what is math assignment

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Assigning More Meaningful Math Homework

A small set of problems or even one substantial problem can be enough to supplement classroom instruction.

As a math teacher of more than 23 years, I have had the habit, almost like a muscle memory repetition, of assigning daily math homework for my middle school students. It wasn’t until recently that I paused to reflect, “Why am I assigning this?” The easy answer is, “My students need to practice to develop their skills.”

If I dig a bit deeper into the “why,” I wonder, “Are all of my students benefiting from this assignment? Did I assign an appropriate amount and level of problems? What resources do my students have or not have to be successful with this assignment? Is the assignment meaningful or busywork?”

Consider the following suggestions for making math homework more meaningful.

3 Ways to Create More Meaningful Math Homework

1. Think quality over quantity. The National Council for Teachers of Mathematics Homework page of tips for teachers suggests, “Only assign what’s necessary to augment instruction. If you can get sufficient information by assigning only five problems, then don’t assign fifty.”

Worksheets and problem sets from textbook publishers might contain dozens of problems that repeat the same concepts. It is OK to assign part of a page, such as “p. 34 #s 3, 5, and 17.” I tell my middle school students, “I handpicked these particular problems because they capture the objective of today’s lesson.” When students know that their teacher carefully “handpicked” a problem set, they might pay more attention to the condensed assignment because it was tailored for them.

Even one problem can be sufficient. Robert Kaplinsky, cofounder of Open Middle , routinely shares on X (formerly Twitter) single problems that are really engaging and give students a good chance to practice skills.

The depth and exploration that can come from one single problem can be richer than 20 routine problems. You might be surprised by how much depth can be inspired by a single problem.

2. Consider choice and variety. It’s unrealistic to create a personalized daily homework assignment for each student in your class. Student voice and choice can be applied to your preexisting assignments without your having to re-create the homework wheel.

Traditional assignments can be modified by offering students choice. This might look like “ Choose any five of these problems ,” or take this tip from educator Peter Liljedahl and designate problems as “mild, medium, or spicy” and let students pick their level for that assignment.

Offering homework level choice also promotes a culture of growth mindset through messaging like “You might choose mild problems for this lesson; however, tomorrow you might feel you’re ready for a medium level.” Level choice can vary day to day—your math level is not fixed.

Daily homework can also be spiced up by offering a variety of types of assignments. Consider assigning problems that go beyond providing a single number answer. Here are a few examples to get students thinking beyond just getting a particular problem right:

• When simplifying (4 + 5) x 5 - 3, what is the first step?
• When simplifying (4 + 5) x 5 - 3, Ali got the answer 18. What advice do you have for her?
• Write your own order of operations problem with a solution of 42.

Check out these websites for even more creative ways to vary homework:

• Three-Acts Math Tasks
• Open Middle
• Would You Rather Math

3. Remember, accountability doesn’t have to result in a grade. There is a major difference between assigning homework for a grade and assigning homework purely for practice. When a grade is the result of an assignment, the stakes get higher for the student.

In the February 2023 Washington Post article “ A deep dive into whether—and how—homework should be graded ,“ former teacher Rick Wormeli wrote, “When early attempts at mastery are not used against them, and accountability comes in the form of actually learning content, adolescents flourish.” If homework is truly for practice, this is an opportunity for students to make mistakes and take risks without the fear of a penalty.

Even if homework is graded as a completion grade, there are considerations of equity and meaningfulness of the practice.

Consider the following questions when deciding to give a completion grade for a homework assignment: Do all students have a home environment that is supportive of homework? Do some students have additional support, such as tutors or parents, to help them get the homework completed? Would students copy homework assignments from each other just to earn the completion grade?

If not grades, then how do we hold students accountable for practicing outside of class?

Student presentations and discussions are a way to check for understanding of an assignment and to let students know you expect them to attempt the problems. This might look like a debate in which students take sides on how to approach a problem . Alternatively, students could post their work on the board to share their strategies with the class or discuss their solutions in small groups. Communicating their mathematical thinking deepens their understanding .

Education consultants Ashley Marlow and Katie Novak write in their Edutopia article “ Making Math Accessible for All Students ” (July 2022), “When students have opportunities to think, reason, explain, and model their thinking, they are more readily able to develop a deep understanding of mathematics beyond rote memorization. The goal is for all students to experience success in higher learning of mathematics—requiring those reasoning and sense-making skills and increasing engagement.”

The next time you’re planning your lessons and assignments, pause and reflect on the meaningfulness of the homework assignment. Could it be shorter but more in-depth? Can students have a choice in their work? Will students find value in doing the work even if it is not for a grade? You might find that students take more ownership and care in their homework if it’s more meaningful to them.

• Instasolving

Top-Rated Tips From Experts: How to Write a Maths Assignment

Mathematics Mastery: Trust Our Experts for Impeccable Assignment Writing, Ensuring Precision and Academic Excellence.

Writing a mathematics assignment is rewarding yet challenging, requiring precision, clarity, and a deep understanding of mathematical concepts. Navigating this academic challenge successfully can be exhaustive for students. Under this circumstance, consider the insights and tips our expert professionals share. These tips enhance your approach to creating high-quality and accurate math assignments.

From effective problem-solving strategies to the importance of clear communication, these expert-recommended guidelines offer a comprehensive framework for achieving excellence in math homework. Whether you're a student looking to improve your assignment skills or an educator guiding others through the process, incorporating these tips can elevate the quality and impact of your math assignments.

Mastering Math Assignments: Strategies for Success

Embracing the world of mathematics can be enjoyable and intellectually stimulating when approached with the right mindset. Here are some valuable tips to enhance your ability to solve math assignments efficiently.

Eliminate Distractions: Dedicate focused time to complete your math assignments, as they often involve multi-step problem-solving. Minimize distractions by putting away your phone and notifying others of your study plans. Seeking assistance from math assignment help services can provide expert guidance and support you require for math assignments.

Utilize Mathematical Tricks: Learn and apply quick mathematical tricks from your professors for calculations. You can also opt for assistance from a math assignment helper to ace different techniques. Familiarity with these tricks can significantly expedite the problem-solving of math writing assignment.

Thoroughly Read and Follow Instructions: Carefully read the instructions provided by your instructor before diving into the assignment. Use a notepad to jot down essential information and create an outline for a better understanding. Clearly articulate your problem-solving process, ensuring you address all aspects mentioned in the instructions.

Start with Challenging Problems: Prioritize your math problems based on difficulty and importance. Tackle the most challenging problems first and gradually work your way down. This approach ensures efficient utilization of your energy and time.

Take Regular, Short Breaks: Maintain concentration by taking short breaks to alleviate stress and prevent burnout. Breaks of 15 to 20 minutes allow you to recharge without losing focus on your assignments.

Study with a Friend: Collaborating with a study companion can provide motivation and a fresh perspective. Explaining concepts to each other enhances understanding and promotes effective learning.

Seek Expert Assistance: Don't hesitate to ask for help from experts at any stage of your assignment. Technology facilitates instant access to professionals who can assist with complex problems, ensuring academic success.

Practice Regularly: Building a strong foundation in mathematics requires consistent practice. Regular practice eases complex computations and explores various methods. It also helps you to analyze shortcuts for rapid problem-solving.

Common Worries About Math Assignments:

Many students experience common worries about math assignments. They often feel anxious about understanding complex concepts and solving difficult problems. Time management is another concern, as math assignments can be time-consuming and require focused attention. The fear of making mistakes and the pressure to achieve high grades can also contribute to stress. Additionally, some students worry about not having adequate resources or support to complete their assignments successfully. These concerns can impact their confidence and performance, but with the right strategies and help, such as tutoring and practice, students can overcome these challenges and excel in their math assignments.

Understanding why students may struggle with math assignments is essential. Common reasons include:

Problems in Math Concept Understanding

Difficulty in understanding math arises from a combination of factors.

Some learners find the abstract nature of mathematical concepts challenging. Others face problems due to confusion from previously perceived knowledge. Most students acknowledged they often find it hard to visualize and relate math problems to real-world scenarios. Foundational knowledge gaps hinder comprehension. Effective math understanding requires a supportive learning environment, including several techniques that we have explained in the visual. These steps emphasize building a solid foundation for understanding and solving mathematical problems.

Challenges Due to Teaching

Students face diverse study challenges due to teaching methodologies. One of them is the lack of effective communication between educators and students. If teachers fail to articulate concepts clearly, students may struggle to grasp subject materials. Inadequate teacher-student interaction can hinder the resolution of doubts or concept clarification. Outdated teaching methods fail to engage students, resulting in a lack of interest and motivation to study. Effective teaching involves adapting to the diverse needs of students. It fosters a positive and interactive study environment for a better learning experience.

Inconsistency in Practice:

Inconsistency in practice poses a significant challenge in various contexts, ranging from skill development to routine tasks. Irregular practice can impede progress. Inconsistency in practicing new skills or habits can hinder the cultivation of expertise or the formation of positive routines. The key to overcoming this challenge is establishing and adhering to a consistent, structured practice regimen. By doing so, individuals can enhance their learning, skill acquisition, and personal growth, fostering a more sustainable and effective path toward mastery and success.

Quick Tips for Improved Math Skills

Improve your math skills with these quick tips: practice regularly, review and understand basic concepts, seek help when needed, use online resources, and solve various problem types. Stay organized, work on time management, and maintain a positive attitude towards learning. Consistency and persistence are key to mastering math.

Enhance your arithmetic abilities and speed up problem-solving with these additional tips:

Master the Basics:

Proficiency in fundamental operations is crucial for tackling more advanced topics. It ensures a solid foundation by mastering the core principles of arithmetic.

Understand Concepts, Don't Memorize:

Focus on understanding the underlying concepts rather than memorizing procedures. It enables flexible problem-solving and a deeper grasp of mathematical principles.

Use Online Resources:

Explore online platforms offering interactive math exercises and tutorials. Math assignment help websites provide valuable resources for offering academic assistance.

Break Down Complex Problems:

Break down complex problems into smaller, manageable steps. This approach makes problem-solving more approachable and helps prevent feeling overwhelmed.

Teach Others:

Teaching a concept to someone else is an effective way to solidify your understanding. This process forces you to articulate ideas and reinforces your knowledge.

Seek Help When Needed:

Don't hesitate to ask for help when you encounter difficulties. Teachers, classmates, or online communities can provide valuable insights and clarification.

Review and Reflect:

Regularly review previously learned material and reflect on your progress. Identify areas of weakness and focus on reinforcing those concepts.

Utilize Educational Apps and Games:

Explore educational apps and games that make learning math enjoyable. These approaches can add an element of fun to your practice sessions.

Assignments are integral to academic growth, and mastering math is a journey that requires dedication and effective time management. Seek assistance when needed, and you'll develop strong math skills with consistent practice.

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Introduction to Math 444 Assignments

This sheet is about the regular individual assignments. there may be other written work stemming from group activity in class, etc., that is not covered by this description., what is the purpose of the written assignments.

As in most subjects, one masters math by doing. This means that all one's studying should involve drawing, working examples on paper, etc. But the written assignments provide the impetus to work out in polished form problems and constructions and proofs that develop understanding of geometry. They also provide a running assessment for student and instructor of what is going on in the class as to levels of understanding, areas of success, areas that need work, etc.

How often are assignments given?

In the beginning part of the course, shorter assignments will be due two or three times a week. This will gradually move to longer and less freqent assignments of about one per week. Students who don't like daily assignments are free to work ahead and treat assignments 1a, 1B, 1C, etc. as a single weekly assignment.

What kind of written assignments?

Assignment items to turn in will be of several forms.

• Fairly routine exercises or constructions.
• Problems to solve - some challenging and some less so. These can be construction problems, numerical or ratio problems, or existence of special geometrical relationships such as similarity, concurrence, etc.
• Major or minor theorems to prove.
• Physical models and drawings will sometines be assigned.
• Investigations and open-ended problems. As the course moves on, such problems will appear on the assignments.

When are assignments due?

• Assignments are due at the beginning of class on the due date .
• No further writing is allowed once you walk into the room. Experience has shown that this leads to poor quality presentation, and sometimes to copying from other students.
• Students may discuss problems outside of clas s, so long as each student writes up her/his work independently. However, "sharing" answers in the 10 minutes before class is too late for group reflection.
• Late Assignments cause problem, so they may not be accepted. Often questions on an assignment will be answered during class. Also, the grading will sometimes be scheduled on the day the assignments are due. We do realize that nobody is perfect (least of all us) and that crises occur. If it happens that a student does not have an assignment on time once or even twice and asks to turn it in late, we will probably say OK, at least for partial credit. Once or twice also will not make or break one's grade. But don't make a habit of it.

Grading of Assignments will Follow several Strategies

We feel it is important that assignments be graded and feedback provided. However, some of the problems assigned in this course are not quickly graded. A bit of calculation will show that any problem that just takes 10 minutes per student will add up to about 5 or 6 hours of grading time. Thus the instructors will adopt a number of strategies to make grading possible. In some cases, students will self-report and then groups will discuss answers.

Quality of Presentation Counts

Since Math 444 is a course for students who intend to make a career explaining mathematics, the assignments turned in should be well-written as to language and thought. Answer in complete sentences. Use white space. Break up long explanations or proofs into paragraphs. Write neatly or word process. Presentation can definitely be part of the grade (plus or minus). Egregious examples of illegibility or bad writing will not be graded

What this means is that one should often write down a draft page for brainstorming and then write a completely new page to turn in.

Good and Bad News about Assignment Grades

This section gives two contradictory messages. The good news is that in Math 444 one does not have to get 100% to get a high grade. The problems are not routine, so perfection is not required, though a high standard is expected. The bad news (for some) is that skipping the homework and trying to get a high grade by being brilliant on the tests will not work because of the weight put on assignments. Only part of the course experience shows up on the tests, so high test grades alone do not ensure a high course grade in the absence of strong assignment grades.

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Difference between $:=$ and $=$

I am sorry if this is quite elementary question. But I always think, that why we use $:=$ at some places, instead of $=$. Is there any fundamental difference between these two? Before reading Terry Tao's blog (4 months ago), I had never seen a symbol like this (:=).

• 7 $\begingroup$ The former is usually used for definitions. This is adapted from some programming languages. The notation is deliberately unsymmetrical... $\endgroup$ –  J. M. ain't a mathematician Commented Sep 11, 2011 at 14:41
• $\begingroup$ @J. M. Oh.. I too thought the same, but wasn't sure. Because my first year math books have definition notation of this type: $\displaystyle {{=}^{\mathrm{def}}}$. $\endgroup$ –  Gaurav Tiwari Commented Sep 11, 2011 at 14:49
• 3 $\begingroup$ en.wikipedia.org/wiki/List_of_logic_symbols $\endgroup$ –  user9464 Commented Sep 11, 2011 at 15:03
• 3 $\begingroup$ I've also seen $\stackrel{\Delta}{=}$ used for definitions. I don't think there is an established notation in general; most just use $=$ and indicate that it is a definition rather than a relation in the surrounding text. $\endgroup$ –  user856 Commented Sep 11, 2011 at 15:32

3 Answers 3

Some authors like to distinguish the assertion "A is equal to B" from "define A as an object equal to B".

In math, the first one is always $A=B$ (I've never seen anything else, at least). But for definitions, I've seen

• $A \triangleq B$
• $A \equiv B$
• $A \stackrel{\mathrm{def}}{=} B$

Often the triple equal sign is used for strong notions of equivalence (such as in binary relations) or with the "mod" symbol.

In computer programming, we are typographically limited of course. The assignment operator is usually distinguished from the comparison operator, and this is done in different ways depending on the language. For example in Pascal you assign x := 5 and compare x = 5 . However in C you assign x = 5 and compare x == 5 . (It should be noted that in C this syntax has caused untold confusion and a few famous bugs.)

My favorite (meaning "most awful") example is in PHP and javascript where one can do a "strict compare" with x === 5 . Finally, some languages have ridiculous vagaries like distinguishing which comparision operator is used based on the type of variable (usually strings being different from numerics), with syntaxes like .eq. and related.

• $\begingroup$ better answer! Thanks. $\endgroup$ –  Gaurav Tiwari Commented Sep 11, 2011 at 15:37
• $\begingroup$ "syntaxes like .eq. and related." - Heh... man, that sure was old... ;) $\endgroup$ –  J. M. ain't a mathematician Commented Sep 11, 2011 at 15:41
• $\begingroup$ @J.M. Yeah, .eq. is FORTRAN77 (I'm showing my age here). $\endgroup$ –  Fixee Commented Sep 11, 2011 at 15:42
• $\begingroup$ I know; I've had the pleasure of re-implementing old code... :D $\endgroup$ –  J. M. ain't a mathematician Commented Sep 11, 2011 at 15:43

The notation := was used in some early programming languages. The meaning of x := 3 was "assign 3 to the variable x". So, the assignment could be distinguished from the the test if (x = 3) ... . In mathematics the obvious meaning is "Let x be 3".

Modern programming languages use x = 3 for the assignment, and == for the equality test if (x == 3) ... .

• 7 $\begingroup$ "Early" and "modern", eh? You may find some disagreement with those characterizations. $\endgroup$ –  GEdgar Commented Sep 11, 2011 at 16:59
• $\begingroup$ @Edgar: I know ... I did't find a better characterization at the moment of writing. I thought of Algol, Pascal with ':=', and Java, Javascript, C++, Python with '='. But I know the old C, Fortran with "=", etc. I wanted to say that I don't see ':=' so often today but surely there are some. Well, perhaps I should have written in some ones is ':=' and in other ones is '='. $\endgroup$ –  Jiri Kriz Commented Sep 11, 2011 at 19:03
• 3 $\begingroup$ Mathematica uses := and = as (delayed and immediate) assignment operators, and == for equality comparisons. $\endgroup$ –  J. M. ain't a mathematician Commented Sep 11, 2011 at 21:37
• $\begingroup$ Early and modern seems pretty accurate. Languages that don't use "==" for comparison or "=" for assignment could be called post-modern. :) $\endgroup$ –  Andres Riofrio Commented Mar 20, 2012 at 18:48
• 3 $\begingroup$ Actually, = and := for assignment both date back to the dawn of high-level languages: FORTRAN used = in 1957, and ALGOL 58 used := . Today the main distinction is between C-like syntax (with = and == ) and Pascal-like syntax (with := and = ); C and Pascal were both initially developed in the early 1970s. It doesn't seem really meaningful do call that a distinction between "early" and "modern". $\endgroup$ –  hmakholm left over Monica Commented Nov 11, 2015 at 17:52

The equality symbol = denotes an operator that is part of the object theory. The definition symbol $\triangleq$ (and its variants) denotes a syntactic replacement, and is part of the metatheory .

This distinction is emphasized in the formal syntax of TLA+ , whose axioms are those of Zermelo-Fraenkel set theory + Hilbert's $\varepsilon$, and functions. The theory has some axiom schemata for equality, as is usually the case with theories that include equality. In order to refer to symbols, they first have to be declared. For example:

Symbol a is declared to be a constant, and x as a variable (constants stay unchanged though a behavior, variables can change). The above theorem holds due to the indiscernibility of identicals (i.e., equal things "look the same").

A symbol can be defined , instead of declared. For example:

The symbol Foo literally stands for a \/ b . The definition symbol == (typeset as $\triangleq$) is not an operator of the object theory , as \in , = , + , and other symbols are. It is metatheoretic notation that says that the string Foo abbreviates the expression a \/ b . So wherever Foo appears later, it is syntactically replaced by the expression a \/ b before any reasoning takes place.

Interestingly, there is another way of writing the last example:

What we have done is declare the identifier Foo as a constant, so some value in the object theory (here some set), and added an axiom that it so happens that the value of Foo is the same as the value of a \/ b ( \/ denotes disjunction).

A defined symbol can be formally understood as shorthand for an extension by definition . The last example resembles an extension by definition that would correspond to the middle example. A clear introduction to the concept of extending a theory with definitions is given in Kunen's "Foundations of mathematics" , Sec. II.15 on pages 148--152.

The theorem prover TLAPS has several examples of how definitions and declarations are used in its library of modules .

As a side note, $a \equiv b$ in TLA+ means $(a \in \mathrm{BOOLEAN}) \wedge (b \in \mathrm{BOOLEAN}) \wedge (a = b)$, and := means assignment in the algorithm language PlusCal , which translates to TLA+.

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Math Assignment

 What is an annotated bibliography?

Assignments in math courses are usually a list of problems to be solved. Each question may be of various difficulties and types, which are marked and returned to the student to be used as a source of feedback. Math assignments are designed to provide opportunities for 'doing math' and to consolidate students’ understanding of the content. The questions often come from the most recent week of learned material, but some questions may require students to synthesize concepts from further back. This is because learning math is cumulative by nature; you continuously build upon what you've learned before. Because math assignments are typically used as a checkpoint for understanding, they tend to weigh less than other assessments in terms of grade, therefore, it's important to treat them as learning opportunities instead of a tool for maximizing course grades.

Click on the Timeline for a visual representation of the timeline. Click on the Checklist for a document containing the checklist items for a math assignment.

According to your start and end dates ( 2024-08-21 to 2024-09-05 ), you have 15 days to finish your assignment.

Step 1: Understand the questions Complete by Wed Aug 21, 2024

Read through the assignment once it’s released and understand the questions. Read the questions early to prime your brain; too many students start too late.

• For some courses, assignments come out before or as content is covered, so having the questions in your mind as you see the content can help prime you to actively learn the content and will also help you know where to look for relevant material in the lecture notes and/or videos. Active learning of content means you are asking yourself questions as you are learning, actively monitoring your understanding of the topic, and anticipating next steps.
• Read all the questions to estimate the range of tested material, make note of anything you don’t understand. Understanding the question correctly reduces the chance of getting stuck.

The hardest thing being with a mathematician is that they always have problems.

– Tendai Chitewere

Step 2: Gather materials and review Complete by Sat Aug 24, 2024

• Gather required materials (lecture notes and/or videos, textbook sections).
• Study the material (definitions, concepts, solved problems, etc.) until you are familiar with it. Identify connections across concepts/topics and practice additional problems. Regularly reviewing your material will also help you to find relevant information quickly if you get stuck on a question.

Step3: Solve questions and get help Complete by Tue Sep 03, 2024

Part a: attempt to solve each question.

• Rule of thumb for time allocation: one hard question may take significantly longer than all easy questions combined. Consider how long you’ve taken to solve all the easy questions. It may take at least twice as long to solve one hard question. Try all easy questions as soon as possible to gauge how long the rest could take.
• Try giving yourself a break for a day before attempting them again to allow time for your mind to continue working subconsciously.
• Still stuck? Check out Math problems: What to do when you're stuck for more strategies and tips.
• Keep track of unsolved questions and what stages you were stuck at. Don’t throw away any work in progress.Your instructor or TA can provide better help if you can narrow down on the issue.

The only way to learn mathematics is to do mathematics.

– Paul Halmos

Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

– Paul Halmos  

Part B: Get help on unsolved questions

• The sooner you ask for support, the more time you and the instructor/TA will have for follow-up conversations.
• Focus on asking questions to help you close the gap between your understanding, and the understanding you need to complete the question.  Questions such as “Can I get a hint on #4?” or “What is the answer?” are not typically helpful in increasing your problem-solving abilities.
• For questions you were challenged by, but were able to solve after receiving help from the instructor or TA, make sure that you fully understand how to come up with the solution in case a similar question is asked on a test.
• Many students find it helpful to study together. Make sure you understand what your instructor’s expectations are around group work (if you’re not sure, ask!). Be mindful about how much you post about an assignment solution on the course’s discussion board and if you’re in doubt, consider posting privately. Familiarize yourself with the academic integrity standards expected in your courses.

The goal of seeking help is to try to find out how to approach questions that you are stuck on. From this extra help you still may not see the complete picture of how to solve the problem. To see this may require that you go back to Part A , now with the instructor/TA’s suggestions in mind.

I have not failed. I've just found 10,000 ways that won't work.

– Thomas Alva Edison

Step 4: Check your solutions Complete by Thu Sep 05, 2024

• It’s important that you spend some time away from your solutions before you check them. For problems you did solve, try to figure out if you can check your own answer like you would have to on a test (i.e. don’t look up answers in a book or online).
• You could ask the instructor/TA for guidance, but avoid questions such as “Did I do this right?” Instead, ask yourself whether you are asking the right questions about your solution. 
• Remember not to focus only on the final answer, but also on how you communicated your answer. Ask yourself if a typical student in the class could follow along with what you’ve written. 
• Remember to cite any external sources that you used in your work, including work completed with others (if this was permitted).

Step 5: Review Marked Assignments Complete by Thu Sep 05, 2024

After receiving your marked assignment check over incorrect questions as well as those you got right. Review posted solutions and read them critically. Often, there are different approaches to a problem and you may learn about some of these new approaches through this review.

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Stanford Online

Introduction to mathematical thinking.

HSTAR-Y0001

Stanford Graduate School of Education

The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years.

Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today's world. This course helps to develop that crucial way of thinking.

The course is offered in two versions. The eight-week-long Basic Course is designed for people who want to develop or improve mathematics-based, analytic thinking for professional or general life purposes. The ten-week-long Extended Course is aimed primarily at first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course.

Subtitles for all video lectures available in: Portuguese (provided by  The Lemann Foundation ), English

Course Syllabus

Instructor's welcome and introduction

•  Introductory material
•  Analysis of language – the logical combinators
•  Analysis of language – implication
•  Analysis of language – equivalence
•  Analysis of language – quantifiers
•  Working with quantifiers
•  Proofs
•  Proofs involving quantifiers
•  Elements of number theory
• Beginning real analysis

Recommended Background

High school mathematics. Specific requirements are familiarity with elementary symbolic algebra, the concept of a number system (in particular, the characteristics of, and distinctions between, the natural numbers, the integers, the rational numbers, and the real numbers), and some elementary set theory (including inequalities and intervals of the real line). Students whose familiarity with these topics is somewhat rusty typically find that with a little extra effort they can pick up what is required along the way. The only heavy use of these topics is in the (optional) final two weeks of the Extended Course.

A good way to assess if your  basic  school background is adequate (even if currently rusty) is to glance at the topics in the book  Adding It Up: Helping Children Learn Mathematics  (free download), published by the US National Academies Press in 2001. Though aimed at K-8 mathematics teachers and teacher educators, it provides an excellent coverage of what constitutes a good basic mathematics education for life in the Twenty-First Century (which was the National Academies' aim in producing it).

Dr Keith Devlin, Co-founder and Executive Director H-STAR Institute

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