenture critical reasoning questions with answers 2025.

The test assesses your ability to critically think and develop logical conclusions on the basis of textual material. In employment evaluations in the engineering industry, critical thinking exams are commonly used to evaluate the analytical critical thinking abilities of the candidate.

You will find the most recent Accenture Critical Reasoning and Problem Solving Questions 2025 as well as practice questions based on these questions right here. The following chapters will be covered for the Accenture Critical Thinking Questions 2025

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July 26, 2024

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Classroom Q&A

With larry ferlazzo.

In this EdWeek blog, an experiment in knowledge-gathering, Ferlazzo will address readers’ questions on classroom management, ELL instruction, lesson planning, and other issues facing teachers. Send your questions to [email protected]. Read more from this blog.

Eight Instructional Strategies for Promoting Critical Thinking

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(This is the first post in a three-part series.)

The new question-of-the-week is:

What is critical thinking and how can we integrate it into the classroom?

This three-part series will explore what critical thinking is, if it can be specifically taught and, if so, how can teachers do so in their classrooms.

Today’s guests are Dara Laws Savage, Patrick Brown, Meg Riordan, Ph.D., and Dr. PJ Caposey. Dara, Patrick, and Meg were also guests on my 10-minute BAM! Radio Show . You can also find a list of, and links to, previous shows here.

You might also be interested in The Best Resources On Teaching & Learning Critical Thinking In The Classroom .

Current Events

Dara Laws Savage is an English teacher at the Early College High School at Delaware State University, where she serves as a teacher and instructional coach and lead mentor. Dara has been teaching for 25 years (career preparation, English, photography, yearbook, newspaper, and graphic design) and has presented nationally on project-based learning and technology integration:

There is so much going on right now and there is an overload of information for us to process. Did you ever stop to think how our students are processing current events? They see news feeds, hear news reports, and scan photos and posts, but are they truly thinking about what they are hearing and seeing?

I tell my students that my job is not to give them answers but to teach them how to think about what they read and hear. So what is critical thinking and how can we integrate it into the classroom? There are just as many definitions of critical thinking as there are people trying to define it. However, the Critical Think Consortium focuses on the tools to create a thinking-based classroom rather than a definition: “Shape the climate to support thinking, create opportunities for thinking, build capacity to think, provide guidance to inform thinking.” Using these four criteria and pairing them with current events, teachers easily create learning spaces that thrive on thinking and keep students engaged.

One successful technique I use is the FIRE Write. Students are given a quote, a paragraph, an excerpt, or a photo from the headlines. Students are asked to F ocus and respond to the selection for three minutes. Next, students are asked to I dentify a phrase or section of the photo and write for two minutes. Third, students are asked to R eframe their response around a specific word, phrase, or section within their previous selection. Finally, students E xchange their thoughts with a classmate. Within the exchange, students also talk about how the selection connects to what we are covering in class.

There was a controversial Pepsi ad in 2017 involving Kylie Jenner and a protest with a police presence. The imagery in the photo was strikingly similar to a photo that went viral with a young lady standing opposite a police line. Using that image from a current event engaged my students and gave them the opportunity to critically think about events of the time.

Here are the two photos and a student response:

F - Focus on both photos and respond for three minutes

In the first picture, you see a strong and courageous black female, bravely standing in front of two officers in protest. She is risking her life to do so. Iesha Evans is simply proving to the world she does NOT mean less because she is black … and yet officers are there to stop her. She did not step down. In the picture below, you see Kendall Jenner handing a police officer a Pepsi. Maybe this wouldn’t be a big deal, except this was Pepsi’s weak, pathetic, and outrageous excuse of a commercial that belittles the whole movement of people fighting for their lives.

I - Identify a word or phrase, underline it, then write about it for two minutes

A white, privileged female in place of a fighting black woman was asking for trouble. A struggle we are continuously fighting every day, and they make a mockery of it. “I know what will work! Here Mr. Police Officer! Drink some Pepsi!” As if. Pepsi made a fool of themselves, and now their already dwindling fan base continues to ever shrink smaller.

R - Reframe your thoughts by choosing a different word, then write about that for one minute

You don’t know privilege until it’s gone. You don’t know privilege while it’s there—but you can and will be made accountable and aware. Don’t use it for evil. You are not stupid. Use it to do something. Kendall could’ve NOT done the commercial. Kendall could’ve released another commercial standing behind a black woman. Anything!

Exchange - Remember to discuss how this connects to our school song project and our previous discussions?

This connects two ways - 1) We want to convey a strong message. Be powerful. Show who we are. And Pepsi definitely tried. … Which leads to the second connection. 2) Not mess up and offend anyone, as had the one alma mater had been linked to black minstrels. We want to be amazing, but we have to be smart and careful and make sure we include everyone who goes to our school and everyone who may go to our school.

As a final step, students read and annotate the full article and compare it to their initial response.

Using current events and critical-thinking strategies like FIRE writing helps create a learning space where thinking is the goal rather than a score on a multiple-choice assessment. Critical-thinking skills can cross over to any of students’ other courses and into life outside the classroom. After all, we as teachers want to help the whole student be successful, and critical thinking is an important part of navigating life after they leave our classrooms.

‘Before-Explore-Explain’

Patrick Brown is the executive director of STEM and CTE for the Fort Zumwalt school district in Missouri and an experienced educator and author :

Planning for critical thinking focuses on teaching the most crucial science concepts, practices, and logical-thinking skills as well as the best use of instructional time. One way to ensure that lessons maintain a focus on critical thinking is to focus on the instructional sequence used to teach.

Explore-before-explain teaching is all about promoting critical thinking for learners to better prepare students for the reality of their world. What having an explore-before-explain mindset means is that in our planning, we prioritize giving students firsthand experiences with data, allow students to construct evidence-based claims that focus on conceptual understanding, and challenge students to discuss and think about the why behind phenomena.

Just think of the critical thinking that has to occur for students to construct a scientific claim. 1) They need the opportunity to collect data, analyze it, and determine how to make sense of what the data may mean. 2) With data in hand, students can begin thinking about the validity and reliability of their experience and information collected. 3) They can consider what differences, if any, they might have if they completed the investigation again. 4) They can scrutinize outlying data points for they may be an artifact of a true difference that merits further exploration of a misstep in the procedure, measuring device, or measurement. All of these intellectual activities help them form more robust understanding and are evidence of their critical thinking.

In explore-before-explain teaching, all of these hard critical-thinking tasks come before teacher explanations of content. Whether we use discovery experiences, problem-based learning, and or inquiry-based activities, strategies that are geared toward helping students construct understanding promote critical thinking because students learn content by doing the practices valued in the field to generate knowledge.

An Issue of Equity

Meg Riordan, Ph.D., is the chief learning officer at The Possible Project, an out-of-school program that collaborates with youth to build entrepreneurial skills and mindsets and provides pathways to careers and long-term economic prosperity. She has been in the field of education for over 25 years as a middle and high school teacher, school coach, college professor, regional director of N.Y.C. Outward Bound Schools, and director of external research with EL Education:

Although critical thinking often defies straightforward definition, most in the education field agree it consists of several components: reasoning, problem-solving, and decisionmaking, plus analysis and evaluation of information, such that multiple sides of an issue can be explored. It also includes dispositions and “the willingness to apply critical-thinking principles, rather than fall back on existing unexamined beliefs, or simply believe what you’re told by authority figures.”

Despite variation in definitions, critical thinking is nonetheless promoted as an essential outcome of students’ learning—we want to see students and adults demonstrate it across all fields, professions, and in their personal lives. Yet there is simultaneously a rationing of opportunities in schools for students of color, students from under-resourced communities, and other historically marginalized groups to deeply learn and practice critical thinking.

For example, many of our most underserved students often spend class time filling out worksheets, promoting high compliance but low engagement, inquiry, critical thinking, or creation of new ideas. At a time in our world when college and careers are critical for participation in society and the global, knowledge-based economy, far too many students struggle within classrooms and schools that reinforce low-expectations and inequity.

If educators aim to prepare all students for an ever-evolving marketplace and develop skills that will be valued no matter what tomorrow’s jobs are, then we must move critical thinking to the forefront of classroom experiences. And educators must design learning to cultivate it.

So, what does that really look like?

Unpack and define critical thinking

To understand critical thinking, educators need to first unpack and define its components. What exactly are we looking for when we speak about reasoning or exploring multiple perspectives on an issue? How does problem-solving show up in English, math, science, art, or other disciplines—and how is it assessed? At Two Rivers, an EL Education school, the faculty identified five constructs of critical thinking, defined each, and created rubrics to generate a shared picture of quality for teachers and students. The rubrics were then adapted across grade levels to indicate students’ learning progressions.

At Avenues World School, critical thinking is one of the Avenues World Elements and is an enduring outcome embedded in students’ early experiences through 12th grade. For instance, a kindergarten student may be expected to “identify cause and effect in familiar contexts,” while an 8th grader should demonstrate the ability to “seek out sufficient evidence before accepting a claim as true,” “identify bias in claims and evidence,” and “reconsider strongly held points of view in light of new evidence.”

When faculty and students embrace a common vision of what critical thinking looks and sounds like and how it is assessed, educators can then explicitly design learning experiences that call for students to employ critical-thinking skills. This kind of work must occur across all schools and programs, especially those serving large numbers of students of color. As Linda Darling-Hammond asserts , “Schools that serve large numbers of students of color are least likely to offer the kind of curriculum needed to ... help students attain the [critical-thinking] skills needed in a knowledge work economy. ”

So, what can it look like to create those kinds of learning experiences?

Designing experiences for critical thinking

After defining a shared understanding of “what” critical thinking is and “how” it shows up across multiple disciplines and grade levels, it is essential to create learning experiences that impel students to cultivate, practice, and apply these skills. There are several levers that offer pathways for teachers to promote critical thinking in lessons:

1.Choose Compelling Topics: Keep it relevant

A key Common Core State Standard asks for students to “write arguments to support claims in an analysis of substantive topics or texts using valid reasoning and relevant and sufficient evidence.” That might not sound exciting or culturally relevant. But a learning experience designed for a 12th grade humanities class engaged learners in a compelling topic— policing in America —to analyze and evaluate multiple texts (including primary sources) and share the reasoning for their perspectives through discussion and writing. Students grappled with ideas and their beliefs and employed deep critical-thinking skills to develop arguments for their claims. Embedding critical-thinking skills in curriculum that students care about and connect with can ignite powerful learning experiences.

2. Make Local Connections: Keep it real

At The Possible Project , an out-of-school-time program designed to promote entrepreneurial skills and mindsets, students in a recent summer online program (modified from in-person due to COVID-19) explored the impact of COVID-19 on their communities and local BIPOC-owned businesses. They learned interviewing skills through a partnership with Everyday Boston , conducted virtual interviews with entrepreneurs, evaluated information from their interviews and local data, and examined their previously held beliefs. They created blog posts and videos to reflect on their learning and consider how their mindsets had changed as a result of the experience. In this way, we can design powerful community-based learning and invite students into productive struggle with multiple perspectives.

3. Create Authentic Projects: Keep it rigorous

At Big Picture Learning schools, students engage in internship-based learning experiences as a central part of their schooling. Their school-based adviser and internship-based mentor support them in developing real-world projects that promote deeper learning and critical-thinking skills. Such authentic experiences teach “young people to be thinkers, to be curious, to get from curiosity to creation … and it helps students design a learning experience that answers their questions, [providing an] opportunity to communicate it to a larger audience—a major indicator of postsecondary success.” Even in a remote environment, we can design projects that ask more of students than rote memorization and that spark critical thinking.

Our call to action is this: As educators, we need to make opportunities for critical thinking available not only to the affluent or those fortunate enough to be placed in advanced courses. The tools are available, let’s use them. Let’s interrogate our current curriculum and design learning experiences that engage all students in real, relevant, and rigorous experiences that require critical thinking and prepare them for promising postsecondary pathways.

Critical Thinking & Student Engagement

Dr. PJ Caposey is an award-winning educator, keynote speaker, consultant, and author of seven books who currently serves as the superintendent of schools for the award-winning Meridian CUSD 223 in northwest Illinois. You can find PJ on most social-media platforms as MCUSDSupe:

When I start my keynote on student engagement, I invite two people up on stage and give them each five paper balls to shoot at a garbage can also conveniently placed on stage. Contestant One shoots their shot, and the audience gives approval. Four out of 5 is a heckuva score. Then just before Contestant Two shoots, I blindfold them and start moving the garbage can back and forth. I usually try to ensure that they can at least make one of their shots. Nobody is successful in this unfair environment.

I thank them and send them back to their seats and then explain that this little activity was akin to student engagement. While we all know we want student engagement, we are shooting at different targets. More importantly, for teachers, it is near impossible for them to hit a target that is moving and that they cannot see.

Within the world of education and particularly as educational leaders, we have failed to simplify what student engagement looks like, and it is impossible to define or articulate what student engagement looks like if we cannot clearly articulate what critical thinking is and looks like in a classroom. Because, simply, without critical thought, there is no engagement.

The good news here is that critical thought has been defined and placed into taxonomies for decades already. This is not something new and not something that needs to be redefined. I am a Bloom’s person, but there is nothing wrong with DOK or some of the other taxonomies, either. To be precise, I am a huge fan of Daggett’s Rigor and Relevance Framework. I have used that as a core element of my practice for years, and it has shaped who I am as an instructional leader.

So, in order to explain critical thought, a teacher or a leader must familiarize themselves with these tried and true taxonomies. Easy, right? Yes, sort of. The issue is not understanding what critical thought is; it is the ability to integrate it into the classrooms. In order to do so, there are a four key steps every educator must take.

• Integrating critical thought/rigor into a lesson does not happen by chance, it happens by design. Planning for critical thought and engagement is much different from planning for a traditional lesson. In order to plan for kids to think critically, you have to provide a base of knowledge and excellent prompts to allow them to explore their own thinking in order to analyze, evaluate, or synthesize information.
• SIDE NOTE – Bloom’s verbs are a great way to start when writing objectives, but true planning will take you deeper than this.

QUESTIONING

• If the questions and prompts given in a classroom have correct answers or if the teacher ends up answering their own questions, the lesson will lack critical thought and rigor.
• Script five questions forcing higher-order thought prior to every lesson. Experienced teachers may not feel they need this, but it helps to create an effective habit.
• If lessons are rigorous and assessments are not, students will do well on their assessments, and that may not be an accurate representation of the knowledge and skills they have mastered. If lessons are easy and assessments are rigorous, the exact opposite will happen. When deciding to increase critical thought, it must happen in all three phases of the game: planning, instruction, and assessment.

TALK TIME / CONTROL

• To increase rigor, the teacher must DO LESS. This feels counterintuitive but is accurate. Rigorous lessons involving tons of critical thought must allow for students to work on their own, collaborate with peers, and connect their ideas. This cannot happen in a silent room except for the teacher talking. In order to increase rigor, decrease talk time and become comfortable with less control. Asking questions and giving prompts that lead to no true correct answer also means less control. This is a tough ask for some teachers. Explained differently, if you assign one assignment and get 30 very similar products, you have most likely assigned a low-rigor recipe. If you assign one assignment and get multiple varied products, then the students have had a chance to think deeply, and you have successfully integrated critical thought into your classroom.

Thanks to Dara, Patrick, Meg, and PJ for their contributions!

Please feel free to leave a comment with your reactions to the topic or directly to anything that has been said in this post.

Consider contributing a question to be answered in a future post. You can send one to me at [email protected] . When you send it in, let me know if I can use your real name if it’s selected or if you’d prefer remaining anonymous and have a pseudonym in mind.

You can also contact me on Twitter at @Larryferlazzo .

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Critical Thinking and Decision-Making  - What is Critical Thinking?

Critical thinking and decision-making  -, what is critical thinking, critical thinking and decision-making what is critical thinking.

Critical Thinking and Decision-Making: What is Critical Thinking?

Lesson 1: what is critical thinking, what is critical thinking.

Critical thinking is a term that gets thrown around a lot. You've probably heard it used often throughout the years whether it was in school, at work, or in everyday conversation. But when you stop to think about it, what exactly is critical thinking and how do you do it ?

Watch the video below to learn more about critical thinking.

Simply put, critical thinking is the act of deliberately analyzing information so that you can make better judgements and decisions . It involves using things like logic, reasoning, and creativity, to draw conclusions and generally understand things better.

This may sound like a pretty broad definition, and that's because critical thinking is a broad skill that can be applied to so many different situations. You can use it to prepare for a job interview, manage your time better, make decisions about purchasing things, and so much more.

The process

As humans, we are constantly thinking . It's something we can't turn off. But not all of it is critical thinking. No one thinks critically 100% of the time... that would be pretty exhausting! Instead, it's an intentional process , something that we consciously use when we're presented with difficult problems or important decisions.

Improving your critical thinking

In order to become a better critical thinker, it's important to ask questions when you're presented with a problem or decision, before jumping to any conclusions. You can start with simple ones like What do I currently know? and How do I know this? These can help to give you a better idea of what you're working with and, in some cases, simplify more complex issues.  

Real-world applications

Let's take a look at how we can use critical thinking to evaluate online information . Say a friend of yours posts a news article on social media and you're drawn to its headline. If you were to use your everyday automatic thinking, you might accept it as fact and move on. But if you were thinking critically, you would first analyze the available information and ask some questions :

• What's the source of this article?
• Is the headline potentially misleading?
• What are my friend's general beliefs?
• Do their beliefs inform why they might have shared this?

After analyzing all of this information, you can draw a conclusion about whether or not you think the article is trustworthy.

Critical thinking has a wide range of real-world applications . It can help you to make better decisions, become more hireable, and generally better understand the world around you.

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Critical thinking and problem-solving, jump to: , what is critical thinking, characteristics of critical thinking, why teach critical thinking.

• Teaching Strategies to Help Promote Critical Thinking Skills

References and Resources

When examining the vast literature on critical thinking, various definitions of critical thinking emerge. Here are some samples:

• "Critical thinking is the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action" (Scriven, 1996).
• "Most formal definitions characterize critical thinking as the intentional application of rational, higher order thinking skills, such as analysis, synthesis, problem recognition and problem solving, inference, and evaluation" (Angelo, 1995, p. 6).
• "Critical thinking is thinking that assesses itself" (Center for Critical Thinking, 1996b).
• "Critical thinking is the ability to think about one's thinking in such a way as 1. To recognize its strengths and weaknesses and, as a result, 2. To recast the thinking in improved form" (Center for Critical Thinking, 1996c).

Perhaps the simplest definition is offered by Beyer (1995) : "Critical thinking... means making reasoned judgments" (p. 8). Basically, Beyer sees critical thinking as using criteria to judge the quality of something, from cooking to a conclusion of a research paper. In essence, critical thinking is a disciplined manner of thought that a person uses to assess the validity of something (statements, news stories, arguments, research, etc.).

Back        

Wade (1995) identifies eight characteristics of critical thinking. Critical thinking involves asking questions, defining a problem, examining evidence, analyzing assumptions and biases, avoiding emotional reasoning, avoiding oversimplification, considering other interpretations, and tolerating ambiguity. Dealing with ambiguity is also seen by Strohm & Baukus (1995) as an essential part of critical thinking, "Ambiguity and doubt serve a critical-thinking function and are a necessary and even a productive part of the process" (p. 56).

Another characteristic of critical thinking identified by many sources is metacognition. Metacognition is thinking about one's own thinking. More specifically, "metacognition is being aware of one's thinking as one performs specific tasks and then using this awareness to control what one is doing" (Jones & Ratcliff, 1993, p. 10 ).

In the book, Critical Thinking, Beyer elaborately explains what he sees as essential aspects of critical thinking. These are:

• Dispositions: Critical thinkers are skeptical, open-minded, value fair-mindedness, respect evidence and reasoning, respect clarity and precision, look at different points of view, and will change positions when reason leads them to do so.
• Criteria: To think critically, must apply criteria. Need to have conditions that must be met for something to be judged as believable. Although the argument can be made that each subject area has different criteria, some standards apply to all subjects. "... an assertion must... be based on relevant, accurate facts; based on credible sources; precise; unbiased; free from logical fallacies; logically consistent; and strongly reasoned" (p. 12).
• Argument: Is a statement or proposition with supporting evidence. Critical thinking involves identifying, evaluating, and constructing arguments.
• Reasoning: The ability to infer a conclusion from one or multiple premises. To do so requires examining logical relationships among statements or data.
• Point of View: The way one views the world, which shapes one's construction of meaning. In a search for understanding, critical thinkers view phenomena from many different points of view.
• Procedures for Applying Criteria: Other types of thinking use a general procedure. Critical thinking makes use of many procedures. These procedures include asking questions, making judgments, and identifying assumptions.

Oliver & Utermohlen (1995) see students as too often being passive receptors of information. Through technology, the amount of information available today is massive. This information explosion is likely to continue in the future. Students need a guide to weed through the information and not just passively accept it. Students need to "develop and effectively apply critical thinking skills to their academic studies, to the complex problems that they will face, and to the critical choices they will be forced to make as a result of the information explosion and other rapid technological changes" (Oliver & Utermohlen, p. 1 ).

As mentioned in the section, Characteristics of Critical Thinking , critical thinking involves questioning. It is important to teach students how to ask good questions, to think critically, in order to continue the advancement of the very fields we are teaching. "Every field stays alive only to the extent that fresh questions are generated and taken seriously" (Center for Critical Thinking, 1996a ).

Beyer sees the teaching of critical thinking as important to the very state of our nation. He argues that to live successfully in a democracy, people must be able to think critically in order to make sound decisions about personal and civic affairs. If students learn to think critically, then they can use good thinking as the guide by which they live their lives.

Teaching Strategies to Help Promote Critical Thinking

The 1995, Volume 22, issue 1, of the journal, Teaching of Psychology , is devoted to the teaching critical thinking. Most of the strategies included in this section come from the various articles that compose this issue.

• CATS (Classroom Assessment Techniques): Angelo stresses the use of ongoing classroom assessment as a way to monitor and facilitate students' critical thinking. An example of a CAT is to ask students to write a "Minute Paper" responding to questions such as "What was the most important thing you learned in today's class? What question related to this session remains uppermost in your mind?" The teacher selects some of the papers and prepares responses for the next class meeting.
• Cooperative Learning Strategies: Cooper (1995) argues that putting students in group learning situations is the best way to foster critical thinking. "In properly structured cooperative learning environments, students perform more of the active, critical thinking with continuous support and feedback from other students and the teacher" (p. 8).
• Case Study /Discussion Method: McDade (1995) describes this method as the teacher presenting a case (or story) to the class without a conclusion. Using prepared questions, the teacher then leads students through a discussion, allowing students to construct a conclusion for the case.
• Using Questions: King (1995) identifies ways of using questions in the classroom:
• Reciprocal Peer Questioning: Following lecture, the teacher displays a list of question stems (such as, "What are the strengths and weaknesses of...). Students must write questions about the lecture material. In small groups, the students ask each other the questions. Then, the whole class discusses some of the questions from each small group.
• Reader's Questions: Require students to write questions on assigned reading and turn them in at the beginning of class. Select a few of the questions as the impetus for class discussion.
• Conference Style Learning: The teacher does not "teach" the class in the sense of lecturing. The teacher is a facilitator of a conference. Students must thoroughly read all required material before class. Assigned readings should be in the zone of proximal development. That is, readings should be able to be understood by students, but also challenging. The class consists of the students asking questions of each other and discussing these questions. The teacher does not remain passive, but rather, helps "direct and mold discussions by posing strategic questions and helping students build on each others' ideas" (Underwood & Wald, 1995, p. 18 ).
• Use Writing Assignments: Wade sees the use of writing as fundamental to developing critical thinking skills. "With written assignments, an instructor can encourage the development of dialectic reasoning by requiring students to argue both [or more] sides of an issue" (p. 24).
• Written dialogues: Give students written dialogues to analyze. In small groups, students must identify the different viewpoints of each participant in the dialogue. Must look for biases, presence or exclusion of important evidence, alternative interpretations, misstatement of facts, and errors in reasoning. Each group must decide which view is the most reasonable. After coming to a conclusion, each group acts out their dialogue and explains their analysis of it.
• Spontaneous Group Dialogue: One group of students are assigned roles to play in a discussion (such as leader, information giver, opinion seeker, and disagreer). Four observer groups are formed with the functions of determining what roles are being played by whom, identifying biases and errors in thinking, evaluating reasoning skills, and examining ethical implications of the content.
• Ambiguity: Strohm & Baukus advocate producing much ambiguity in the classroom. Don't give students clear cut material. Give them conflicting information that they must think their way through.
• Angelo, T. A. (1995). Beginning the dialogue: Thoughts on promoting critical thinking: Classroom assessment for critical thinking. Teaching of Psychology, 22(1), 6-7.
• Beyer, B. K. (1995). Critical thinking. Bloomington, IN: Phi Delta Kappa Educational Foundation.
• Center for Critical Thinking (1996a). The role of questions in thinking, teaching, and learning. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
• Center for Critical Thinking (1996b). Structures for student self-assessment. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univclass/trc.nclk
• Center for Critical Thinking (1996c). Three definitions of critical thinking [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
• Cooper, J. L. (1995). Cooperative learning and critical thinking. Teaching of Psychology, 22(1), 7-8.
• Jones, E. A. & Ratcliff, G. (1993). Critical thinking skills for college students. National Center on Postsecondary Teaching, Learning, and Assessment, University Park, PA. (Eric Document Reproduction Services No. ED 358 772)
• King, A. (1995). Designing the instructional process to enhance critical thinking across the curriculum: Inquiring minds really do want to know: Using questioning to teach critical thinking. Teaching of Psychology, 22 (1) , 13-17.
• McDade, S. A. (1995). Case study pedagogy to advance critical thinking. Teaching Psychology, 22(1), 9-10.
• Oliver, H. & Utermohlen, R. (1995). An innovative teaching strategy: Using critical thinking to give students a guide to the future.(Eric Document Reproduction Services No. 389 702)
• Robertson, J. F. & Rane-Szostak, D. (1996). Using dialogues to develop critical thinking skills: A practical approach. Journal of Adolescent & Adult Literacy, 39(7), 552-556.
• Scriven, M. & Paul, R. (1996). Defining critical thinking: A draft statement for the National Council for Excellence in Critical Thinking. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
• Strohm, S. M., & Baukus, R. A. (1995). Strategies for fostering critical thinking skills. Journalism and Mass Communication Educator, 50 (1), 55-62.
• Underwood, M. K., & Wald, R. L. (1995). Conference-style learning: A method for fostering critical thinking with heart. Teaching Psychology, 22(1), 17-21.
• Wade, C. (1995). Using writing to develop and assess critical thinking. Teaching of Psychology, 22(1), 24-28.

Other Reading

• Bean, J. C. (1996). Engaging ideas: The professor's guide to integrating writing, critical thinking, & active learning in the classroom. Jossey-Bass.
• Bernstein, D. A. (1995). A negotiation model for teaching critical thinking. Teaching of Psychology, 22(1), 22-24.
• Carlson, E. R. (1995). Evaluating the credibility of sources. A missing link in the teaching of critical thinking. Teaching of Psychology, 22(1), 39-41.
• Facione, P. A., Sanchez, C. A., Facione, N. C., & Gainen, J. (1995). The disposition toward critical thinking. The Journal of General Education, 44(1), 1-25.
• Halpern, D. F., & Nummedal, S. G. (1995). Closing thoughts about helping students improve how they think. Teaching of Psychology, 22(1), 82-83.
• Isbell, D. (1995). Teaching writing and research as inseparable: A faculty-librarian teaching team. Reference Services Review, 23(4), 51-62.
• Jones, J. M. & Safrit, R. D. (1994). Developing critical thinking skills in adult learners through innovative distance learning. Paper presented at the International Conference on the practice of adult education and social development. Jinan, China. (Eric Document Reproduction Services No. ED 373 159)
• Sanchez, M. A. (1995). Using critical-thinking principles as a guide to college-level instruction. Teaching of Psychology, 22(1), 72-74.
• Spicer, K. L. & Hanks, W. E. (1995). Multiple measures of critical thinking skills and predisposition in assessment of critical thinking. Paper presented at the annual meeting of the Speech Communication Association, San Antonio, TX. (Eric Document Reproduction Services No. ED 391 185)
• Terenzini, P. T., Springer, L., Pascarella, E. T., & Nora, A. (1995). Influences affecting the development of students' critical thinking skills. Research in Higher Education, 36(1), 23-39.

On the Internet

• Carr, K. S. (1990). How can we teach critical thinking. Eric Digest. [On-line]. Available HTTP: http://ericps.ed.uiuc.edu/eece/pubs/digests/1990/carr90.html
• The Center for Critical Thinking (1996). Home Page. Available HTTP: http://www.criticalthinking.org/University/
• Ennis, Bob (No date). Critical thinking. [On-line], April 4, 1997. Available HTTP: http://www.cof.orst.edu/cof/teach/for442/ct.htm
• Montclair State University (1995). Curriculum resource center. Critical thinking resources: An annotated bibliography. [On-line]. Available HTTP: http://www.montclair.edu/Pages/CRC/Bibliographies/CriticalThinking.html
• No author, No date. Critical Thinking is ... [On-line], April 4, 1997. Available HTTP: http://library.usask.ca/ustudy/critical/
• Sheridan, Marcia (No date). Internet education topics hotlink page. [On-line], April 4, 1997. Available HTTP: http://sun1.iusb.edu/~msherida/topics/critical.html

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7 Module 7: Thinking, Reasoning, and Problem-Solving

This module is about how a solid working knowledge of psychological principles can help you to think more effectively, so you can succeed in school and life. You might be inclined to believe that—because you have been thinking for as long as you can remember, because you are able to figure out the solution to many problems, because you feel capable of using logic to argue a point, because you can evaluate whether the things you read and hear make sense—you do not need any special training in thinking. But this, of course, is one of the key barriers to helping people think better. If you do not believe that there is anything wrong, why try to fix it?

The human brain is indeed a remarkable thinking machine, capable of amazing, complex, creative, logical thoughts. Why, then, are we telling you that you need to learn how to think? Mainly because one major lesson from cognitive psychology is that these capabilities of the human brain are relatively infrequently realized. Many psychologists believe that people are essentially “cognitive misers.” It is not that we are lazy, but that we have a tendency to expend the least amount of mental effort necessary. Although you may not realize it, it actually takes a great deal of energy to think. Careful, deliberative reasoning and critical thinking are very difficult. Because we seem to be successful without going to the trouble of using these skills well, it feels unnecessary to develop them. As you shall see, however, there are many pitfalls in the cognitive processes described in this module. When people do not devote extra effort to learning and improving reasoning, problem solving, and critical thinking skills, they make many errors.

As is true for memory, if you develop the cognitive skills presented in this module, you will be more successful in school. It is important that you realize, however, that these skills will help you far beyond school, even more so than a good memory will. Although it is somewhat useful to have a good memory, ten years from now no potential employer will care how many questions you got right on multiple choice exams during college. All of them will, however, recognize whether you are a logical, analytical, critical thinker. With these thinking skills, you will be an effective, persuasive communicator and an excellent problem solver.

The module begins by describing different kinds of thought and knowledge, especially conceptual knowledge and critical thinking. An understanding of these differences will be valuable as you progress through school and encounter different assignments that require you to tap into different kinds of knowledge. The second section covers deductive and inductive reasoning, which are processes we use to construct and evaluate strong arguments. They are essential skills to have whenever you are trying to persuade someone (including yourself) of some point, or to respond to someone’s efforts to persuade you. The module ends with a section about problem solving. A solid understanding of the key processes involved in problem solving will help you to handle many daily challenges.

7.1. Different kinds of thought

7.2. Reasoning and Judgment

7.3. Problem Solving

READING WITH PURPOSE

Remember and understand.

By reading and studying Module 7, you should be able to remember and describe:

• Concepts and inferences (7.1)
• Procedural knowledge (7.1)
• Metacognition (7.1)
• Characteristics of critical thinking:  skepticism; identify biases, distortions, omissions, and assumptions; reasoning and problem solving skills  (7.1)
• Reasoning:  deductive reasoning, deductively valid argument, inductive reasoning, inductively strong argument, availability heuristic, representativeness heuristic  (7.2)
• Fixation:  functional fixedness, mental set  (7.3)
• Algorithms, heuristics, and the role of confirmation bias (7.3)
• Effective problem solving sequence (7.3)

By reading and thinking about how the concepts in Module 6 apply to real life, you should be able to:

• Identify which type of knowledge a piece of information is (7.1)
• Recognize examples of deductive and inductive reasoning (7.2)
• Recognize judgments that have probably been influenced by the availability heuristic (7.2)
• Recognize examples of problem solving heuristics and algorithms (7.3)

Analyze, Evaluate, and Create

By reading and thinking about Module 6, participating in classroom activities, and completing out-of-class assignments, you should be able to:

• Use the principles of critical thinking to evaluate information (7.1)
• Explain whether examples of reasoning arguments are deductively valid or inductively strong (7.2)
• Outline how you could try to solve a problem from your life using the effective problem solving sequence (7.3)

7.1. Different kinds of thought and knowledge

• Take a few minutes to write down everything that you know about dogs.
• Do you believe that:
• Psychic ability exists?
• Hypnosis is an altered state of consciousness?
• Magnet therapy is effective for relieving pain?
• Aerobic exercise is an effective treatment for depression?
• UFO’s from outer space have visited earth?

On what do you base your belief or disbelief for the questions above?

Of course, we all know what is meant by the words  think  and  knowledge . You probably also realize that they are not unitary concepts; there are different kinds of thought and knowledge. In this section, let us look at some of these differences. If you are familiar with these different kinds of thought and pay attention to them in your classes, it will help you to focus on the right goals, learn more effectively, and succeed in school. Different assignments and requirements in school call on you to use different kinds of knowledge or thought, so it will be very helpful for you to learn to recognize them (Anderson, et al. 2001).

Factual and conceptual knowledge

Module 5 introduced the idea of declarative memory, which is composed of facts and episodes. If you have ever played a trivia game or watched Jeopardy on TV, you realize that the human brain is able to hold an extraordinary number of facts. Likewise, you realize that each of us has an enormous store of episodes, essentially facts about events that happened in our own lives. It may be difficult to keep that in mind when we are struggling to retrieve one of those facts while taking an exam, however. Part of the problem is that, in contradiction to the advice from Module 5, many students continue to try to memorize course material as a series of unrelated facts (picture a history student simply trying to memorize history as a set of unrelated dates without any coherent story tying them together). Facts in the real world are not random and unorganized, however. It is the way that they are organized that constitutes a second key kind of knowledge, conceptual.

Concepts are nothing more than our mental representations of categories of things in the world. For example, think about dogs. When you do this, you might remember specific facts about dogs, such as they have fur and they bark. You may also recall dogs that you have encountered and picture them in your mind. All of this information (and more) makes up your concept of dog. You can have concepts of simple categories (e.g., triangle), complex categories (e.g., small dogs that sleep all day, eat out of the garbage, and bark at leaves), kinds of people (e.g., psychology professors), events (e.g., birthday parties), and abstract ideas (e.g., justice). Gregory Murphy (2002) refers to concepts as the “glue that holds our mental life together” (p. 1). Very simply, summarizing the world by using concepts is one of the most important cognitive tasks that we do. Our conceptual knowledge  is  our knowledge about the world. Individual concepts are related to each other to form a rich interconnected network of knowledge. For example, think about how the following concepts might be related to each other: dog, pet, play, Frisbee, chew toy, shoe. Or, of more obvious use to you now, how these concepts are related: working memory, long-term memory, declarative memory, procedural memory, and rehearsal? Because our minds have a natural tendency to organize information conceptually, when students try to remember course material as isolated facts, they are working against their strengths.

One last important point about concepts is that they allow you to instantly know a great deal of information about something. For example, if someone hands you a small red object and says, “here is an apple,” they do not have to tell you, “it is something you can eat.” You already know that you can eat it because it is true by virtue of the fact that the object is an apple; this is called drawing an  inference , assuming that something is true on the basis of your previous knowledge (for example, of category membership or of how the world works) or logical reasoning.

Procedural knowledge

Physical skills, such as tying your shoes, doing a cartwheel, and driving a car (or doing all three at the same time, but don’t try this at home) are certainly a kind of knowledge. They are procedural knowledge, the same idea as procedural memory that you saw in Module 5. Mental skills, such as reading, debating, and planning a psychology experiment, are procedural knowledge, as well. In short, procedural knowledge is the knowledge how to do something (Cohen & Eichenbaum, 1993).

Metacognitive knowledge

Floyd used to think that he had a great memory. Now, he has a better memory. Why? Because he finally realized that his memory was not as great as he once thought it was. Because Floyd eventually learned that he often forgets where he put things, he finally developed the habit of putting things in the same place. (Unfortunately, he did not learn this lesson before losing at least 5 watches and a wedding ring.) Because he finally realized that he often forgets to do things, he finally started using the To Do list app on his phone. And so on. Floyd’s insights about the real limitations of his memory have allowed him to remember things that he used to forget.

All of us have knowledge about the way our own minds work. You may know that you have a good memory for people’s names and a poor memory for math formulas. Someone else might realize that they have difficulty remembering to do things, like stopping at the store on the way home. Others still know that they tend to overlook details. This knowledge about our own thinking is actually quite important; it is called metacognitive knowledge, or  metacognition . Like other kinds of thinking skills, it is subject to error. For example, in unpublished research, one of the authors surveyed about 120 General Psychology students on the first day of the term. Among other questions, the students were asked them to predict their grade in the class and report their current Grade Point Average. Two-thirds of the students predicted that their grade in the course would be higher than their GPA. (The reality is that at our college, students tend to earn lower grades in psychology than their overall GPA.) Another example: Students routinely report that they thought they had done well on an exam, only to discover, to their dismay, that they were wrong (more on that important problem in a moment). Both errors reveal a breakdown in metacognition.

The Dunning-Kruger Effect

In general, most college students probably do not study enough. For example, using data from the National Survey of Student Engagement, Fosnacht, McCormack, and Lerma (2018) reported that first-year students at 4-year colleges in the U.S. averaged less than 14 hours per week preparing for classes. The typical suggestion is that you should spend two hours outside of class for every hour in class, or 24 – 30 hours per week for a full-time student. Clearly, students in general are nowhere near that recommended mark. Many observers, including some faculty, believe that this shortfall is a result of students being too busy or lazy. Now, it may be true that many students are too busy, with work and family obligations, for example. Others, are not particularly motivated in school, and therefore might correctly be labeled lazy. A third possible explanation, however, is that some students might not think they need to spend this much time. And this is a matter of metacognition. Consider the scenario that we mentioned above, students thinking they had done well on an exam only to discover that they did not. Justin Kruger and David Dunning examined scenarios very much like this in 1999. Kruger and Dunning gave research participants tests measuring humor, logic, and grammar. Then, they asked the participants to assess their own abilities and test performance in these areas. They found that participants in general tended to overestimate their abilities, already a problem with metacognition. Importantly, the participants who scored the lowest overestimated their abilities the most. Specifically, students who scored in the bottom quarter (averaging in the 12th percentile) thought they had scored in the 62nd percentile. This has become known as the  Dunning-Kruger effect . Many individual faculty members have replicated these results with their own student on their course exams, including the authors of this book. Think about it. Some students who just took an exam and performed poorly believe that they did well before seeing their score. It seems very likely that these are the very same students who stopped studying the night before because they thought they were “done.” Quite simply, it is not just that they did not know the material. They did not know that they did not know the material. That is poor metacognition.

In order to develop good metacognitive skills, you should continually monitor your thinking and seek frequent feedback on the accuracy of your thinking (Medina, Castleberry, & Persky 2017). For example, in classes get in the habit of predicting your exam grades. As soon as possible after taking an exam, try to find out which questions you missed and try to figure out why. If you do this soon enough, you may be able to recall the way it felt when you originally answered the question. Did you feel confident that you had answered the question correctly? Then you have just discovered an opportunity to improve your metacognition. Be on the lookout for that feeling and respond with caution.

concept :  a mental representation of a category of things in the world

Dunning-Kruger effect : individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

inference : an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

metacognition :  knowledge about one’s own cognitive processes; thinking about your thinking

Critical thinking

One particular kind of knowledge or thinking skill that is related to metacognition is  critical thinking (Chew, 2020). You may have noticed that critical thinking is an objective in many college courses, and thus it could be a legitimate topic to cover in nearly any college course. It is particularly appropriate in psychology, however. As the science of (behavior and) mental processes, psychology is obviously well suited to be the discipline through which you should be introduced to this important way of thinking.

More importantly, there is a particular need to use critical thinking in psychology. We are all, in a way, experts in human behavior and mental processes, having engaged in them literally since birth. Thus, perhaps more than in any other class, students typically approach psychology with very clear ideas and opinions about its subject matter. That is, students already “know” a lot about psychology. The problem is, “it ain’t so much the things we don’t know that get us into trouble. It’s the things we know that just ain’t so” (Ward, quoted in Gilovich 1991). Indeed, many of students’ preconceptions about psychology are just plain wrong. Randolph Smith (2002) wrote a book about critical thinking in psychology called  Challenging Your Preconceptions,  highlighting this fact. On the other hand, many of students’ preconceptions about psychology are just plain right! But wait, how do you know which of your preconceptions are right and which are wrong? And when you come across a research finding or theory in this class that contradicts your preconceptions, what will you do? Will you stick to your original idea, discounting the information from the class? Will you immediately change your mind? Critical thinking can help us sort through this confusing mess.

But what is critical thinking? The goal of critical thinking is simple to state (but extraordinarily difficult to achieve): it is to be right, to draw the correct conclusions, to believe in things that are true and to disbelieve things that are false. We will provide two definitions of critical thinking (or, if you like, one large definition with two distinct parts). First, a more conceptual one: Critical thinking is thinking like a scientist in your everyday life (Schmaltz, Jansen, & Wenckowski, 2017).  Our second definition is more operational; it is simply a list of skills that are essential to be a critical thinker. Critical thinking entails solid reasoning and problem solving skills; skepticism; and an ability to identify biases, distortions, omissions, and assumptions. Excellent deductive and inductive reasoning, and problem solving skills contribute to critical thinking. So, you can consider the subject matter of sections 7.2 and 7.3 to be part of critical thinking. Because we will be devoting considerable time to these concepts in the rest of the module, let us begin with a discussion about the other aspects of critical thinking.

Let’s address that first part of the definition. Scientists form hypotheses, or predictions about some possible future observations. Then, they collect data, or information (think of this as making those future observations). They do their best to make unbiased observations using reliable techniques that have been verified by others. Then, and only then, they draw a conclusion about what those observations mean. Oh, and do not forget the most important part. “Conclusion” is probably not the most appropriate word because this conclusion is only tentative. A scientist is always prepared that someone else might come along and produce new observations that would require a new conclusion be drawn. Wow! If you like to be right, you could do a lot worse than using a process like this.

A Critical Thinker’s Toolkit 

Now for the second part of the definition. Good critical thinkers (and scientists) rely on a variety of tools to evaluate information. Perhaps the most recognizable tool for critical thinking is  skepticism (and this term provides the clearest link to the thinking like a scientist definition, as you are about to see). Some people intend it as an insult when they call someone a skeptic. But if someone calls you a skeptic, if they are using the term correctly, you should consider it a great compliment. Simply put, skepticism is a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided. People from Missouri should recognize this principle, as Missouri is known as the Show-Me State. As a skeptic, you are not inclined to believe something just because someone said so, because someone else believes it, or because it sounds reasonable. You must be persuaded by high quality evidence.

Of course, if that evidence is produced, you have a responsibility as a skeptic to change your belief. Failure to change a belief in the face of good evidence is not skepticism; skepticism has open mindedness at its core. M. Neil Browne and Stuart Keeley (2018) use the term weak sense critical thinking to describe critical thinking behaviors that are used only to strengthen a prior belief. Strong sense critical thinking, on the other hand, has as its goal reaching the best conclusion. Sometimes that means strengthening your prior belief, but sometimes it means changing your belief to accommodate the better evidence.

Many times, a failure to think critically or weak sense critical thinking is related to a  bias , an inclination, tendency, leaning, or prejudice. Everybody has biases, but many people are unaware of them. Awareness of your own biases gives you the opportunity to control or counteract them. Unfortunately, however, many people are happy to let their biases creep into their attempts to persuade others; indeed, it is a key part of their persuasive strategy. To see how these biases influence messages, just look at the different descriptions and explanations of the same events given by people of different ages or income brackets, or conservative versus liberal commentators, or by commentators from different parts of the world. Of course, to be successful, these people who are consciously using their biases must disguise them. Even undisguised biases can be difficult to identify, so disguised ones can be nearly impossible.

Here are some common sources of biases:

• Personal values and beliefs.  Some people believe that human beings are basically driven to seek power and that they are typically in competition with one another over scarce resources. These beliefs are similar to the world-view that political scientists call “realism.” Other people believe that human beings prefer to cooperate and that, given the chance, they will do so. These beliefs are similar to the world-view known as “idealism.” For many people, these deeply held beliefs can influence, or bias, their interpretations of such wide ranging situations as the behavior of nations and their leaders or the behavior of the driver in the car ahead of you. For example, if your worldview is that people are typically in competition and someone cuts you off on the highway, you may assume that the driver did it purposely to get ahead of you. Other types of beliefs about the way the world is or the way the world should be, for example, political beliefs, can similarly become a significant source of bias.
• Racism, sexism, ageism and other forms of prejudice and bigotry.  These are, sadly, a common source of bias in many people. They are essentially a special kind of “belief about the way the world is.” These beliefs—for example, that women do not make effective leaders—lead people to ignore contradictory evidence (examples of effective women leaders, or research that disputes the belief) and to interpret ambiguous evidence in a way consistent with the belief.
• Self-interest.  When particular people benefit from things turning out a certain way, they can sometimes be very susceptible to letting that interest bias them. For example, a company that will earn a profit if they sell their product may have a bias in the way that they give information about their product. A union that will benefit if its members get a generous contract might have a bias in the way it presents information about salaries at competing organizations. (Note that our inclusion of examples describing both companies and unions is an explicit attempt to control for our own personal biases). Home buyers are often dismayed to discover that they purchased their dream house from someone whose self-interest led them to lie about flooding problems in the basement or back yard. This principle, the biasing power of self-interest, is likely what led to the famous phrase  Caveat Emptor  (let the buyer beware) .  

Knowing that these types of biases exist will help you evaluate evidence more critically. Do not forget, though, that people are not always keen to let you discover the sources of biases in their arguments. For example, companies or political organizations can sometimes disguise their support of a research study by contracting with a university professor, who comes complete with a seemingly unbiased institutional affiliation, to conduct the study.

People’s biases, conscious or unconscious, can lead them to make omissions, distortions, and assumptions that undermine our ability to correctly evaluate evidence. It is essential that you look for these elements. Always ask, what is missing, what is not as it appears, and what is being assumed here? For example, consider this (fictional) chart from an ad reporting customer satisfaction at 4 local health clubs.

Clearly, from the results of the chart, one would be tempted to give Club C a try, as customer satisfaction is much higher than for the other 3 clubs.

There are so many distortions and omissions in this chart, however, that it is actually quite meaningless. First, how was satisfaction measured? Do the bars represent responses to a survey? If so, how were the questions asked? Most importantly, where is the missing scale for the chart? Although the differences look quite large, are they really?

Well, here is the same chart, with a different scale, this time labeled:

Club C is not so impressive any more, is it? In fact, all of the health clubs have customer satisfaction ratings (whatever that means) between 85% and 88%. In the first chart, the entire scale of the graph included only the percentages between 83 and 89. This “judicious” choice of scale—some would call it a distortion—and omission of that scale from the chart make the tiny differences among the clubs seem important, however.

Also, in order to be a critical thinker, you need to learn to pay attention to the assumptions that underlie a message. Let us briefly illustrate the role of assumptions by touching on some people’s beliefs about the criminal justice system in the US. Some believe that a major problem with our judicial system is that many criminals go free because of legal technicalities. Others believe that a major problem is that many innocent people are convicted of crimes. The simple fact is, both types of errors occur. A person’s conclusion about which flaw in our judicial system is the greater tragedy is based on an assumption about which of these is the more serious error (letting the guilty go free or convicting the innocent). This type of assumption is called a value assumption (Browne and Keeley, 2018). It reflects the differences in values that people develop, differences that may lead us to disregard valid evidence that does not fit in with our particular values.

Oh, by the way, some students probably noticed this, but the seven tips for evaluating information that we shared in Module 1 are related to this. Actually, they are part of this section. The tips are, to a very large degree, set of ideas you can use to help you identify biases, distortions, omissions, and assumptions. If you do not remember this section, we strongly recommend you take a few minutes to review it.

skepticism :  a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

bias : an inclination, tendency, leaning, or prejudice

• Which of your beliefs (or disbeliefs) from the Activate exercise for this section were derived from a process of critical thinking? If some of your beliefs were not based on critical thinking, are you willing to reassess these beliefs? If the answer is no, why do you think that is? If the answer is yes, what concrete steps will you take?

7.2 Reasoning and Judgment

• What percentage of kidnappings are committed by strangers?
• Which area of the house is riskiest: kitchen, bathroom, or stairs?
• What is the most common cancer in the US?
• What percentage of workplace homicides are committed by co-workers?

An essential set of procedural thinking skills is  reasoning , the ability to generate and evaluate solid conclusions from a set of statements or evidence. You should note that these conclusions (when they are generated instead of being evaluated) are one key type of inference that we described in Section 7.1. There are two main types of reasoning, deductive and inductive.

Deductive reasoning

Suppose your teacher tells you that if you get an A on the final exam in a course, you will get an A for the whole course. Then, you get an A on the final exam. What will your final course grade be? Most people can see instantly that you can conclude with certainty that you will get an A for the course. This is a type of reasoning called  deductive reasoning , which is defined as reasoning in which a conclusion is guaranteed to be true as long as the statements leading to it are true. The three statements can be listed as an  argument , with two beginning statements and a conclusion:

Statement 1: If you get an A on the final exam, you will get an A for the course

Statement 2: You get an A on the final exam

Conclusion: You will get an A for the course

This particular arrangement, in which true beginning statements lead to a guaranteed true conclusion, is known as a  deductively valid argument . Although deductive reasoning is often the subject of abstract, brain-teasing, puzzle-like word problems, it is actually an extremely important type of everyday reasoning. It is just hard to recognize sometimes. For example, imagine that you are looking for your car keys and you realize that they are either in the kitchen drawer or in your book bag. After looking in the kitchen drawer, you instantly know that they must be in your book bag. That conclusion results from a simple deductive reasoning argument. In addition, solid deductive reasoning skills are necessary for you to succeed in the sciences, philosophy, math, computer programming, and any endeavor involving the use of logic to persuade others to your point of view or to evaluate others’ arguments.

Cognitive psychologists, and before them philosophers, have been quite interested in deductive reasoning, not so much for its practical applications, but for the insights it can offer them about the ways that human beings think. One of the early ideas to emerge from the examination of deductive reasoning is that people learn (or develop) mental versions of rules that allow them to solve these types of reasoning problems (Braine, 1978; Braine, Reiser, & Rumain, 1984). The best way to see this point of view is to realize that there are different possible rules, and some of them are very simple. For example, consider this rule of logic:

therefore q

Logical rules are often presented abstractly, as letters, in order to imply that they can be used in very many specific situations. Here is a concrete version of the of the same rule:

I’ll either have pizza or a hamburger for dinner tonight (p or q)

I won’t have pizza (not p)

Therefore, I’ll have a hamburger (therefore q)

This kind of reasoning seems so natural, so easy, that it is quite plausible that we would use a version of this rule in our daily lives. At least, it seems more plausible than some of the alternative possibilities—for example, that we need to have experience with the specific situation (pizza or hamburger, in this case) in order to solve this type of problem easily. So perhaps there is a form of natural logic (Rips, 1990) that contains very simple versions of logical rules. When we are faced with a reasoning problem that maps onto one of these rules, we use the rule.

But be very careful; things are not always as easy as they seem. Even these simple rules are not so simple. For example, consider the following rule. Many people fail to realize that this rule is just as valid as the pizza or hamburger rule above.

if p, then q

therefore, not p

Concrete version:

If I eat dinner, then I will have dessert

I did not have dessert

Therefore, I did not eat dinner

The simple fact is, it can be very difficult for people to apply rules of deductive logic correctly; as a result, they make many errors when trying to do so. Is this a deductively valid argument or not?

Students who like school study a lot

Students who study a lot get good grades

Jane does not like school

Therefore, Jane does not get good grades

Many people are surprised to discover that this is not a logically valid argument; the conclusion is not guaranteed to be true from the beginning statements. Although the first statement says that students who like school study a lot, it does NOT say that students who do not like school do not study a lot. In other words, it may very well be possible to study a lot without liking school. Even people who sometimes get problems like this right might not be using the rules of deductive reasoning. Instead, they might just be making judgments for examples they know, in this case, remembering instances of people who get good grades despite not liking school.

Making deductive reasoning even more difficult is the fact that there are two important properties that an argument may have. One, it can be valid or invalid (meaning that the conclusion does or does not follow logically from the statements leading up to it). Two, an argument (or more correctly, its conclusion) can be true or false. Here is an example of an argument that is logically valid, but has a false conclusion (at least we think it is false).

Either you are eleven feet tall or the Grand Canyon was created by a spaceship crashing into the earth.

You are not eleven feet tall

Therefore the Grand Canyon was created by a spaceship crashing into the earth

This argument has the exact same form as the pizza or hamburger argument above, making it is deductively valid. The conclusion is so false, however, that it is absurd (of course, the reason the conclusion is false is that the first statement is false). When people are judging arguments, they tend to not observe the difference between deductive validity and the empirical truth of statements or conclusions. If the elements of an argument happen to be true, people are likely to judge the argument logically valid; if the elements are false, they will very likely judge it invalid (Markovits & Bouffard-Bouchard, 1992; Moshman & Franks, 1986). Thus, it seems a stretch to say that people are using these logical rules to judge the validity of arguments. Many psychologists believe that most people actually have very limited deductive reasoning skills (Johnson-Laird, 1999). They argue that when faced with a problem for which deductive logic is required, people resort to some simpler technique, such as matching terms that appear in the statements and the conclusion (Evans, 1982). This might not seem like a problem, but what if reasoners believe that the elements are true and they happen to be wrong; they will would believe that they are using a form of reasoning that guarantees they are correct and yet be wrong.

deductive reasoning :  a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

argument :  a set of statements in which the beginning statements lead to a conclusion

deductively valid argument :  an argument for which true beginning statements guarantee that the conclusion is true

Inductive reasoning and judgment

Every day, you make many judgments about the likelihood of one thing or another. Whether you realize it or not, you are practicing  inductive reasoning   on a daily basis. In inductive reasoning arguments, a conclusion is likely whenever the statements preceding it are true. The first thing to notice about inductive reasoning is that, by definition, you can never be sure about your conclusion; you can only estimate how likely the conclusion is. Inductive reasoning may lead you to focus on Memory Encoding and Recoding when you study for the exam, but it is possible the instructor will ask more questions about Memory Retrieval instead. Unlike deductive reasoning, the conclusions you reach through inductive reasoning are only probable, not certain. That is why scientists consider inductive reasoning weaker than deductive reasoning. But imagine how hard it would be for us to function if we could not act unless we were certain about the outcome.

Inductive reasoning can be represented as logical arguments consisting of statements and a conclusion, just as deductive reasoning can be. In an inductive argument, you are given some statements and a conclusion (or you are given some statements and must draw a conclusion). An argument is  inductively strong   if the conclusion would be very probable whenever the statements are true. So, for example, here is an inductively strong argument:

• Statement #1: The forecaster on Channel 2 said it is going to rain today.
• Statement #2: The forecaster on Channel 5 said it is going to rain today.
• Statement #3: It is very cloudy and humid.
• Statement #4: You just heard thunder.
• Conclusion (or judgment): It is going to rain today.

Think of the statements as evidence, on the basis of which you will draw a conclusion. So, based on the evidence presented in the four statements, it is very likely that it will rain today. Will it definitely rain today? Certainly not. We can all think of times that the weather forecaster was wrong.

A true story: Some years ago psychology student was watching a baseball playoff game between the St. Louis Cardinals and the Los Angeles Dodgers. A graphic on the screen had just informed the audience that the Cardinal at bat, (Hall of Fame shortstop) Ozzie Smith, a switch hitter batting left-handed for this plate appearance, had never, in nearly 3000 career at-bats, hit a home run left-handed. The student, who had just learned about inductive reasoning in his psychology class, turned to his companion (a Cardinals fan) and smugly said, “It is an inductively strong argument that Ozzie Smith will not hit a home run.” He turned back to face the television just in time to watch the ball sail over the right field fence for a home run. Although the student felt foolish at the time, he was not wrong. It was an inductively strong argument; 3000 at-bats is an awful lot of evidence suggesting that the Wizard of Ozz (as he was known) would not be hitting one out of the park (think of each at-bat without a home run as a statement in an inductive argument). Sadly (for the die-hard Cubs fan and Cardinals-hating student), despite the strength of the argument, the conclusion was wrong.

Given the possibility that we might draw an incorrect conclusion even with an inductively strong argument, we really want to be sure that we do, in fact, make inductively strong arguments. If we judge something probable, it had better be probable. If we judge something nearly impossible, it had better not happen. Think of inductive reasoning, then, as making reasonably accurate judgments of the probability of some conclusion given a set of evidence.

We base many decisions in our lives on inductive reasoning. For example:

Statement #1: Psychology is not my best subject

Statement #2: My psychology instructor has a reputation for giving difficult exams

Statement #3: My first psychology exam was much harder than I expected

Judgment: The next exam will probably be very difficult.

Decision: I will study tonight instead of watching Netflix.

Some other examples of judgments that people commonly make in a school context include judgments of the likelihood that:

• A particular class will be interesting/useful/difficult
• You will be able to finish writing a paper by next week if you go out tonight
• Your laptop’s battery will last through the next trip to the library
• You will not miss anything important if you skip class tomorrow
• Your instructor will not notice if you skip class tomorrow
• You will be able to find a book that you will need for a paper
• There will be an essay question about Memory Encoding on the next exam

Tversky and Kahneman (1983) recognized that there are two general ways that we might make these judgments; they termed them extensional (i.e., following the laws of probability) and intuitive (i.e., using shortcuts or heuristics, see below). We will use a similar distinction between Type 1 and Type 2 thinking, as described by Keith Stanovich and his colleagues (Evans and Stanovich, 2013; Stanovich and West, 2000). Type 1 thinking is fast, automatic, effortful, and emotional. In fact, it is hardly fair to call it reasoning at all, as judgments just seem to pop into one’s head. Type 2 thinking , on the other hand, is slow, effortful, and logical. So obviously, it is more likely to lead to a correct judgment, or an optimal decision. The problem is, we tend to over-rely on Type 1. Now, we are not saying that Type 2 is the right way to go for every decision or judgment we make. It seems a bit much, for example, to engage in a step-by-step logical reasoning procedure to decide whether we will have chicken or fish for dinner tonight.

Many bad decisions in some very important contexts, however, can be traced back to poor judgments of the likelihood of certain risks or outcomes that result from the use of Type 1 when a more logical reasoning process would have been more appropriate. For example:

Statement #1: It is late at night.

Statement #2: Albert has been drinking beer for the past five hours at a party.

Statement #3: Albert is not exactly sure where he is or how far away home is.

Judgment: Albert will have no difficulty walking home.

Decision: He walks home alone.

As you can see in this example, the three statements backing up the judgment do not really support it. In other words, this argument is not inductively strong because it is based on judgments that ignore the laws of probability. What are the chances that someone facing these conditions will be able to walk home alone easily? And one need not be drunk to make poor decisions based on judgments that just pop into our heads.

The truth is that many of our probability judgments do not come very close to what the laws of probability say they should be. Think about it. In order for us to reason in accordance with these laws, we would need to know the laws of probability, which would allow us to calculate the relationship between particular pieces of evidence and the probability of some outcome (i.e., how much likelihood should change given a piece of evidence), and we would have to do these heavy math calculations in our heads. After all, that is what Type 2 requires. Needless to say, even if we were motivated, we often do not even know how to apply Type 2 reasoning in many cases.

So what do we do when we don’t have the knowledge, skills, or time required to make the correct mathematical judgment? Do we hold off and wait until we can get better evidence? Do we read up on probability and fire up our calculator app so we can compute the correct probability? Of course not. We rely on Type 1 thinking. We “wing it.” That is, we come up with a likelihood estimate using some means at our disposal. Psychologists use the term heuristic to describe the type of “winging it” we are talking about. A  heuristic   is a shortcut strategy that we use to make some judgment or solve some problem (see Section 7.3). Heuristics are easy and quick, think of them as the basic procedures that are characteristic of Type 1.  They can absolutely lead to reasonably good judgments and decisions in some situations (like choosing between chicken and fish for dinner). They are, however, far from foolproof. There are, in fact, quite a lot of situations in which heuristics can lead us to make incorrect judgments, and in many cases the decisions based on those judgments can have serious consequences.

Let us return to the activity that begins this section. You were asked to judge the likelihood (or frequency) of certain events and risks. You were free to come up with your own evidence (or statements) to make these judgments. This is where a heuristic crops up. As a judgment shortcut, we tend to generate specific examples of those very events to help us decide their likelihood or frequency. For example, if we are asked to judge how common, frequent, or likely a particular type of cancer is, many of our statements would be examples of specific cancer cases:

Statement #1: Andy Kaufman (comedian) had lung cancer.

Statement #2: Colin Powell (US Secretary of State) had prostate cancer.

Statement #3: Bob Marley (musician) had skin and brain cancer

Statement #4: Sandra Day O’Connor (Supreme Court Justice) had breast cancer.

Statement #5: Fred Rogers (children’s entertainer) had stomach cancer.

Statement #6: Robin Roberts (news anchor) had breast cancer.

Statement #7: Bette Davis (actress) had breast cancer.

Judgment: Breast cancer is the most common type.

Your own experience or memory may also tell you that breast cancer is the most common type. But it is not (although it is common). Actually, skin cancer is the most common type in the US. We make the same types of misjudgments all the time because we do not generate the examples or evidence according to their actual frequencies or probabilities. Instead, we have a tendency (or bias) to search for the examples in memory; if they are easy to retrieve, we assume that they are common. To rephrase this in the language of the heuristic, events seem more likely to the extent that they are available to memory. This bias has been termed the  availability heuristic   (Kahneman and Tversky, 1974).

The fact that we use the availability heuristic does not automatically mean that our judgment is wrong. The reason we use heuristics in the first place is that they work fairly well in many cases (and, of course that they are easy to use). So, the easiest examples to think of sometimes are the most common ones. Is it more likely that a member of the U.S. Senate is a man or a woman? Most people have a much easier time generating examples of male senators. And as it turns out, the U.S. Senate has many more men than women (74 to 26 in 2020). In this case, then, the availability heuristic would lead you to make the correct judgment; it is far more likely that a senator would be a man.

In many other cases, however, the availability heuristic will lead us astray. This is because events can be memorable for many reasons other than their frequency. Section 5.2, Encoding Meaning, suggested that one good way to encode the meaning of some information is to form a mental image of it. Thus, information that has been pictured mentally will be more available to memory. Indeed, an event that is vivid and easily pictured will trick many people into supposing that type of event is more common than it actually is. Repetition of information will also make it more memorable. So, if the same event is described to you in a magazine, on the evening news, on a podcast that you listen to, and in your Facebook feed; it will be very available to memory. Again, the availability heuristic will cause you to misperceive the frequency of these types of events.

Most interestingly, information that is unusual is more memorable. Suppose we give you the following list of words to remember: box, flower, letter, platypus, oven, boat, newspaper, purse, drum, car. Very likely, the easiest word to remember would be platypus, the unusual one. The same thing occurs with memories of events. An event may be available to memory because it is unusual, yet the availability heuristic leads us to judge that the event is common. Did you catch that? In these cases, the availability heuristic makes us think the exact opposite of the true frequency. We end up thinking something is common because it is unusual (and therefore memorable). Yikes.

The misapplication of the availability heuristic sometimes has unfortunate results. For example, if you went to K-12 school in the US over the past 10 years, it is extremely likely that you have participated in lockdown and active shooter drills. Of course, everyone is trying to prevent the tragedy of another school shooting. And believe us, we are not trying to minimize how terrible the tragedy is. But the truth of the matter is, school shootings are extremely rare. Because the federal government does not keep a database of school shootings, the Washington Post has maintained their own running tally. Between 1999 and January 2020 (the date of the most recent school shooting with a death in the US at of the time this paragraph was written), the Post reported a total of 254 people died in school shootings in the US. Not 254 per year, 254 total. That is an average of 12 per year. Of course, that is 254 people who should not have died (particularly because many were children), but in a country with approximately 60,000,000 students and teachers, this is a very small risk.

But many students and teachers are terrified that they will be victims of school shootings because of the availability heuristic. It is so easy to think of examples (they are very available to memory) that people believe the event is very common. It is not. And there is a downside to this. We happen to believe that there is an enormous gun violence problem in the United States. According the the Centers for Disease Control and Prevention, there were 39,773 firearm deaths in the US in 2017. Fifteen of those deaths were in school shootings, according to the Post. 60% of those deaths were suicides. When people pay attention to the school shooting risk (low), they often fail to notice the much larger risk.

And examples like this are by no means unique. The authors of this book have been teaching psychology since the 1990’s. We have been able to make the exact same arguments about the misapplication of the availability heuristics and keep them current by simply swapping out for the “fear of the day.” In the 1990’s it was children being kidnapped by strangers (it was known as “stranger danger”) despite the facts that kidnappings accounted for only 2% of the violent crimes committed against children, and only 24% of kidnappings are committed by strangers (US Department of Justice, 2007). This fear overlapped with the fear of terrorism that gripped the country after the 2001 terrorist attacks on the World Trade Center and US Pentagon and still plagues the population of the US somewhat in 2020. After a well-publicized, sensational act of violence, people are extremely likely to increase their estimates of the chances that they, too, will be victims of terror. Think about the reality, however. In October of 2001, a terrorist mailed anthrax spores to members of the US government and a number of media companies. A total of five people died as a result of this attack. The nation was nearly paralyzed by the fear of dying from the attack; in reality the probability of an individual person dying was 0.00000002.

The availability heuristic can lead you to make incorrect judgments in a school setting as well. For example, suppose you are trying to decide if you should take a class from a particular math professor. You might try to make a judgment of how good a teacher she is by recalling instances of friends and acquaintances making comments about her teaching skill. You may have some examples that suggest that she is a poor teacher very available to memory, so on the basis of the availability heuristic you judge her a poor teacher and decide to take the class from someone else. What if, however, the instances you recalled were all from the same person, and this person happens to be a very colorful storyteller? The subsequent ease of remembering the instances might not indicate that the professor is a poor teacher after all.

Although the availability heuristic is obviously important, it is not the only judgment heuristic we use. Amos Tversky and Daniel Kahneman examined the role of heuristics in inductive reasoning in a long series of studies. Kahneman received a Nobel Prize in Economics for this research in 2002, and Tversky would have certainly received one as well if he had not died of melanoma at age 59 in 1996 (Nobel Prizes are not awarded posthumously). Kahneman and Tversky demonstrated repeatedly that people do not reason in ways that are consistent with the laws of probability. They identified several heuristic strategies that people use instead to make judgments about likelihood. The importance of this work for economics (and the reason that Kahneman was awarded the Nobel Prize) is that earlier economic theories had assumed that people do make judgments rationally, that is, in agreement with the laws of probability.

Another common heuristic that people use for making judgments is the  representativeness heuristic (Kahneman & Tversky 1973). Suppose we describe a person to you. He is quiet and shy, has an unassuming personality, and likes to work with numbers. Is this person more likely to be an accountant or an attorney? If you said accountant, you were probably using the representativeness heuristic. Our imaginary person is judged likely to be an accountant because he resembles, or is representative of the concept of, an accountant. When research participants are asked to make judgments such as these, the only thing that seems to matter is the representativeness of the description. For example, if told that the person described is in a room that contains 70 attorneys and 30 accountants, participants will still assume that he is an accountant.

inductive reasoning :  a type of reasoning in which we make judgments about likelihood from sets of evidence

inductively strong argument :  an inductive argument in which the beginning statements lead to a conclusion that is probably true

heuristic :  a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

availability heuristic :  judging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

representativeness heuristic:   judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

Type 1 thinking : fast, automatic, and emotional thinking.

Type 2 thinking : slow, effortful, and logical thinking.

• What percentage of workplace homicides are co-worker violence?

Many people get these questions wrong. The answers are 10%; stairs; skin; 6%. How close were your answers? Explain how the availability heuristic might have led you to make the incorrect judgments.

• Can you think of some other judgments that you have made (or beliefs that you have) that might have been influenced by the availability heuristic?

7.3 Problem Solving

• Please take a few minutes to list a number of problems that you are facing right now.
• Now write about a problem that you recently solved.
• What is your definition of a problem?

Mary has a problem. Her daughter, ordinarily quite eager to please, appears to delight in being the last person to do anything. Whether getting ready for school, going to piano lessons or karate class, or even going out with her friends, she seems unwilling or unable to get ready on time. Other people have different kinds of problems. For example, many students work at jobs, have numerous family commitments, and are facing a course schedule full of difficult exams, assignments, papers, and speeches. How can they find enough time to devote to their studies and still fulfill their other obligations? Speaking of students and their problems: Show that a ball thrown vertically upward with initial velocity v0 takes twice as much time to return as to reach the highest point (from Spiegel, 1981).

These are three very different situations, but we have called them all problems. What makes them all the same, despite the differences? A psychologist might define a  problem   as a situation with an initial state, a goal state, and a set of possible intermediate states. Somewhat more meaningfully, we might consider a problem a situation in which you are in here one state (e.g., daughter is always late), you want to be there in another state (e.g., daughter is not always late), and with no obvious way to get from here to there. Defined this way, each of the three situations we outlined can now be seen as an example of the same general concept, a problem. At this point, you might begin to wonder what is not a problem, given such a general definition. It seems that nearly every non-routine task we engage in could qualify as a problem. As long as you realize that problems are not necessarily bad (it can be quite fun and satisfying to rise to the challenge and solve a problem), this may be a useful way to think about it.

Can we identify a set of problem-solving skills that would apply to these very different kinds of situations? That task, in a nutshell, is a major goal of this section. Let us try to begin to make sense of the wide variety of ways that problems can be solved with an important observation: the process of solving problems can be divided into two key parts. First, people have to notice, comprehend, and represent the problem properly in their minds (called  problem representation ). Second, they have to apply some kind of solution strategy to the problem. Psychologists have studied both of these key parts of the process in detail.

When you first think about the problem-solving process, you might guess that most of our difficulties would occur because we are failing in the second step, the application of strategies. Although this can be a significant difficulty much of the time, the more important source of difficulty is probably problem representation. In short, we often fail to solve a problem because we are looking at it, or thinking about it, the wrong way.

problem :  a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

problem representation :  noticing, comprehending and forming a mental conception of a problem

Defining and Mentally Representing Problems in Order to Solve Them

So, the main obstacle to solving a problem is that we do not clearly understand exactly what the problem is. Recall the problem with Mary’s daughter always being late. One way to represent, or to think about, this problem is that she is being defiant. She refuses to get ready in time. This type of representation or definition suggests a particular type of solution. Another way to think about the problem, however, is to consider the possibility that she is simply being sidetracked by interesting diversions. This different conception of what the problem is (i.e., different representation) suggests a very different solution strategy. For example, if Mary defines the problem as defiance, she may be tempted to solve the problem using some kind of coercive tactics, that is, to assert her authority as her mother and force her to listen. On the other hand, if Mary defines the problem as distraction, she may try to solve it by simply removing the distracting objects.

As you might guess, when a problem is represented one way, the solution may seem very difficult, or even impossible. Seen another way, the solution might be very easy. For example, consider the following problem (from Nasar, 1998):

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 miles per hour. At the same time, a fly that travels at a steady 15 miles per hour starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner until he is crushed between the two front wheels. Question: what total distance did the fly cover?

Please take a few minutes to try to solve this problem.

Most people represent this problem as a question about a fly because, well, that is how the question is asked. The solution, using this representation, is to figure out how far the fly travels on the first leg of its journey, then add this total to how far it travels on the second leg of its journey (when it turns around and returns to the first bicycle), then continue to add the smaller distance from each leg of the journey until you converge on the correct answer. You would have to be quite skilled at math to solve this problem, and you would probably need some time and pencil and paper to do it.

If you consider a different representation, however, you can solve this problem in your head. Instead of thinking about it as a question about a fly, think about it as a question about the bicycles. They are 20 miles apart, and each is traveling 10 miles per hour. How long will it take for the bicycles to reach each other? Right, one hour. The fly is traveling 15 miles per hour; therefore, it will travel a total of 15 miles back and forth in the hour before the bicycles meet. Represented one way (as a problem about a fly), the problem is quite difficult. Represented another way (as a problem about two bicycles), it is easy. Changing your representation of a problem is sometimes the best—sometimes the only—way to solve it.

Unfortunately, however, changing a problem’s representation is not the easiest thing in the world to do. Often, problem solvers get stuck looking at a problem one way. This is called  fixation . Most people who represent the preceding problem as a problem about a fly probably do not pause to reconsider, and consequently change, their representation. A parent who thinks her daughter is being defiant is unlikely to consider the possibility that her behavior is far less purposeful.

Problem-solving fixation was examined by a group of German psychologists called Gestalt psychologists during the 1930’s and 1940’s. Karl Dunker, for example, discovered an important type of failure to take a different perspective called  functional fixedness . Imagine being a participant in one of his experiments. You are asked to figure out how to mount two candles on a door and are given an assortment of odds and ends, including a small empty cardboard box and some thumbtacks. Perhaps you have already figured out a solution: tack the box to the door so it forms a platform, then put the candles on top of the box. Most people are able to arrive at this solution. Imagine a slight variation of the procedure, however. What if, instead of being empty, the box had matches in it? Most people given this version of the problem do not arrive at the solution given above. Why? Because it seems to people that when the box contains matches, it already has a function; it is a matchbox. People are unlikely to consider a new function for an object that already has a function. This is functional fixedness.

Mental set is a type of fixation in which the problem solver gets stuck using the same solution strategy that has been successful in the past, even though the solution may no longer be useful. It is commonly seen when students do math problems for homework. Often, several problems in a row require the reapplication of the same solution strategy. Then, without warning, the next problem in the set requires a new strategy. Many students attempt to apply the formerly successful strategy on the new problem and therefore cannot come up with a correct answer.

The thing to remember is that you cannot solve a problem unless you correctly identify what it is to begin with (initial state) and what you want the end result to be (goal state). That may mean looking at the problem from a different angle and representing it in a new way. The correct representation does not guarantee a successful solution, but it certainly puts you on the right track.

A bit more optimistically, the Gestalt psychologists discovered what may be considered the opposite of fixation, namely  insight . Sometimes the solution to a problem just seems to pop into your head. Wolfgang Kohler examined insight by posing many different problems to chimpanzees, principally problems pertaining to their acquisition of out-of-reach food. In one version, a banana was placed outside of a chimpanzee’s cage and a short stick inside the cage. The stick was too short to retrieve the banana, but was long enough to retrieve a longer stick also located outside of the cage. This second stick was long enough to retrieve the banana. After trying, and failing, to reach the banana with the shorter stick, the chimpanzee would try a couple of random-seeming attempts, react with some apparent frustration or anger, then suddenly rush to the longer stick, the correct solution fully realized at this point. This sudden appearance of the solution, observed many times with many different problems, was termed insight by Kohler.

Lest you think it pertains to chimpanzees only, Karl Dunker demonstrated that children also solve problems through insight in the 1930s. More importantly, you have probably experienced insight yourself. Think back to a time when you were trying to solve a difficult problem. After struggling for a while, you gave up. Hours later, the solution just popped into your head, perhaps when you were taking a walk, eating dinner, or lying in bed.

fixation :  when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

functional fixedness :  a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

mental set :  a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

insight :  a sudden realization of a solution to a problem

Solving Problems by Trial and Error

Correctly identifying the problem and your goal for a solution is a good start, but recall the psychologist’s definition of a problem: it includes a set of possible intermediate states. Viewed this way, a problem can be solved satisfactorily only if one can find a path through some of these intermediate states to the goal. Imagine a fairly routine problem, finding a new route to school when your ordinary route is blocked (by road construction, for example). At each intersection, you may turn left, turn right, or go straight. A satisfactory solution to the problem (of getting to school) is a sequence of selections at each intersection that allows you to wind up at school.

If you had all the time in the world to get to school, you might try choosing intermediate states randomly. At one corner you turn left, the next you go straight, then you go left again, then right, then right, then straight. Unfortunately, trial and error will not necessarily get you where you want to go, and even if it does, it is not the fastest way to get there. For example, when a friend of ours was in college, he got lost on the way to a concert and attempted to find the venue by choosing streets to turn onto randomly (this was long before the use of GPS). Amazingly enough, the strategy worked, although he did end up missing two out of the three bands who played that night.

Trial and error is not all bad, however. B.F. Skinner, a prominent behaviorist psychologist, suggested that people often behave randomly in order to see what effect the behavior has on the environment and what subsequent effect this environmental change has on them. This seems particularly true for the very young person. Picture a child filling a household’s fish tank with toilet paper, for example. To a child trying to develop a repertoire of creative problem-solving strategies, an odd and random behavior might be just the ticket. Eventually, the exasperated parent hopes, the child will discover that many of these random behaviors do not successfully solve problems; in fact, in many cases they create problems. Thus, one would expect a decrease in this random behavior as a child matures. You should realize, however, that the opposite extreme is equally counterproductive. If the children become too rigid, never trying something unexpected and new, their problem solving skills can become too limited.

Effective problem solving seems to call for a happy medium that strikes a balance between using well-founded old strategies and trying new ground and territory. The individual who recognizes a situation in which an old problem-solving strategy would work best, and who can also recognize a situation in which a new untested strategy is necessary is halfway to success.

Solving Problems with Algorithms and Heuristics

For many problems there is a possible strategy available that will guarantee a correct solution. For example, think about math problems. Math lessons often consist of step-by-step procedures that can be used to solve the problems. If you apply the strategy without error, you are guaranteed to arrive at the correct solution to the problem. This approach is called using an  algorithm , a term that denotes the step-by-step procedure that guarantees a correct solution. Because algorithms are sometimes available and come with a guarantee, you might think that most people use them frequently. Unfortunately, however, they do not. As the experience of many students who have struggled through math classes can attest, algorithms can be extremely difficult to use, even when the problem solver knows which algorithm is supposed to work in solving the problem. In problems outside of math class, we often do not even know if an algorithm is available. It is probably fair to say, then, that algorithms are rarely used when people try to solve problems.

Because algorithms are so difficult to use, people often pass up the opportunity to guarantee a correct solution in favor of a strategy that is much easier to use and yields a reasonable chance of coming up with a correct solution. These strategies are called  problem solving heuristics . Similar to what you saw in section 6.2 with reasoning heuristics, a problem solving heuristic is a shortcut strategy that people use when trying to solve problems. It usually works pretty well, but does not guarantee a correct solution to the problem. For example, one problem solving heuristic might be “always move toward the goal” (so when trying to get to school when your regular route is blocked, you would always turn in the direction you think the school is). A heuristic that people might use when doing math homework is “use the same solution strategy that you just used for the previous problem.”

By the way, we hope these last two paragraphs feel familiar to you. They seem to parallel a distinction that you recently learned. Indeed, algorithms and problem-solving heuristics are another example of the distinction between Type 1 thinking and Type 2 thinking.

Although it is probably not worth describing a large number of specific heuristics, two observations about heuristics are worth mentioning. First, heuristics can be very general or they can be very specific, pertaining to a particular type of problem only. For example, “always move toward the goal” is a general strategy that you can apply to countless problem situations. On the other hand, “when you are lost without a functioning gps, pick the most expensive car you can see and follow it” is specific to the problem of being lost. Second, all heuristics are not equally useful. One heuristic that many students know is “when in doubt, choose c for a question on a multiple-choice exam.” This is a dreadful strategy because many instructors intentionally randomize the order of answer choices. Another test-taking heuristic, somewhat more useful, is “look for the answer to one question somewhere else on the exam.”

You really should pay attention to the application of heuristics to test taking. Imagine that while reviewing your answers for a multiple-choice exam before turning it in, you come across a question for which you originally thought the answer was c. Upon reflection, you now think that the answer might be b. Should you change the answer to b, or should you stick with your first impression? Most people will apply the heuristic strategy to “stick with your first impression.” What they do not realize, of course, is that this is a very poor strategy (Lilienfeld et al, 2009). Most of the errors on exams come on questions that were answered wrong originally and were not changed (so they remain wrong). There are many fewer errors where we change a correct answer to an incorrect answer. And, of course, sometimes we change an incorrect answer to a correct answer. In fact, research has shown that it is more common to change a wrong answer to a right answer than vice versa (Bruno, 2001).

The belief in this poor test-taking strategy (stick with your first impression) is based on the  confirmation bias   (Nickerson, 1998; Wason, 1960). You first saw the confirmation bias in Module 1, but because it is so important, we will repeat the information here. People have a bias, or tendency, to notice information that confirms what they already believe. Somebody at one time told you to stick with your first impression, so when you look at the results of an exam you have taken, you will tend to notice the cases that are consistent with that belief. That is, you will notice the cases in which you originally had an answer correct and changed it to the wrong answer. You tend not to notice the other two important (and more common) cases, changing an answer from wrong to right, and leaving a wrong answer unchanged.

Because heuristics by definition do not guarantee a correct solution to a problem, mistakes are bound to occur when we employ them. A poor choice of a specific heuristic will lead to an even higher likelihood of making an error.

algorithm :  a step-by-step procedure that guarantees a correct solution to a problem

problem solving heuristic :  a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

confirmation bias :  people’s tendency to notice information that confirms what they already believe

An Effective Problem-Solving Sequence

You may be left with a big question: If algorithms are hard to use and heuristics often don’t work, how am I supposed to solve problems? Robert Sternberg (1996), as part of his theory of what makes people successfully intelligent (Module 8) described a problem-solving sequence that has been shown to work rather well:

• Identify the existence of a problem.  In school, problem identification is often easy; problems that you encounter in math classes, for example, are conveniently labeled as problems for you. Outside of school, however, realizing that you have a problem is a key difficulty that you must get past in order to begin solving it. You must be very sensitive to the symptoms that indicate a problem.
• Define the problem.  Suppose you realize that you have been having many headaches recently. Very likely, you would identify this as a problem. If you define the problem as “headaches,” the solution would probably be to take aspirin or ibuprofen or some other anti-inflammatory medication. If the headaches keep returning, however, you have not really solved the problem—likely because you have mistaken a symptom for the problem itself. Instead, you must find the root cause of the headaches. Stress might be the real problem. For you to successfully solve many problems it may be necessary for you to overcome your fixations and represent the problems differently. One specific strategy that you might find useful is to try to define the problem from someone else’s perspective. How would your parents, spouse, significant other, doctor, etc. define the problem? Somewhere in these different perspectives may lurk the key definition that will allow you to find an easier and permanent solution.
• Formulate strategy.  Now it is time to begin planning exactly how the problem will be solved. Is there an algorithm or heuristic available for you to use? Remember, heuristics by their very nature guarantee that occasionally you will not be able to solve the problem. One point to keep in mind is that you should look for long-range solutions, which are more likely to address the root cause of a problem than short-range solutions.
• Represent and organize information.  Similar to the way that the problem itself can be defined, or represented in multiple ways, information within the problem is open to different interpretations. Suppose you are studying for a big exam. You have chapters from a textbook and from a supplemental reader, along with lecture notes that all need to be studied. How should you (represent and) organize these materials? Should you separate them by type of material (text versus reader versus lecture notes), or should you separate them by topic? To solve problems effectively, you must learn to find the most useful representation and organization of information.
• Allocate resources.  This is perhaps the simplest principle of the problem solving sequence, but it is extremely difficult for many people. First, you must decide whether time, money, skills, effort, goodwill, or some other resource would help to solve the problem Then, you must make the hard choice of deciding which resources to use, realizing that you cannot devote maximum resources to every problem. Very often, the solution to problem is simply to change how resources are allocated (for example, spending more time studying in order to improve grades).
• Monitor and evaluate solutions.  Pay attention to the solution strategy while you are applying it. If it is not working, you may be able to select another strategy. Another fact you should realize about problem solving is that it never does end. Solving one problem frequently brings up new ones. Good monitoring and evaluation of your problem solutions can help you to anticipate and get a jump on solving the inevitable new problems that will arise.

Please note that this as  an  effective problem-solving sequence, not  the  effective problem solving sequence. Just as you can become fixated and end up representing the problem incorrectly or trying an inefficient solution, you can become stuck applying the problem-solving sequence in an inflexible way. Clearly there are problem situations that can be solved without using these skills in this order.

Additionally, many real-world problems may require that you go back and redefine a problem several times as the situation changes (Sternberg et al. 2000). For example, consider the problem with Mary’s daughter one last time. At first, Mary did represent the problem as one of defiance. When her early strategy of pleading and threatening punishment was unsuccessful, Mary began to observe her daughter more carefully. She noticed that, indeed, her daughter’s attention would be drawn by an irresistible distraction or book. Fresh with a re-representation of the problem, she began a new solution strategy. She began to remind her daughter every few minutes to stay on task and remind her that if she is ready before it is time to leave, she may return to the book or other distracting object at that time. Fortunately, this strategy was successful, so Mary did not have to go back and redefine the problem again.

Pick one or two of the problems that you listed when you first started studying this section and try to work out the steps of Sternberg’s problem solving sequence for each one.

a mental representation of a category of things in the world

an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

knowledge about one’s own cognitive processes; thinking about your thinking

individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

Thinking like a scientist in your everyday life for the purpose of drawing correct conclusions. It entails skepticism; an ability to identify biases, distortions, omissions, and assumptions; and excellent deductive and inductive reasoning, and problem solving skills.

a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

an inclination, tendency, leaning, or prejudice

a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

a set of statements in which the beginning statements lead to a conclusion

an argument for which true beginning statements guarantee that the conclusion is true

a type of reasoning in which we make judgments about likelihood from sets of evidence

an inductive argument in which the beginning statements lead to a conclusion that is probably true

fast, automatic, and emotional thinking

slow, effortful, and logical thinking

a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

udging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

noticing, comprehending and forming a mental conception of a problem

when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

a sudden realization of a solution to a problem

a step-by-step procedure that guarantees a correct solution to a problem

The tendency to notice and pay attention to information that confirms your prior beliefs and to ignore information that disconfirms them.

a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

Introduction to Psychology Copyright © 2020 by Ken Gray; Elizabeth Arnott-Hill; and Or'Shaundra Benson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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What is Critical Thinking?

Critical thinking is the ability to think clearly and rationally, understanding the logical connection between ideas.  Critical thinking has been the subject of much debate and thought since the time of early Greek philosophers such as Plato and Socrates and has continued to be a subject of discussion into the modern age, for example the ability to recognise fake news .

Critical thinking might be described as the ability to engage in reflective and independent thinking.

In essence, critical thinking requires you to use your ability to reason. It is about being an active learner rather than a passive recipient of information.

Critical thinkers rigorously question ideas and assumptions rather than accepting them at face value. They will always seek to determine whether the ideas, arguments and findings represent the entire picture and are open to finding that they do not.

Critical thinkers will identify, analyse and solve problems systematically rather than by intuition or instinct.

Someone with critical thinking skills can:

Understand the links between ideas.

Determine the importance and relevance of arguments and ideas.

Recognise, build and appraise arguments.

Identify inconsistencies and errors in reasoning.

Approach problems in a consistent and systematic way.

Reflect on the justification of their own assumptions, beliefs and values.

Critical thinking is thinking about things in certain ways so as to arrive at the best possible solution in the circumstances that the thinker is aware of. In more everyday language, it is a way of thinking about whatever is presently occupying your mind so that you come to the best possible conclusion.

Critical Thinking is:

A way of thinking about particular things at a particular time; it is not the accumulation of facts and knowledge or something that you can learn once and then use in that form forever, such as the nine times table you learn and use in school.

The Skills We Need for Critical Thinking

The skills that we need in order to be able to think critically are varied and include observation, analysis, interpretation, reflection, evaluation, inference, explanation, problem solving, and decision making.

Specifically we need to be able to:

Think about a topic or issue in an objective and critical way.

Identify the different arguments there are in relation to a particular issue.

Evaluate a point of view to determine how strong or valid it is.

Recognise any weaknesses or negative points that there are in the evidence or argument.

Notice what implications there might be behind a statement or argument.

Provide structured reasoning and support for an argument that we wish to make.

The Critical Thinking Process

You should be aware that none of us think critically all the time.

Sometimes we think in almost any way but critically, for example when our self-control is affected by anger, grief or joy or when we are feeling just plain ‘bloody minded’.

On the other hand, the good news is that, since our critical thinking ability varies according to our current mindset, most of the time we can learn to improve our critical thinking ability by developing certain routine activities and applying them to all problems that present themselves.

Once you understand the theory of critical thinking, improving your critical thinking skills takes persistence and practice.

Try this simple exercise to help you to start thinking critically.

Think of something that someone has recently told you. Then ask yourself the following questions:

Who said it?

Someone you know? Someone in a position of authority or power? Does it matter who told you this?

What did they say?

Did they give facts or opinions? Did they provide all the facts? Did they leave anything out?

Where did they say it?

Was it in public or in private? Did other people have a chance to respond an provide an alternative account?

When did they say it?

Was it before, during or after an important event? Is timing important?

Why did they say it?

Did they explain the reasoning behind their opinion? Were they trying to make someone look good or bad?

How did they say it?

Were they happy or sad, angry or indifferent? Did they write it or say it? Could you understand what was said?

What are you Aiming to Achieve?

One of the most important aspects of critical thinking is to decide what you are aiming to achieve and then make a decision based on a range of possibilities.

Once you have clarified that aim for yourself you should use it as the starting point in all future situations requiring thought and, possibly, further decision making. Where needed, make your workmates, family or those around you aware of your intention to pursue this goal. You must then discipline yourself to keep on track until changing circumstances mean you have to revisit the start of the decision making process.

However, there are things that get in the way of simple decision making. We all carry with us a range of likes and dislikes, learnt behaviours and personal preferences developed throughout our lives; they are the hallmarks of being human. A major contribution to ensuring we think critically is to be aware of these personal characteristics, preferences and biases and make allowance for them when considering possible next steps, whether they are at the pre-action consideration stage or as part of a rethink caused by unexpected or unforeseen impediments to continued progress.

The more clearly we are aware of ourselves, our strengths and weaknesses, the more likely our critical thinking will be productive.

The Benefit of Foresight

Perhaps the most important element of thinking critically is foresight.

Almost all decisions we make and implement don’t prove disastrous if we find reasons to abandon them. However, our decision making will be infinitely better and more likely to lead to success if, when we reach a tentative conclusion, we pause and consider the impact on the people and activities around us.

The elements needing consideration are generally numerous and varied. In many cases, consideration of one element from a different perspective will reveal potential dangers in pursuing our decision.

For instance, moving a business activity to a new location may improve potential output considerably but it may also lead to the loss of skilled workers if the distance moved is too great. Which of these is the more important consideration? Is there some way of lessening the conflict?

These are the sort of problems that may arise from incomplete critical thinking, a demonstration perhaps of the critical importance of good critical thinking.

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In Summary:

Critical thinking is aimed at achieving the best possible outcomes in any situation. In order to achieve this it must involve gathering and evaluating information from as many different sources possible.

Critical thinking requires a clear, often uncomfortable, assessment of your personal strengths, weaknesses and preferences and their possible impact on decisions you may make.

Critical thinking requires the development and use of foresight as far as this is possible. As Doris Day sang, “the future’s not ours to see”.

Implementing the decisions made arising from critical thinking must take into account an assessment of possible outcomes and ways of avoiding potentially negative outcomes, or at least lessening their impact.

• Critical thinking involves reviewing the results of the application of decisions made and implementing change where possible.

It might be thought that we are overextending our demands on critical thinking in expecting that it can help to construct focused meaning rather than examining the information given and the knowledge we have acquired to see if we can, if necessary, construct a meaning that will be acceptable and useful.

After all, almost no information we have available to us, either externally or internally, carries any guarantee of its life or appropriateness.  Neat step-by-step instructions may provide some sort of trellis on which our basic understanding of critical thinking can blossom but it doesn’t and cannot provide any assurance of certainty, utility or longevity.

Continue to: Critical Thinking and Fake News Critical Reading

See also: Analytical Skills Understanding and Addressing Conspiracy Theories Introduction to Neuro-Linguistic Programming (NLP)

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Critical Thinking Is About Asking Better Questions

• John Coleman

Six practices to sharpen your inquiry.

Critical thinking is the ability to analyze and effectively break down an issue in order to make a decision or find a solution. At the heart of critical thinking is the ability to formulate deep, different, and effective questions. For effective questioning, start by holding your hypotheses loosely. Be willing to fundamentally reconsider your initial conclusions — and do so without defensiveness. Second, listen more than you talk through active listening. Third, leave your queries open-ended, and avoid yes-or-no questions. Fourth, consider the counterintuitive to avoid falling into groupthink. Fifth, take the time to stew in a problem, rather than making decisions unnecessarily quickly. Last, ask thoughtful, even difficult, follow-ups.

Are you tackling a new and difficult problem at work? Recently promoted and trying to both understand your new role and bring a fresh perspective? Or are you new to the workforce and seeking ways to meaningfully contribute alongside your more experienced colleagues? If so, critical thinking — the ability to analyze and effectively break down an issue in order to make a decision or find a solution — will be core to your success. And at the heart of critical thinking is the ability to formulate deep, different, and effective questions.

• JC John Coleman is the author of the HBR Guide to Crafting Your Purpose . Subscribe to his free newsletter, On Purpose , or contact him at johnwilliamcoleman.com . johnwcoleman

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Defining Critical Thinking and Problem Solving

Critical thinking, what are critical thinking and problem-solving rubrics.

We identify three constructs of critical thinking and problem solving that all of our students develop: effective reasoning, decision making, and problem solving. Each of these constructs is defined in a way to be applicable across disciplines, but tied to a cognitive routine that gives structure to thinking processes. By defining these skills in this way, we allow students to develop habits of mind for thinking that they can transfer to a wide variety of settings.

How do you use critical thinking and problem-solving rubrics?

We utilize these rubrics in two core ways: to define the constructs and as a tool for formative assessment.

​The rubrics define the constructs of effective reasoning, problem-solving, and decision making for teachers, students, and families. Talking about critical thinking and problem solving can feel like an amorphous thing, so we have found that it is important to provide language that describes the specific steps and types of thinking we want students to accomplish. Rubrics provide that language.

Beyond just defining the construct, the rubric gives a finer grained detail of where a student is in their development of the cognitive skills that we are targeting. By breaking down specific descriptors of where a student is on a continuum of growth in each component of their cognitive skills, teachers can direct instruction to best meet a student where they are and push them to improve their thinking.

How did you create these rubrics?

To create the critical thinking and problem-solving rubrics, we completed a review of the relevant literature around 21st Century Skills and Deeper Learning. As a community, we identified the three areas of critical thinking and problem-solving on which we would focus our efforts: effective reasoning, problem-solving, and decision making.

With those three constructs identified, we reviewed rubrics and research from the Partnership for 21st Century Skills , the Buck Institute for Education , Next Generation Learning Challenges: MyWays Reports , Catalina Foothills School District’s Resources for Deep Learning , Laura Greenstein’s work in Assessing 21st Century Skills , and many others. Building on that work, we crafted our own rubrics aligned with each of our constructs. We then tried them out in classrooms with teachers giving assignments that aligned to the rubrics. We refined the rubrics based on teacher feedback. Finally in partnership with the Stanford Center on Assessment, Learning, and Equity (SCALE), we further refined the rubrics to best define the constructs.

Home > Blog > Tips for Online Students > Why Is Critical Thinking Important and How to Improve It

Tips for Online Students , Tips for Students

Why Is Critical Thinking Important and How to Improve It

Updated: July 8, 2024

Published: April 2, 2020

Why is critical thinking important? The decisions that you make affect your quality of life. And if you want to ensure that you live your best, most successful and happy life, you’re going to want to make conscious choices. That can be done with a simple thing known as critical thinking. Here’s how to improve your critical thinking skills and make decisions that you won’t regret.

What Is Critical Thinking?

Critical thinking is the process of analyzing facts to form a judgment. Essentially, it involves thinking about thinking. Historically, it dates back to the teachings of Socrates , as documented by Plato.

Today, it is seen as a complex concept understood best by philosophers and psychologists. Modern definitions include “reasonable, reflective thinking focused on deciding what to believe or do” and “deciding what’s true and what you should do.”

The Importance Of Critical Thinking

Why is critical thinking important? Good question! Here are a few undeniable reasons why it’s crucial to have these skills.

1. Critical Thinking Is Universal

Critical thinking is a domain-general thinking skill. What does this mean? It means that no matter what path or profession you pursue, these skills will always be relevant and will always be beneficial to your success. They are not specific to any field.

2. Crucial For The Economy

Our future depends on technology, information, and innovation. Critical thinking is needed for our fast-growing economies, to solve problems as quickly and as effectively as possible.

3. Improves Language & Presentation Skills

In order to best express ourselves, we need to know how to think clearly and systematically — meaning practice critical thinking! Critical thinking also means knowing how to break down texts, and in turn, improve our ability to comprehend.

4. Promotes Creativity

By practicing critical thinking, we are allowing ourselves not only to solve problems but also to come up with new and creative ideas to do so. Critical thinking allows us to analyze these ideas and adjust them accordingly.

5. Important For Self-Reflection

Without critical thinking, how can we really live a meaningful life? We need this skill to self-reflect and justify our ways of life and opinions. Critical thinking provides us with the tools to evaluate ourselves in the way that we need to.

Photo by Marcelo Chagas from Pexels

6. the basis of science & democracy.

In order to have a democracy and to prove scientific facts, we need critical thinking in the world. Theories must be backed up with knowledge. In order for a society to effectively function, its citizens need to establish opinions about what’s right and wrong (by using critical thinking!).

Benefits Of Critical Thinking

We know that critical thinking is good for society as a whole, but what are some benefits of critical thinking on an individual level? Why is critical thinking important for us?

1. Key For Career Success

Critical thinking is crucial for many career paths. Not just for scientists, but lawyers , doctors, reporters, engineers , accountants, and analysts (among many others) all have to use critical thinking in their positions. In fact, according to the World Economic Forum, critical thinking is one of the most desirable skills to have in the workforce, as it helps analyze information, think outside the box, solve problems with innovative solutions, and plan systematically.

2. Better Decision Making

There’s no doubt about it — critical thinkers make the best choices. Critical thinking helps us deal with everyday problems as they come our way, and very often this thought process is even done subconsciously. It helps us think independently and trust our gut feeling.

3. Can Make You Happier!

While this often goes unnoticed, being in touch with yourself and having a deep understanding of why you think the way you think can really make you happier. Critical thinking can help you better understand yourself, and in turn, help you avoid any kind of negative or limiting beliefs, and focus more on your strengths. Being able to share your thoughts can increase your quality of life.

4. Form Well-Informed Opinions

There is no shortage of information coming at us from all angles. And that’s exactly why we need to use our critical thinking skills and decide for ourselves what to believe. Critical thinking allows us to ensure that our opinions are based on the facts, and help us sort through all that extra noise.

5. Better Citizens

One of the most inspiring critical thinking quotes is by former US president Thomas Jefferson: “An educated citizenry is a vital requisite for our survival as a free people.” What Jefferson is stressing to us here is that critical thinkers make better citizens, as they are able to see the entire picture without getting sucked into biases and propaganda.

6. Improves Relationships

While you may be convinced that being a critical thinker is bound to cause you problems in relationships, this really couldn’t be less true! Being a critical thinker can allow you to better understand the perspective of others, and can help you become more open-minded towards different views.

7. Promotes Curiosity

Critical thinkers are constantly curious about all kinds of things in life, and tend to have a wide range of interests. Critical thinking means constantly asking questions and wanting to know more, about why, what, who, where, when, and everything else that can help them make sense of a situation or concept, never taking anything at face value.

8. Allows For Creativity

Critical thinkers are also highly creative thinkers, and see themselves as limitless when it comes to possibilities. They are constantly looking to take things further, which is crucial in the workforce.

9. Enhances Problem Solving Skills

Those with critical thinking skills tend to solve problems as part of their natural instinct. Critical thinkers are patient and committed to solving the problem, similar to Albert Einstein, one of the best critical thinking examples, who said “It’s not that I’m so smart; it’s just that I stay with problems longer.” Critical thinkers’ enhanced problem-solving skills makes them better at their jobs and better at solving the world’s biggest problems. Like Einstein, they have the potential to literally change the world.

10. An Activity For The Mind

Just like our muscles, in order for them to be strong, our mind also needs to be exercised and challenged. It’s safe to say that critical thinking is almost like an activity for the mind — and it needs to be practiced. Critical thinking encourages the development of many crucial skills such as logical thinking, decision making, and open-mindness.

11. Creates Independence

When we think critically, we think on our own as we trust ourselves more. Critical thinking is key to creating independence, and encouraging students to make their own decisions and form their own opinions.

12. Crucial Life Skill

Critical thinking is crucial not just for learning, but for life overall! Education isn’t just a way to prepare ourselves for life, but it’s pretty much life itself. Learning is a lifelong process that we go through each and every day.

How To Improve Your Critical Thinking

Now that you know the benefits of thinking critically, how do you actually do it?

• Define Your Question: When it comes to critical thinking, it’s important to always keep your goal in mind. Know what you’re trying to achieve, and then figure out how to best get there.
• Gather Reliable Information: Make sure that you’re using sources you can trust — biases aside. That’s how a real critical thinker operates!
• Ask The Right Questions: We all know the importance of questions, but be sure that you’re asking the right questions that are going to get you to your answer.
• Look Short & Long Term: When coming up with solutions, think about both the short- and long-term consequences. Both of them are significant in the equation.
• Explore All Sides: There is never just one simple answer, and nothing is black or white. Explore all options and think outside of the box before you come to any conclusions.

How Is Critical Thinking Developed At School?

Critical thinking is developed in nearly everything we do, but much of this essential skill is encouraged and practiced in school. Fostering a culture of inquiry is crucial, encouraging students to ask questions, analyze information, and evaluate evidence.

Teaching strategies like Socratic questioning, problem-based learning, and collaborative discussions help students think for themselves. When teachers ask questions, students can respond critically and reflect on their learning. Group discussions also expand their thinking, making them independent thinkers and effective problem solvers.

How Does Critical Thinking Apply To Your Career?

Critical thinking is a valuable asset in any career. Employers value employees who can think critically, ask insightful questions, and offer creative solutions. Demonstrating critical thinking skills can set you apart in the workplace, showing your ability to tackle complex problems and make informed decisions.

In many careers, from law and medicine to business and engineering, critical thinking is essential. Lawyers analyze cases, doctors diagnose patients, business analysts evaluate market trends, and engineers solve technical issues—all requiring strong critical thinking skills.

Critical thinking also enhances your ability to communicate effectively, making you a better team member and leader. By analyzing and evaluating information, you can present clear, logical arguments and make persuasive presentations.

Incorporating critical thinking into your career helps you stay adaptable and innovative. It encourages continuous learning and improvement, which are crucial for professional growth and success in a rapidly changing job market.

Photo by Oladimeji Ajegbile from Pexels

Critical thinking is a vital skill with far-reaching benefits for personal and professional success. It involves systematic skills such as analysis, evaluation, inference, interpretation, and explanation to assess information and arguments.

By gathering relevant data, considering alternative perspectives, and using logical reasoning, critical thinking enables informed decision-making. Reflecting on and refining these processes further enhances their effectiveness.

The future of critical thinking holds significant importance as it remains essential for adapting to evolving challenges and making sound decisions in various aspects of life.

What are the benefits of developing critical thinking skills?

Critical thinking enhances decision-making, problem-solving, and the ability to evaluate information critically. It helps in making informed decisions, understanding others’ perspectives, and improving overall cognitive abilities.

How does critical thinking contribute to problem-solving abilities?

Critical thinking enables you to analyze problems thoroughly, consider multiple solutions, and choose the most effective approach. It fosters creativity and innovative thinking in finding solutions.

What role does critical thinking play in academic success?

Critical thinking is crucial in academics as it allows you to analyze texts, evaluate evidence, construct logical arguments, and understand complex concepts, leading to better academic performance.

How does critical thinking promote effective communication skills?

Critical thinking helps you articulate thoughts clearly, listen actively, and engage in meaningful discussions. It improves your ability to argue logically and understand different viewpoints.

How can critical thinking skills be applied in everyday situations?

You can use critical thinking to make better personal and professional decisions, solve everyday problems efficiently, and understand the world around you more deeply.

What role does skepticism play in critical thinking?

Skepticism encourages questioning assumptions, evaluating evidence, and distinguishing between facts and opinions. It helps in developing a more rigorous and open-minded approach to thinking.

What strategies can enhance critical thinking?

Strategies include asking probing questions, engaging in reflective thinking, practicing problem-solving, seeking diverse perspectives, and analyzing information critically and logically.

In this article

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The Peak Performance Center

The pursuit of performance excellence, critical thinking.

Critical thinking refers to the process of actively analyzing, assessing, synthesizing, evaluating and reflecting on information gathered from observation, experience, or communication. It is thinking in a clear, logical, reasoned, and reflective manner to solve problems or make decisions. Basically, critical thinking is taking a hard look at something to understand what it really means.

Critical Thinkers

Critical thinkers do not simply accept all ideas, theories, and conclusions as facts. They have a mindset of questioning ideas and conclusions. They make reasoned judgments that are logical and well thought out by assessing the evidence that supports a specific theory or conclusion.

When presented with a new piece of new information, critical thinkers may ask questions such as;

“What information supports that?”

“How was this information obtained?”

“Who obtained the information?”

“How do we know the information is valid?”

“Why is it that way?”

“What makes it do that?”

“How do we know that?”

“Are there other possibilities?”

Combination of Analytical and Creative Thinking

Many people perceive critical thinking just as analytical thinking. However, critical thinking incorporates both analytical thinking and creative thinking. Critical thinking does involve breaking down information into parts and analyzing the parts in a logical, step-by-step manner. However, it also involves challenging consensus to formulate new creative ideas and generate innovative solutions. It is critical thinking that helps to evaluate and improve your creative ideas.

Elements of Critical Thinking

Critical thinking involves:

• Gathering relevant information
• Evaluating information
• Asking questions
• Assessing bias or unsubstantiated assumptions
• Making inferences from the information and filling in gaps
• Using abstract ideas to interpret information
• Formulating ideas
• Weighing opinions
• Reaching well-reasoned conclusions
• Considering alternative possibilities
• Testing conclusions
• Verifying if evidence/argument support the conclusions

Developing Critical Thinking Skills

Critical thinking is considered a higher order thinking skills, such as analysis, synthesis, deduction, inference, reason, and evaluation. In order to demonstrate critical thinking, you would need to develop skills in;

Interpreting : understanding the significance or meaning of information

Analyzing : breaking information down into its parts

Connecting : making connections between related items or pieces of information.

Integrating : connecting and combining information to better understand the relationship between the information.

Evaluating : judging the value, credibility, or strength of something

Reasoning : creating an argument through logical steps

Deducing : forming a logical opinion about something based on the information or evidence that is available

Inferring : figuring something out through reasoning based on assumptions and ideas

Generating : producing new information, ideas, products, or ways of viewing things.

Blooms Taxonomy

Bloom’s Taxonomy Revised

Mind Mapping

Chunking Information

Brainstorming

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What is the Critical Thinking Test?

Critical thinking practice test, take a free practice critical thinking test, practice critical thinking test.

Updated November 16, 2023

The Critical Thinking Test is a comprehensive evaluation designed to assess individuals' cognitive capacities and analytical prowess.

This formal examination, often referred to as the critical thinking assessment, is a benchmark for those aiming to demonstrate their proficiency in discernment and problem-solving.

In addition, this evaluative tool meticulously gauges a range of skills, including logical reasoning, analytical thinking, and the ability to evaluate and synthesize information.

This article will embark on an exploration of the Critical Thinking Test, elucidating its intricacies and elucidating its paramount importance. We will dissect the essential skills it measures and clarify its significance in gauging one's intellectual aptitude.

We will examine examples of critical thinking questions, illuminating the challenging scenarios that candidates encounter prompting them to navigate the complexities of thought with finesse.

Before going ahead to take the critical thinking test, let's delve into the realm of preparation. This segment serves as a crucible for honing the skills assessed in the actual examination, offering candidates a chance to refine their analytical blades before facing the real challenge. Here are some skills that will help you with the critical thinking assessment: Logical Reasoning: The practice test meticulously evaluates your ability to deduce conclusions from given information, assess the validity of arguments, and recognize patterns in logic. Analytical Thinking: Prepare to dissect complex scenarios, identify key components, and synthesize information to draw insightful conclusions—a fundamental aspect of the critical thinking assessment. Problem-Solving Proficiency: Navigate through intricate problems that mirror real-world challenges, honing your capacity to approach issues systematically and derive effective solutions. What to Expect: The Critical Thinking Practice Test is crafted to mirror the format and complexity of the actual examination. Expect a series of scenarios, each accompanied by a set of questions that demand thoughtful analysis and logical deduction. These scenarios span diverse fields, from business and science to everyday scenarios, ensuring a comprehensive evaluation of your critical thinking skills. Examples of Critical Thinking Questions Scenario: In a business context, analyze the potential impacts of a proposed strategy on both short-term profitability and long-term sustainability. Question: What factors would you consider in determining the viability of the proposed strategy, and how might it affect the company's overall success? Scenario: Evaluate conflicting scientific studies on a pressing environmental issue.

Question: Identify the key methodologies and data points in each study. How would you reconcile the disparities to form an informed, unbiased conclusion?

Why Practice Matters

Engaging in the Critical Thinking Practice Test familiarizes you with the test format and cultivates a mindset geared towards agile and astute reasoning. This preparatory phase allows you to refine your cognitive toolkit, ensuring you approach the assessment with confidence and finesse.

We'll navigate through specific examples as we proceed, offering insights into effective strategies for tackling critical thinking questions. Prepare to embark on a journey of intellectual sharpening, where each practice question refines your analytical prowess for the challenges ahead.

This is a practice critical thinking test.

The test consists of three questions . 

After you have answered all the questions, you will be shown the correct answers and given full explanations.

Make sure you read and fully understand each question before answering. Work quickly, but don't rush. You cannot afford to make mistakes on a real test .

If you get a question wrong, make sure you find out why and learn how to answer this type of question in the future. 

Six friends are seated in a restaurant across a rectangular table. There are three chairs on each side. Adam and Dorky do not have anyone sitting to their right and Clyde and Benjamin do not have anyone sitting to their left. Adam and Benjamin are not sitting on the same side of the table.

If Ethan is not sitting next to Dorky, who is seated immediately to the left of Felix?

You might also be interested in these other PRT articles:

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Critical Thinking: A Simple Guide and Why It’s Important

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Strong critical thinking skills are crucial for career success, regardless of educational background. It embodies the ability to engage in astute and effective decision-making, lending invaluable dimensions to professional growth.

At its essence, critical thinking is the ability to analyze, evaluate, and synthesize information in a logical and reasoned manner. It’s not merely about accumulating knowledge but harnessing it effectively to make informed decisions and solve complex problems. In the dynamic landscape of modern careers, honing this skill is paramount.

The Impact of Critical Thinking on Your Career

☑️ problem-solving mastery.

Visualize critical thinking as the Sherlock Holmes of your career journey. It facilitates swift problem resolution akin to a detective unraveling a mystery. By methodically analyzing situations and deconstructing complexities, critical thinkers emerge as adept problem solvers, rendering them invaluable assets in the workplace.

☑️ Refined Decision-Making

Navigating dilemmas in your career path resembles traversing uncertain terrain. Critical thinking acts as a dependable GPS, steering you toward informed decisions. It involves weighing options, evaluating potential outcomes, and confidently choosing the most favorable path forward.

☑️ Enhanced Teamwork Dynamics

Within collaborative settings, critical thinkers stand out as proactive contributors. They engage in scrutinizing ideas, proposing enhancements, and fostering meaningful contributions. Consequently, the team evolves into a dynamic hub of ideas, with the critical thinker recognized as the architect behind its success.

☑️ Communication Prowess

Effective communication is the cornerstone of professional interactions. Critical thinking enriches communication skills, enabling the clear and logical articulation of ideas. Whether in emails, presentations, or casual conversations, individuals adept in critical thinking exude clarity, earning appreciation for their ability to convey thoughts seamlessly.

☑️ Adaptability and Resilience

Perceptive individuals adept in critical thinking display resilience in the face of unforeseen challenges. Instead of succumbing to panic, they assess situations, recalibrate their approaches, and persist in moving forward despite adversity.

☑️ Fostering Innovation

Innovation is the lifeblood of progressive organizations, and critical thinking serves as its catalyst. Proficient critical thinkers possess the ability to identify overlooked opportunities, propose inventive solutions, and streamline processes, thereby positioning their organizations at the forefront of innovation.

☑️ Confidence Amplification

Critical thinkers exude confidence derived from honing their analytical skills. This self-assurance radiates during job interviews, presentations, and daily interactions, catching the attention of superiors and propelling career advancement.

So, how can one cultivate and harness this invaluable skill?

✅ developing curiosity and inquisitiveness:.

Embrace a curious mindset by questioning the status quo and exploring topics beyond your immediate scope. Cultivate an inquisitive approach to everyday situations. Encourage a habit of asking “why” and “how” to deepen understanding. Curiosity fuels the desire to seek information and alternative perspectives.

✅ Practice Reflection and Self-Awareness:

Engage in reflective thinking by assessing your thoughts, actions, and decisions. Regularly introspect to understand your biases, assumptions, and cognitive processes. Cultivate self-awareness to recognize personal prejudices or cognitive biases that might influence your thinking. This allows for a more objective analysis of situations.

✅ Strengthening Analytical Skills:

Practice breaking down complex problems into manageable components. Analyze each part systematically to understand the whole picture. Develop skills in data analysis, statistics, and logical reasoning. This includes understanding correlation versus causation, interpreting graphs, and evaluating statistical significance.

✅ Engaging in Active Listening and Observation:

Actively listen to diverse viewpoints without immediately forming judgments. Allow others to express their ideas fully before responding. Observe situations attentively, noticing details that others might overlook. This habit enhances your ability to analyze problems more comprehensively.

✅ Encouraging Intellectual Humility and Open-Mindedness:

Foster intellectual humility by acknowledging that you don’t know everything. Be open to learning from others, regardless of their position or expertise. Cultivate open-mindedness by actively seeking out perspectives different from your own. Engage in discussions with people holding diverse opinions to broaden your understanding.

✅ Practicing Problem-Solving and Decision-Making:

Engage in regular problem-solving exercises that challenge you to think creatively and analytically. This can include puzzles, riddles, or real-world scenarios. When making decisions, consciously evaluate available information, consider various alternatives, and anticipate potential outcomes before reaching a conclusion.

✅ Continuous Learning and Exposure to Varied Content:

Read extensively across diverse subjects and formats, exposing yourself to different viewpoints, cultures, and ways of thinking. Engage in courses, workshops, or seminars that stimulate critical thinking skills. Seek out opportunities for learning that challenge your existing beliefs.

✅ Engage in Constructive Disagreement and Debate:

Encourage healthy debates and discussions where differing opinions are respectfully debated.

This practice fosters the ability to defend your viewpoints logically while also being open to changing your perspective based on valid arguments. Embrace disagreement as an opportunity to learn rather than a conflict to win. Engaging in constructive debate sharpens your ability to evaluate and counter-arguments effectively.

✅ Utilize Problem-Based Learning and Real-World Applications:

Engage in problem-based learning activities that simulate real-world challenges. Work on projects or scenarios that require critical thinking skills to develop practical problem-solving approaches. Apply critical thinking in real-life situations whenever possible.

This could involve analyzing news articles, evaluating product reviews, or dissecting marketing strategies to understand their underlying rationale.

In conclusion, critical thinking is the linchpin of a successful career journey. It empowers individuals to navigate complexities, make informed decisions, and innovate in their respective domains. Embracing and honing this skill isn’t just an advantage; it’s a necessity in a world where adaptability and sound judgment reign supreme.

So, as you traverse your career path, remember that the ability to think critically is not just an asset but the differentiator that propels you toward excellence.

Ivy Exec is the premier resource for professionals seeking career advancement. Whether you are on the job, or looking for your next one - Ivy Exec has the tools you need.

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Accenture Critical Reasoning and Problem Solving Previous Year Questions

Explanation:  As per the Venn diagram, conclusion I and II follows

Explanation:  In a new study, researchers led by Peak identify a likely cause for the missing rocks: the breakup of Earth's ancient supercontinent Rodina approximately 700 million years ago. The upheaval was so violent it likely washed rocks and sediment into the ocean — one billion years’ worth, in the case of the Grand Canyon, option D is correct.

Question: What is the Cost Price of flour?  Statements:  I. Kevin mixes flour and sugar in the ratio 3:5 to make pretzels. He sells pretzels at $10 per kg.  II. The ratio of price of sugar and flour is 9:4 (per kg) and he earns 66(?) % profit.  

A. Statements I and II together are not sufficient to answer the question asked and additional data to the problem is needed  B. Each statement alone is sufficient to answer the question  C. Only one of the statements, alone, is sufficient to answer the question but other statement is not  D. Both statements I and II together are sufficient to answer the question asked but neither statement alone is sufficient

Using statement III. Sheldon is sitting to the immediate left of Jack.

By statement II. Justin is the only person sitting between Bruce and Peter.

Hence, Nicholas and Justin are not immediate neighbours.

Explanation:  From, N got the minimum rating and the rating was an even number, therefore the minimum even number between 1 to 10 is 2, N rank is 2

III. Only two participants got ratings in even numbers.

IV. The rating of O was double the rating of Q.

Possible values for

O    Q

4      2

6      3

8      4

10    5

Participants from New York got the highest ranking, but was not O, as O is not highest 10 and 5 isn’t possible, and only two participants got even N is already even rank hence 4 and 8 , 2 and 4 not possible

So the rank of O is 6 and Q is 3

O got a higher rating than M, values possible for M less than 6 are 5,4,3,2,1 as N is the minimum rank 2 and 1 can’t be taken and Q is 3 , 4 is even so only possibility for M is 5

R was from London. Participants from New York got the highest ranking, Hence R is not highest possible ranks left are 9,7 ( even numbers can’t be taken) R will be 7 and the left P will be 9 and belongs to New York.

The second highest rating given is 7.

Explanation:  P belongs to New York as per above table.

Explanation:  the rating of coffee prepared by Q is 3 as per above table.

Explanation:  M got a rating of 5 is definitely true.

Explanation:  As per the statement gatherings could turn  in to COVID super spread , so only Conclusion I follows, decision on Conclusion II can’t be taken as the statement doesn’t discuss about vaccination.

Explanation:  Only Conclusion II follows , anything about pollution isn’t discussed in the statement hence conclusion I doesn’t follow

If all cats are mammals and some mammals are birds, which of the following must be true?

A. Some cats are birds. B. All mammals are cats. C. Some birds are cats. D. None of the above.

Explanation:  The statements only tell us that all cats are mammals and some mammals are birds. They do not imply any direct relationship between cats and birds.

John, Kate, and Alex are sitting in a row. Kate is not next to John. John is not next to Alex. How many possible orders can they be seated in?

A. 1 B. 2 C. 3 D. 4

Explanation: T he valid arrangements are Alex-Kate-John and John-Kate-Alex.

In a group of people, 70% have a car, 60% have a house, and 50% have both. What percentage of people have either a car or a house or both?

A. 80% B. 70% C. 60% D. 50%

Explanation:  Using the principle of inclusion and exclusion: P(A∪B)=P(A)+P(B)−P(A∩B)=70P(A \cup B) = P(A) + P(B) - P(A \cap B) = 70% + 60% - 50% = 80%P(A∪B)=P(A)+P(B)−P(A∩B)=70.

A man walks 3 km north, then turns right and walks 4 km, then turns left and walks 5 km. Which direction is he facing now?

A. North B. South C. East D. West

Explanation:  He walks north, then turns right (east), then turns left (north) and continues walking north.

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Critical Thinking Test: Online Preparation & Free Practice Questions – 2024

• Information
• Free Example Questions

What Is Critical Thinking?

Critical thinking is a form of decision making and reasoning using data and observations. Someone who is a strong critical thinker can find quality solutions efficiently and can evaluate issues objectively.

What Is a Critical Thinking Test?

Critical thinking tests provide companies valuable insight into the leadership, reasoning, and overall capabilities of candidates. Because strong critical thinking skills are highly sought after, the critical thinking test can be applicable to any field and discipline across multiple levels of expertise from recent graduate to executive. However, it is commonly administered to those applying for criminal justice and business-related occupations.

Job seekers with upcoming critical thinking tests will be evaluated on more than their ability to rationalize, critical thinking tests also measure the following subsets:

• Organizing & Planning
• Strategizing
• Decision Making
• Problem Solving

The format of the critical thinking uses hypothetical scenarios to assess candidates. The scenarios are typically relevant to the field you are interested in to assess your knowledge of the role. There will also be general questions concerning more basic issues or problems that commonly occur in a workplace environment.

The critical thinking test is multiple-choice with thirty minutes to complete the assessment. Candidates will receive a notification stating whether or not they passed within a week of completion.

How Is the Critical Thinking Test Scored?

The critical reasoning test is scored based on your raw score and your percentile in comparison with your norm group. It’s important to note that these will not be the same number.

A norm group is a collection of scores from individuals in your field at your level of experience. The percentile score is used to alert employers if you exceed, meet or miss the benchmark for the average expectations of candidates. You will be rated on a scale of one to one hundred with fifty consisting of the mean and median scores.

A raw score is simply the number of correct answers. The critical thinking test comprises your raw score based on the performance in the following areas:

• Recognizing Assumptions The candidate must be able to understand when a statement is made with no supporting evidence and how this can affect a decision. Further, candidates are asked to identify these discrepancies, whether they are stated explicitly or implicitly, and assess its relevance to the given scenario.
• Evaluating Arguments Candidates must evaluate arguments without considering inferences or being subjective. Beyond that, candidates must assess the supporting evidence, the structure of the argument and the degree of its influence. It is very important to dismiss emotions for this portion of the critical thinking test.
• Drawing Conclusions Drawing conclusions puts a large emphasis on reasoning. In this section, it’s important to assess all of the available evidence and data to form a plausible conclusion that accurately applies to all the given information. Employers also want to see candidates that will consider all possible solutions rather than making the evidence fit a desired narrative.

Employers will receive all of this information in a performance report construed by the assessment company. Employers will also be given insight into your overall potential, job knowledge, creativity and job performance per the report.

Where Will I Take a Critical Thinking Test?

Critical thinking tests are non-proctored online assessments that are typically sent via email after an initial screening. For some occupations, the company may ask that the candidate take the critical thinking test again on-site either before their final interview or during an assessment day. The most common test candidates are asked to take is the Watson Glaser Critical Thinking Appraisal (WGCTA) created by the popular assessment company, Pearson . This assessment company is on their third edition with new scoring and subsets described above. The WGCTA gained popularity because of its ability to assess a candidate’s potential alongside their aptitude. Another established assessment is the SHL Critical Reasoning Battery that contains sixty questions with a thirty-minute time limit. Both of the aforementioned critical thinking tests are multiple choice.

How to Prepare for the Critical Thinking Test?

The critical thinking test is difficult to study for because the test is designed to assess your bare knowledge and raw skills. In order to prepare successfully, it is important to focus on the areas of the test that you can equip yourself for. One aspect of the test that demands preparation is the time limit. Many candidates’ scores are negatively impacted because they skip or guess too many of the questions in an attempt to beat the clock. If you want to optimize your chances of achieving a good score, use online practice tests to acquaint yourself with the time constraint and the general theme of the questions. By utilizing the online practice tests, you can find the pace that works best for you. Another helpful way to prepare is running through sample questions. This way, you can warm-up your brain and gain an understanding of the expectations that both the test and the company have of you.

Free Sample Questions to Practice

• Look over her past quizzes to see what she missed.
• Set aside more time during the week to review the material for the quiz.
• Get to class on early Wednesday and briefly look over the chapters.
• Get a good night’s sleep.
• Parents should find an alternative way to get their kids to school next week.
• The premiums must be over-priced.
• Collective bargaining is no longer a feasible solution.
• Their employers are being unreasonable.
• People in Hawaii dislike living on an island.
• Colder climates induce more happiness than warmer climates.
• The high scores on the Alaska survey were produced by people who enjoy snow.
• People in Hawaii should move to Alaska.
• Jenny’s credit card was declined at the mall.
• Jenny’s bank keeps charging her $30 overdraft fees.
• Jenny’s check bounced when she attempted to purchase a new TV.
• Jenny spends more money than she makes.
• Lori has thirty cans of soda in a refrigerator in her garage and another fourteen sitting on the counter. Lori does not have anymore cans of soda. Therefore, Lori has 44 cans of soda.
• The accounting department loves math. My friend works in the accounting department. My friend loves math.
• Everyone southbound on the freeway yesterday was late to work. Jackie was southbound on the freeway. Jackie was late to work.
• Adrian lives in either Springfield, California, or Springfield, Illinois. If he lives in Illinois, then he is an American.

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Critical Thinking Test: Sample Questions with Explanations (2024)

Employers value and seek candidates who demonstrate advanced critical thinking skills. They often administer critical thinking tests as part of their hiring process. Critical thinking tests can be very difficult for those who don’t prepare. A great way to start practicing is by taking our critical thinking free practice test.

What Does The Critical Thinking Test Include?

The Critical Thinking Test assesses your capacity to think critically and form logical conclusions when given written information. Critical thinking tests are generally used in job recruitment processes, in the legal sector. These tests measure the analytical critical thinking abilities of a candidate.

Why Is Critical Thinking Useful?

Critical thinking is put into action in various stages of decision-making and problem-solving tasks:

• Identify the problem
• Choose suitable information to find the solution
• Identify the assumptions that are implied and written in the text
• Form hypotheses and choose the most suitable and credible answers
• Form well-founded conclusions and determine the soundness of inferences

What is Watson Glaser Test and what Critical Thinking Skills it Measures?

The most common type of critical thinking test is the Watson-Glaser Critical Thinking Appraisal (W-GCTA). Typically used by legal and financial organizations, as well as management businesses, a Watson Glaser test is created to assess candidates’ critical thinking skills.

The test consists of 10 questions to be answered in 10 minutes approx (although there is no timer on the test itself). Our test is slightly harder than the real thing, to make it sufficiently challenging practice.

You need to get 70% correct to pass the test. Don’t forget to first check out the test techniques section further down this page beforehand.

Questions          25

Pass percentage          70%.

The test is broken down into five central areas:

• Assumptions
• Interpretation

Critical Thinking Course

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The Five Critical Thinking Skills Explained

1. recognition of assumption.

You’ll be presented with a statement. The statement is then followed by several proposed assumptions. When answering, you must work out if an assumption was made or if an assumption was not made in the statement. An assumption is a proclamation that an individual takes for granted. This section of the tests measures your ability to withhold from forming assumptions about things that are not necessarily correct.

• 1: Assumption Made
• 2: Assumption Not Made

Although the passage does state that Charlie’s fundraising team is doing its best so that the charity event can meet its goal, nowhere did it state that their team is leading the event.

2. Evaluation of Arguments

You will be presented with an argument. You will then be asked to decide whether the argument is strong or weak. An argument is considered strong if it directly connects to the statement provided, and is believed to be significant.

No, participation awards should not be given in every competition because studies have shown that this would cause the participants to put in less effort because they will get a prize no matter what the outcome is.

• 1: Strong Argument
• 2: Weak Argument

This is a strong argument as it provides evidence as to why participation awards should not be given in every competition

3. Deductions

In deduction questions, you will need to form conclusions based solely on the information provided in the question and not based on your knowledge. You will be given a small passage of information and you will need to evaluate a list of deductions made based on that passage. If the conclusion cannot be formed for the information provided, then the conclusion does not follow. The answer must be entirely founded on the statements made and not on conclusions drawn from your knowledge.

In a surprise party for Donna, Edna arrived after Felix and Gary did. Kelly arrived before Felix and Gary did.

• 1: Conclusion Follows
• 2: Conclusion Does not Follow

For questions like this, jot down the clues to help you out. Use initials as a quick reference.

K | F&G | E

Looking at the simple diagram, “K”, which stands for “Kelly,” arrived before Edna “E” did. The answer is A.

4. Interpretation

In these questions, you are given a passage of information followed by a list of possible conclusions. You will need to interpret the information in the paragraph and determine whether or not each conclusion follows, based solely on the information given.

A number of students were given the following advice:

“The use of powerful words is a technique, which makes you a better writer. Your choice of words is very important in molding the way people interaction with the article. You should use powerful words to spice up your article. Power words should be used liberally to enhance the flavor of what you write! ”

In the fourth sentence, it is stated, “Power words should be used liberally to enhance the flavor of what you write!”

Thus, if you were to write an essay, using powerful words can give more flavor to it.

5. Inferences

An inference is a conclusion made from observed or supposed facts and details. It is information that is not apparent in the information provided but rather is extracted from it. In this section, you will be provided with a passage of information about a specific scene or event. A list of possible inferences will then be given, and you will need to decide if they are ‘true’, ‘false’, ‘possibly true’, ‘possibly false’, or whether it is not possible to say based on the information provided.

With the advancement of technology, the need for more infrastructure has never been higher. According to the plan of the current U.S. Administration, it aims to put a $1 trillion investment on improving infrastructure, a portion of which will include priority projects and technologies that can strengthen its economic competitiveness such as transportation, 5G wireless communication technology, rural broadband technologies, advanced manufacturing technologies, and even artificial intelligence.

It stated that it expects to work with Congress to develop a comprehensive infrastructure package, which is expected to have a budget of $200 billion for certain priorities.

• 2: Probably True
• 3: Not Enough Information
• 4: Probably False

Although it was mentioned in the passage that the U.S. government is to allocate $200 billion on certain priorities, it did not specify if these certain priorities were for ‘transportation, 5G wireless communication technology, rural broadband technologies, advanced manufacturing technologies, and artificial intelligence’ or if the aforementioned priorities will have a different allocation.

What we can be sure of, however, is that at least a portion of the $1 trillion infrastructure budget will be used on the mentioned priorities regardless, meaning that there is a chance that $200 billion will be used on those aforementioned areas.

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HYPOTHESIS AND THEORY article

Teaching abductive reasoning for use as a problem-solving tool in organic chemistry and beyond.

• 1 Chemistry Program, Department of Natural Sciences, Central College, Pella, IA, United States
• 2 Department of Chemistry, Augsburg University, Minneapolis, MN, United States
• 3 Department of Chemistry, University of Saint Joseph, West Hartford, CT, United States
• 4 Department of Chemical and Environmental Sciences, United States Coast Guard Academy, New London, CT, United States

The second-year undergraduate Organic Chemistry course sequence is often cited as one of the most, if not the most, challenging for students in the US. Thus, a persistent question remains: What is it about Organic Chemistry that makes the course so difficult for students? Herein, we put forward the hypothesis that a new mode of thinking and problem solving is expected of the students; these skills have not yet been developed in their prior scientific coursework and are often not deliberately taught in Organic Chemistry. This form of reasoning and problem solving, known as abductive reasoning, is highlighted for its connection to medical diagnosis and scientific thinking. We provide examples to showcase how instructors could explicitly foreground the reasoning process in their classroom. Ultimately, we argue that teaching how to reason using abduction may benefit students in both the short term (in the course) and the long term (in their careers as scientists and medical practitioners).

“What changes must be made in the kind of science that we teach and the way that we teach it so that the fundamental ideas of our discipline can be used outside the classroom?” – Herron & Greenbowe

1 Introduction

1.1 background.

Organic Chemistry, as traditionally taught in the US as a primarily second-year undergraduate course sequence, is often considered a course for “weeding out pre-meds” ( Moran, 2013 ) that “strik[es] fear in the hearts of students” ( Garg, 2019 ). This socially constructed barrier adds an additional level of pedagogical challenge for instructors. We, the authors, are instructors of Organic Chemistry and also write and review questions for standardized exams that are required for entrance into specialized medical programs; 1 thus, we are at a position in both the content delivery and assessment where we find ourselves continually asking the question: What do we want students to learn in the Organic Chemistry course sequence?

While some students may think the answer to this question is “to know, understand, and recite back the course material,” this is an unsatisfying response for a number of reasons. First, such a response would imply that only memorization and algorithmic problem-solving skills are necessary for success in Organic Chemistry ( Stowe and Cooper, 2017 ). 2 However, expert organic chemists recognize that the interconnected complexities within chemical systems means that simply following basic rules (i.e., deductive inference) will not necessarily lead to a set outcome (e.g., bulky bases do not always react via E2) ( Achet et al., 1986 ). Second, while the students enter our classrooms as novices, some of them will go on to become practicing, expert organic chemists. We owe it to them, and the future of scientific discovery, to build a sound foundation of both fundamental (e.g., understanding the aldol condensation) and higher order (e.g., performing retrosynthetic analysis) skills within the discipline. Third, most US health professions (e.g., MD, DO, PA, DDS, DMD, OD, PharmD) require this course to be taken as a prerequisite for admission into their graduate programs ( Kovac, 2002 ). These students should be presented, within their undergraduate education, the chance to improve their scientific reasoning and critical thinking skills. We think that these three features, which might not be clear to all students entering the course, illustrate that students are expected to learn and problem solve in new ways—essentially to begin to “think like a chemist” (e.g., Platt, 1964 ).

While certain ideas within this article were presented in a preceding paper ( Wackerly, 2021 ), we intend to flesh out and expand upon some of those initial assertions in this manuscript and craft a more detailed hypothesis that the use of abductive reasoning is critical in the learning of organic chemistry concepts. Herein we provide support for this hypothesis by viewing it from a few different conceptual angles. First, we provide a science education overview on why learning certain organic chemistry concepts is considered challenging for students. Then, we briefly summarize the medical education viewpoint on the teaching of diagnosis and why this is important to many students in Organic Chemistry. Finally, using the lens of the Organic Chemistry curriculum we provide problem-solving examples of how abductive reasoning can assist in the teaching and learning of organic chemistry.

1.2 Why is science difficult to learn?

Johnstone asked this titular question in his seminal 1991 paper ( Johnstone, 1991 ). One conclusion that he drew, which has since been supported by a variety of other work (e.g., Graulich, 2015 ; Tiettmeyer et al., 2017 ; Reid, 2020 ; Dood and Watts, 2022 ), is that the nature and complexity of scientific concepts strain the working memory of students. To assist instructors in conceptualizing the strain of a given concept, he created the “triangle model” which illustrated three levels of thought ( Figure 1 ). He argued that the more levels a concept included the more cognitive load was placed on students.

Figure 1 . Reproduction of Johnstone’s model: “Triangle of Levels of Thought”.

One feature that might make learning science difficult is that the instructor, or expert, may not be aware of the extent of cognitive load they are placing on students, or novices. When “multicomponent phenomena that are invisible, dynamic, and interdependent” are presented to students, a large demand is placed on the working memory of novices ( Hmelo-Silver et al., 2007 ). However, experts are able to easily connect two or more cognitive components by “chunking several pieces of information together” ( Overton and Potter, 2008 ) and through years of practice ( Randles and Overton, 2015 ). Specialization within a discipline that requires connecting multiple levels will lower cognitive load for such repetitive tasks over time ( Tiettmeyer et al., 2017 ; Price et al., 2021 ). However, students have typically not been exposed to such tasks, let alone have the opportunity to consistently repeat them, and thus instructors need to disentangle new concepts that might cause cognitive overload for students so they can process and incorporate new material starting from their present knowledge base and scientific models. 3

“[R]easoning [is the] knowledge of some facts [which] leads to a belief in others not directly observed.” – C. S. Peirce

1.3 Why is organic chemistry so difficult to learn?

Here we argue that it should come as no surprise when former and current students of organic chemistry cite that organic chemistry is difficult to learn, because they are asked to problem solve and reason in new ways utilizing new content without prior exposure to, or repetition of, these scientific tasks. 4 Naturally, when a student enters a course they are expected to be ignorant of the course content since they enroll to learn it. However, students might feel that a bait-and-switch has occurred in Organic Chemistry because not only is the content new, but the logical processes required to be successful are also typically new to the students as well.

In prior scientific courses, which for most pre-health ( vide infra ) US students are two courses in general biology and two in general chemistry, students are typically required to perform recall (memorization) or reason algorithmically on summative assessment items ( Raker and Towns, 2010 ). While these skills hold value in organic chemistry, current organic chemistry education research shows that skills such as multivariate ( Kraft et al., 2010 ; Christian and Talanquer, 2012 ) and mechanistic reasoning ( Bhattacharyya, 2013 ) are more important. 5 Thus, inspired by the work in chemistry education research, the philosophy of science, and Johnstone’s seminal triangle, here we propose a tetrahedron model of layered reasoning strategies that are important for consideration by instructors when teaching novice organic chemistry students.

The bottom-most point of the tetrahedron ( Figure 2 ) was chosen to be memorization because it is not a reasoning skill. However, terms and chemical facts still need to be learned by students, which is often not a problem because they have developed this skill during their general biology and chemistry coursework. Algorithmic reasoning is a skill many students leaving General Chemistry assume they will utilize in Organic Chemistry because it was employed so frequently in that course. For example, if a student knows the pressure, temperature, and number of moles of an ideal gas, these students will likely be able to provide the volume of the gas’s container. While these mathematical and deductive reasoning skills remain relevant in the laboratory portion of Organic Chemistry and even for the IUPAC naming of organic molecules (i.e., there is a definitive rule set), they start to break down when chemical systems become more complex and chemical formulas evolve to contain more meaning in the form of chemical structures.

Figure 2 . Tetrahedron model of problem-solving in Organic Chemistry.

The right corner of the tetrahedron is for the set of competencies required to interpret diagrams in organic chemistry, such as visualization ( Gilbert, 2005 ), visuo-spatial reasoning ( Pribyl and Bodner, 1987 ; Habraken, 1996 ), and representational competence ( Kozma and Russell, 1997 ). In lieu of individually listing these skills, we designate this corner as perceptual learning, which integrates conceptual knowledge with a broad set of skills, including those related to visualization and representational competence ( Van Dantzig et al., 2008 ; Kellman and Massey, 2013 ). Perceptual learning “refers, roughly, to the long-lasting changes in perception that result from practice or experience” ( Connolly, 2017 ), and is beginning to be more deeply explored in organic chemistry pedagogy (e.g., Kim et al., 2019 ).

We briefly illustrate how changes associated with perceptual learning might take place with students. Consider, for example, that in General Chemistry students might be asked to calculate the heat of combustion of hexane (denoted at C 6 H 14 ). For most students at that stage, the sole association they would have with the compound’s name is its molecular formula, whereas its “zig-zag” structure might represent nothing more than a crooked line. As these students progress into Organic Chemistry and learn about different representational systems and constitutional isomers, the verbal representation “hexane” changes, this is because the term is now associated with five unique isomers each with unique connectivity, properties, and reactivity (e.g., radical reaction with Br 2 ). Through this process, the students’ perception for the term “hexane” changes from representing a single molecular formula to representing a family of five constitutional isomers each with a unique bond-line structure. This process continues as students advance to more complex structures (e.g., stereochemistry) and learn additional concepts like three-dimensionality, IMFs, physical properties, etc. We propose that the three corners of the tetrahedron discussed thus far are often directly connected to abductive reasoning which focuses on solving problems by generating the most likely most likely outcome of a chemical situation.

Our hypothesis includes the postulation that abductive reasoning is a complex reasoning skill for students in Organic Chemistry and should explicitly be taught in the classroom. While this idea has been presented by us previously ( Wackerly, 2021 ), here we will just provide a brief overview so we can move on to discuss the relevance of this reasoning skill within the Organic Chemistry classroom and to highlight some examples. Firstly, the term “abduction” ( Douven, 2021 ) is often used interchangeably with the terms “inference to the best explanation” ( Lipton, 2017 ) and “scientific hypothesis”—and below we will argue “diagnosis.” All of these terms hold common ground in that they use reasoning that connects various (similar or dissimilar) pieces of evidence/observations together in a way where a plausible conclusion can causally describe the collection of phenomena. 6 For example, say you are inside of grain windmill by the grindstone, and then you begin to see the stone rotating and producing flour. You will abduce that the weather outside has become windy. While this is a simple example only requiring you to understand that outside wind turns the sails and the sails, via a series of machinery, turn the grindstone, it is similar to the reasoning employed by expert organic chemists. Leaving the windmill and heading into your synthetic laboratory, let us say you wish to publish a new compound in the Journal of Organic Chemistry . According to the journal, to conclude that you have made this new compound you must “establish both identity and degree of purity.” Minimally, this means you will need to obtain a 1 H NMR spectrum, 13 C NMR spectrum, and HRMS spectrum then interpret the data present in the spectra to abduce the molecular structure of your new compound. This exact same skill that is required of expert organic chemists, is typically required of students in Organic Chemistry ( Stowe and Cooper, 2019a ). Thus, these students should be taught how to reason like expert scientists in order for them to develop into scientists ( Cartrette and Bodner, 2010 ). Just as the spectroscopic analysis example highlights, instructors of Organic Chemistry often profess a goal is for students to develop critical thinking and scientific problem-solving skills: Our hypothesis presented here is that instructors must explicitly utilize the abductive reasoning process within their teaching and assessment.

Solving problems that require abductive reasoning will also require skills from the three other points of the tetrahedron, which will render them cognitively complex. Teaching abductive reasoning in the classroom should not require additional formal training for instructors/experts since abductive reasoning skills have already been developed over the course of their careers. Further, philosophers have long held ( Harman, 1965 ) that humans utilize abductive reasoning as a matter of course in their day-to-day lives. Paralleling human logic, abductive reasoning has likely been utilized ( Pareschi, 2023 ) and will continue to be ( Dai and Muggleton, 2021 ) an integral part of artificial intelligence. This reasoning skill is particularly important for students required to take Organic Chemistry. It might be obvious that future scientists will need the skills to create new hypotheses and design experiments that could potentially refute current hypotheses, but in our experience, it seems less obvious to pre-health students that using abductive reasoning for problem solving in Organic Chemistry will play a critical role in their desired careers.

2 Framing for pre-health students (diagnosis)

2.1 why is organic chemistry relevant for pre-health students.

In a post-COVID world where test-optional admissions are on the rise and the future of post-graduate education feels increasingly uncertain, convincing students of the importance of Organic Chemistry goes beyond just passing the course. This is especially true for the majority of students taking Organic Chemistry who are pre-health majors. Instructors need to show students the connection between organic chemistry and the health field.

Thus, problem solving in Organic Chemistry can be framed as a diagnostic problem-solving tool–similar to what medical practitioners do when making a diagnosis ( Stowe and Cooper, 2019b ). By overtly showing students the parallels between medical diagnosis and organic chemistry problem solving, instructors demonstrate that students are not just being taught a bunch of facts–they are developing critical thinking skills they can use in the real world. Bridging the gap between theory and practice helps students see the bigger picture and gives them the tools they need to succeed in both their studies and future careers.

The parallels between medical diagnosis and organic chemistry problem solving should be readily apparent ( Table 1 ). Both involve analyzing complex systems (human body/chemical reactions) to identify patterns and relationships, emphasizing the importance of critical thinking and logic-based problem-solving skills, as well as using evidence. Both fields rely on the use of abductive reasoning ( Wackerly, 2021 ; Martini, 2023 ), although typically neither field explicitly states it to students. Table 1 uses simplified language accessible to students that describes the abductive theory of method (ATOM) in clinical diagnosis ( Vertue and Haig, 2008 ), and its parallel to expert thinking in organic chemistry.

Table 1 . Comparison of medical diagnosis to skills developed in Organic Chemistry.

For example, to “diagnose” the product of an organic chemistry reaction, first the background information, including structure, reactivity, and stability of the starting materials and reagents must be analyzed, which is similar to how medical professionals take patient history. Abductive reasoning is then used to generate the most likely answer. Finally, the hypothesis is tested through gathering evidence such as utilizing spectroscopic analysis which is similar to a physician ordering lab work or imaging. This is an iterative process, wherein multiple pieces of spectroscopic evidence are needed to point to the same answer. Similarly, a physician may order additional studies or perform physical exams to support or refute their medical diagnosis. Although the goals appear different, the same skills are developed such as drawing hypothesis based on empirical evidence. By explicitly demonstrating how these thought processes are parallel, instructors of Organic Chemistry may help students to appreciate the mental training they are receiving in the course.

Organic Chemistry has been deemed essential as a prerequisite for medical school by a panel of medical school professors of biochemistry ( Buick, 1995 ). While many current medical students do not think that the material covered in Organic Chemistry was a valuable part of their undergraduate curriculum, the majority agree that the critical thinking skills learned in the course were valuable ( Dixson et al., 2022 ). While there are those in the field of medicine who think that Organic Chemistry should be de-emphasized in the pre-med curriculum, those that defend Organic Chemistry do so for some of the same reasons we discuss herein, namely that the critical thinking and problem-solving skills in the course directly align with patient diagnosis ( Higgins and Reed, 2007 ).

This process of abductive reasoning coupled with framing for the medical field may serve the students better in both the short term and long term. Students who employ more metacognitive strategies such as the type we are advocating for here are better able to solve problems in Organic Chemistry ( Blackford et al., 2023 ). Connecting course material to students’ future career aspirations also leads to better engagement and course performance ( Hulleman et al., 2010 ). Additional benefits of this diagnostic reasoning process include students’ ability to apply this metacognitive strategy in other courses in their majors, such as biology ( Morris Dye and Dangremond Stanton, 2017 ), and their future medical careers ( Friel and Chandar, 2021 ). Therefore, diagnostic reasoning should be explicitly modeled and assessed in Organic Chemistry courses.

2.2 Using “diagnosis” in examples for students

While there are a variety of ways to teach students how to approach organic chemistry problems like an expert, we would like to present how to do this through the lens of “diagnosis.” Other ways of describing argumentation and the process of problem solving have been discussed in the chemical education literature (e.g., Cruz-Ramírez De Arellano and Towns, 2014 ; Stowe and Cooper, 2019a ; Walker et al., 2019 ) as well as the philosophy of chemistry literature (e.g., Kovac, 2002 ; Goodwin, 2003 ). While they differ in the number of steps and what those steps are called, the processes have a similar logical flow. First, gather evidence and make observations ( What you see ), link this to previous knowledge ( What you know ), and finally make a reasoned conclusion ( Hypothesis ) which is a logical consequence—often via abductive inference.

The following examples ( Figures 3 – 6 ) are designed to highlight the use of these three steps to explicitly diagnose problems from across the two semester Organic Chemistry sequence. This process can be used in the classroom as a model to guide students through the abduction process and could be used to explicitly scaffold problems. Moreover, instructors can use this model to ascertain the complexity of their assessments including the required prerequisite factual knowledge and the multiple steps required. The complexity of organic chemistry questions is determined by the number of “subtasks” the student must complete ( Raker et al., 2013 ), factual knowledge required, and facets of perceptual learning ( vide supra ). A number of explicit decisions were made in formulating the below questions. The discussion points are certainly not exhaustive, and practitioners should adapt questions to their own students and situations. The amount of information provided or not provided, such as the exclusion of lone-pairs and inorganic by-products, was chosen to be consistent with the information provided by practicing organic chemists and one goal of teaching organic chemistry is to facilitate the development toward expert-level practice. We intentionally included one example of additional information, Figure 4 entry marked with a *, to highlight that there are many more subtasks that could be utilized to assist with arriving at a probable conclusion, but we tried to exclude all other non-essential explanations. We do not suggest that all students should solve each problem from top to bottom as outlined here; in reality expert chemists often take different routes, based on the same evidence and premises, to reach similar conclusions. Although these problems are multiple-choice, we have modeled how to solve them as either multiple-choice or open format. The complexity of these questions can also be adjusted, for example in Figure 4 the mechanistic arrows could be included in the distractors and answer instead of in the stem. This type of alteration can allow for the assessment of mechanistic thinking (e.g., Bodé et al., 2019 ; Finkenstaedt-Quinn et al., 2020 ; Watts et al., 2020 ; Dood and Watts, 2022 ). The following examples demonstrate that when the diagnosis/abduction process is utilized, students can develop and enhance their problem-solving skills.

Figure 3 . Diagnosis of an aromaticity problem.

Figure 4 . Diagnosis of a mechanism problem.

Figure 5 . Diagnosis of a substitution/elimination question.

Figure 6 . Diagnosis of a predict the product reaction.

The first example shown in Figure 3 is a case of aromaticity ( Jin et al., 2022 ). Students will typically memorize the requirements and check the structure for being cyclic, planar, containing Huckel’s number (4 n  + 2) electrons, and a p orbital at every vertex (i.e., conjugated). However, this problem does not ask for a simple definition of aromaticity, but an application of the ruleset to a structure students would not have typically encountered. The diagnosis requires observations about the structure including recognition of the implicit lone pairs on the nitrogen atoms and the carbon–carbon π bonds, recall of the requirements of aromaticity, and then application of abductive reasoning to the concepts learned (e.g., in class) and perceived by the structural representation. It is easy to see that the 1,4-dihydropyrazine is cyclic, has 8 π electrons, and a p orbital at each vertex. However, this simple analysis would result in the structure being anti-aromatic, so the student must recognize that in order for it to be non-aromatic as the problem states, planarity must be disrupted.

The second example shown in Figure 4 is a curved arrow mechanism problem for a reaction not typically covered in the Organic Chemistry course sequence ( Sarode et al., 2016 ). Students must apply the rules of curved arrows and properly atom map to diagnose the correct product. 7 The pre-existing conditions for a mechanism question with arrows shown include the nature of curved arrows and the examination of the scheme will require atom mapping and keeping track of which bonds are broken and formed.

The third example shown in Figure 5 is a substitution/elimination problem ( Brown et al., 1956 ). Students frequently find these reactions challenging and may employ a variety of heuristic models to approach them. Just as medical diagnosis begins with gathering information (taking patient information), solving this problem begins with direct observation and application of what is known about the structure and reactivity of these molecules. The alkyl halide has a good leaving group and has tertiary electrophilic carbon classification while the t -butoxide reagent is electron-rich, bulky, and reactive. Students must reason abductively how these characteristics interact with each other. This iterative process first eliminates S N 2 due to the nature of the alkyl halide, then identifies E2 as the mechanism with the bulky alkoxide. Next, an understanding of thermodynamics vs. kinetics to differentiate the two possible E2 pathways. Finally, a re-examination of the problem indicates the less stable product is formed preferentially; this is best explained by steric crowding in the transition state of the reaction between the alkyl halide and alkoxide.

The final, and most complex, example shown in Figure 6 is a predict the product, addition reaction problem ( Inoue and Murata, 1997 ) that is analogous to halohydrin formation. The problem requires separate diagnoses as it is layered where advancement to the second part is necessitated by the successful completion of the first addition step. Students would need to differentiate between the nucleophilicity of the alcohol and π bond after recognizing them as potential nucleophiles. After using abduction to recognize the higher reactivity of the π bond, students should then reason that selenium is electrophilic, akin to bromine in Br 2 due to being polarizable and bonded to a leaving group. This diagnosis is supported when taking into account the stereospecificity of the transformation, which precludes carbocation intermediates. The second diagnosis requires that students recall the regioselectivity of reactions with 3-membered cationic rings at the more substituted carbon. The remaining nucleophilic oxygen atom can now react with a higher energy seleniranium ion. However, conformational analysis of the transition state is needed to discern the pseudo axial/equatorial approach of the oxygen atom on the seleniranium ion ( Figure 6 , bottom). Students would then need to apply their knowledge of chair conformations and the lower energy state when having ring substituents equatorial. Thus, the trans -oxacyclohexane is formed.

3 Conclusion and future work

While Organic Chemistry is often regarded as the most challenging undergraduate course in the US, we argue it has gotten a “bad rap” because students are not always prepared for the challenges that lie ahead when they enter the course. Students generally perform better on assessments when they employ metacognitive strategies (i.e., “thinking about thinking”). This has been demonstrated in a variety of courses ( Arslantas et al., 2018 ), including Organic Chemistry (e.g., Graulich et al., 2021 ; Blackford et al., 2023 ). The consensus is that students who employ more metacognitive strategies in Organic Chemistry are more successful in problem-solving tasks and are better able to use those strategies when they are explicitly modeled and scaffolded. We have argued that instructors of Organic Chemistry should teach and demonstrate how to think and problem solve via “diagnosis” (i.e., abductive reasoning) in their classrooms. We hypothesize that students may score higher on metrics that assess scientific learning when these types of diagnostic models are utilized.

As constructors of nationally standardized exams, we fully acknowledge that a lot of growth on organic chemistry knowledge assessment still remains to be achieved. For Organic Chemistry course instructors, we hope the above insight into abductive reasoning can also be used on the assessment side of teaching requirements. Namely, that the cognitive load placed on students when solving each problem be carefully considered when constructing summative assessment items. Though this point has been frequently made previously e.g., (see Raker et al., 2013 ), we believe it is worthwhile for all writers of questions in Organic Chemistry to map out, step-by-step, the logic required to solve each question to determine the cognitive load. This can, in turn, help these instructors teach from a novice-focused perspective—as opposed to the “sage on the stage.” The prior section provided examples with varying levels of complexity and demonstrated that cognitive load can be approximated by the number of reasoning steps (subtasks) required when the assessment piece is broken down. Further, this process could potentially also help the exam writer identify if items require little to no scientific reasoning (e.g., pure memorization questions).

The above manuscript merely outlines a hypothesis that we have generated over the course of our time teaching Organic Chemistry with this “diagnosis” method of abduction. To fully explore its validity, educational research is needed. This will be a precarious endeavor, because measuring the efficacy of teaching abductive reasoning will require assessment of scientific thinking skills in Organic Chemistry, and, as we just pointed out, there are already strong arguments that we are still quite far away from such valid assessments. However, we can be sure that if you are teaching Organic Chemistry from the perspective of your experience and expertise as an organic chemist, then opening a window for your students into how you think and problem solve will benefit your students. Our position is that instructors of Organic Chemistry should not only be explicitly teaching students the abductive reasoning skills to tackle complex problems, but they should also frame it as “diagnosing” the chemical situation.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

JW: Writing – review & editing, Writing – original draft, Conceptualization. MW: Writing – review & editing, Writing – original draft. SZ: Writing – review & editing, Writing – original draft.

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

Acknowledgments

The authors would like to thank Gautam Bhattacharyya for helpful discussions during revisions of this manuscript, specifically regarding perceptual learning theory. JW would like to acknowledge his undergraduate Organic Chemistry professor, Thomas Nalli, on the recent occasion of his 65th birthday for teaching him scientific problem-solving skills and fostering his interest in the discipline.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

1. ^ We wish to keep the focus of this manuscript on the relevant student population of the Organic Chemistry course sequence. Students intending to pursue medically relevant careers which require advanced degrees (e.g., medical, dental, optometry, pharmacy, etc.) are a large portion of this population. However, if the reader is curious, we specifically write for the dental and optometric admissions exams.

2. ^ In this manuscript we attempt to provide the reader a broad overview of important chemical education and philosophy of chemistry publications. Since this is not a review article and the scope is quite a bit smaller, all possible relevant literature has not been cited.

3. ^ Cognitive overload could also stem from misconceptions and oversimplified concepts, such as the oft-stated “breaking bonds in ATP releases energy” from introductory biology courses.

4. ^ This can be contrasted with General Chemistry which repeats some of the content of the high school chemistry.

5. ^ Multivariate and mechanistic reasoning are highlighted as examples because they often require combining features from all four points of the tetrahedron.

6. ^ The conclusion need not explain the entire collection of evidence as some may be irrelevant, and they are unrelated to the conclusion. However, the entire collection may not contain a piece of evidence that refutes the conclusion. Thus, abductive reasoning can be useful in differentiating science from non-science and pseudoscience.

7. ^ While one could argue that the diagnosis/answer to the problem presented in Figure 4 does not require abductive reasoning, we have included it because the skills required here can be applied to more complex problems that, for example, include mechanistic reasoning ( vide infra ).

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Keywords: abduction, abductive reasoning, organic chemistry, diagnosis, metacognition, problem solving, pre-health education

Citation: Wackerly JW, Wentzel MT and Zingales SK (2024) Teaching abductive reasoning for use as a problem-solving tool in organic chemistry and beyond. Front. Educ . 9:1412417. doi: 10.3389/feduc.2024.1412417

Received: 04 April 2024; Accepted: 05 July 2024; Published: 06 August 2024.

Reviewed by:

Copyright © 2024 Wackerly, Wentzel and Zingales. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Jay Wm. Wackerly, [email protected]

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

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Critical Thinking Math Problems

Ability to make decisions by the application of logical, sceptic, and objective analyses and evaluations of data, arguments, and other evidence is known as critical thinking. Regarding mathematics, critical thinking is not only about making calculations but also involves logical reasoning, pattern recognition and problem-solving strategies.

In this article, we dive into various aspects of critical thinking in mathematics, including problems, strategies for solving them, which activities can increase critical thinking, and some examples with solutions. If you want to improve your critical thinking or even know more about getting better at math, then keep reading.

Table of Content

Types of Critical Thinking Math Problems

Strategies for solving critical thinking problems, activities to enhance critical thinking, benefits of critical thinking in math, examples and solutions.

There can be various types of problems. Some may be related to numbers, others to shapes and some even comprehensions. Let’s take a closer look at them:

• Word Problems : Students must convert verbal descriptions into mathematical equations to solve word problems. These exercises improve understanding abilities and encourage applying mathematical ideas to practical settings.
• Logic Puzzles : Logic puzzles, like logic grids or Sudoku, require students to apply deductive reasoning and pattern recognition skills. A methodical approach and multiple steps are frequently needed to solve these puzzles.
• Algebraic Problems : Variables and unknowns are part of algebra. Students must work with equations and inequalities. Working through these kinds of problems promotes the development of abstract thinking as well as proficiency with symbols, formulas, and identities.
• Geometric Problems : Geometric problems relate to the shapes, sizes and properties of figures such as circles, triangles and rhombuses. Solving these problems involves spatial reasoning and the application of theorems and postulates.
• Probability and Statistics : While statistics entails evaluating vast amounts of data, generating hypotheses, and concluding, probability involves determining the possibility that an event will occur. Students learn to deal with ambiguity and generate well-informed predictions from these problems.

The majority of people think they are terrible at math, and when they are asked to solve problems in math, they just don’t know how. The following are some methods that you can apply to resolve an issue:

• Understand Problem : The most important thing is to understand the problem statement. The question always provides some key information, such as relations, conditions or equations. If required, read the problem several times and carefully note the provided information and what is being asked.
• Break Down Problem : Some problems can be more complex than others and may require lengthy procedures and much thinking. Breaking down such problems into smaller parts can make them easier to solve. This step-by-step approach makes it easier to tackle each component individually.
• Visualize Problem : Try to draw diagrams, graphs or charts to simplify the problem and make abstract concepts more concrete. Visualization aids in understanding relationships and patterns.
• Use Logical Reasoning : Logical reasoning involves making connections between known information and new insights. This process includes making assumptions, drawing inferences, and evaluating the validity of conclusions.
• Check Your Work : After arriving at a solution, it is crucial to verify its accuracy. This can involve reworking the problem, checking calculations, and ensuring that the solution makes sense in the context of the problem.

Improving critical thinking is not only about working out problems; there are other ways in which you can boost your critical thinking:

• Math Games: What better way to learn than playing games? Chess or strategy board games can help develop strategic thinking and problem-solving. These games make learning fun and engaging.
• Group Work: Working in a group can be a great and fun way of learning. A problem can be solved in multiple ways. When you work together in a group, you get to know about how others perceive problems and can learn different ways to solve them.
• Real-World Projects: It is really important for students to learn to use their critical thinking skills in practical situation. When students solve real world problems, they get to use their textbook knowledge which can help to deepen their understanding of the subject matter. This also helps students build real-world skills that can benefit their careers.
• Technology Integration: Learning may become more interactive through the use of technology in the classroom. Students’ learning in the classroom has changed dramatically as a result of educational technology, including software, digital whiteboards, and, more recently, artificial intelligence.

Working on critical thinking in math has several benefits, such as:

• Improved Problem-Solving Skills: Critical thinking boosts students ability to solve problems. It equips students with the ability to approach problems with various methods come up with different solutions and choose the most effective one.
• Enhanced Creativity: Critical thinking exercises help students become more creative because they let them see problems from multiple perspectives and come up with unique solutions.
• Better Decision Making: Critical thinking equips students with the ability to take well-informed decisions. It allows them to take action based on logical reasoning and evidence.
• Greater Academic Success: Students with strong critical thinking skills often perform better in their academics as they can handle complex problems across multiple subjects.
• Lifelong Learning: Critical thinking fosters a curiosity and continuous learning mindset essential for personal and professional growth.

In conclusion, critical thinking is the ability to solve problem by analyzing available information, and observations. Enhancing one’s critical thinking abilities can be achieved through tackling an array of problems. Beyond the classroom, critical thinking helps students make better decisions and solve problems throughout their lives.

Let’s take some examples of different types of critical thinking questions to understand better:

Example 1: Students were scheduled for a field trip by a school. The field trip is being attended by 120 students in total. 28 students can fit on a school bus. Find out the number of buses required to hold every student and the number of students on the final bus.

To find the number of buses required, divide the total number of students by the number of students each bus can hold. 120 ÷ 28 ≈ 4.29 You need five buses because one cannot have a fraction of a bus. Multiply the number of full buses (4) by the bus capacity (28) to determine the number of students on the final bus, then deduct that amount from the total number of students: 120 – (4×28) = 120 – 112 = 8 students on the last bus.

Example 2: There are three crates labeled Apples, Oranges, and Apples and Oranges. Each label is incorrect. How can you determine the contents of each crate by picking one fruit from one crate?

Pick a fruit from the crate labeled Apples and Oranges. If you pick an apple, this crate must be Apples (since all labels are wrong). The crate labeled Apples must then be Oranges, and the crate labeled Oranges must be Apples and Oranges. If you pick an orange instead, the crate is Oranges, the one labeled Apples is Apples and Oranges, and the one labeled Oranges is Apples.

Example 3: Solve for x in the equation 3x – 7 = 2x + 5

Subtract 2x from both sides: 3x – 2x – 7 = 5. Simplify: x – 7 = 5. Add 7 to both sides: x = 12.

Example 4: Find the area of a trapezoid with bases of 5 cm and 7 cm, and a height of 4 cm.

Use formula for the area of a trapezoid: Area = 12×(Base 1​ + Base 2 ​)×Height = 12​×(5 + 7)×4 = 24 cm²

Example 5: What is the probability of drawing an ace or a king from a standard deck of 52 playing cards?

In a 52-card deck, there are 4 aces and 4 kings, which results in 8 positive outcomes. Probability is calculated by dividing the total number of outcomes by the number of positive outcomes. 8/52 = 2/13

Example 6: A survey was conducted to find out whether people prefer tea or coffee. 80 out of 200 people said they prefer tea over coffee. What percentage of the surveyed population prefers tea?

To calculate the percentage of people who preferred tea: = 80200 × 100 = 40% So, 40% of the surveyed population prefers tea.

Example 7: What is the next number in the sequence 2, 6, 12, 20, 30, …?

Addition of consecutive even numbers is the pattern: 2 + 4 = 6 6 + 6 = 12 12 + 8 = 20 20 + 10 = 30 Thus, 30 + 12 = 42 would be the following number.

Practice Problems with Solutions

Q1. A library has 400 books. 25% of them are fiction. How many fiction books are there?

0.25×400 = 100 There are 100 fiction books

Q2. There are five houses in a row. Each house is painted in a different color. It is known that the red house is not next to the blue house, and the green house is to the left of the yellow house. If the blue house is on one end and the white house is directly to the right of the red house. Find where is the green house?

Blue house should be the last one since it is the only possible solution to meet all the given criteria. The red house cannot be next to the blue house and must be placed such that the white house is directly to its right. So the possible arrangement is {R, W, G, Y, B}. Green house is the 3rd house.

Q3. Solve for z: 5z – 2 = 3z + 8.

5z – 3z – 2 = 8 2z – 2 = 8 2z = 10 z = 5

Q4. Determine the volume of an 8-cm-long, 5-cm-wide, and 10-cm-tall rectangular prism.

Volume = length×width×height = 8×5×10 = 400 cm³

Q5. What is the probability of drawing a heart from a deck of 52 cards?

There are 13 hearts in the deck. Probability is 13/52 = 1/4

Q6. The sales of a company increased from $150,000 to $180,000. What was the percentage increase in sale?

Percentage Increase = (180,000 – 150,000)/150,000×100 Percentage Increase = 20%

Q7. What is the next number in the sequence 3, 9, 27, 81, …?

The pattern is multiplication by 3: 81×3 = 243 The next number is 243.

Q8. If 4y + 6 = 3y – 2, determine the value of y?

4y + 6 = 3y – 2 4y-3y + 6 = – 2 y + 6 = – 2 y = – 8

Frequently Asked Questions

What is critical thinking in math problems.

In mathematics, critical thinking is a talent that enables pupils to examine, assess, and interpret data in order to come up with a solution. These issues frequently need the application of multiple steps and a profound comprehension of mathematical ideas.

How to improve mathematical critical thinking abilities?

Solve a variety of problems, including word puzzles, algebraic equations, and logic issues, to hone your critical thinking skills in mathematics. Engage in activities like games and puzzles that test your ability to think and analyze information.

Why are critical thinking skills important in math?

Mathematical critical thinking abilities are crucial because they enable students to comprehend difficult subjects, work through difficulties efficiently, and apply their knowledge of mathematical reasoning to practical situations.

Can critical thinking math problems be used in standardized tests?

Yes, analytical thinking Standardized examinations frequently utilize arithmetic questions to evaluate a student’s capacity for critical thought, analysis, and problem-solving. The purpose of these questions is to assess a student’s deeper comprehension of mathematical ideas rather than just rote memory.

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Breakthrough models AlphaProof and AlphaGeometry 2 solve advanced reasoning problems in mathematics

Artificial general intelligence (AGI) with advanced mathematical reasoning has the potential to unlock new frontiers in science and technology.

We’ve made great progress building AI systems that help mathematicians discover new insights , novel algorithms and answers to open problems . But current AI systems still struggle with solving general math problems because of limitations in reasoning skills and training data.

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Breakthrough AI performance solving complex math problems

The IMO is the oldest, largest and most prestigious competition for young mathematicians, held annually since 1959.

Each year, elite pre-college mathematicians train, sometimes for thousands of hours, to solve six exceptionally difficult problems in algebra, combinatorics, geometry and number theory. Many of the winners of the Fields Medal , one of the highest honors for mathematicians, have represented their country at the IMO.

More recently, the annual IMO competition has also become widely recognised as a grand challenge in machine learning and an aspirational benchmark for measuring an AI system’s advanced mathematical reasoning capabilities.

This year, we applied our combined AI system to the competition problems, provided by the IMO organizers. Our solutions were scored according to the IMO’s point-awarding rules by prominent mathematicians Prof Sir Timothy Gowers , an IMO gold medalist and Fields Medal winner, and Dr Joseph Myers , a two-time IMO gold medalist and Chair of the IMO 2024 Problem Selection Committee.

“ The fact that the program can come up with a non-obvious construction like this is very impressive, and well beyond what I thought was state of the art.

Prof Sir Timothy Gowers, IMO gold medalist and Fields Medal winner

First, the problems were manually translated into formal mathematical language for our systems to understand. In the official competition, students submit answers in two sessions of 4.5 hours each. Our systems solved one problem within minutes and took up to three days to solve the others.

AlphaProof solved two algebra problems and one number theory problem by determining the answer and proving it was correct. This included the hardest problem in the competition, solved by only five contestants at this year’s IMO. AlphaGeometry 2 proved the geometry problem, while the two combinatorics problems remained unsolved.

Each of the six problems can earn seven points, with a total maximum of 42. Our system achieved a final score of 28 points, earning a perfect score on each problem solved — equivalent to the top end of the silver-medal category . This year, the gold-medal threshold starts at 29 points, and was achieved by 58 of 609 contestants at the official competition.

Graph showing performance of our AI system relative to human competitors at IMO 2024. We earned 28 out of 42 total points, achieving the same level as a silver medalist in the competition.

AlphaProof: a formal approach to reasoning

AlphaProof is a system that trains itself to prove mathematical statements in the formal language Lean . It couples a pre-trained language model with the AlphaZero reinforcement learning algorithm, which previously taught itself how to master the games of chess, shogi and Go.

Formal languages offer the critical advantage that proofs involving mathematical reasoning can be formally verified for correctness. Their use in machine learning has, however, previously been constrained by the very limited amount of human-written data available.

In contrast, natural language based approaches can hallucinate plausible but incorrect intermediate reasoning steps and solutions, despite having access to orders of magnitudes more data. We established a bridge between these two complementary spheres by fine-tuning a Gemini model to automatically translate natural language problem statements into formal statements, creating a large library of formal problems of varying difficulty.

When presented with a problem, AlphaProof generates solution candidates and then proves or disproves them by searching over possible proof steps in Lean. Each proof that was found and verified is used to reinforce AlphaProof’s language model, enhancing its ability to solve subsequent, more challenging problems.

We trained AlphaProof for the IMO by proving or disproving millions of problems, covering a wide range of difficulties and mathematical topic areas over a period of weeks leading up to the competition. The training loop was also applied during the contest, reinforcing proofs of self-generated variations of the contest problems until a full solution could be found.

Process infographic of AlphaProof’s reinforcement learning training loop: Around one million informal math problems are translated into a formal math language by a formalizer network. Then a solver network searches for proofs or disproofs of the problems, progressively training itself via the AlphaZero algorithm to solve more challenging problems.

A more competitive AlphaGeometry 2

AlphaGeometry 2 is a significantly improved version of AlphaGeometry . It’s a neuro-symbolic hybrid system in which the language model was based on Gemini and trained from scratch on an order of magnitude more synthetic data than its predecessor. This helped the model tackle much more challenging geometry problems, including problems about movements of objects and equations of angles, ratio or distances.

AlphaGeometry 2 employs a symbolic engine that is two orders of magnitude faster than its predecessor. When presented with a new problem, a novel knowledge-sharing mechanism is used to enable advanced combinations of different search trees to tackle more complex problems.

Before this year’s competition, AlphaGeometry 2 could solve 83% of all historical IMO geometry problems from the past 25 years, compared to the 53% rate achieved by its predecessor. For IMO 2024, AlphaGeometry 2 solved Problem 4 within 19 seconds after receiving its formalization.

Illustration of Problem 4, which asks to prove the sum of ∠KIL and ∠XPY equals 180°. AlphaGeometry 2 proposed to construct E, a point on the line BI so that ∠AEB = 90°. Point E helps give purpose to the midpoint L of AB, creating many pairs of similar triangles such as ABE ~ YBI and ALE ~ IPC needed to prove the conclusion.

New frontiers in mathematical reasoning

As part of our IMO work, we also experimented with a natural language reasoning system, built upon Gemini and our latest research to enable advanced problem-solving skills. This system doesn’t require the problems to be translated into a formal language and could be combined with other AI systems. We also tested this approach on this year’s IMO problems and the results showed great promise.

Our teams are continuing to explore multiple AI approaches for advancing mathematical reasoning and plan to release more technical details on AlphaProof soon.

We’re excited for a future in which mathematicians work with AI tools to explore hypotheses, try bold new approaches to solving long-standing problems and quickly complete time-consuming elements of proofs — and where AI systems like Gemini become more capable at math and broader reasoning.

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Traditional approaches to ARC have primarily focused on program synthesis and leveraging large language models (LLMs). While these methods have advanced the field, they often need to catch up due to the logical complexities involved in ARC tasks. The performance of these models has yet to meet expectations, leading researchers to explore alternative approaches fully. Reinforcement learning has emerged as a promising yet underexplored method for tackling ARC, offering a new perspective on addressing its unique challenges.

Researchers from the Gwangju Institute of Science and Technology and Korea University have introduced ARCLE (ARC Learning Environment) to address these challenges. ARCLE is a specialized RL environment designed to facilitate research on ARC. It was developed using the Gymnasium framework, providing a structured platform where RL agents can interact with ARC tasks. This environment enables researchers to train agents using reinforcement learning techniques specifically tailored for the complex tasks presented by ARC.

ARCLE comprises several key components: environments, loaders, actions, and wrappers. The environment component includes a base class and its derivatives, which define the structure of action and state spaces and user-definable methods. The loaders component supplies the ARC dataset to ARCLE environments, defining how datasets should be parsed and sampled. Actions in ARCLE are defined to enable various grid manipulations, such as coloring, moving, and rotating pixels. These actions are designed to reflect the types of manipulations required to solve ARC tasks. The wrappers component modifies the environment’s action or state space, enhancing the learning process by providing additional functionalities.

The research demonstrated that RL agents trained within ARCLE using proximal policy optimization (PPO) could successfully learn individual tasks. The introduction of non-factorial policies and auxiliary losses significantly improved performance. These enhancements effectively mitigated issues related to navigating the vast action space and achieving the hard-to-reach goals of ARC tasks. The research highlighted that agents equipped with these advanced techniques showed marked improvements in task performance. For instance, the PPO-based agents achieved a high success rate in solving ARC tasks when trained with auxiliary loss functions that predicted previous rewards, current rewards, and next states. This multi-faceted approach helped the agents learn more effectively by providing additional guidance during training.

Agents trained with proximal policy optimization (PPO) and enhanced with non-factorial policies and auxiliary losses achieved a success rate exceeding 95% in random settings. The introduction of auxiliary losses, which included predicting previous rewards, current rewards, and next states, led to a marked increase in cumulative rewards and success rates. Performance metrics showed that agents trained with these methods outperformed those without auxiliary losses, achieving a 20-30% higher success rate in complex ARC tasks. 

To conclude, the research underscores the potential of ARCLE in advancing RL strategies for abstract reasoning tasks. By creating a dedicated RL environment tailored to ARC, the researchers have paved the way for exploring advanced RL techniques such as meta-RL, generative models, and model-based RL. These methodologies promise to enhance AI’s reasoning and abstraction capabilities further, driving progress in the field. The integration of ARCLE into RL research addresses the current challenges of ARC and contributes to the broader endeavor of developing AI that can learn, reason, and abstract effectively. This research invites the RL community to engage with ARCLE and explore its potential for advancing AI research.

Check out the Paper . All credit for this research goes to the researchers of this project. Also, don’t forget to follow us on  Twitter and join our  Telegram Channel and  LinkedIn Gr oup . If you like our work, you will love our  newsletter..

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Nikhil is an intern consultant at Marktechpost. He is pursuing an integrated dual degree in Materials at the Indian Institute of Technology, Kharagpur. Nikhil is an AI/ML enthusiast who is always researching applications in fields like biomaterials and biomedical science. With a strong background in Material Science, he is exploring new advancements and creating opportunities to contribute.

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