research paper on problem faced by students in learning mathematics

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• Published: 19 December 2019

Problematizing teaching and learning mathematics as “given” in STEM education

• Yeping Li 1 &
• Alan H. Schoenfeld 2  

International Journal of STEM Education volume  6 , Article number:  44 ( 2019 ) Cite this article

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Mathematics is fundamental for many professions, especially science, technology, and engineering. Yet, mathematics is often perceived as difficult and many students leave disciplines in science, technology, engineering, and mathematics (STEM) as a result, closing doors to scientific, engineering, and technological careers. In this editorial, we argue that how mathematics is traditionally viewed as “given” or “fixed” for students’ expected acquisition alienates many students and needs to be problematized. We propose an alternative approach to changes in mathematics education and show how the alternative also applies to STEM education.

Introduction

Mathematics is commonly perceived to be difficult (e.g., Fritz et al. 2019 ). Moreover, many believe “it is ok—not everyone can be good at math” (Rattan et al. 2012 ). With such perceptions, many students stop studying mathematics soon after it is no longer required of them. Giving up learning mathematics may seem acceptable to those who see mathematics as “optional,” but it is deeply problematic for society as a whole. Mathematics is a gateway to many scientific and technological fields. Leaving it limits students’ opportunities to learn a range of important subjects, thus limiting their future job opportunities and depriving society of a potential pool of quantitatively literate citizens. This situation needs to be changed, especially as we prepare students for the continuously increasing demand for quantitative and computational literacy over the twenty-first century (e.g., Committee on STEM Education 2018 ).

The goal of this editorial is to re-frame issues of change in mathematics education, with connections to science, technology, engineering, and mathematics (STEM) education. We are hardly the first to call for such changes; the history of mathematics and philosophy has seen ongoing changes in conceptualization of the discipline, and there have been numerous changes in the past century alone (Schoenfeld 2001 ). Yet changes in practice of how mathematics is viewed, taught, and learned have fallen far short of espoused aspirations. While there has been an increased focus on the processes and practices of mathematics (e.g., problem solving) over the past half century, the vast majority of the emphasis is still on what content should be presented to students. It is thus not surprising that significant progress has not been made.

We propose a two-fold reframing. The first shift is to re-emphasize the nature of mathematics—indeed, all of STEM—as a sense-making activity. Mathematics is typically conceptualized and presented as a body of content to be learned and processes to be engaged in, which can be seen in both the NCTM Standards volumes and the Common Core Standards. Alternatively, we believe that all of the mathematics studied in K-12 can be viewed as the codification of experiences of both making sense and sense making through various practices including problem solving, reasoning, communicating, and mathematical modeling, and that students can and should experience it that way. Indeed, much of the inductive part of mathematics has been lost, and the deductive part is often presented as rote procedures rather than a form of sense making. If we arrange for students to have the right experiences, the formal mathematics can serve to organize and systematize those experiences.

The second shift is suggested by the first, with specific attention to classroom instruction. Whether mathematics or STEM, the main focus of most instruction has been on the content and practices of the discipline, and what the teacher should do in order to make it accessible to students. Instead, we urge that the main focus should be on the student’s experience of the discipline – on the affordances the environment provides the student for disciplinary sense making. We will introduce the Teaching for Robust Understanding (TRU) Framework, which can be used to problematize instruction and guide needed reframing. The first dimension of TRU (The Discipline) focuses on the re-framing discussed above: is the content conceptualized as something rich and connected that can be experienced and codified in meaningful ways? The second dimension (Cognitive Demand) examines opportunities students have to do that kind of sense-making and codification. The third (Equitable Access to Content) examines who has such opportunities: is there equitable access to the core ideas? Dimension 4 (Agency, Ownership, and Identity) asks, do students encounter the discipline in ways that enable them to see themselves as sense makers, building both agency and positive disciplinary identities? Finally, dimension 5 (Formative Assessment) asks, does instruction routinely use formative assessment, allowing student thinking to become public so that instruction can be adjusted accordingly?

We begin with a historical background, briefly discussing different views regarding the nature of mathematics. We then problematize traditional approaches to mathematics teaching and learning. Finally, we discuss possible changes in the context of STEM education.

Knowing the background: the development of conceptions about the nature of mathematics

The scholarly understanding of the nature of mathematics has evolved over its long history (e.g., Devlin 2012 ; Dossey 1992 ). Explicit discussions regarding the nature of mathematics took place among Greek mathematicians from 500 BC to 300 AD (see, https://en.wikipedia.org/wiki/Greek_mathematics ). In contrast to the primarily utilitarian approaches that preceded them, the Greeks pioneered the study of mathematics for its own sake and pursued the development and use of generalized mathematical theories and proofs, especially in geometry and measurement (Boyer 1991 ). Different perspectives about the nature of mathematics were gradually developed during that time. Plato perceived the study of mathematics as pursuing the truth that exists in external world beyond people’s mind. Mathematics was treated as a body of knowledge, in the ideal forms, that exists on its own, which human’s mind may or may not sense. Aristotle, Plato’s student, believed that mathematicians constructed mathematical ideas as a result of the idealization of their experience with objects (Dossey 1992 ). In this perspective, Aristotle emphasized logical reasoning and empirical realization of mathematical objects that are accessible to the human senses. The two schools of thought that evolved from Plato’s and Aristotle’s contrasting conceptions of the nature of mathematics have had important implications for the ensuing development of mathematics as a discipline, and for mathematics education.

Several more schools of thought were developed as mathematicians tackled new problems in mathematics (Dossey 1992 ). Davis and Hersh ( 1980 ) provides an entertaining and informative account of these developments. Three major schools of thought in the early 1900s dealt with paradoxes in the real number system and the theory of sets: (1) logicism, as an outgrowth of the Platonic school, accepts the external existence of mathematics and emphasizes the form rather than the interpretation in a specific setting; (2) intuitionism, as influenced by Aristotle’s ideas, only accepts the mathematics to be developed from the natural numbers forward via “valid” patterns of mental reasoning (not empirical realization in Aristotle’s thought); and (3) formalism, also aligned with Aristotle’s ideas, builds mathematics upon the formal axiomatic structures to free mathematics from contradictions. These three schools of thought are similar in that they view the contents of mathematics as products , but they differ in whether products are viewed as pre-existing or created through experience. The development of these three schools of thought illustrates that the view of mathematics as products has its long history in mathematics.

With the gradual development of school mathematics since 1900s (Stanic and Kilpatrick 1992 ), the conception of the nature of mathematics has increasingly received attention from mathematics educators. Which notion of mathematics mathematics education adopts and uses has a direct and strong impact on the way of school mathematics being presented and approached in education. Although the history of school mathematics is relatively short in comparison with mathematics itself, we can find ample examples about the influence of different views of mathematics on curriculum and classroom instruction in the USA and other education systems (e.g., Dossey et al. 2016 ; Li and Lappan 2014 ; Li, Silver, and Li 2014 ; Stanic and Kilpatrick 1992 ). For instance, the “New Math” movement of 1950s and 1960s used the formalism school of thought as the core of reform efforts. The content was presented in a structural format, using the set theoretic language and conceptions. But the result was not a successful progression toward a school mathematics that is best for students and teachers (e.g., Kline 1973 ). Alternatively, Dossey ( 1992 ), in his review of the nature of mathematics, identified and selected scholars’ works and ideas applicable to both professional mathematicians and mathematics educators (e.g., Davis and Hersh 1980 ; Hersh 1986 ; Tymoczko 1986 ). Those scholars' ideas rested on what professional mathematicians do, not what mathematicians think about what mathematics is. Dossey ( 1992 ) specifically cited Hersh ( 1986 ) to emphasize mathematics is about ideas and should be accepted as a human activity, not strictly governed by any one school of thought.

Devlin ( 2000 ) argued that mathematics is not a single entity but has four different faces: (1) computation, formal reasoning, and problem solving; (2) a way of knowing; (3) a creative medium; and (4) applications. Further, he contended school mathematics typically focuses on the first face, makes some reference to the fourth face, but pays almost no attention to the other two faces. His conception of mathematics assembles ideas from the history of mathematics and observes mathematical activities occurring across different settings.

Our brief review shows that the nature of mathematics can be understood as having different faces, rather than being governed by any single school of thought. At the same time, the ideas of Plato and Aristotle continue to influence the ways that mathematicians, mathematics educators, and the general public perceive mathematics. Despite nearly a half century of process-oriented research (see below), let alone Pólya’s work on problem solving, mathematics is still perceived of largely as products —a body of knowledge as highlighted in the three schools (logicist, intuitionist, formalist) of thought, rather than ideas that call for active thinking and creation. The evolving conceptions about the nature of mathematics in history suggests there is room for us to decide how mathematics can be perceived, rather than being bounded by a pre-occupied notion of mathematics as “given” or “fixed.” Each and every learner can experience mathematics through different practices and “own” mathematics as a human activity.

Problematizing what is important for students to learn in and through mathematics

The evolving conceptions about the nature of mathematics suggest that choices exist when deciding what and how to teach and learn mathematics but they do not specify what and how to make the choice. Decisions require articulating options for conceptions of what is important for students to learn in and through mathematics and evaluating the advantages and drawbacks for the students for each option.

According to Stanic and Kilpatrick ( 1992 ), the history of school mathematics curricula presents two important and real changes over the years: one is at the turn of the twentieth century when school mathematics was reformed as a unified and applied curriculum to accommodate dramatically increased student populations from diverse backgrounds, and the other is the “New Math” movement of the 1950s and 1960s, intended to integrate modern mathematics into school curriculum. The perceived failure of the “New Math” movement led to the “Back to Basics” movement in the 1970s, followed by “Problem Solving” in the 1980s, and then the Curriculum Standards movement in the 1990s and after. The history shows school mathematics curricula have emphasized teaching and learning mathematical knowledge and skills, together with problem solving and some applications of mathematics, a picture that is consistent with what Devlin ( 2000 ) refers to as the 1st face and some reference to the 4th face of mathematics.

Therefore, although there have been reforms in mathematics curriculum and instruction, there are hardly real changes in how mathematics is conceptualized and presented in school education in the USA (Stanic and Kilpatrick 1992 ) and other education systems (e.g., Leung and Li 2010 ; Li and Lappan 2014 ). The dominant conception remains mathematics as products , frequently referring to a body of static knowledge and skills that need to be learned and acquired (Fisher 1990 ). This continues to be largely the case in practice, despite advances in conceptualization (see below).

It should be noted that conceptualizing mathematics as “a body of knowledge and skills” is not wrong, especially with such a long history of knowledge creation and accumulation in mathematics, but it is not adequate for school mathematics nowadays. The set of concepts and procedures, after years of development, exceeds what could be covered in any school curricula. Moreover, this body of knowledge and skills keeps growing, as the product of human intelligence and scholarship in mathematics. Devlin ( 2012 ) pointed out that school mathematics mainly covers what was developed in the Greek mathematics, plus just two further advances from the seventh century: calculus and probability theory. It is no wonder if someone questions the value of learning such a small set of knowledge and skills developed more than a thousand years ago. Meanwhile, this body of knowledge and skills are often abstract, static, and “foreign” to many students and teachers who learned to perceive mathematics as an external entity in existence (Plato’s notion) rather than Aristotelian emphasis on experimentation (Cooney 1987 ). It is thus not surprising for so many students and teachers to claim that mathematics is difficult (e.g., Fritz et al. 2019 ) and “it is ok—not everyone can be good at math” (Rattan et al. 2012 ).

What can be made meaningful should be critically important to those who want to (or need to) learn and teach mathematics. In fact, there is significant evidence that students often try to make sense of mathematics that is “presented” or “given” to them, although they made numerous errors that can be decoded to study their thinking (e.g., Ashlock 2010 ). Indeed, misconceptions are best thought of not as errors that need to be “fixed,” but as plausible abstractions on the basis of what students have learned—i.e., attempts at sense-making (Smith et al. 1993 ). Conceiving mathematics as about “ideas,” we can help students to play, own, experience, and think about some key ideas just like what they do in many other activities, such as game play (Gee 2005 ). Definitions of concepts and formal languages and procedures can be postponed until students are ready to consider why and how they are needed. Mathematics should be taken and accepted as a human activity (Dossey 1992 ), and developing students’ mathematical thinking (about ideas) should be emphasized in learning mathematics itself (Devlin 2012 ) and in STEM (Li et al. 2019a ).

Along with the shift from products to ideas in mathematics, scholars have already focused on how people work with ideas in mathematics. Elaborated in detail by Schoenfeld ( in press ), the revolution began with George Pólya (1887–1985) who had a fundamental interest in having students learn and understand content via problem solving. For Pólya, mathematics was about inquiry, sense making, and understanding how and why mathematical ideas (instead of content as products) fit together the way they do. The call for problem solving in the 1980s in the USA was (at least partially) inspired by Pólya’s ideas after a decade of “back to basics” in the 1970s. It has been recognized since that the practices of mathematics (including problem solving) are every bit as important as the content itself, and the two shouldn’t be separated. In the follow-up standards movement, the content and practices have been the “warp and weave” of the fabric doing mathematics, as articulated in Principles and Standards for School Standards (NCTM 2000 ). There were five content standards and five process standards (i.e., problem solving, reasoning, connecting, communicating, representing). It is widely acknowledged, also in the Common Core State Standards in the USA (CCSSI 2010 ), that both content and processes/practices are essential and they form the base for next steps.

Problematizing how mathematics is taught and learned, with connections to STEM education

How the ways that mathematics is often taught cause concerns.

Conceiving mathematics as a body of facts and procedures to be “mastered” has been long-standing in mathematics education practice, and it often results in students’ learning by rote memorization. For example, Schoenfeld ( 1988 ) provided a detailed account of the disasters of a “well-taught” mathematics course, documenting a 10th-grade geometry class taught by a confident and experienced teacher. The teacher taught and managed his class well, and his students also did well on standardized examinations, which focused on content and procedures. At the same time, however, Schoenfeld pointed out that the students developed counterproductive views of mathematics. Although the students developed some level of proficiency in content and procedures, they gained (or were reinforced in) the kinds of beliefs about mathematics as being fragmented and disconnected. Schoenfeld argued that the course led students to develop a robust and counterproductive set of beliefs about the nature of geometry.

Seeking possible origins about students’ counterproductive beliefs about mathematics from mathematics instruction motivated Schoenfeld’s study (Schoenfeld 1988 ). Such an intuitive motivation is also evident in other studies. Keitel ( 2006 ) compared the lessons of two teachers (T1 and T2) in Germany who taught their classes very differently. T1 regularly taught the class emphasizing routine individual practice and memorization of specific algebraic rules. T1 emphasized the importance of such practices for test taking, and the students followed his instruction. Even when T1 one day introduced a non-routine problem that connects algebra and geometry, the overwhelming emphasis on mastering routines and algorithms seemed to overshadow in dealing such a non-routine problem. In contrast, T2’s teaching emphasized students’ initiatives and collaboration, although T2 also used formal routine tasks. At the end, students in T2’s class reported positively about their experience, enjoyed working together, and appreciated the opportunities of thinking mathematically. Studies by Schoenfeld ( 1988 ) and Keitel ( 2006 ) indicate how students’ experience in mathematics classes influences their perceptions of mathematics and also imply the importance of learning about teachers’ perceptions of mathematics that likely guide their instructional practice (Cooney 1987 ).

Rattan et al. ( 2012 ) found that teachers with different perceptions of mathematics teach differently. Specifically, Rattan et al. looked at these teachers holding an entity (fixed) theory of mathematics intelligence (G1) versus incremental theory (G2). Through their studies, Rattan and colleagues found that G1 teachers more readily judged students to have low ability, comforted students for low mathematical ability, and used “kind” strategies (e.g., assigning less homework) unlikely to promote their engagement with the field than G2 teachers. Students who received comfort-oriented feedback perceived their teachers’ entity theory and low expectations and reported lowered motivation and expectations for their own performance. The results suggest how teachers’ inadequate perceptions of mathematics and beliefs about the nature of students’ mathematical intelligence contributed to low achievement, diminished self-esteem and viewed mathematics is only a set of static facts and procedures. Further, the results suggest that how mathematics is taught influences more than students’ proficiency with mathematics content in a class. Sun ( 2018 ) made a similar argument after synthesizing existing literature and analyzing classroom observation data.

Based on the 2012 US national survey of science and mathematics education conducted by Horizon Research, Banilower et al. ( 2013 ) reported that a vast majority of mathematics teachers, from 81% at the high school level to 90% at the elementary level, believe that students should be given definitions of new vocabulary at the beginning of instruction on a mathematical idea. Also, many teachers believe that they should explain an idea to students before having them consider evidence for it and that hands-on activities should be used primarily to reinforce ideas students have already learned. The report suggests many teachers emphasized pedagogical practices of “give” and “present,” perhaps influenced by conceptions of mathematics that are more Platonic than Aristotelian, similar to what was reported about teachers’ practices more than two decades ago (Cooney 1987 ).

How to change?

Given that the evidence demonstrates a compelling case for changing how mathematics is taught, we turn our attention to suggesting how to realize this transformation. Changing how mathematics is taught and learned is not a new endeavor for both mathematics educators and mathematicians (e.g., Li, Silver, and Li 2014 ; Schoenfeld in press ). For example, the “Moore Method,” developed and used by Robert Lee Moore (a famous topologist) in the early twentieth century, shifted instruction from teacher-centered lecturing to student-centered mathematical development (Coppin et al. 2009 ). In its purest form, students were presented with mathematical definitions and asked to develop and/or prove theorems from them after class, without reading mathematics books or using other resources. When students returned to the class, they were asked to prove a theorem. As a result, students developed the mathematics themselves, instead of the instructor presenting the proofs and doing the mathematics for students. The method has had its own success in producing many great mathematicians; however, the high-pressure environment also drowned many students who might have been successful otherwise (Schoenfeld in press ).

Although the “Moore Method” was used primarily in advanced mathematics courses at the post-secondary level, it illustrates how a different conception of mathematics led to a different instructional approach in which students developed mathematics. However, it might be the opposite end of a spectrum, in comparison to approaches that present mathematics to students in accommodating and easy-to-digest ways that can be as much easy as possible. Neither extreme is a good option for K-12 students. Again, it becomes important for us to consider options that can support the value of learning mathematics.

Our discussion in the previous section highlights the importance of taking mathematics as a human activity, ensuring it is meaningful to students, and developing students’ mathematical thinking about ideas, rather than simply absorbing a set of static and disconnected knowledge and skills. We call for a shift in teaching mathematics based on Platonic conceptions to approaches based on more of Aristotelian conceptions. In essence, Plato emphasized ideal forms of mathematical objects, perhaps inaccessible through people’s sense making efforts. As a result, learners lack ownership of the ideal forms of mathematical objects, because mathematical objects cannot and should not be created by human reasoning. In contrast, Aristotle emphasized that mathematical objects are developed through logic reasoning and empirical realization. In other words, mathematical objects exist only when they can be sensed and verified by people's efforts. This differs from Plato’s passive perspective, highlights human ownership of mathematical ideas and encourages people to make mathematics make sense, termed as making sense by McCallum ( 2018 ). Aristotelian conceptions view mathematics as objects that learners can actively develop and structure as mathematically meaningful, which is more in line with what research mathematicians do. McCallum ( 2018 ) argued that both sense-making and making-sense stances are needed for a complete view of mathematics and learning, recognizing that not attending to both stances carries risks. “Just as it is a risk of the sense-making stance that the mathematics gets ignored, it is a risk of the making-sense stance that the sense-maker gets ignored.” (McCallum 2018 ).

In addition, there is the issue of personal identity: if students come to avoid mathematics because they are uncomfortable with it (in fact, mathematics anxiety has become a widespread problem for all ages across the globe, see Luttenberger et al. 2018 ) then mathematics instruction has failed them, regardless of test scores.

In the following, we discuss sense-making and making-sense stances first with specific examples from mathematics. Then, we discuss connections to STEM education.

Sense making is much more than the acquisition of knowledge and skills

Sense making has long been emphasized in mathematics education community. William A. Brownell is a well-known, early 20 th century scholar who advocated the value of sense making in the learning of mathematics. For example, Brownell ( 1945 ) discussed how arithmetic can and should be taught and learned not only as procedures, but also as a meaningful system of thinking. He shared many examples like the following division,

Brownell suggested to ask questions: what does the 5 of 576 mean? Why must 57 be the first partial dividend? Do you actually divide 8 into 57, or into 57…’s? etc., instead of simply letting students memorize how to carry out the procedure. What Brownell advocated has been commonly accepted and emphasized in mathematics education nowadays as sense making (e.g., Schoenfeld 1992 ).

There can be different ways of sense making of the same computation. As an example, the sense making process for the above long division can come out with mental math as: I am looking to see how close I can get to 570 with multiples of 80; 7 multiples of 80 gives me 560, which is close. Of course, given base 10 notation, that’s the same as 8 multiples of 70, which is why the 7 goes over the 57. And when I subtract 560, there are 16 left over, so that’s another 2 8 s. Such a sense-making process also works, as finding the answer (quotient, k ) of 576 ÷ 8 is the same operation as to find k that satisfies 576 = k × 8. In mathematics, division and multiplication are alternate but equivalent ways of doing the same operation.

To help students build numerical reasoning and make sense of computations, many teachers use number talks in their classrooms for students to practice and share these mental math and computation strategies (e.g., Parrish 2011 ). In fact, new terms are being created and used in mathematics education about sense making, such as number sense (e.g., Sowder 1992 ) and symbol sense (Arcavi, 1994 ). Some instructional programs, such as Cognitively Guided Instruction (see, e.g., Carpenter et al., 1997 , 1998 ), make sense making the core of instructional activities. We argue that such activities should be more widely adopted.

Making sense makes the other side of mathematical practice visible, and values idea development and ownership

The making-sense stance, as termed by McCallum ( 2018 ), is not commonly practiced as it is pertinent to expert mathematician’s practices. Where sense making (as discussed previously) emphasizes the process of making sense of what is being learned, making sense emphasizes the process of making mathematics make sense. Making sense highlights the importance for students to experience mathematics through creating, designing, developing, and connecting mathematical ideas. As an example, for the above division computation, 8 \( \overline{\Big)576\ } \) , students may wonder why the division procedure is performed from left to right, which is different from the other operations (addition, subtraction, and multiplication) that are all performed from right to left. In fact, students can be encouraged to explore if the division can also be performed from right to left (i.e., starting from the one’s place). They may discover, with possible support from the teacher, that the division can be done in this way. However, once the division is moved to the high-value places, it will require the process to go back down to the low-value places for completion. In other words, the division process starting from the low-value place would require repeated processes of returning to the low-value places; as a result, it is inefficient. As mathematical procedure is designed to improve efficiency, the division procedure is thus set to be carried out from the high-value place to low-value place (i.e., from left to right). Students who work this out experience mathematics more deeply than the sense-making described by Brownell ( 1945 ).

There are plenty of making-sense opportunities in classroom instruction. For example, kindergarten children are often given opportunities to play with manipulatives like cube trains and snub cubes, to explore and learn about patterns, numbers, and measurement through various connections. The recording of such activities typically results in numerical expressions or operations of these connections. In addition, such activities can also serve as a context to encourage students to design and create a way of “recording” these connections directly with a drawing line next to the connected train cubes. Such a design activity will help students to develop the concept of a number line that includes the original/starting point, unit, and direction (i.e., making mathematics make sense), instead of introducing the number line to students as a mathematical concept being “given” years later.

Learning how to provide students with opportunities to develop mathematics may occur with experience. Huang et al. ( 2010 ) found that expert and novice teachers in China both valued students’ mastering of mathematical knowledge and skills and their development in mathematical thinking methods and abilities. However, novice teachers were particularly concerned about the effectiveness of their guidance, in contrast to expert teachers who emphasized the development of students’ mathematical thinking and higher-order thinking abilities and properly dealing with important and difficult content points. The results suggest that teachers’ perceptions and pedagogical practices can change and improve over time. However, it may be worth asking if support for teacher development would accelerate the process.

Connecting changes in mathematics and STEM education

Although it is commonly acknowledged that mathematics is foundational to STEM, mathematics is being related to STEM education at a distance in practice and also in scholarship development (English 2016 , see additional notes at the end of this editorial). Holding the conception of mathematics as products does not support integrating mathematics with other STEM disciplines, as mathematics can be perceived simply as a set of tools for these disciplines. At the same time, mathematics and science have often proceeded along parallel tracks, with mathematics focused on “problem solving” while science has focused on “inquiry.” To better connect mathematics and other disciplines in STEM, we should focus on ideas and thinking development in mathematics (Li et al. 2019a ), unifying instruction from the student perspective (the Teaching for Robust Understanding framework, discussed below).

Emphasizing both sense making and making sense in mathematics education opens opportunities for connections with similar practices in other STEM disciplines. For example, sense making is very much emphasized in science education (Hogan 2019 ; Kapon 2017 ; Odden and Russ 2019 ), often combined with reflections in engineering (Kilgore et al. 2013 ; Turns et al. 2014 ), and also in the context of using technology (e.g., Antonietti and Cantoia 2000 ; Dick and Hollebrands 2011 ). Science is fundamentally about discovery and understanding of the natural world. This notion provides a natural link to mathematical modeling (e.g., Burkhardt 1981 ). Beyond that, in science education, sense making places a heavy focus on the construction and evaluation of explanation (Kapon 2017 ), and can even be defined as a process of constructing an explanation to resolve a perceived gap or conflict in knowledge (Odden and Russ 2019 ). Design and making play vital roles in engineering and technology education (Dym et al., 2005 ), as is student reflection on these experiences (e.g., Turns et al. 2014 ). Indeed, STEM disciplines share the same conceptual process of sense making as learners, individually or in a group, actively engage with the natural or man-made world, explore it, and then develop, test, refine, and use ideas together with specific explanation. If mathematics was conceived as an “empirical” discipline, connections with other STEM disciplines would be strengthened. In philosophical terms, Lakatos ( 1976 ) made similar claims Footnote 1 .

Similar to the emphasis on sense making placed in the Mathematics Curriculum Standards (e.g., NCTM, 1989 , 2000 ), Next Generation Science Standards (NGSS Lead States 2013 ) prompted a shift in science education away from simply knowing science content and procedures to practicing and using science, together with engineering, to make sense of the world and create the future. In a review, Fitzgerald and Palincsar ( 2019 ) concluded sense making is a productive lens for investigating and characterizing great teaching across multiple disciplines.

Mathematics has stronger linkages to creation and design than traditionally imagined. Therefore, its connections to engineering and technology could be much stronger. However, the deep-rooted conception of mathematics as products has traditionally discouraged students and teachers from considering and valuing design and design thinking (Li et al. 2019b ). Conceiving mathematics as making sense should help promote conceptual changes in mathematical practice to value idea generation and design activity. Connections generated from such a shift will support teaching and learning not only in individual STEM disciplines, but also in integrated STEM education.

At the same time, although STEM education as a commonly recognized field does not have a long history (Li 2014 , 2018a ), its rapid development can help introduce ideas for exploring how mathematics can be taught and learned. For example, the concept of projects is common in engineering professional practice, and the project-based learning (PjBL) as an instructional approach is a key component in some engineering programs (e.g., Berger 2016 ; de los Ríos et al. 2010 ; Mills and Treagust 2003 ). de los Ríos et al. ( 2010 ) highlighted three main advantages of PjBL: (1) development in technical, personal, and contextual competences; (2) students’ engagement with real problems from professional contexts; and (3) collaborative learning facilitated through the integration of teaching and research. Such advantages are important for students’ learning of mathematics and are aligned well with efforts to develop 21 st century skills, including problem solving, communication, collaboration, and critical thinking.

Design-based learning (DBL) is another instructional approach commonly used in engineering and technology fields. Gómez Puente et al. ( 2013 ) conducted a sampled review and concluded that DBL projects consist of open-ended, hands-on, authentic, and multidisciplinary design tasks. Teachers using DBL facilitate both the process for students to gain domain-specific knowledge and thinking activities to generate innovative solutions. Such features could be adapted for mathematics education, especially integrated STEM education, in concert with design and design thinking. In addition to a few examples discussed above about making sense in mathematics, there is a growing body of publications developed by and for mathematics teachers with specific examples of investigations, design projects, and instructional activities associated with STEM (Li et al. 2019b ).

A framework for helping students to gain important experiences in and through mathematics, as connected to other disciplines in STEM

For observing and evaluating classroom instruction in general and mathematics classroom instruction in specific, there are several widely used frameworks and rubrics available. However, a trial use of selected frameworks with sampled mathematics classroom instruction episodes suggested their disagreements on what counts as high-quality instruction, especially with aspects on disciplinary thinking being valued and relevant classroom practices (Schoenfeld et al. 2018 ). The results suggest the importance of choice making, when we consider a framework in discussing and evaluating teaching practices.

Our discussion above highlights the importance of shifting away from viewing mathematics simply as a set of static knowledge and skills, to focusing on ideas and thinking development in teaching and learning mathematics. Further discussion of several aspects of changes specifies the needs of developing and using practices associated with sense making, making sense, and connecting mathematics and STEM education for changes.

To support effective mathematics instruction, we propose the use of the Teaching for Robust Understanding (TRU) framework to help characterize powerful learning environments. With the conception of mathematics as “empirical” and a focus on students’ experience, then the focus of instruction should also be changed. We argue that shift is from instruction conceived as “what should the teacher do” to instruction conceived as “what mathematical experiences should students have in order for them to develop into powerful thinkers?” It is the shift in the frame of TRU that makes it so powerful and pertinent for all these proposed changes. Moreover, TRU only uses a small number of actionable dimensions after distilling the literature on teaching for robust or powerful understanding. That makes TRU a practical mechanism for problematizing instruction.

Figure 1 presents the TRU Math framework that identifies five key dimensions along which powerful classroom environments can be characterized: the mathematics; cognitive demand; equitable access; agency, ownership, and identity; and formative assessment. These five dimensions were distilled from an extensive literature review, thus capturing what the literature considers to be essential. They were tested against classroom videotapes and data on student performance, and the results indicated that classrooms that did well on the TRU dimensions produced students who did correspondingly well on tests of mathematical knowledge, thinking, and problem solving (e.g., Schoenfeld 2014 , 2019 ). In brief, the argument regarding the importance of the five dimensions of TRU Math is as follows. First, the quality of the mathematics discussed (dimension 1) is critical. What individual students learn is unlikely to be richer than what they experience in the classroom. Whether individual students’ understanding rises to the level of what is discussed/presented in the classroom depends on other factors, which are captured in the remaining four dimensions. For example, you surely have had the experience, at a lecture, of hearing beautiful content presented, and then not being able to do any of the assigned problems! The remaining four dimensions capture aspects needed to support the development of all students with respect to sense making, making sense, ownership, and feedback loop. Dimension 2: Cognitive Demand. Are students engaged in sense making and making sense? Are they engaged in “productive struggle”? Dimension 3: Equitable Access. Are all students fully engaged with the central content and practices of the domain so that every student can profit from it? Dimension 4: Agency, Ownership, and Identity. Do all students have opportunities to develop idea ownership and mathematical agency? Dimension 5: Formative Assessment. Are students encouraged and supported to share their thinking with a meaningful feedback loop for instructional adjustment and improvement?

The TRU Mathematics Framework: The five dimensions of powerful mathematics classrooms

The first key point about TRU is that students learn more in classrooms that are powerful along the five TRU dimensions. Second, the shift of attention from the teacher to the environment is fundamentally important. The key question is not “Is the teacher doing particular things to support learning?”; instead, it is, “Are students experiencing instruction so that it is conducive to their growth as mathematical thinkers and learners?” Third, the framework is not prescriptive; it respects teacher autonomy. There are many ways to be an excellent teacher. The question is, Does the learning environment created by the teacher provide each student rich opportunities along the five dimensions of the framework? Specifically, in describing the dimensions of powerful instruction, the framework serves to problematize instruction. Asking “how am I doing along each dimension; how can I improve?” can lead to richer instruction without prescribing or imposing a particular style or particular norms on teachers.

Extending to STEM education

Now, we suggest the following. If you teach biology, chemistry, physics, engineering, or any other STEM field, replace “mathematics” in Fig. 1 with your discipline. The first dimension is about rich content and practices in your field. And the remaining four dimensions are about necessary aspects of your students’ classroom engagement with the discipline. Practices associated with sense making, making sense, and STEM education are all be reflected in these five dimensions, with central attention on students’ experience in such classroom environments. Although the TRU framework was originally developed for characterizing effective mathematics classroom environments, it has been carefully framed in a way that is applicable to many different disciplines (Schoenfeld 2014 ). Our discussion above already specified why sense making, making sense, and specific instructional approaches like PjBL and DBL are shared across disciplines in STEM education. Thus, the TRU framework is applicable to other STEM disciplines. The natural analogue of the TRU framework for any field is given in Fig. 2 .

The domain-general version of the TRU framework

Both the San Francisco Unified School District and the Chicago Public Schools adopted the TRU Math framework and found results within mathematics sufficiently promising that they expanded their efforts to all subject areas for professional development and instruction, using the domain-general TRU framework. Work is still in its early stages. Current efforts might be best conceptualized as a laboratory for exploration rather than a promissory note for improvement across all different disciplines. It will take time to accumulate data to show effectiveness. For further information about the domain-general TRU framework and tools for professional development are available at the TRU framework website, https://truframework.org/

Finally, as a framework, TRU is not a set of specific tools or guidelines, although it can be used to guide their development. To help lead our discussion to something more practical, we can use the framework to check and identify aspects that are typically under-emphasized and move them to center stage in order to improve classroom instruction. Specifically, the following is a list of sample under-emphasized norms and practices that can be identified (Schoenfeld in press ).

Establishing a climate of inquiry, in which mathematics is experienced as a discipline of exploration and sense making.

Developing students’ ownership of ideas through the process of developing, sharing, refining, and using ideas; concepts and language can come later.

Focusing on big ideas, and not losing the forest for the trees.

Making student thinking central to classroom discourse.

Ensuring that classroom discourse is respectful and inviting.

Where to start? Begin by problematizing teaching and the nature of learning environments

Here we start by stipulating that STEM disciplines as practiced, are living, breathing fields of inquiry. Knowledge is important; ideas are important; practices are important. The list given above applied to all STEM disciplines, not just mathematics.

The issue, then, is developing teacher capacity to craft environments that have the properties described immediately above. Here we share some thoughts, and the topic itself can well be discussed extensively in another paper. To make changes in teaching, it should start with assessing and changing teaching practice itself (Hiebert and Morris 2012 ). Opening up teachers’ perceptions of teaching practices should not be done by telling teachers what to do!—the same rules of learning apply to teachers as they apply to students. Learning environments for teachers should offer teachers the same opportunities for rich engagement, challenge, equitable access, and ownership as we hope students will experience (Schoenfeld 2015 ). Working together with teachers to study and reflect on their teaching practices in light of the TRU framework, we can help teachers to find out what their students are experiencing and why changes are needed. The framework can also help guide teachers to learn what changes would be needed, and to try out changes to learn how their students’ learning may differ. It is this iterative and concrete process that can hopefully help shift participating teachers’ perceptions of mathematics. Many tools for problematizing teaching are available at the TRU web site (see https://truframework.org/ ). If teachers can work together with a focus on selected lessons like what teachers often do in China, the process would help form a school-based learning community that can contribute to not only participating teachers’ practice change but also their expertise improvement (Huang et al. 2011 ; Li and Huang 2013 ).

As reported before (Li 2018b ), publications in the International Journal of STEM Education show a mix of individual-disciplinary and multidisciplinary education in STEM over the past several years. Although one journal’s publications are limited in its scope of providing a picture about the scholarship development related to mathematics and STEM education, it can allow us to get a sense of related development.

If taking a closer look at the journal’s publications over the past three years from 2016 to 2018, we found that the number of articles published with a clear focus on mathematics is relatively small: three (out of 21) in 2016, six (out of 34) in 2017, and five (out of 56) in 2018. At the same time, we should point out that these publications from 2016 to 2018 seem to reflect a trend, over these three years, of moving toward issues that can go beyond mathematics itself, as what was noted before (Li 2018b ). Specifically, for these three articles published in 2016, they are all about mathematics education at either elementary school (Ding 2016 ; Zhao et al. 2016 ) or university levels (Schoenfeld et al. 2016 ). Out of the six published in 2017, three are on mathematics education (Hagman et al. 2017 ; Keller et al. 2017 ; Ulrich and Wilkins 2017 ) and the other three on either teacher professional development (Borko et al. 2017 ; Jacobs et al. 2017 ) or connection with engineering (Jehopio and Wesonga 2017 ). For the five published in 2018, two are on mathematics education (Beumann and Wegner 2018 ; Wilkins and Norton 2018 ) and the other three have close association with other disciplines in STEM (Blotnicky et al. 2018 ; Hayward and Laursen 2018 ; Nye et al. 2018 ). This trend likely reflects a growing interest, with close connection to mathematics, in both mathematics education community and a broader STEM education community of developing and sharing multidisciplinary and interdisciplinary scholarship.

Availability of data and materials

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Many thanks to Jeffrey E. Froyd for his careful review and detailed comments on an earlier version of this editorial that help improve its readability and clarity. Thanks also go to Marius Jung for his valuable feedback.

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• Design-based learning
• Making sense
• Mathematics
• Project-based learning
• Sense making
• STEM education

ORIGINAL RESEARCH article

Perceived causes of students’ poor performance in mathematics: a case study at ba and tavua secondary schools.

• 1 Ministry of Education, Heritage and Arts, Suva, Fiji
• 2 School of Information Technology, Engineering, Mathematics and Physics, The University of the South Pacific, Suva, Fiji

Poor achievement in mathematics is an issue of great concern for many countries across the globe. Fiji is one of the countries in the South Pacific experiencing the same trends, pressures, and concerns. This study aims to seek the views of stakeholders (students, teachers, heads of departments, and school heads) with regards to the causes of poor achievement in mathematics at the senior grades of secondary schools in the districts of Ba and Tavua, Fiji. A descriptive design using both quantitative and qualitative approaches were utilized whereby data were collected from 201 upper secondary school respondents comprising 171 students, 16 mathematics teachers, 7 department heads, and 7 school heads from seven randomly selected schools in the districts of Ba and Tavua. The study found that the students had a negative attitude toward mathematics. It was also found that an ineffective mathematics curriculum in secondary schools was the reason behind poor performance in the subject. Moreover, many of the primary school teachers lacked potential and competence to teach mathematics at primary school levels, and this largely contributed toward the lack of interest amongst students, hence translating into poor achievement at both upper and lower secondary levels. On the other hand, however, it was gathered that secondary school teachers were rather positive, good quality, performing, and fully qualified as far as the teaching of mathematics and delivery of the subject matter was concerned. Review and amendments to the year 12 and 13 mathematics curriculum, use of technologies to teach mathematics, improving the quality of primary school mathematics teachers, reducing the emphasis on exams, introducing internal assessments, projects, and field work in the mathematics curriculum were a few of the significant recommendations made from this study.

Introduction

Globally, mathematics is regarded as one of the most important subjects in the school curriculum [ 1 ]. It is the foundation of scientific and technological knowledge that contributes significantly toward the socioeconomic development of a nation [ 1 – 6 ].

Mathematics plays a vital role in everyday life of so many people [ 7 , 8 ]. According to [ 2 ], mathematics is one subject that affects all aspects of human life at different levels. A study by [ 9 ] claimed that both education and human life do not effectively function without the knowledge of mathematics. In formal education, mathematics forms the basis of many of the sciences such as physics, chemistry, biology, engineering, and IT disciplines as well as the nonscience disciplines such as accounting, economics, geography, and even physical education, music, and art [ 1 , 4 , 6 , 7 , 10 – 15 ]. It is one of the most important subjects in the school curriculum, which acts as a bridge for all knowledge [ 3 ]. Studies by [ 16 , 17 ] stressed that mathematics is the bedrock and a tool for the scientific, technological, and economic advancement of any country. It is a common belief of educationists that no one can make progress in any field without having the basic knowledge of mathematics [ 18 ]. According to [ 1 , 8 ], mathematics is the foundation of science and technology without which a nation will not prosper and achieve economic independence. That is why mathematics is one of the leading core subjects in the secondary schools’ curriculum.

Personnels require mathematical skills in various disciplines, workplace, and sectors. Even things like the hydrogen bomb, missiles, space crafts, and satellites would not have been possible without the knowledge of mathematics [ 19 ]. Mathematics has its application in a wide range of informal settings, including vegetable selling, sewing, fishing, construction work, shopping, purchasing, carpet laying, video games, cabs and buses, farming, entertainment, sports, and everyday family activities [ 20 , 21 ]. Ultimately, the survival of any human being in this competitive world is almost impossible without the knowledge and skill in mathematics.

Despite the highly decorated and recognized importance of mathematics and the fact that it is the prerequisite for most of the subjects, poor achievement and lack of interest in mathematics (and STEM) among students remains as an issue of concern in schools, colleges, and universities in developed and developing countries alike [ 22 – 25 ]. Mathematics continues to be one of the most challenging subjects in schools as perceived by students [ 7 , 26 – 28 ]. There is a general impression that its very nature complicates mathematics. Because of this impression, majority of students have a phobia for this subject [ 9 , 29 – 31 ]. Besides, mathematics students of the 21st century enter mathematics classrooms with a serious lack of fluency and reliability in numerical and algebraic manipulation and simplification, problem-solving, and negative attitude [ 28 , 32 , 33 ].

It is quite evident that students with good mathematical skills can think analytically and have better reasoning abilities. That is why mathematics is used as an essential entry requirement for most of the courses at the higher education institutes, especially for courses relating to science, technology, and engineering disciplines [ 22 ]. Reference [ 34 ] claimed that the number of students enrolling in higher level mathematics courses had declined significantly. Due to this, there was an increase in mathematically underprepared students enrolling in undergraduate courses leading to curtailed enrollments and low pass rates in higher education (HE) institutes. Fiji with three major higher education institutions, namely, The University of the South Pacific, The Fiji National University, and The University of Fiji face the same challenge of decline in the quantity and quality of applicants enrolling for Science, Technology, Engineering, and Mathematics (STEM) courses due to low pass rates in mathematics at years 12 and 13 national examinations [ 22 , 25 , 28 , 35 ]. Many Fijian students fail to meet the basic entry requirements for HE institutions in STEM courses that require either a pass or a higher cutoff mark in mathematics [ 36 ]. The domino effect of this over the years has forced HE institutes to remove the high cutoff marks for specific disciplines in order to avoid losing on students [ 37 ].

The Fijian government and its academic stakeholders have long been investing profoundly in the education sector. The government over the past few years has been providing initiatives such as transport assistance (bus fare and boat fare subsidies), free textbooks, and grants to uplift the standard of education in Fiji [ 11 , 38 ]. Despite such massive investments in education and the important role that mathematics plays in society, there has been a continuous trend of poor achievement in mathematics, especially at the years 12 and 13 grades of secondary schools in Fiji. The national examination results of FY12CE and FY13CE is demonstrated in Table 1 .

TABLE 1 . Performance in years 12 and 13 mathematics national examinations.

Studies by [ 16 , 39 , 40 ] claimed that the continual trend of poor achievement in mathematics is a function of cross-factors related to students, teachers, and schools. It is evident from several studies that student, teacher, and curriculum factors seem to have a significant effect on mathematics achievement [ 1 , 16 , 33 , 41 , 42 ].

While there are anecdotal pieces of evidence on why we are facing low achievement, there has been a dearth of formal and high-quality research in this area. The present study intends to carry out a thorough investigation on the student, teacher, and curriculum factors by cross-examining the views and perceptions of students, teachers, heads of the mathematics department, and the school heads. The article analyses and discusses the views of the respondents on the factors contributing to students’ poor achievement in mathematics, especially at the senior grades of selected secondary schools in the west of Fiji Islands. The findings of this research would provide an empirical insight to the Curriculum Development Unit (CDU), Ministry of Education, Heritage and Arts (MEHA), Higher Education Institutes (HE), and other relevant academic stakeholders to bring about effective reviews and reforms in the education system in order to improve the achievement of students in mathematics at the senior secondary grades. It is anticipated that the recommendations of this study would bring about a positive mental attitude and perception of students toward mathematics. Moreover, the way mathematics is taught in both primary and secondary schools has to be changed.

Secondary School Mathematics Reforms in Fiji

The secondary school mathematics in Fiji has not seen any significant structural changes in the past 3 decades. Whereas almost all areas of the curriculum have changed to fit better with the context of Fiji, it is still an academic system that is driven by examinations [ 43 ]. The examination system in mathematics at the secondary level, which currently has external examinations at years 10, 12, and 13, is an entirely written examination in mathematics with no form of internal assessments.

However, for several years, the Overseas School Certificate and the General School Certificate Examinations from the United Kingdom were adopted in the Fijian education system. Then in the 1960s, came the switch to the New Zealand syllabi and examinations—the School Certificate and the University Entrance Examinations. The former was dropped and the latter replaced by local examinations in 1989 [ 44 ].

There were no significant changes in the mathematics curriculum for the next 2 decades until internal assessments came into effect. In 2011, Fiji Junior Certificate Examination was abolished, and internal assessment was implemented in all the secondary schools in Fiji [ 45 ]. It was anticipated that the reform in the curriculum would allow teachers to adopt a student-centered approach, shifting the focus of instruction from the teacher to the student. The shift from the teacher-centered approach would have allowed a student to be free from the constant pressure and trauma of external examinations. Form six (year 12) and form seven (year 13) examinations remained since they play an important function in the selection of students for further education and employment opportunities.

However, a report presented to the cabinet by the Education Minister in 2015 stated that the raw results for the external examinations showed very low mean marks and percentage pass rates in years 12 and 13 examinations which portrayed a failure in the education system. Mathematics recorded a percentage pass of 7.5%, one of the lowest performing subjects’ among all the other subjects in Fiji Year 13 external examination in 2014 [ 35 ]. The predicament was seen to be due to the removal of external exams up to year 11, and thus, poorly prepared students passed on from one year to the other without their teachers and parents knowing the true status of the students’ level of attainment that year. Removal of scaling was further proposed and passed by the cabinet to reflect a student’s true ability as results in mathematics in the past showed exaggerated percentages and averages that did not correctly portray the true stock of knowledge that the student had acquired [ 46 ].

In 2015, the honorable minister for education, Dr. Mahendra Reddy, further stressed that the Fijian curriculum was below the standard of some of the countries, whose graduates were more competitive at an equivalent level [ 47 , 48 ]. Dr. Reddy claimed that the graduates from HE institutes were fraught with lack of soft skills, lack of competency in English proficiencies, unwilling to think outside the box, and had poor research skills [ 47 ].

In the year 2018, the repercussions of poor achievement in mathematics were felt when the Ministry of Education, Heritage and Arts identified an immediate shortage of mathematics, physics, biology, chemistry, and industrial arts teachers anticipating the shortage to continue in the foreseeable future [ 49 ]. The shortage of teachers in STEM disciplines is attributed to poor achievement in mathematics at senior secondary grades since very few students are able to qualify for such courses. Most of these elite students who qualify and graduate prefer joining the private sector rather than teaching, contemplating better pay scale, and faster promotion chances, the trend shared by other countries in the South Pacific region [ 22 ]. To add on, MEHA has gone to the extent of hiring retired industrial arts teachers who wish to rejoin the service as assistant teachers. In few cases, teachers of nonengineering discipline are even appointed by the school administrators to take up the role of teaching engineering subjects at secondary schools due to the shortage of industrial arts teachers in the country. Also, some graduates who do make it to the teaching programs for STEM courses prefer to migrate to neighboring countries after few years of service, for attractive and better salary packages in comparison of what is paid to teachers locally.

Removal of scaling in national exams; preparation of localized and prescribed textbooks; reintroduction of national exams in year 10; introducing standard exams for years 9, 10, and 11; upgrading the quality; and providing detailed solutions of past year national exam papers were few of the reforms that took place over the past 4 years. Still, the result in mathematics at years 12 and 13 grades, the number of students enrolling at universities for STEM programs, and the number of graduates in mathematics, science, and technology continued to decline significantly.

Literature Review

The continuing trend of poor achievement in mathematics in Fiji secondary schools raises concerns to the Fijian government and the stakeholders on whether or not the Fijian education system can supply graduates who possess the essential skills to enable them to cope with the ever-evolving technological society. Several studies have attributed students’ low achievement in mathematics to student, teacher, and curriculum factors. For this study, students’ attitude and perception toward mathematics, teachers’ attitude, and perception toward mathematics, teaching methodologies of mathematics teachers, quality and performance of mathematics teachers, and the effectiveness and relevance of mathematics curriculum were the five factors identified to be influencing students’ achievement in mathematics at the senior grades of secondary schools in Fiji. The following review summarizes from the literature the above five factors that contributed to the low achievement of students in mathematics.

Attitude and Perception of Students Toward Mathematics

First, attitude determines the effort a student is likely to put in his or her learning of a subject. It refers to someone’s basic liking or disliking of a subject [ 13 , 50 ]. Several studies have been carried out in many countries to find the factors that influence the students’ performance in mathematics. Among these factors, student attitude and perception is one significant factor that has been consistently studied [ 13 , 51 – 55 ]. Studies such as [ 2 , 3 , 43 , 55 ] attributed challenges to teaching mathematics to the negative attitudes and perception of students as they perceive mathematics as a difficult subject to pass. A recent study by [ 1 ] found out that 92.50% of students hated mathematics, whereas 86.25% had unjust fear toward mathematics. The prolonged fear and anxiety of students in mathematics ultimately generates a negative attitude of students that becomes relatively permanent in future [ 56 ].

On the contrary, a study by [ 2 ] on the three colleges of Ghana found that students had a positive attitude toward mathematics with a willingness to learn. However, they are uncomfortable due to the conditions around them. These conditions do not necessarily mean that a student is always liable for his or her poor achievement. However, to date, while there have been local studies assessing school teachers’ preparedness for mathematics [ 57 ] and secondary students’ attitude in science [ 25 ] and ICT [ 58 , 59 ], there has been no research carried out locally to assess students’ attitude and perception toward mathematics. This requires views from students, teachers, heads of departments, and school heads to gain deeper insights into students’ lack of interest and low achievement in mathematics at the senior grades of secondary schools in Fiji.

Attitude and Perception of Teachers Teaching Mathematics

Second, the question that arises here is can the students be blamed for the poor attitude toward mathematics? According to [ 60 ], teachers’ negative beliefs about mathematics have a strong influence on students’ attitude and achievement in mathematics. Studies such as [ 6 , 53 , 54 , 61 , 62 ] have stressed on teachers’ attitude in mathematics being the significant determinant of negative attitude among students. The way students perceive teachers’ characteristics will affect their attitude toward mathematics [ 5 , 57 ]. Teachers’ personal and professional characteristics play a significant role in students’ liking or disliking of mathematics. Studies by [ 53 , 62 ] show that boring teachers, teachers’ lack of commitment, teachers’ personality, students’ failure to understand the topic, and the poor performance of students in exams relate to teachers’ negative attitude. While there is a dearth of relevant studies in Fiji, an international study by [ 6 ] has found out that the majority of the mathematics teachers in secondary schools display a positive attitude toward teaching mathematics. However, there are no recorded observations of this issue in Fiji. Therefore, an in-depth and comprehensive formal research needs to be conducted to find the general trend of local teachers’ attitude toward teaching mathematics and if this attitude affects their students’ attitude toward performance in mathematics.

Teaching Methods Used by Mathematics Teachers

Third, several studies have attributed poor academic achievement of students to the deficiency in teaching method(s) used by mathematics teachers [ 1 – 3 , 63 – 65 ]. According to [ 65 ], teachers employ wrong teaching methods of learning, which results in general hatred for the subject by the students. The author further concluded that if mathematics is to be appreciated by students, teachers must use new pedagogies and technologies that can stimulate students to gain interest in mathematics classes. A recent study by [ 1 ] found that 85.63% of students claimed that poor teaching methods of some mathematics teachers scare students from the subject. According to [ 66 , 67 ], in the current era of education, students are encouraged to discover and build their knowledge through active participation. Teachers should incorporate methods that involve active participation of students, considering students’ interest. A local study by [ 68 ] justified that due to the exam-oriented system, teachers are too much concerned with finishing the syllabus and drilling the students with the exam questions and answers. He further stressed that teachers are reluctant and sometimes hesitant to use other approaches to the teaching and learning of mathematics as it would take up too much time and are deemed irrelevant to passing exams.

Quality, Performance, and Qualification of Mathematics Teachers

Moreover, great teachers are quality and better performing teachers who tend to inspire people around regardless of any challenges or barriers. Quality, performance, and qualification of mathematics teachers are other important factors that significantly influence the attitude and achievement of mathematics students. It is evident through research that the achievement of students is strongly linked to high-quality and qualified teachers [ 68 ]. A recent study by [ 1 ] revealed that the majority of the students indicated that their teachers did not have enough potential to teach mathematics. Most of the mathematics teachers do not make the teaching of mathematics practical and exciting due to inadequate training at HE institutions or lack of training for preservice teachers on the 21st-century pedagogies in mathematics, which ultimately leads to negative attitude and poor achievement in mathematics among students. It is, therefore, important that both preservice and in-service training are essential for the quality professional development of the teacher [ 2 ]. Studies by [ 28 , 69 ] have emphasized that technology is essential in teaching and learning mathematics. Some secondary schools in Fiji, such as Nadi Sangam Kuppuswamy Memorial College, Swami Viveka Nanda College, Tilak High School, and Vunimono High School, have already blended ICT entirely in years 12 and 13 of the school curriculum. A recent local study by [ 70 ] emphasized that ICT in this modern era allows various innovative and creating assessments to be incorporated in lessons, which were not possible using traditional assessment methods. He further added that the workload of teachers is significantly reduced by the use of ICT, allowing teachers to utilize more time to focus on the key role, that is, to enhance learning among students. Many primary and secondary schools have plans underway to integrate ICT in every classroom [ 10 , 13 , 72 ]; however, investing in such initiatives still proves to be an expensive affair for many schools in Fiji. Another local study conducted by [ 10 ] shows that together with the implementation of ICT in the teaching and learning curriculum, students need to have relevant skills such as computer competencies and computer self-efficacies in order to successfully and effectively utilize these tools for their learning processes. Additionally, students also need to have relevant digital literacy skills in order to survive and thrive in this digital world [ 71 , 72 ]; hence, the teachers as mentors of the students need to have relevant digital literacy skills themselves.

Also, teachers play a very crucial role in integrating ICT in the school curriculum, and without proper training, knowledge, and competency of teachers, ICT may fail to deliver its expected outcome in education. Use of ICT, mobiles, laptops, podcasts, videos, Internets, and other assistive technologies improve the way mathematics is taught and enhance students’ understanding of the basic concepts more rapidly and effectively. However, a study by [ 73 ] found that mathematics teachers are not fully utilizing these facilities in their classroom teaching. According to [ 9 ], most of the mathematics teachers do not even make the teaching of mathematics practical and exciting. They are not competent enough to teach mathematics dynamically, which leads to negative attitude among pupils implying improper guidance by the teachers as well. A study by [ 74 ] concluded that the lack of competent mathematics teachers leads to the failure of students in mathematics in Nigerian secondary schools. Teacher’s language and background knowledge of the content contributes significantly toward academic achievements [ 75 ]. A study by [ 72 , 76 ] shows that linguistic and conceptual comprehension is a matter of concern. Mathematics teachers need to give a clear explanation to students about mathematical concepts where both language and a basic understanding of the concept is required to ensure each student understands rather than left confused. A study by [ 77 ] proved that teachers’ clarity, communication skills, content knowledge, and assessment procedures significantly impact students’ achievement in mathematics. To add on, studies such as [ 1 , 74 , 78 , 79 ] have attributed students’ low achievement in mathematics to lack of qualified mathematics teachers teaching at secondary schools. To address such issues in the South Pacific, a new cohort-taught pedagogical model known as the Science Teachers Accelerated Program (STAP) was introduced by The University of the South Pacific (USP) for those in-service science teachers outside the vicinity of USP campuses have to upskill and upgrade their qualifications through cohort teaching [ 22 ]. The program has mixed delivery modes and leverages heavily on ICT tools and technologies, including tablets and virtual classrooms [ 23 ], which have proven to be statistically significantly effective and productive in terms of quality and qualification of science teachers teaching at secondary schools in the South Pacific.

Effectiveness and Relevance of Mathematics Curriculum

Finally, a study by [ 80 ] described the curriculum in developing countries as too compact and exam-oriented. For teachers and stakeholders, the exam results of the schools are of great concern to them. Thus, due to the exam-oriented system, teachers are too much concerned with finishing the syllabus and drilling students with the exam questions and answers [ 68 ]. In the same view, [ 81 ] claimed that curriculum and assessment in Fijian schools do not serve the actual purpose effectively and efficiently. Examinations are not able to assess the attitude of students, leaving an important facet of life underdeveloped and probably the reason for not attaining quality. He further claimed that the gap in the curriculum content and the forms of assessment to achieve the outcomes has labeled the Fijian education system hapless. The Education Commission Report 2000 even reflected that the exam-oriented curriculum does not allow for outcome-based teaching and learning to progress. In many developing countries, several studies and researches have been carried out on curriculum and examinations influencing students’ interest and achievements in mathematics [ 7 , 81 – 84 ]. Local studies by [ 85 , 86 ] recommended that the Ministry of Education should review the curriculum to make it relevant and flexible to the diverse needs of different regions and background of the students. Reference [ 5 ] emphasized that the curriculum that currently exists focuses primarily on impoverished ideas about student learning or are based on no model of learning at all. It is quite evident that the mathematics content and assessments at years 11, 12, and 13 are dominated by arithmetic and is broad, non-contextualized, and irrelevant to real life when compared to years 9 and 10.

The majority of the local research works from the literature were conducted in primary schools, which focused on limited factors affecting performance in mathematics. At the same time, there are several factors responsible for students’ poor achievement in mathematics. Therefore, the study intends to contribute to the existing literature investigating the above five factors contributing to poor achievement in mathematics at the senior grades of secondary schools in the Western Division of Viti Levu, Fiji.

Research Objectives

The aim of this study was to examine and assess the factors that contribute to students’ poor achievement in mathematics at the senior grade (years 12 and 13) of secondary schools.

The study sought to:

a) assess students’ attitude and perception toward mathematics at senior grades of Tavua and Ba secondary schools

b) assess student perception on teachers’ attitude toward teaching mathematics at Tavua and Ba secondary schools

c) evaluate the qualification of mathematics teachers of Tavua and Ba secondary schools

d) identify teaching methods used by mathematics teachers of Tavua and Ba secondary schools

e) student and teacher perception on the effectiveness of the current mathematics curriculum at the senior secondary grades.

Research Questions

Specifically, this study aims to answer the following research questions:

a) What is the students’ attitude and perception toward mathematics at senior secondary grades?

b) What is the student perception on teachers’ attitude toward teaching mathematics at senior secondary grades?

c) What are the teaching methods used by mathematics teachers at senior secondary grades?

d) What are the qualifications of mathematics teachers in Tavua and Ba schools?

e) What is the student and teacher perception on the current mathematics curriculum at the senior secondary grades effective?

Methodology

This study is a descriptive study in which a cross-sectional survey research design was adopted. The data for the research were collected by the use of questionnaires, interviews, and student focus group discussion. The target population was 201 respondents which comprised 171 students, 16 mathematics teachers, 7 department heads, and 7 school heads from seven randomly selected secondary schools in the districts of Tavua and Ba. Random Sampling technique was used to select the seven secondary schools from a population of 14 secondary schools within the districts of Ba and Tavua. The sample, therefore, represented 50% of the population of Ba and Tavua secondary schools. The mathematics teachers, heads of departments, and the school heads were a part of the sample, who answered the questionnaires and also took part in the individual interviews as per the schedule. The stratified random sampling technique was then used for the selection of students from years 12 and 13 by obtaining a list containing recent overall academic results of each student in order to group them with varied abilities. This was done to ensure that the views of all the students with different abilities are equally represented. Furthermore, the purposive sampling method was used to select the students for the focus group discussion. Students within the Ba community were identified by the principal researcher, who were very inquisitive about the study’s objective and were outspoken to give personal and true opinions for the study. All the respondents were assured of confidentiality and their identity anonymity to protect the privacy of each respondent and to get the required information, which are the true opinions of each respondent. The appointments with the school heads were made and the consent of each respondent was also taken prior to the field research.

Research Tool Development and Pilot Study

There were four sets of questionnaires designed for each group of respondents (students, teachers, heads of departments, and school heads). The questionnaires were almost the same except for the content being rephrased to suit the opinion of the different groups of respondents. The questionnaire utilized the Likert scale to collect quantitative data for the research along with a section for suggestions and recommendations to curb the issue of poor achievement in mathematics. Three sets of interview questions were then designed. This was only for the mathematics teachers, heads of departments, and school heads. The students were not considered to be interviewed due to time constraints and a busy schedule for students after the reopening of schools post–COVID-19 lockdown in the country. Students were rather selected for the focus group discussion that was held at one of the libraries in the Ba town. The interviews and the focus group discussion only collected the qualitative data for the research. Pilot testing of these tools was also done in the two secondary schools in the district of Ba and Lautoka, which were not part of the sample. This was done to establish the clarity, meaning, and comprehensibility of each item in the tools. After the pilot study, the research tools along with the responses were discussed among the co-researchers for further review and amendment for its reliability and validity. A Cronbach alpha test using Statistical Package for the Social Sciences (SPSS) was carried out. The alpha value of 0.86 indicated that the questionnaire was valid and reliable for the study.

Demographic Characteristics of the Respondents

From the target population of 201 respondents, 181 respondents comprising of 151 years 12 and 13 students, 16 mathematics teachers, 7 heads of mathematics department, and 7 school heads answered the questionnaire. The same 16 mathematics teachers, 7 heads of the mathematics department, and 6 school heads from the 181 respondents group were the respondents who were also interviewed. The remaining 20 respondents from the target population were the years 12 and 13 students from the four secondary schools in the Ba district. They volunteered to be part of the student focus group discussion. From the 13 secondary schools in the districts of Tavua and Ba, 7 schools were randomly chosen to be the sample of this study. Data on Table 2 indicate the gender distribution of the participants in the study.

TABLE 2 . Gender of respondents.

Results and Discussion

a) Research question 1: What is the attitude and perception of students toward mathematics at senior secondary grades?

Students, teachers, heads of the mathematics department, and the school heads selected for the study were asked to give opinions on years 12 and 13 students’ attitude and perception toward mathematics. Each of the students selected expressed views on their own attitude and perception toward mathematics while the teachers, heads of mathematics department, and the school heads expressed their opinion on students’ attitude and perception toward the subject. The responses obtained are presented in Table 3 and Table 4 , as shown below.

TABLE 3 . Students’ attitude and perception toward mathematics—student perception.

TABLE 4 . Students’ attitude and perception toward mathematics—educators perception.

Table 3 shows that majority of the students perceived mathematics as a difficult subject. Students’ responses for each item showed that more than 50% of the students had a fear of mathematics as a subject and preferred learning other subjects, with the majority not wishing to continue with mathematics at the university level.

Table 4 shows responses from the 30 educators. More than 50% of the educators also perceived that students found mathematics a difficult subject and mostly failed because they had mathematics phobia. Looking at the educators’ responses, more than 50% believed that students lacked mathematics basics and hardly participated in any classroom activity. The responses from both the students and teachers were similar and derived from the “SA” and “A” columns. The following were a few of the responses from the interviews and student focus group discussions on how students perceive mathematics.

“I enjoyed and liked mathematics in my first three years of primary school only. Now I hate this subject. I do not see any reason why should we study mathematics? Where is it used in real life?” Student FG 13.

“I was really doing well in mathematics till year 4. Then I was taught by a teacher who always confused me. The explanations were not clear and understandable. The same teacher taught me in year 5 and from then I have lost interest in the subject.” Student FG 2.

“Students have a preconceived idea that mathematics is difficult. Till we change their attitude, we will never be able to achieve a better result in mathematics. Mathematics has to be made compulsory along with English in order to make them realise that they have to study and pass the subject if they want to achieve something in life”. Principal 2.

“Mathematics is a scoring subject. My teacher teaches us so well. She always motivates us to learn, but I do not know the basics. When now I am eager to study, I still find mathematics going over my head. I can answer few simple questions but when it comes to complex exercises, I just lose hope again.” Student FG 19.

“Students have a negative attitude and perception from primary school. Due to the ministry’s policy on compulsory education till year 12, they are just getting promoted. A child not knowing the previous year work is rarely able to grasp the concepts in the current year. It becomes very difficult for teachers in a classroom of over 30 students to go over basics and then teach them the concept.” Teacher 5.

“Mathematics is just numbers. It is so boring. Why are there no projects in mathematics like other technical subjects? I love to do technical drawing and computer studies as it has projects. In technical drawing we do practicals and projects which makes me enjoy the subject.” Student FG 12.

“My mathematics teachers work really hard. Some even take extra classes such as afternoon classes, Saturday classes and evening classes. Teachers go to the extent of going to students home and teach. Despite these efforts, some students do not bother. They do not even show interest and take advantage of extra efforts by our department teachers. Fact is that it is not their fault totally. They do not have a good foundation. By the time they reach year 12 and 13, mathematics is perceived to be a foreign language to them. They know that no matter how hard they try, nothing would change as they would still fail.”

b) Research question 2: What is the teachers’ attitude toward teaching mathematics at senior secondary grades?

The students were asked to give opinions on teachers’ attitude toward teaching mathematics at senior secondary grades.

Table 5 shows the student perception of the teachers' attitude in Tavua and Ba secondary schools. From the results, close to 85% of the students perceived that the teachers had a positive attitude toward teaching mathematics and always motivated them to learn. This is derived from the percentage of responses given under the “SA” and “A” columns. Similarly, teachers had been positively conditioning students at the senior grades; however, students’ prolonged negative mindset about mathematics from primary school failed to gain positive predilection for the subject. The teachers provided the students with summary notes for easier understanding and provided recaps before beginning new lessons. About 50% of the students indicated that their teachers' incorporated games, fun, and technology while teaching mathematics. Overall, the teachers’ attitude was positive in the delivery of mathematics lessons to the students.

c) Research question 3: What are the teaching methods used by mathematics teachers at senior secondary grades?

TABLE 5 . Student perception of teachers’ attitude toward teaching mathematics.

For this question, Table 6 was used as a guideline for the type of teaching methods used by the educators. In total, 23 educators answered this question and the results are presented below.

TABLE 6 . Teaching methods.

Data obtained from analyses show that 46.4% of the mathematics teachers used interactive lecture method, 24.3% use learner-centered method, 16.6% used teacher-centered method and 12.7% use collaborative learning method in their mathematics lessons. There were mixed reactions to the type of methods employed by the mathematics teachers of Tavua and Ba secondary schools. From the results it was evident that few of the teachers still preferred teacher-centered method (lecture method) of teaching their mathematics lessons. Many researchers have argued that the lecture method is a passive, ineffective, and antiquated teaching method used by teachers that would soon become obsolete [ 87 ]. However, few teachers find lecture method to be useful in covering a substantial amount of content, especially with large class sizes [88].

d) Research question 3: What are the qualifications attained by the mathematics teachers?

The survey also captured the mathematics and teacher training qualifications. The results are shown in Figures 1 and 2 .

FIGURE 1 . Highest level of mathematics teachers’ qualification.

FIGURE 2 . Teacher training qualification of mathematics teachers.

Figure 1 shows that majority of the teachers at secondary schools have degree qualifications with 24% having post graduate qualifications. The teachers with Diploma are upgrading their qualifications to degree. Figure 2 shows the teacher training qualifications and 100% of the teachers’ have teacher training qualification ranging from secondary teacher training certificate to post graduate diploma in education.

e) Research 5: Is the mathematics curriculum in senior secondary grades effective and relevant?

A 14-item Likert scale was developed to assist in detecting the nature and effectiveness of the mathematics curriculum at years 12 and 13 grades as opined by the respondents of Tavua and Ba secondary schools. The responses obtained are presented in Table 7 , as shown below.

TABLE 7 . Effectiveness and relevance of mathematics curriculum.

Out of 181 respondents, 145 (79.6%) have indicated that mathematics textbooks are very much dominated by arithmetic. It mostly deals with numbers, calculations, and complex computations. Also, 124 (68.5%) respondents agreed that the current mathematics curriculum at the senior secondary grades focuses only on examinations. In comparison, 101 (55.8%) respondents have shown that the mathematics curriculum in the senior secondary grades focuses mainly on the product (performance in exams) instead of the process (learning and understanding). This strongly agrees with the study by [ 43 ] who also identified the exam-oriented curriculum as one of the challenges in the senior grades of secondary schools in Fiji. Furthermore, the data obtained showed that 100 (55.2%) respondents have indicated that the mathematics curriculum at senior secondary grades is broad and lengthy compared to the other subjects. It was quite evident that majority of the teachers, heads of mathematics department, and school heads in the interviews have expressed disappointments regarding the current mathematics curriculum at the senior grades of secondary schools in Fiji.

“Curriculum is broad and lengthy and does not address the needs of students who wish to pursue further studies outside mathematics, science, and technical subjects.” (HOD Interview 5).

“Content of Year 13 has very less relevance to the real life.” (Teacher 13).

“People are not interested in certain topics because they do not find it relevant to real life.” (Student FG 5).

“Years 12 and 13 mathematics curriculum needs to be reviewed and the numbers of strands need to be reduced to incorporate more time for project work/class-based assessments.” (HOD Interview 5).

“Experienced teachers or department heads are the best stakeholders in terms of consultation and amendment of mathematics curriculum. Furthermore, there has to be consistency in external exam papers from year to year” (HOD Interview 5).

“The mathematics curriculum needs to be realigned to suit the Fijian context and the need of students.” (HOD Interview 2).

“Some students totally lose interest in mathematics upon reaching years 12 and 13 and therefore focus on subjects with projects to get a good aggregate. They ignore mathematics as they know that there is no chance of passing mathematics purely through exams.” (HOD Interview 6).

“External exams need not to be abolished but the weighting should be inclusive of projects and class internal assessments.” (HOD Interview 5).

There had been very poor results over the years in year 12 and 13 external exams. This means both the examination and the curriculum do not serve its purpose.” (HOD Interview 7).

• The overall mean response of the students, teachers, heads of the mathematics department, and school heads indicates that the mathematics curriculum at the senior grades of secondary schools is ineffective and irrelevant and therefore needs to be reviewed.

The data below show the rating of respondents’ perception of factors that contribute to poor achievement in mathematics. Out of 181 respondents, only 93 entries were analyzed since the remaining 88 entries were invalid. The responses obtained are analyzed in Figure 3 below.

FIGURE 3 . Factors that contribute to poor achievement in mathematics (A) . Students’ attitude and perception toward mathematics (B) . Teachers’ attitude toward teaching mathematics (C) . Teaching methods used by mathematics teachers (D) . Quality, performance, and qualification of mathematics teachers ( E) . Poorly developed curriculum and examinations.

Figure 3 revealed that students’ attitude and perception toward mathematics (58.1%) and poorly developed curriculum and examinations (34.3%) were the factors perceived to be significantly contributing to students’ poor achievement in mathematics at the senior grades of secondary schools. The respondents perceived that teacher attitude (2.2%); teaching methodologies (2.2%); and teacher quality, performance, and qualification (4.3%) had the least impact on students’ poor achievement in mathematics.

Limitations and Strengths

There was a dearth of local literature on poor achievement of students in mathematics and as such international literature was mostly referred to as a guide. Furthermore, time constraint was a factor since the principal researcher holds a full-time academic position during the time of this project. Hence, the sample schools chosen were around the vicinity of the principal researchers’ district origin. Despite these limitations, the study utilized an expansive approach to study different dynamics contributing to students’ poor achievement in mathematics from the views of students, teachers, heads of departments, and the school heads. The findings of the study also depict the notion of the problem faced in the teaching and learning of mathematics.

Conclusion and Recommendations

The study was carried out to examine and assess the factors contributing to the poor achievement of students at the senior grades of Tavua and Ba secondary schools in Western Fiji. Students’ attitude and perception toward mathematics, student perception on teachers’ attitude toward mathematics, teacher methodologies, teacher qualification, and student and teacher perception on the current curriculum in mathematics were the factors studied for this research. The study found that students had a negative attitude and perception toward mathematics. Furthermore, students perceived that mathematics teachers had a positive attitude toward teaching mathematics and are fully qualified to teach mathematics at secondary school levels as far as the teaching of mathematics and delivery of the subject matter was concerned. The method of teaching by the mathematics teachers was also appropriate and was fairly justified; however, limited use of technologies by the mathematics teachers in teaching mathematics was a matter of concern among most of the students. Furthermore, the study revealed the students and educators perceive that the current mathematics curriculum for years 12 and 13 are ineffective. This implied that students’ negative attitude and perception toward mathematics and the ineffective mathematics curriculum are the significant factors perceived to be significantly contributing to poor achievement of students in mathematics at the senior secondary grades. Moreover, many of the primary school teachers lacked potential and competence to teach mathematics at primary school levels, and this largely contributed toward the lack of interest among students, hence translating into poor achievement at both upper and lower secondary levels were found to be the reasons for students’ negative attitude and poor performance at secondary schools. The following recommendations are made based on the findings of the study: The mathematics curriculum at both years 12 and 13 need to be reviewed and amended in order to allow outcome-based teaching and learning to take place. The relevance and application of mathematics in real life should also be reflected in the curriculum.

The teachers, heads of departments, and the school heads have strongly emphasized ( via interviews) the need for MEHA and CDU to involve all the academic stakeholders including even the students and mathematics teachers in regards to any consultation, reviews, and amendments to the school curriculum. Exams should not be the only method of assessing students’ performance in mathematics. Internal assessments/field work/projects need to be a part of mathematics curriculum to understand mathematics better and at the same time develop interest among students with diverse needs. Students tend to learn better with technologies. There is a need for teachers to incorporate 21st century teaching tools, gadgets, and technology in teaching mathematics. Technology provides additional opportunities for students to see and interact with mathematics concepts and develop a positive attitude and perception toward the subject. Teacher quality should not be compromised at any cost, especially teachers who are responsible to teach the foundation of mathematics in primary schools. Content-focused teacher training to be implemented for primary school teachers in Fiji to teach specialized subjects in schools in order to build a good foundation among students and maintain positive attitude and perception of students toward mathematics across all levels.

Data Availability Statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author Contributions

Study conception and design: SC and KC; data collection: SC; analysis and interpretation of results: SC, AP, and VC; draft manuscript preparation: SC and KC. All authors reviewed the results and approved the final version of the manuscript.

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.

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Keywords: low academic achievement, teacher quality, curriculum, mathematics in schools, teacher attitude

Citation: Chand S, Chaudhary K, Prasad A and Chand V (2021) Perceived Causes of Students’ Poor Performance in Mathematics: A Case Study at Ba and Tavua Secondary Schools. Front. Appl. Math. Stat. 7:614408. doi: 10.3389/fams.2021.614408

Received: 06 October 2020; Accepted: 04 February 2021; Published: 23 April 2021.

Reviewed by:

Copyright © 2021 Chand, Chaudhary, Prasad and Chand. 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: Samlesh Chand, [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.

• Open access
• Published: 16 October 2021

Mathematics self-concept and challenges of learners in an online learning environment during COVID-19 pandemic

• Rex Bringula   ORCID: orcid.org/0000-0002-1789-9601 1 ,
• Jon Jester Reguyal 1 ,
• Don Dominic Tan 1 &
• Saida Ulfa 2  

Smart Learning Environments volume  8 , Article number:  22 ( 2021 ) Cite this article

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In this mixed-methods research, the relationship between four factors of individual online learners and their mathematics self-concept was explored. In addition, the challenges the students faced in learning mathematics online during the Coronavirus disease (COVID-19) pandemic were determined. The participant students were from two mathematics classes offered online during the summer of 2020. Pure online classes were first offered during this period because face-to-face learning sessions were suspended due to the COVID-19 pandemic. It was found that students owned the devices they were using for online classes. Internet connection and power interruption were the most problematic aspects of online learning. Students had positive as well as negative mathematics online learning self-concepts. Individual factors were partly related to mathematics self-concept. Qualitative data shows that students faced technological, personal, domestic, assessment, pedagogical, consultation, and test anxiety challenges. Implications and recommendations for teaching mathematics in an online environment are offered.

Introduction

Academic self-concept (ASC) is one's academic self-perceptions of one's general ability in school (Shavelson et al., 1976 ). It also refers to the interests of the students towards a particular course (Joyce & Yates, 2007 ). This idea has been widely investigated in the field of mathematics that eventually forms a field of study named mathematics self-concept (Lee & Kung, 2018 ; Pajares & Miller, 1994 ; Reyes, 1984 ). Mathematics self-concept has “to do with how sure a person is of being able to learn new topics in mathematics, perform well in mathematics class, and do well on mathematics tests” (p. 560, Reyes, 1984 ). It also refers to an individual's perception of his/her abilities related to mathematics as compared to others (Bong & Skaalvik, 2003 ). It is related to mathematics achievement (Lee & Kung, 2018 ). Thus, it is important to understand how students perceived their mathematics learning abilities.

However, these studies (e.g., Bong & Skaalvik, 2003 ; Lee & Kung, 2018 ; Pajares & Miller, 1994 ; Reyes, 1984 ) were investigated in the context of face-to-face learning where students can seek immediate learning interventions from teachers or their classmates. Moreover, while prior works (e.g., Baticulon et al., 2021 ; Fabito et al., 2021 ) outlined the barriers to online learning in the context of the COVID-19 pandemic, the mathematics self-concept in the context of online learning is not yet investigated. The sudden shift from face-to-face to online learning platforms posed challenges to students. Shifting to an online learning platform requires access to a device, Internet, and physical learning space, and a strong habit of learner’s autonomy. Due to the digital divide, device ownership is a persistent challenge in the effective implementation of online learning. For developing countries, intermittent Internet connection remains a barrier to effective online learning (Salac & Kim, 2016 ).

The shift from a classroom environment to a home learning environment raises another concern for students. Students that have no access to a personal physical online learning environment could be disrupted by noise and other distractions (Baticulon et al., 2021 ; Bringula et al., 2021 ). The pedagogical style also changed. In particular, there are parts of the content of the syllabus that students have to learn on their own (i.e., asynchronous session)—tasks that may not be a practice in a face-to-face setup. The ability of the students to learn and study course material during asynchronous sessions poses difficulties to online learners. These barriers are found to hinder the effective implementation of online learning. However, it is unclear whether these variables have a significant relationship with the students’ mathematics online learning self-concept (i.e., perceived abilities of the students to learn online mathematics online courses).

In other words, while technological, personal, institutional, and community barriers are identified in online learning in this time of the COVID-19 pandemic (Baticulon et al., 2021 ; Fabito et al., 2021 ), the relationship of these constraints to the mathematics self-concept of the students in a fully online class is still unknown. In addition, mathematics self-concept in an online learning setup is not considered in the existing guidelines on improving mathematics online education (e.g., Lee & Kung, 2018 ). Understanding the mathematics self-concept of the students at the early stages of the implementation of online learning could inform educators to improve or diversify their mathematics online teaching strategies. Hence, it is imperative to understand the relationship of these variables with the mathematics self-concept of the students in an online learning environment.

The findings of this study will benefit the teachers, the students, and the schools. Teachers may apply the recommended approach derived from the findings of the study in improving their online pedagogies. Schools may utilize the findings of the study in providing institutionalized strategies in addressing students’ online learning mathematics self-concept. In turn, students' online mathematics self-concept may be improved and students' mathematics learning may be achieved. Consequently, students' attrition to mathematics online learning courses may be lessened.

This study attempted to contribute to the existing threads of discussion of mathematics self-concept in the context of a fully online learning environment. The study investigated the mathematics self-concept of two classes of computing students during the summer of 2020. The results of the study served as a basis in the formulation of a checklist of suggestions for online mathematics teachers. Specifically, the study aims to answer the following research questions (RQ). (RQ) (1) What are the online learners-related factors in terms of device ownership, perceived Internet speed, personal physical learning space access, and mathematics learning autonomy? (2) What is the mathematics self-concept of online learners in terms of mathematics ability, mathematics interest, and perceived mathematics performance? (3) Is there a significant relationship between learner-related factors and mathematics self-concept? (4) Is there a significant difference in the mathematics self-concept of online learners between those with personal learning space and those without personal learning space?, and (5) What are the experiences of the respondents in terms of challenges in online learning?

Literature review

Online mathematics education.

Berge et al. ( 2000 ) set forth the following recommendations for teachers to ensure effective implementation of online learning courses: (1) state the hardware and software requirements of the course, (2) be available for consultation, (3) be creative in interacting with students online, (4) provide course performance feedback, (5) listen to student concerns and encourage class participation, (6) establish clear policies, goals, course objectives, and course expectations, (7) be knowledgeable with the online learning software, (8) use different pedagogical styles, (9) encourage collaboration among students, and (10) be proactive and solve problems to avoid escalation. Meanwhile, Herrington et al. ( 2004 ) suggested that online activities should be composed of relevant, ill-defined, varied, and complex tasks that could be integrated across different subject areas. Moreover, teachers may also provide activities supporting opportunities for collaboration and reflection.

Different studies reported the factors that influenced the achievement of online mathematics education students. The results of these studies served as focal points on improving online mathematics teaching. The study of Wadsworth et al. ( 2007 ) disclosed that four learning strategies (motivation, concentration, information processing, and self-testing) and self-efficacy predicted online mathematics grade achievement. They suggested that online mathematics educators provided real-world examples and conduct meetings with students regarding learning strategies. In a similar study, Glass and Sue ( 2008 ) showed that assignments were the most preferred learning object and had the most impact on learning. Thus, adequate practice drills and timely feedback were necessary for online mathematics education. The findings of Wadsworth et al. ( 2007 ) and Glass and Sue ( 2008 ) are consistent with the guidelines of Herrington et al. ( 2004 ).

Güzeller and Akin ( 2012 ) compared the mathematics achievement, attitudes, anxiety, and self-efficacy of students in the web-based mathematics instructions (WBMI) and traditional mathematics instructions (TMI). It was found that there was a significant difference between the WBMI and TMI in terms of achievement, attitudes, anxiety, and self-efficacy, having more favorable results on the WBMI. Students were advised to take the WBMI to familiarize themselves with this platform. Kim et al. ( 2014 ) extended the study of Güzeller and Akin (2012) by investigating the impact of anxiety and other forms of academic emotions (anger, shame, boredom, enjoyment, and pride) on online mathematics achievement. Kim et al. ( 2014 ) further hypothesized that motivation, self-efficacy, and cognitive processes (e.g., cognitive strategy, self-regulated learning) influenced online mathematics achievement. Academic emotions were accounted for 37% of the variation in student achievement.

Finally, Cho and Heron ( 2015 ) determined the impact of self-regulated learning (SRL), learning strategies, and emotions on satisfaction with online learning and online mathematics achievement. The study revealed that motivation influenced achievement, and both motivation and emotion were related to satisfaction. Based on these findings, the study formulated the following recommendations for teachers: enhance students' self-efficacy, design supporting tools in online courseware, provide course orientation, provide SRL support through social media, and restructure the format of the course.

Academic self-concept, academic self-efficacy, and mathematics self-concept

Arens et al. ( 2020 ) discussed the similarity and the difference between academic self-concept and academic self-efficacy. In general, both academic self-concept and academic self-efficacy address students' competence (Bong & Skaalvik, 2003 cited in Arens et al., 2020 ). However, academic self-concept is related to the self-perceived competence of a student in an academic domain in general (e.g., math; Marsh & Craven, 2006 ). Meanwhile, academic self-efficacy is self-perceived confidence to perform successfully a given task in a specific domain (e.g., Bandura, 2001 ; Zimmerman, 2000 ). In other words, self-concept is a domain-specific construct while self-efficacy is a domain- and task-specific construct (Arens et al., 2020 ).

Self-concept is investigated in the field of mathematics. Mathematics self-concept is attributed to students’ aspirations to pursue degree programs in STEM (science, technology, engineering, and mathematics) (Sax et al., 2015 ). Students with positive mathematics self-concept stay and finish their chosen degree program which, in turn, contributes to improving the school's student retention (Ackerman et al., 2013 ). This can be explained by the findings that mathematics self-concept is positively related to mathematics achievement (Kung, 2009 ; Lee & Kung, 2018 ). Moreover, mathematics self-concept and mathematics self-efficacy (i.e., a belief of student’s ability in solving mathematical problems or tasks related to mathematics; Masitoh & Firtriyani, 2018 ) predicted mathematics achievement (Kung, 2009 ). Thus, it is necessary to understand the mathematics self-concept of the students since it serves as a basis to cultivate the students’ learning interests in the program.

The measurement of mathematics self-concept is primarily based on the prior works of self-concept (e.g., Marsh & O'Neill, 1984 ; Marsh et al., 1983 , 2005 ). The questionnaire was composed of different areas including mathematics, verbal abilities, academic capabilities, problem-solving/creativity, physical abilities/sports, physical appearance, relations with same-sex peers, relations with opposite-sex peers, relations with parents, religion/spirituality, honesty/reliability, emotional stability/security, and general self-concept. The mathematics factor was composed of items such as “I find many mathematical problems interesting and challenging”, “Mathematics makes me feel inadequate”, “I am quite good at mathematics”, “I have trouble understanding anything that is based upon mathematics”, and “I have always done well in mathematics classes”. Recent studies on mathematics self-concept based their works on self-concept questionnaire developed by Marsh and colleagues (Marsh & O'Neill, 1984 ; Marsh et al., 1983 , 2005 ). For example, Lee and Kung ( 2018 ) investigated the mathematics self-concept and mathematics achievement of junior high school students in Taiwan. The authors devised a 13-item questionnaire measuring competence in mathematics, affection towards mathematics, and comparison of mathematics abilities.

Barriers to online learning

Cavanaugh et al. ( 2009 ) reported the barriers in the existing literature to online learning implementation as well as the benefits and challenges of online learning. Higher levels of motivation, expanding educational access, providing high-quality learning opportunities, improving student outcomes and skills, allowing educational choice, and administrative efficiency are notable benefits of online learning. Meanwhile, the high cost of start-ups, digital divide issues, governmental approval, and student readiness are the challenges raised in the implementation of online learning.

In a recent study, Binti Abd Aziz et al. ( 2020 ) investigated the barriers to online learning. According to the authors, addressing these barriers could lead to effective online learning practices. They identified the barriers in terms of attitudes, interruptions, technology skills, and personal skills. Attitudes towards online learning refer to the feelings of the people towards online learning. Computer, online, and computer application literacy skills are the components of technology skills. Interruptions to online learning are defined as the limits to technological access because students may be living in rural areas, being part of a minority group, having disabilities, or due to mature age. Personal skills are skills relating to prior experience of using online learning. Path analysis disclosed that attitudes toward online learning and technology skills are the main barriers to online learning.

The inability of the students to study at their own pace has also posed a barrier to students' online learning. Learner autonomy is the ability of learners to assume control or to take charge of their learning (Benson, 2001 ). Autonomous learners were able to understand the online learning process (Fotiadou et al., 2017 ), which enabled them to achieve high grades in online learning classes (Yen & Liu, 2009 ). In other words, mathematics learner autonomy is the ability of learners to understand and assume control learning of the online materials with little supervision (Benson, 2001 ; Fotiadou et al., 2017 ).

In the Philippines, Pena-Bandalaria ( 2009 ) reported that personal concerns (e.g., difficulty to interact and contact teachers, difficulty to seek help, difficulty in understanding the topics), technical concerns (e.g., problems accessing the course site), and the digital divide were barriers to online learning. This finding is consistent with the results of the study of Gledhill et al. ( 2017 ). Gledhill et al. ( 2017 ) also revealed that limited or poor access to the Internet, technology, and networks were the constraints of e-learning in less developed countries. Perceived Internet speed is the subjective evaluation of the speed of the Internet in supporting online learning sessions (Gledhill et al., 2017 ). Natividad ( 2021 ), and Salac and Kim ( 2016 ) explained the slow Internet connection in the Philippines. They agreed that Internet connection in the Philippines is slow due to limited Internet infrastructure which is brought by outdated laws and heavy bureaucratic processes for the development of Internet infrastructure. Furthermore, intermittent power supply was a major problem that hinders e-learning implementation in less developed countries (Bhuasiri et al., 2012 ).

In this time of COVID-19 pandemic, Philippine higher education institutions also implemented emergency online learning programs (Murphy, 2020 ). The emergency implementation may caught students unprepared (Aguilera-Hermida, 2020 ; Daniel, 2020 ). Baticulon et al. ( 2021 ) reported the major barriers of Filipino medical students to adopt online learning. These barriers can be classified as technological (lack of devices, issues with the online platform, Internet connectivity), individual (students’ learning style, physical and mental health), domestic (concerns at home, financial distress), institutional (school curriculum), and community barriers (lockdown restrictions, infrastructure challenges, and sociopolitical issues). Students found it difficult to understand the learning materials on their own. It was also reported that students had difficulty studying at home because of noise, distractions, and small space. Personal physical learning space refers to the space dedicated to online learning that is free from distraction or noise (Baticulon et al., 2021 ).

In a similar study of Fabito et al. ( 2021 ), they found that difficulty of clarifying topics or discussions with the professors, lack of study or working area dedicated for online activities, and lack of good Internet connection were the top three barriers and challenges that the 300 computing students (Computer Science and Information Technology) experienced. The study concluded that students and teachers were both not prepared to undergo full online learning. In a similar study, Bringula et al. ( 2021 ) found that the number of owned devices had a positive influence on the perceived academic online learning performance of computing students. Device ownership refers to the number of devices students own in accessing the LMS (Bringula et al., 2021 ). It was shown that students that own multiple devices are more likely to have positive dispositions towards their academic online learning performance than those students who have difficulty access to online learning devices.

Synthesis of literature review

There is a wealth of studies proposing the effective delivery of mathematics online education. These recommendations did not consider students' mathematic online self-concept. Moreover, the recommended teaching strategies are not set forth in the context of COVID-19 pandemic where students faced physical and psychological challenges (Baticulon et al., 2021 ; Bringula et al., 2021 ; Gledhill et al., 2017 ; Pena-Bandalaria, 2009 ). In particular, the continuous lockdowns in the Philippines exacerbate the existing phenomenon of digital divide, e.g., students cannot utilize public pay-for-access of computers and Internet in computer shops (Baticulon et al., 2021 ). Students may experience difficulty on engaging to mathematics online learning due to limited access to basic online resources. In turn, students may feel less capable of learning the mathematics online materials at their own pace. Determining the possible connection between barriers to online learning and students’ mathematics self-concept may help teachers, parents, and educational institutions to formulate pedagogical interventions to achieve desired online mathematics achievement.

Methodology

Research design, research setting, participants, sample, and data gathering procedure.

This mixed-method study was conducted in one department of a university in Manila. In the quantitative part, the participants of the study were second-year college students (subsequently referred to as online learners) enrolled in two classes in Numerical Analysis. There were 69 online learners enrolled in the said course. This was the only mathematics course offered at the time the study was conducted. The study was conducted after the first week of full implementation of pure online learning through a learning management system (LMS). Learning materials and lecture sessions were both delivered in synchronous and asynchronous methods, but mostly delivered in asynchronous methods (36 h out of 54 h). Online learners were informed at the beginning of the online class sessions about this setup. The survey form was distributed in the LMS. An online survey form was distributed to all online learners but only 54 students participated in the study. Online learners in the study are composed of male (59%) and female (41%) students with an average age of 20 years old.

Research instrument

The study utilized a content-validated survey form that served as a research instrument. The survey form consisted of two parts. The first part gathered information about the online learners profiles such as device ownership, perceived Internet speed, personal physical learning space ownership, and mathematics learning autonomy. They were also asked whether they have personal/private physical space for online learning (Baticulon et al., 2021 ). Perceived Internet speed can be answered using the responses “Very slow”, “Slow”, “Sometimes fast, sometimes slow”, “Fast”, and “Very fast”.

The mathematics learning autonomy intends to determine the level of independence to learn mathematics. It was measured using a 5-point scale in which the most negative response (i.e., total dependence to teachers or classmates) had an assigned value of 1 while the most positive response had a value of 5 (i.e., can independently learn the course content). These variables were selected because these were deemed relevant to the participants of the study and these were believed to influence engagement in online learning in the context of the COVID-19 pandemic (Pynos, 2016 ).

The second part solicited data on the mathematics self-concept of online learners. Mathematics self-concept consisted of mathematics ability (12 items), interest (2 items), and perceived mathematics performance (1 item). Online learners were asked about their perceived abilities and interest in mathematics learning when the course is delivered in an online setting. All items of ability and interest were preceded by the phrase “Considering that the course is delivered online,…”. The items could be answered using the responses “very untrue to me”, “untrue to me”, “unsure”, “true to me”, and “very true to me”. These verbal responses had assigned values from -2 to 2, where the most negative response has a value of -2 while the most positive response has a value of 2. Students were asked to complete the sentence “Considering that the course is delivered online, your mathematics grade will be…” to measure their perceived mathematics performance. This question could be answered using the responses “higher than face-to-face”, “about the same with face-to-face”, “lower than face-to-face”, and “not sure/cannot tell”.

The definition of academic self-concept relating to ability (Shavelson et al., 1976 ) and interest (Joyce & Yates, 2007 ) served as the basis in the construction of the research instrument. The items of the research instrument were adapted from Joyce and Yates ( 2007 ), Marsh et al. ( 1983 ), and Lee and Kung ( 2018 ). Only the ability component in the questionnaire of Lee and Kung ( 2018 ) was adopted in this study. The items applicable to online learning were retained. The adapted research instrument was then pilot-tested to 50 students who were not part of the study. Factor and Cronbach’s alpha analyses revealed that all items were found valid (factor loading ≥ 0.50) and reliable (Cronbach’s alpha α ≥ 0.70).

Statistical treatment of data

Frequency counts, means, and cross-tabulation were used to describe the data. Spearman Rank correlation was employed to determine the relationship between the profile of the online learners and mathematics self-concept. Mann–Whitney U test was used to determine whether there is a significant difference between the mathematics self-concept of online learners when categorized by personal physical learning space access. A 0.05 level of significance was used to determine the significance of the results.

Interview sessions, selection of interview participants, and participants

In the qualitative part, a series of separate interviews were conducted with three Information Technology (IT) students to further explain the findings of the study. The participants were selected based on their mathematics abilities and access to personal learning space. The authors sought the recommendations of the teachers who handled the course to identify the possible participants. The teachers identified and categorized the mathematics abilities of the students. These classifications are reliable because teachers know their students’ capabilities (Cheong et al., 2004 ; Lambert, 2002 ; Reeve, 2006 ). The participants consisted of 3 male, third-year students with an average age of 20 years old and they had varying degrees of mathematics abilities (i.e., struggling, average, and high performing). One of the informants (i.e., respondents) has no personal space dedicated to online learning. Two female participants were invited but they refused to participate in the study.

The interviews were conducted through Google Meet. Informants were interviewed one-by-one on different occasions to protect their identities. Students were asked about their challenges in online learning. The students were asked about their study practices in a face-to-face class (e.g., attending classes, review preparations for quizzes and exams, practicing solving math problems, reading materials, and taking lecture notes), challenges experienced in an online learning class, their perceived abilities in a face-to-face and online learning class, and their recommendations relating to the improvement of online pedagogy.

Qualitative data analysis

The interviews were transcribed and were tabulated in a word processor. The tabulated results were then presented for validation. The purpose of the validation process was to determine whether other students had the same experiences. The validation of interview results was done through a presentation to another set of students (i.e., the validators) with the same level of mathematics abilities and also enrolled in the mathematics class during the summer period. Throughout these procedures, all identities of the participants were kept confidential. The validators were asked whether they agree or disagree with the responses collected from the interviews. The validators were composed of IT students (3 males with an average age of 20 years old; 2 third-year students and 1 s-year student; one of the respondents has no personal learning space). After the validation process, the tabulated responses were analyzed through qualitative content analysis (Hsieh & Shannon, 2005 ; Mayring, 2014 ).

The responses were coded and categorized based on Baticulon et al.'s ( 2021 ) classifications of barriers to online learning. The codes were keywords or phrases that represented the challenges of learning mathematics in an online environment or their recommendations to their teachers about teaching mathematics in an online setup. The authors also made their classifications if an item did not fit on the Baticulon et al.'s ( 2021 ) classification. The codes were then assigned to themes (i.e., challenges of online learning and the recommendations to improve online learning). One of the authors coded the responses. When coding the text, the coder was guided by the themes. The process was repeated until all keywords and phrases were assigned to the themes. Afterward, the research team deliberated whether they agree (or disagree) with the themes. In case of disagreement, the deliberation process was repeated until a consensus was reached (Bringula et al., 2019 ).

Both the quantitative and qualitative results of the study were presented during the department general faculty meeting to elicit feedback and inputs on how mathematics online teaching practices be improved and to validate the themes proposed in this study. The attendees of the meeting served as the external validators. The external validators involved two mathematics teachers (both female with at least 25 years of teaching experience), one LMS administrator (female with 20 years of work experience; conducts LMS training and develops University-wide online course materials), and one academic administrator. All external validators agreed with the themes and recommendations of the study.

RQ1: Learners-related factors

It is found that 98% of the online learners own 1 or 2 devices (see Table 1 ). More than half of the respondents do not have personal learning space during online learning sessions. Seventy percent reported that intermittent Internet connection is the most problematic aspect of online learning. They perceived that mathematics delivered in an online platform is harder to learn than in a face-to-face setup. A large percentage of online learners (89%) mainly rely on lectures from teachers or from the help of their classmates to understand mathematical concepts. Fifty-six percent of the students rely on their teacher’s or classmates’ consultation to understand the lessons.

RQ2: Mathematics self-concept in an online learning setup

Online learners have negative notions about their capabilities in terms of understanding the lessons, solving problems easily, finishing the course, performing better relative to their classmates’ abilities, and performing better relative to their schoolmates’ abilities (Table 2 ). They feel it is more enjoyable to learn in a face-to-face setting than online. In terms of mathematics performance, 43% of the students feel that they will get lower grades in the online course than when it is done in a face-to-face session. Another 43% reported that they are unsure of what grade they will get at the end of the semester (Fig.  1 ). This perceived academic performance could be attributed to the unfamiliarity of the new learning environment. Students feel unprepared for online learning (Daniel, 2020 ; Fabito et al., 2021 ; Murphy, 2020 ) and they may feel it undesirable (Aguilera-Hermida, 2020 ).

Perceived mathematics performance in an online learning setup

Despite these negative notions, students have positive outlooks in terms of achieving good grades, doing well in the course, attending classes, doing assignments, helping their classmates in their assignments, recalling the lessons, and passing the course. They perceive that learning through the LMS is interesting. These findings suggest that they believe that their abilities can still meet the demands of the course. They are confident that they can still perform well despite the challenges and uncertainties they are facing.

RQ3: Relationship between learners-related factors and mathematics self-concept

It was also determined the relationship between the profile of online learners and their mathematics self-concept (Table 3 ). The number of devices they can use has positive relationship with understanding ( r  = 0.27, p  < 0.05) and recalling ( r  = 0.34, p  < 0.05) the lesson. Perceived Internet speed is positively related to the ease of attending class ( r  = 0.29, p  < 0.05). Statistical analyses found that correlations exist between mathematics learning autonomy and mathematics self-concept. Out of the 15 correlation analyses, six variables have significant correlations with mathematics learning autonomy. Mathematics learning autonomy have positive relationship with the abilities to get good grades ( r  = 0.42), solve problems easily ( r  = 0.40), do assignments easily ( r  = 0.31), recall the lessons ( r  = 0.34), and perform better than their classmates ( r  = 0.40) or schoolmates ( r  = 0.39).

RQ4: Difference in the mathematics self-concept of online learners between with and without personal learning spaces

Further analysis was conducted to determine if mathematics self-concept differs between online learners with and without personal learning spaces. Both online learners (with personal learning space, 30%; without learning space, 33%) agreed that mathematics learning in an online environment is harder than face-to-face. However, they have different opinions in terms of their mathematics grades. Online learners with no learning space perceived that they might have a lower grade than in a face-to-face course (30%) while those who have personal learning spaces are unsure of what grades they will get at the end of the semester (28%). The test of the difference between the means of self-concept of learners explains these results.

Mann Whitney U test revealed that the mathematics self-concept of the two groups of learners differ significantly in terms of achieving good grades ( U (52) = 252.0), solving problems easily ( U (52) = 243.5), doing well in the course ( U (52) = 249.5), answering assignments ( U (52) = 256.0), and recalling the lessons ( U (52) = 225.0) (Table 4 ). The results are unlikely to have arisen from sampling error ( p  < 0.05). Online learners with no personal learning space had lower mathematics concepts than privileged online learners.

RQ5: Challenges on online learning

Table 5 shows the challenges that the informants faced in learning mathematics in an online environment. All validators agreed that they experienced these challenges. All the informants and validators alike agreed that technological challenges are the most pressing concern in online learning. Only one of the informants reported power interruption. This informant is in the province and his province has been experiencing regular power interruptions. This statement confirms the study of Bhuasiri et al. ( 2012 ). The other informants and validators may not experience this because they are all in the National Capital Region where power interruption is rare.

The second challenge involves a problem that can only be solved by the students themselves. One of the informants said:

It is hard to focus on my studies. Unlike in a classroom setup, the environment is conducive to learning. All you need to do is to listen to the teacher. When you are at school, your mind is conditioned to study. When you are at home, you are in a comfort zone. I tend to do other things and delay doing my assignments. I admit: I become less productive and do not manage my time well when I am at home.

However, it must be noted that one of the validators did not agree that he procrastinates. This is the only item that the validators disagree with. Domestic challenges contribute to the distraction of students' online learning. Even while during class sessions, students were asked to run for errands or do simple household chores. Some situations are beyond the control of the students and their families (e.g., visitors). Noise and distractions contribute to domestic challenges. This finding confirms the study of Baticulon et al. ( 2021 ) and Fabito et al. ( 2021 ). One informant commented:

I have other responsibilities at home. Sometimes, I feel guilty because they are all busy doing household chores while I am on just on my computer throughout the day. Sometimes, I have to run errands. There are times that I have to respond to our neighbors' calls who are looking for my parents.

One validator agreed and said:

I agree with this. I want to also add that I only used the living room for my online learning sessions. Sometimes, the people in the house forgot that I am having an online learning session. They play music while walking around the living room. There was even an incident that they look at my laptop thinking that I was only watching movies.

Informants and validators reported that they experienced assessment challenges. Assessment challenges involve few practice drills, design of the online examination, clarity of instructions, and assessment feedback. One informant commented that teachers only provide about three questions and let them study and solve the other problems. This practice is construed as ineffective because, according to the informant, practice is an important activity that builds up their mathematics skills. The informant explained that this could be attributed to the desire of the teacher to cover the whole content of the course syllabus. When he was asked whether he prefers quality over quantity of the content, he chose the former. The validator stated: “There is no point in covering the whole syllabus when you did not understand any of them.”

The design of the examination refers to the way the questions are presented on an online platform. These include questions that require answers with long inputs of formula that are susceptible to typographical errors, multiple types of questions that are prone to guessing, time allotment, unclear instructions, and familiarity with the system itself. Informants and validators pointed out those problems that need to input their solutions in a text box entail a lot of time. They explained that they solve the problems on a piece of paper and then transfer them to the online submission system. They also raised their concerns on submission deadlines as they also have other courses with the same course requirements to satisfy. They also requested feedback on their activities and quiz results so that they would not commit the same mistakes.

The informants and validators further pointed out that they need more time as they familiarized themselves using the system. This is consistent with the study of Binti Abd Aziz et al. ( 2020 ). One informant said: “In an online exam, you are not only concerned with the correctness of your answers. You are also concerned about how you will input your answers correctly in the system. Teachers have to take into account that we also need time to familiarize ourselves with the system.”

The informants and validators also raised a pedagogical issue. The study of Baticulon et al. ( 2021 ) classified understanding the content of the course as a personal barrier. However, this is not the case based on the interviews with the informants and validators. According to informants and validators, it is difficult to understand the topics in online learning because of its delivery. When it is delivered appropriately, they can understand the topics and have better chances of passing the quizzes and exams. They prefer a combination of content delivery strategies including discussion of PowerPoint slides with step-by-step solutions through online meeting apps (e.g., Google Meet), and recorded videos of step-by-step solutions. One of the informants emphasized this comment: “Please do not let us study mathematics on our own. You do not just give the materials to us and let us understand the content.”

Before online learning, informants and validators seek the assistance of their teachers, classmates, or friends. The informants and validators understand that it is difficult to seek consultation because it is difficult to find a common time for consultation. Teachers and students have other responsibilities to attend to after online classroom sessions.

Students developed test anxiety because of the aforementioned challenges. According to the statement of one informant,

There is always a nervous factor when taking the quizzes or examinations since these are time-based activities. I am anxious since my Internet connection could suddenly become unstable or there might be a power interruption. Some teachers do not allow returning to the questions. Once you skipped the question, it will be given a zero mark. Unlike in a paper-based test, you can skip the questions and go back working on it if there is still time. I understand my teachers. They are thinking that a time-limited quiz/exam is a way to deter cheating.

This study investigated the classroom experience of online learners in a mathematics class during the summer of 2020. Toward this goal, the study attempted to determine the relationship between the online learners-related factors and their mathematics self-concept. Moreover, interviews were conducted to determine the challenges they faced in learning mathematics delivered on an online platform. The online-related factors in terms of device ownership revealed that they own 1 or 2 devices. Access to the device is not a problem to this set of participants relative to the general student population that may experience the digital divide (Cavanaugh et al., 2009 ; Pena-Bandalaria, 2009 ). This can be explained by the fact that the participants of this study are IT students, where learning activities, even before the pandemic, are highly dependent on devices.

The quantitative result shows that Internet connection is the most problematic aspect of online learning. An intermittent Internet connection can greatly affect the attendance of the students in online classes. This finding is consistent with the interview results in terms of technological challenges. This is a national problem since the Philippines has slow Internet connectivity (Chiu et al., 2017 ). According to Natividad ( 2021 ), and Salac and Kim ( 2016 ), the Philippines has a slow Internet connection because of the outdated Philippine law and red tape that hinders the quick installations of cell towers. This result confirms the findings of Bhuasiri et al. ( 2012 ) and Baticulon et al. ( 2021 ). Although only one of the informants reported an issue of power interruption, his concern is valid. His concern might not be similar to other informants or validators simply because the other informants and validators are all living in Metro Manila.

Almost half of the participants have no personal learning space during online learning sessions. Online learners with no personal learning space had lower mathematics concepts than privileged online learners. The lack of personal learning space during online sessions puts online learners in a disadvantaged position to attain an optimal learning experience. This is consistent with the findings of Baticulon et al. ( 2021 ), and Fabito et al. ( 2021 ). As explained in the interviews, students who lack personal learning space are more susceptible to distractions during, and even after online learning sessions. Noise and running errands are the most common forms of distraction. The interview results show that other members of the family may simply forget the students are in an online class. In short, as one student commented, access to personal physical learning space can create an environment conducive to online learning.

Online learners disclosed that they understood the content through lectures and constant consultation with teachers. This is consistent with the interview results that students dislike studying the course content on their own. This mathematics learning autonomy is the exact opposite of the nature of asynchronous learning. In asynchronous learning sessions, students have to study a lecture on their own. In other words, learners who are consultation-dependent will resist this educational shift.

As shown in Table 3 , students with low learning autonomy are expected to have lower dispositions of their mathematics abilities. Students who feel inferior about their mathematical abilities tend to have lower mathematics performance (Lee & Kung, 2018 ). In students’ point of view, asynchronous session activities (e.g., reading materials, assignments, practice drills, and quizzes) are challenging in the aspects of assessment, pedagogy, and consultation. These challenges explain why students experience test anxiety. Consequently, these difficulties contributed to their feeling of uncertain or low grade perceptions. The quantitative results provide insights to address this issue, i.e., students have to be gradually introduced into the concept of learners’ autonomy. Furthermore, family members may dedicate a place in the house that will serve as an online learning space.

The shift to an educational setting had a negative impact on the mathematics self-concept of learners. More than 80% of the respondents perceived that they will have a lower grade in mathematics. They also have negative notions of their mathematics self-concept in terms of understanding the lesson, solving problems, finishing the course, performing better relative to their classmates or schoolmates, and enjoying the online class. These negative notions on their capabilities and interest in online learning can be explained by the fact that full online learning is just implemented recently. While online learners have experience using the LMS before the COVID-19 pandemic, they are not yet fully familiar with a fully online learning setup. This is evident in one of the narratives of the informants. On one hand, the positive mathematics self-concept indicates that they are hopeful in the aspects of achieving good grades, attending classes, doing assignments, helping their classmates in their assignments, recalling lectures, passing the course, having the interest to learn, and doing well overall in the course. Teachers have to sustain these positive outlooks to achieve the course outcomes.

Device ownership has a positive relationship with understanding and recalling the lecture. Multiple devices such as laptops and mobile devices are dependable for students' online learning (Muyinda et al., 2010 ). Multiple device ownership allows online learners to view multiple screens and to store multiple copies of learning materials (Pynos, 2016 ). Multiple device usage in learning also creates seamless connectivity that enables the continuity of the learning experience (Milrad et al., 2013 ). This practice allows easy access to information that is useful for solving problems. For example, an online learner may be looking at his/her laptop screen for the given problem while he/she is looking into another device (e.g., cellphone) that displays the formula and the sample solved problems. Furthermore, multiple devices can address accessibility or installation issues.

Perceived Internet speed is positively related to the ease of attending class. This finding is expected. What is more interesting is that perceived Internet speed does not relate to the other items of mathematics self-concept. The results imply that a fast Internet connection is only necessary to attend the class but not necessarily related to the online learners' perceptions about their mathematics abilities. Their perceptions about their abilities and interest in mathematics are not related to the speed of Internet access. In other words, there is no link between the confidence of online learners in their mathematics abilities and their speed of Internet access.

Meanwhile, mathematics learning autonomy is correlated with most of the mathematics self-concept. This vivid finding denotes that mathematics self-concept is mostly related to the perceived abilities of online learners to study at their own pace, i.e., as students become more independent learners, they tend to have a higher mathematics self-concept. Teachers have to emphasize to the online learners that online learning is different from face-to-face where teachers can intervene when confusion or challenges arise in understanding the lessons. Teachers, at the onset of the course, are encouraged to orient online learners that they are expected to be independent learners. Problem sets and learning materials may be given in advance to develop the habit of independent learning.

Consistent with the literature, the respondents of this study experienced technological, personal, and domestic challenges. There is a challenge that students can be addressed by themselves (e.g., procrastination) but most are beyond their control. Domestic challenges require the support and understanding of family members. The students and their family members must have open communication. They should set house rules in terms of household chores and running errands.

The study of Baticulon et al. ( 2021 ) categorized the inability to understand the content of the course as a personal barrier to online learning. In this study, it was shown that this is a pedagogical challenge than a personal problem. Furthermore, it was disclosed that teachers have direct responsibilities on four out of the seven identified challenges. Challenges in the teaching and assessment had the most number of concerns. These results guide teachers to devise creative teaching and fair assessment strategies that could address these concerns. For teaching and learning activities, teachers are advised to provide ample time for lectures and deliver the contents through different forms of multimedia. At the end of each lecture, teachers may elicit feedback from students to assess if the students understood the lessons. It is advisable to gather feedback from struggling as well as high-performing students to understand the challenges of the students with diverse mathematical abilities. Group learning activities may be conducted using the Group Discussion function of the LMS. Teachers may provide practice drills that are not yet included for grade computation. Provide 2–3 days to allow students to do their assignments. The asynchronous sessions may also be utilized as consultation time. Teachers may use the randomized function of the LMS to pick random questions from its databank. Teachers may also request students to show their computer windows during the quiz (see “ Appendix A ”).

Another important role of the teachers is to sustain the positive and counter the negative mathematics self-concept of the students. Teachers at the onset of the course, the questionnaire here may be utilized to determine the mathematics self-concept of the students. Students should be oriented about the course expectations. Teachers should introduce independent learning gradually (“ Appendix A ”).

Conclusions, recommendations, limitations, and implications

This study investigated the profile of online learners and its influence on their mathematics self-concept. It is revealed that online learners in this study have access to devices. Physical learning space is one important aspect of an online learning environment. However, some online learners have physical learning space limitations which make online learning inconvenient. This limitation contributed to their low academic self-concept.

The majority had reported an intermittent Internet connection. Online learners have mixed notions about their mathematics capabilities and interest in learning mathematics in an online environment. They expressed uncertainties about the possible grades they will get at the end of the semester. The ability of online learners to study mathematics at their own pace is the most desired skill for online learners. Moreover, online learners with limited learning space are more likely to experience a lower mathematics self-concept because they cannot focus on the course. Thus, it can be concluded that the profile of online learners partly influences their mathematics self-concept.

Teachers play a significant role in improving and sustaining the mathematics self-concept of online learners. At the beginning of the class, teachers must inform online learners that having a habit of self-paced learning is a highly desirable discipline. Teachers have to sustain the positive mathematics self-concepts of online learners. They may assure online learners that online consultations are available when needed. Timely feedback on the works of online learners is highly encouraged to sustain their positive outlook about their capabilities. Individualized feedback can be provided to inform online learners that they are performing well (or not performing well) relative to his/her classmates.

The negative mathematics self-concepts of online learners serve as a basis for teachers to find ways to address these negative notions. Teachers have to be creative in delivering the content of the course (O’Doherty et al., 2018 ). For instance, PowerPoint slides with a voice recording or a previous video recording of the lesson may be utilized for lecture sessions. These materials may be accessed anytime and students with a slow Internet connection can still follow the phase of the course. Teachers may conduct synchronous learning sessions to answer questions or clarifications. An unwavering teacher’s dedication and understanding are suggested to assist online learners to finish the course.

The study is limited in terms of the participants and sample size. These limitations existed because of the timing of the shift of mode instructions in the university. There were only limited courses offered and a small number of students were enrolled when the study was conducted. Thus, the findings of the study may not be widely applicable beyond this population. Despite these limitations, this study provides clear insights into the students' mathematics self-concept and the challenges they faced in an online learning environment. The realities discovered in this study cannot be denied and deserves the attention of mathematics teachers. Nevertheless, a university-wide investigation of mathematics self-concept may be initiated to improve further the findings of the study.

There are issues raised in the study that cannot be solved by teachers. The members of the family must understand that online learners need physical learning space and minimal disruptions. To address this concern, teachers, or schools may send letters to parents about online learning to observe the online learning schedule of their children. Cooperation and understanding from family members are necessary for providing an environment conducive to online learning. It is strongly recommended that family members dedicate a physical learning space for online learners.

Educational institutions have to select an LMS that can support the demands of the course. The institution needs to understand the online learning requirements of the different degree programs. It is imperative to understand the strengths and limitations of the different LMS. A selection criteria committee may be instituted to select an LMS and to review its effectiveness relative to the needs of the students and faculty. Usability testing of the LMS may be done after the implementation. This will identify the ease of use and satisfaction of use of the LMS. The evaluation process may also evaluate whether the LMS supported the pedagogical requirements of the faculty (Pipan et al., 2008 ).

Another challenge the institution facing is the possibility of students that might be left behind because of inadequate access to devices. The formation of a technical support group is also desirable. Educational institutions may extend their help to online learners by lending laptops, tablets, and mobile Wi-Fi. Local government units may also offer assistance to underprivileged students. For example, a city local government unit in the Philippines provided online learning devices (e.g., laptops or tablets) to students (Casinas, 2020 ) and installed Internet centers to support online learning (Kabagani, 2020 ).

Lastly, the government may reinforce fast Internet connections through legislation. One of the possible legislations is to shorten the application of business process applications of constructing Internet facilities (Natividad, 2021 ; Salac & Kim, 2016 ). The government may allocate funds for the development of Internet infrastructures. These funds may be directed to rural areas. A government-private partnership may also be initiated. With this partnership, lengthy bureaucratic procedures will be avoided. Finally, the government may promote a market of competitiveness through the inclusions of other Internet providers (Salac & Kim, 2016 ).

Availability of data and materials

Data cannot be shared because of existing laws in the country of the authors.

Abbreviations

Academic self-concept

Coronavirus disease

Information technology

• Learning management system

Self-regulated learning

Traditional mathematics instructions

Mann–Whitney U test

Web-based mathematics instructions

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Appendix A: Checklist of suggestions for online mathematics teaching

Mathematics Self-concept in an Online Setting

Determine the mathematics self-concept of the students at the start of the class.

Inform the students of the setup of the course.

Introduce learners to the concept of independent learning.

Assist students in re-enforcing positive self-concepts.

Teaching Learning Activities

Provide ample time for lectures.

Immediately elicit feedback after the lecture.

Gather feedback from high-performing and struggling students about the phase and clarity of the lecture.

Use the combination of PowerPoint slides, online meetings, and videos for course content delivery.

Make the due dates reasonable (2–3 days).

Use the Group Discussion function of the LMS to encourage group study among students.

Assessment and Consultation

Provide individualized feedback on students’ activities.

Inform students about their class performance.

Dedicate synchronous sessions intended for consultation or feedback.

Provide more seatwork and practice drills.

Provide assessment activities that are not recorded.

Balance the types of the assessment – e.g., minimize multiple-choice, more on problems showing solutions, and discourage providing answers that are format-sensitive.

Balance the difficulty levels of the assessment.

Use the randomized function of the LMS to generate quizzes and exam questions.

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Bringula, R., Reguyal, J.J., Tan, D.D. et al. Mathematics self-concept and challenges of learners in an online learning environment during COVID-19 pandemic. Smart Learn. Environ. 8 , 22 (2021). https://doi.org/10.1186/s40561-021-00168-5

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Problem solving in mathematics education: tracing its foundations and current research-practice trends

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In tracing recent research trends and directions in mathematical problem-solving, it is argued that advances in mathematics practices occur and take place around two intertwined activities, mathematics problem formulation and ways to approach and solve those problems. In this context, a problematizing principle emerges as central activity to organize mathematics curriculum proposals and ways to structure problem-solving learning environments. Subjects’ use of concrete, abstract, symbolic, or digital tools not only influences the ways to pose and pursue mathematical problems; but also shapes the type of representation, exploration, and reasoning they engage to work and solve problems. Problem-solving foundations that privilege learners’ development of habits of mathematical practices that involve an inquiry method to formulate conjectures, to look for different ways to represent and approach problems, and to support and communicate results shed light on directions of current research trends and the relevance of rethinking curriculum proposals and extending problem-solving environments in terms of teachers/students’ consistent use of digital tools and online developments.

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1 Introduction and rationale

Mathematical problem solving has been a prominent theme and research area in the mathematics education agenda during the last four decades. Problem-solving perspectives have influenced and shaped mathematics curriculum proposals and ways to support learning environments worldwide (Törner et al., 2007 ; Toh et al., 2023 ). Various disciplinary communities have identified and contributed to connect problem-solving approaches with the students’ learning, construction, and application of mathematical knowledge. The mathematics community recognizes that the formulation and resolution of problems are central activities in the development of the discipline (Halmos, 1980 , Polya, 1945 ). Indeed, the identification and presentation of lists of unsolved mathematical problems have been a tradition that has inspired the mathematics community to approach mathematical problems and to generate mathematical knowledge (Hilbert, 1902 ; Devlin, 2002 ). Thus, mathematical problems, results, and solution attempts provide information regarding what areas and contents were studied at different times during the development of the discipline (Santos-Trigo, 2020a , b ). Cai et al. ( 2023 ) stated that “ …[E]ngaging learners in the activity of problem posing reflects a potentially strong link to the discipline of mathematics” (p. 5). Thurston ( 1994 ) recognized that understanding and applying a mathematical concept implies analysing, coordinating, and integrating diverse meanings (geometric, visual, intuitive, and formal definition) associated with such concept and ways to carry out corresponding procedures and operations in problematic situations.

The centrality of problem-solving in mathematicians’ own work and in their teaching, is incontrovertible. Problem-solving is also a central topic for mathematics educators, who have developed conceptual frameworks to formulate general ideas about problem-solving (as opposed to the specific ideas needed for solving specific problems) (Fried, 2014 ; p.17).

That is, the mathematics education community is interested in analysing and documenting the students’ cognitive and social behaviours to understand and develop mathematical knowledge and problem-solving competencies. “…the idea of understanding how mathematicians treat and solve problems, and then implementing this understanding in instruction design, was pivotal in mathematics education research and practice” (Koichu, 2014 ). In addition, other disciplines such as psychology, cognitive science or artificial intelligence have provided tools and methods to delve into learners’ ways to understand mathematical concepts and to work on problem situations. Thus, members of various communities have often worked in collaboration to identify and relate relevant aspects of mathematical practices with the design and implementation of learning scenarios that foster and enhance students’ mathematical thinking and the development of problem-solving competencies.

2 Methods and procedures

Research focus, themes, and inquiry methods in the mathematical problem-solving agenda have varied and been influenced and shaped by theoretical and methodological developments of mathematics education as a discipline (English & Kirshner, 2016 ; Liljedahl & Cai, 2021 ). Further, research designs and methods used in cognitive, social, and computational fields have influenced the ways in which mathematical problem-solving research are framed. An overarching question to capture shifts and foundations in problem-solving developments was: How has mathematical problem-solving research agenda varied and evolved in terms of ways to frame, pose, and pursue research questions? In addressing this question, it was important to identify and contrast the structure and organization around some published problem-solving reviews (Lester, 1994 ; Törner et al., 2007 ; Rott et al., 2021 ; Liljedahl & Cai, 2021 ; Toh et al., 2023 ) to shed light on a possible route to connect seminal developments in the field with current research trends and perspectives in mathematical problem-solving developments. The goal was to identify common problem-solving principles that have provided a rational and foundations to support recent problem-solving approaches for learners to construct mathematical knowledge and to develop problem-solving competencies. The criteria to select the set of published peer-reviewed studies, to consider in this review, involved choosing articles published in indexed journals (ZDM-Mathematics Education, Educational Studies in Mathematics, Mathematical Thinking and Learning, Journal of Mathematical Behavior, and Journal for Research in Mathematics Education); contributions that appear in International Handbooks in Mathematics Education; and chapters published in recent mathematical problem-solving books. The initial search included 205 publications whose number was reduced to 55, all published in English, based on reviewing their abstracts and conclusions. Around 100 of the initial selection appeared in the references of an ongoing weekly mathematical problem-solving doctoral seminar that has been implemented during the last six years in our department. In addition, some well-known authors in the field were asked to identify their most representative publications to include in the review list. Here, some suggestions were received, but at the end the list of contributions, that appears in the references section, was chosen based on my vision and experience in the field. The goal was to identify main issues or dimensions to frame and analyse recent research trends and perspectives in mathematical problem-solving developments. Thus, seminal reviews in the field (Schoenfeld, 1992 ; Lester, 1994 ; Törner et al., 2007 ) provided directions on ways to structure and select the questions used to analyse the selected contributions. Table  1 shows chosen issues that resemble features of an adjusted framework that Lester ( 1994 ) proposed to organize, summarize, and analyse problem-solving developments in terms of research emphasis (themes and research questions), methodologies (research designs and methods), and achieved results that the problem-solving community addressed during the 1970–1994 period. Furthermore, relevant shifts in the mathematical problem-solving agenda could be identified and explained in terms of what the global mathematics education and other disciplines pursue at different periods.

It is important to mention that the content and structure of this paper involve a narrative synthesis of selected articles that includes contributions related to mathematical problem-solving foundations and those that address recent developments published in the last 9 years that involve the use of digital technologies. Table  1 shows themes, issues, and overarching questions that were used to delve into problem-solving developments.

To contextualize the current state of art in the field, it is important to revisit problem-solving principles and tenets that provide foundations and a rationale to centre and support the design and implementation of learning environments around problem-solving activities (Santos-Trigo, 2020a , b ). The identification of mathematical problem-solving foundations also implies acknowledging what terms, concepts, and language or discourse that the problem-solving community has used to refer to and frame problem-solving approaches. For example, routine and nonroutine tasks, heuristic and metacognitive strategies, students’ beliefs, mathematical thinking and practices, resources, orientations, etc. are common terms used to explain, foster, and characterize students’ problem-solving behaviours and performances. Recently, the consistent use of digital technologies in educational tasks has extended the problem-solving language to include terms such as subjects’ tool appropriation, dynamic models, dragging or moving orderly objects, tracing loci, visual or empirical solution, ChatGPT prompts, etc.

3 On mathematical problem-solving foundations and the problematizing principle

There might be different ways to interpret and implement a problem-solving approach for students to understand concepts and to solve problems (Törner, Schoenfeld, & Reiss, 2007 ; Toh et al., 2023 ); nevertheless, there are common principles or tenets that distinguish and support a problem-solving teaching/learning environment. A salient feature in any problem-solving approach to learn mathematics is a conceptualization of the discipline that privileges and enhance the students’ development of mathematical practices or reasoning habits of mathematical thinking (Cuoco, et, al., 1996 ; Dick & Hollebrands, 2011 ; Schoenfeld, 2022 ). In this context, students need to conceptualize and think of their own learning as a set of dilemmas that are represented, explored, and solved in terms of mathematical resources and strategies (Santos-Trigo, 2023 ; Hiebert et al., 1996 ).

Furthermore, students’ problem-solving experiences and behaviours reflect and become a way of thinking that is consistent with mathematics practices and is manifested in terms of the activities they engage throughout all problem-solving phases. Thus, they privilege the development of mathematics habits such as to always look for different ways to model and explore mathematical problems, to formulate conjectures, and to search for arguments to support them, share problem solutions, defend their ideas, and to develop a proper language to communicate results. In terms of connecting ways of developing mathematical knowledge and the design of learning environments to develop mathematical thinking and problem-solving competencies, Polya ( 1945 ) identifies an inquiry approach for students to understand, make sense, and apply mathematical concepts. He illustrated the importance for students to pose and pursue different questions around four intertwined problem-solving phases: Understanding and making sense of the problem statement (what is the problem about? What data are provided? What is asked to find? etc.), the design of a solution plan (how the problem can be approached? ), the implementation of such plan (how the plan can be achieved? ), and the looking-back phase that involves reviewing the solution process (data used, checking the involved operations, consistency of units, and partial and global solution), generalizing the solution methods and posing new problems. Indeed, the looking-back phase involves the formulation of new or related problems (Toh et al., 2023 ). “For Pólya, mathematics was about inquiry; it was about sense making; it was about understanding how and why mathematical ideas fit together the ways they do” (cited in Schoenfeld, 2020 , p. 1167).

Likewise, the Nobel laureate I. I. Rabi mentioned that, when he came home from school, “while other mothers asked their kids ‘ Did you learn anything today ?’ [my mother] would say, ‘ Izzy, did you ask a good question today ?’” (Berger, 2014 , p.67).

Thus, the problematizing principle is key for students to engage in mathematical problem-solving activities, and it gets activated by an inquiry or inquisitive method that is expressed in terms of questions that students pose and pursue to delve into concepts meaning, representations, explorations, operations, and to work on mathematical tasks (Santos-Trigo, 2020a , b ).

4 The importance of mathematical tasks and the role of tools in problem-solving perspectives

In a problem-solving approach, learners develop a way of thinking to work on different types of tasks that involve a variety of context and aims (Cai & Hwang, 2023 ). A task might require students to formulate a problem from given information, to estimate how much water a family spend in one year, to prove a geometry theorem, to model genetic sequences or to understand the interplay between climate and geography. In this process, students identify mathematical resources, concepts, and strategies to model and explore partial and global solutions, and ways to extend solution methods and results. Furthermore, mathematical tasks or problems are essential for students to engage in mathematical practice and to develop problem-solving competencies. Task statements should be situated in different contexts including realistic, authentic, or mathematical domains, and prompts or questions to solve or respond or even provide information or data for students to formulate and solve their own problems (problem posing). Current events or problematic situations such as climate change, immigration, or pandemics not only are part of individuals concerns; but also, a challenge for teachers and students to model and analyze those complex problems through mathematics and others disciplines knowledge (English, 2023 ). Santos-Trigo ( 2019 ) proposed a framework to transform exercises or routine textbook problems into a series of nonroutine tasks in which students have an opportunity to dynamically model, explore, and extend, the initial problem. Here, the use of technology becomes important to explore the behavior of some elements within the model to find objects’ mathematical relationships. That is, students work on tasks in such a way that even routine problems become a starting point for them to engage in mathematical reflection to extend the initial nature of the task (Santos-Trigo & Reyes-Martínez, 2019 ). Recently, the emergence of tools such as the ChatGPT has confirmed the importance for learners to problematize situations, including complex problems, in terms of providing prompts or inputs that the tool processes and answers. Here, students analyze the tool’ responses and assess its pertinence to work and solve the task. Indeed, a way to use ChatGPT involves that students understand or make sense of the problem statement and pose questions (inputs or prompts) to ask the tool for concept information or ways to approach or solve the task. Then, students analyze the relevance, viability, and consistency of the tool’s answer and introduce new inputs to continue with the solution process or to look for another way to approach the task. Based on the ChatGPT output or task solution, students could always ask whether the tool can provide other ways to solve the task.

5 Main problem-solving research themes and results

In this section the focus will be on identifying certain problem-solving developments that have permeated recent directions of the field. One relates to the importance of extending research designs to analyse and characterize learners’ problem-solving process to work on different types of tasks. Another development involves ways in which theoretical advances in mathematics education have shaped the mathematical problem-solving research agenda and the extent to which regional or national educational systems or traditions influence the developments of conceptual frameworks in the field and ways to implement problem-solving activities within the corresponding system. Finally, research results in the field have provided directions to design and implement curriculum proposals around the world and these proposals have evolved in terms of both content structure and classroom dynamics including the use of digital technologies. Santos-Trigo ( 2023 ) stated that the teachers and students’ systematic use of digital technologies not only expands their ways of reasoning and solving mathematical problems; but also opens new research areas that aim to analyse the integration of several digital tools in curriculum proposals and learning scenarios. The focus of this review will be on presenting problem-solving directions and results in the last 9 years; however, it became relevant to identify and review what principles and tenets provided bases or foundations to support and define current research trends and directions in the field. That is, accumulated research that has contributed to advance and expand the problem-solving research agenda included shifts in the tools used to delve into learners’ problem approaches, the development of conceptual frameworks to explain and characterize students’ mathematical thinking, the tools used to work on mathematical tasks (from paper and pencil, ruler and compass or semiotic tools to digital apps), and in the design of curriculum proposals and the implementation of problem-solving learning scenarios.

5.1 Relevant shifts in problem-solving developments and results

Questions used to analyse important developments in the field include: What research designs and tools are used to foster and analyse learners’ problem-solving performances? How have conceptual frameworks evolved to pose and frame research questions in the field? How have accumulated research results in the field been used to support curriculum proposals and their implementation?

5.1.1 Methodological and research paradigms

Research designs in problem-solving studies have gradually moved from quantitative or statistical paradigms to qualitative perspectives that involve data collection from different sources such as task-based interviews, fieldnotes from observations, students’ written reports, etc. to analyse students’ problem-solving approaches and performances. Trustworthiness of results included triangulating and interpreting data sources from students’ videotapes transcriptions, outside observer notes, class observations, etc. (Stake, 2000 ). Hence, the work of Krutestkii ( 1976 ) was seminal in providing tools to delve into the students’ thinking while solving mathematical tasks. His research program aimed to study the nature and structure of children’ mathematical abilities. His methodological approach involved the use of student’s task-based interviews, teachers, and mathematicians’ questionaries to explore the nature of mathematical abilities, the analysis of eminent mathematicians and physicists regarding their nature and emergence of their talents and case studies of gifted children in mathematics. A major contribution of his research was the variety of mathematical tasks used to explore and analyse the mathematical abilities of school children. Recently, the mathematical problem-posing agenda has been revisited to advance conceptual frameworks to enhance the students’ formulation of problems to learn concepts and to develop problem-solving competencies (Cai et al., 2023 ). In general, the initial qualitative research tendency privileged case studies where individual students were asked to work on mathematical tasks to document their problem-solving performances. Later, research designs include the students’ participation in small groups and the analysis of students’ collaboration with the entire group (Brady et al., 2023 ). Bricolage frameworks that share tenets and information from different fields have become a powerful tool for researchers to understand complex people’ problem-solving proficiency (Lester, 2005 ; English, 2023 ).

5.1.2 Theoretical developments in mathematics education

In mathematics education, the constructivism perspective became relevant to orient and support research programs. Specifically, the recognition that students construct mathematical concepts and ideas through active participation as a part of a learning community that fosters and values what they bring into the classroom (eliciting students’ understanding) and sharing and discussing with peers their ways to work on mathematical activities. Further, it was recognized that students’ learning of mathematics takes place within a sociocultural environment (situated learning) that promotes the students’ interaction in small groups, pairs, and whole group discussions. Thus, problem-solving environments transited from teachers being a main figure to organize learning activities and to model problem-solving behaviours to being centred on students’ active participation to work on a variety of mathematical tasks as a part of a learning community (Lester & Cai, 2016 ). English ( 2023 ) proposed A STEM-based problem-solving framework that addresses the importance of a multidisciplinary approach and experiences to work on complex problems. Here, students develop a system of inquiry that integrates critical thinking, mathematical modelling, and a creative and innovative approach to deal with problematic situations situated in contexts beyond school problems. The STEM-based problem-solving framework enhances and favours the students’ development of multidisciplinary thinking to formulate and approach challenging problematic situations. To this end, they need to problematize information to characterize local and global problems and to collaboratively work on feasible approaches and solutions. It integrates 21st century skills that include an inquiry problem-solving approach to develop and exhibit critical thinking, creativity, and innovative solutions.

5.1.3 Countries or regional education traditions and their influence on the problem-solving agenda

The emergence of problem-solving frameworks takes place within an educational and socio-cultural context that provides conditions for their development and dissemination, but also limitations in their applications inside the mathematics education community. Brady et al. ( 2023 ) pointed out that:

…shifts in the theoretical frameworks of mathematics education researchers favored a widening of the view on problem solving from information-processing theories toward sociocultural theories that encouraged a conception of problem-solving as situated cognition unfolding within a community of practice (p. 34).

In addition, regional or national educational systems and research traditions also shape the problem-solving research and practice agenda. For example, in France, problem-solving approaches and research are framed in terms of two relevant theoretical and practical frameworks: Theory of Didactic Situation and the Anthropological Theory of Didactics (Artigue & Houdement, 2007 ). While, in the Netherlands, problem-solving approaches are situated within the theory of Realistic Mathematics that encourages and supports the students’ construction of meaning of concepts and methods in terms of modelling real-life and mathematical situations (Doorman et al., 2007 ). Ding et al. ( 2022 ) stated that the Chinese educational system refers to problem solving as an instructional goal and an approach to learn mathematics. Here, students deal with different types of problem-solving activities that include finding multiple solutions to one problem, one solution to multiple problems, and one problem multiple changes. Thus, ‘teaching with variation’ is emphasized in Chinese instruction in terms of “variations in solutions, presentations, and conditions/conclusions” (p. 482). Cai and Rott ( 2023 ) proposed a general problem-posing process model that distinguishes four problem-posing phases: Orientation (understanding the situation and what is required or is asked to pose); Connection that involves finding out or generating ideas and strategies to pose problems in different ways such as varying the given situation, or posing new problems; Generation refers to making the posed problem visible for others to understand it; and Reflection involves reflecting on her/his own process to pose the problem including ways to improve problem statements. The challenge in this model is to make explicit how the use of digital technologies can contribute to providing conditions for students to engage in all phases around problem- posing process.

5.1.4 Curriculum proposals and problem-solving teaching/learning scenarios

In the USA, the Common Core State Mathematics Standards curriculum proposal (CCSMS) identifies problem solving as a process standard that supports core mathematical practices that involve reasoning and proof, communication, representation, and connections. Thus, making sense of problems and persevering in solving them, reasoning abstractly and quantitatively, constructing viable arguments and critiquing the reasoning of others, modelling with mathematics, etc. are essential activities for students to develop mathematics proficiency and problem-solving approaches (Schoenfeld, 2023 ). In Singapore, the curriculum proposal identifies problem solving as the centre of its curriculum framework that relates its development with the study of concepts, skills, processes, attitudes, and metacognition (Lee et al., 2019 ). Recently, educational systems have begun to reform curriculum proposals to relate what the use of digital technologies demands in terms of selecting and structuring mathematical contents and ways to extend instructional settings (Engelbrecht & Borba, 2023 ). Indeed, Engelbrecht et al. ( 2023 ) identify what they call a classroom in movement or a distributed classroom - that transforms traditional cubic spaces to study the discipline into a movable setting that might combine remote and face-to-face students work.

It is argued that previous results in mathematical problem-solving research not only have contributed to recognize what is relevant and what common tenets distinguish and support problem-solving approaches; but also have provided bases to identify and pursue current problem-solving developments and directions. Hence, the consistent and coordinated use of several digital technologies and online developments (teaching and learning platforms) has opened new routes for learners to represent, explore, and work on mathematical problems; and to engage them in mathematical discussions beyond formal class settings. How does the students’ use of digital technologies expand the ways they reason and solve mathematical problems? What changes in classroom environments and physical settings are needed to recognize and include students’ face-to-face and remote work? (Engelbrecht et al., 2023 ).

In the next sections, the goal is to characterize the extent to which the consistent use of digital technologies and online developments provides affordances to restructure mathematical curriculum proposals and classrooms or learning settings and to enhance and expand students’ mathematical reasoning.

6 Current mathematical problem-solving trends and developments: the use of digital technologies

Although the use of technologies has been a recurrent theme in research studies, curriculum proposals, and teaching practices in mathematics education; during the COVID-pandemic lockdown, all teachers and students relied on digital technologies to work on mathematical tasks. At different phases, they developed and implemented not only novel paths to present, discuss, and approach teaching/learning activities; but also, ways to monitor and assess students’ problem-solving performances. When schools returned to teachers and students’ face-to-face activities, some questions emerged: What adjustments or changes in school practices are needed to consider and integrate those learning experiences that students developed during the social confinement? What digital tools should teachers and students use to work on mathematical tasks? How should teaching/learning practices reconcile students remote and face-to-face work? To address these questions, recent studies that involve ways to integrate technology in educational practices were reviewed, and their main themes and findings are organized and problematized to shed light on what the use of digital technologies contributes to frame and support learning environments.

6.1 The use of technology to reconceptualize students mathematical learning

There are different studies that document the importance and ways in which the students’ use of tools such as CAS or Excel offers an opportunity for them to think of concepts and problems in terms of different representations to transit from intuitive, visual, or graphic to formal or analytical reasoning (Arcavi et al., 2017 ). Others digital technologies, such as a Dynamic Geometry System Footnote 1 DGS, provide affordances for students to dynamically represent and explore mathematical problems. In students’ use of digital technologies, the problematizing principle becomes relevant to transform the tool into an instrument to work on mathematical tasks. Santos-Trigo ( 2019 ) provides examples where students rely on GeoGebra affordances to reconstruct figures that are given in problem statements; to transform routine problem into an investigation task; to model and explore tasks that involve variational reasoning; and to construct dynamic configurations to formulate and support mathematical relations. In this process, students not only exhibit diverse problem-solving strategies; but also, identify and integrate and use different concepts and resources that are studied in algebra, geometry, and calculus. That is, the use of technology provides an opportunity for students to integrate and connect knowledge from diverse areas or domains. For instance, Sinclair and Ferrara ( 2023 ) used the multi-touch application (TouchCounts) for children to work on mathematical challenging tasks.

6.2 The use of digital technologies to design a didactic route

There is indication, that the use of digital technologies offers different paths for students to learn mathematics (Leung & Bolite-Frant, 2015 ; Leung & Baccaglini-Frank, 2017 ). For instance, in the construction of a dynamic model of a problem, they are required to think of concepts and information embedded in the problem in terms of geometric representation or meaning. Thus, focusing on ways for students to represent and explore concepts geometrically could be the departure point to understand concepts and to solve mathematical problems. In addition, students can explore problems’ dynamic models (dragging schemes) in terms of visual, empirical, and graphic representations to initially identify relations that become relevant to approach and solve the problems. Thus, tool affordances become relevant for students to detect patterns, to formulate conjectures and to transit from empirical to formal argumentation to support problem solutions (Pittalis & Drijvers, 2023 ). Engelbrecht and Borba ( 2023 ) recognized that the prominent use of digital technologies in school mathematics has produced pedagogical shifts in teaching and learning practices to “encourage more active students learning, foster greater engagement, and provide more flexible access to learning’ (p. 1). Multiple use technologies such as internet, communication apps (ZOOM, Teams, Google Meet, etc.) become essential tools for teachers and students to present, communicate, and share information or to collaborate with peers. While tools used to represent, explore, and delve into concepts and to work and solve mathematical problems (Dynamic Geometry Systems, Wolframalpha, etc.) expand the students’ ways of reasoning and solving problems. Both types of technologies are not only important for teachers and students to continue working on school tasks beyond formal settings, but they also provide students with an opportunity to consult online resources such as Wikipedia or KhanAcademy to review or extend their concepts understanding, to analyse solved problems, and to contrast their teachers’ explanation of themes or concepts with those provided in learning platforms.

6.3 Students’ access to mathematics learning

Nowadays, cell phones are essential tools for people or students to interact or to approach diverse tasks and an educational challenge is how teachers/students can use them to work on mathematical tasks. During the COVID-19 social confinement, students relied on communication apps not only to interact with their teachers during class lectures; but also, to keep discussing tasks with peers beyond formal class meetings. That is, students realized that with the use of technology they could expand their learning space to include sharing and discussing ideas and problem solutions with peers beyond class sessions, consulting online learning platforms or material to review or extend their concepts understanding, and to watch videos to contrast experts’ concepts explanations and those provided by their teachers. In this perspective, the use of digital technologies increases the students’ access to different resources and the ways to work on mathematical tasks. Thus, available digital developments seem to extend the students collaborative work in addition to class activities. Furthermore, the flipped classroom model seems to offer certain advantages for students to learn the discipline and this model needs to be analysed in terms of what curriculum changes and ways to assess or monitor students learning are needed in its design and implementation (Cevikbas & Kaiser, 2022 ).

6.4 Changes in curriculum and mathematical assessment

It is recognized that the continuous development and availability of digital technologies is not only altering the ways in which individuals interact and face daily activities; but is also transforming educational practices and settings. Likewise, people’s concerns about multiple events or global problems such climate change, immigration, educational access, renewable resources, or racial conflicts or wars are themes that permeate the educational arena. Thus, curriculum reforms should address ways to connect students’ education with the analysis of these complex problems. English ( 2023 ) stated that:

The ill-defined problems of today, coupled with unexpected disruptions across all walks of life, demand advanced problem-solving by all citizens. The need to update outmoded forms of problem solving, which fail to take into account increasing global challenges, has never been greater (p.5).

In this perspective, mathematics curriculum needs to be structured around essential contents and habits of mathematical thinking for students to understand and make sense of real-world events that lead them to formulate, represent, and deal with a variety of problem situations. “Educators now increasingly seek to emphasise the practical applications of mathematics, such as modelling real-life scenarios and understanding statistical data (Engelbrecht & Borba, 2023 , p. 7). For instance, during the pandemic it was important to problematize the available data to follow, analyze and predict its spread behavior and to propose health measures to reduce people contagion. Thus, exponential functions, graphics, and their interpretations, data analysis, etc. were important mathematics content to understand the pandemic phenomena. Drijvers and Sinclair ( 2023 ) recognized that features of computational thinking share common grounds with mathematical thinking in terms of problem-solving activities that privilege model construction, the use of algorithms, abstraction processes and generalization of results. Thus, “a further integration of computational thinking in the mathematics curriculum is desirable”. In terms of ways to assess and monitor students’ learning, the idea is that with the use of a digital tool (digital wall or log), students could organize, structure, register, and monitor their individual and group work and learning experiences. That is, they could periodically report and share what difficulties they face to understand concepts or to work on a task, what questions they posed, what sources consult, etc. The information that appears in the digital wall is shared within the group and the teacher and students can provide feedback or propose new ideas or solutions (Santos-Trigo et al., 2022 ).

6.5 The integration of technologies and the emergence of conceptual frameworks

Institutions worldwide, in general, are integrating the use of different technologies in their educational practices, and they face the challenge to reconcile previous pandemic models and post confinement learning scenarios. “A pedagogical reason for using technology is to empower learners with extended or amplified abilities to acquire knowledge…technology can empower their cognitive abilities to reason in novice ways (Leung, 2011 , p. 327). Drijvers and Sinclair ( 2023 ) proposed a five-dimensional framework to delve into the rationale and purposes for the mathematics education community to integrate the use of digital technologies in mathematical teaching environments and students learning. The five interrelated categories address issues regarding how teachers and students’ use of digital technology contributes to reconceptualize and improve mathematics learning; to understand and explain how students’ mathematics learning develops; to design environments for mathematics learning; to foster and provide equitable access to mathematics learning; and to change mathematics curricula and teaching and assessment practices (Drijvers & Sinclair, 2023 ). Schoenfeld ( 2022 ) stated that “The challenge is to create robust learning environments that support every student in developing not only the knowledge and practices that underlie effective mathematical thinking, but that help them develop the sense of agency to engage in sense making” (p. 764). Højsted et al. ( 2022 ) argue about the importance of adjusting theoretical frameworks to explicitly integrate the use of digital technologies such as DGS and Computer Algebra Systems (CAS) in teaching practices. They referred to the Danish “Competencies and Mathematical Learning framework” (KOM) that gets articulated through tenets associated with the Theory of Instrumental Orchestration (TIO) and the notion of Justification Mediation (JM). In general terms, the idea is that learners get explicitly involved in a tool’ appropriation process that transforms the artifact into an instrument to understand concepts and to solve mathematical problems. That is, learners’ tool appropriation involves the development of cognitive schemata to rely on technology affordances to work on mathematical tasks. Koichu et al. ( 2022 ) pointed out that the incorporation of problem-solving approaches in instruction should be seen as a specific case of implementing innovation. To this end, they proposed a framework of problem-solving implementation chain that involves “a sequence of actions and interactions beginning with the development of a PS resource by researchers, which teachers then engage with in professional development (PD), and finally, teachers and students make use of in classrooms” (p. 4). In this case, problem-solving resources include the design of problematic situations (tasks) to engage students in mathematical discussions to make sense of problem statements or to ask them to pose a task.

7 Reflections and concluding remarks

Throughout different periods, the research and practice mathematical problem-solving agenda has contributed significantly to understand not only essentials in mathematical practices; but also, the development of conceptual frameworks to explain and document subjects’ cognitive, social, and affective behaviours to understand mathematical concepts and to develop problem-solving competencies. Leikin and Guberman ( 2023 ) pointed out that “…problem-solving is an effective didactical tool that allows pupils to mobilize their existing knowledge, construct new mathematical connections between known concepts and properties, and construct new knowledge in the process of overcoming challenges embedded in the problems” (p. 325). The study of people cognitive functioning to develop multidisciplinary knowledge and to solve problems involves documenting ways in which individuals make decisions regarding ways to organize their subject or disciplinary learning (how to interact with teachers or experts and peers; what material to consult, what tools to use, how to monitor their own learning, etc.) and to engage in disciplinary practices to achieve their learning goals. Both strategic and tactic decisions shape teachers and students’ ways to work on mathematical tasks. Kahneman ( 2011 ) shed light on how human beings make decisions to deal with questions and problematic situations. He argues that individuals rely on two systems to make decisions and engage in thinking processes; system one (fast thinking) that involves automatic, emotional, instinctive reasoning and system two (slow thinking) that includes logical, deliberative, effortful, or conscious reasoning. In educational tasks, the idea is that teachers and students develop experiences based on the construction and activation of system two. Thus, how teachers/students decide what tools or digital developments to use to work on mathematical problems becomes a relevant issue to address in the mathematics education agenda. Recent and consistent developments and the availability of digital technologies open novel paths for teachers and students to represent, explore, and approach mathematical tasks and, provide different tools to extend students and teachers’ mathematical discussions beyond classroom settings. In this perspective, it becomes important to discuss what changes the systematic use of digital technologies bring to the mathematics contents and to the ways to frame mathematical instruction. For example, the use of a Dynamic Geometry System to model and explore calculus, geometry or algebra classic problems dynamically not only offer students an opportunity to connect foundational concepts such as rate of change or the perpendicular bisector concept to geometrically study variational phenomena or conic sections; but also, to engage them in problem-posing activities (Santos-Trigo et al., 2021 ). Thus, teachers need to experience themselves different ways to use digital technologies to work on mathematical tasks and to identify instructional paths for students to internalize the use of digital apps as an instrument to understand concepts and to pose and formulate mathematical problems. Specifically, curriculum proposal should be structured around the development of foundational concepts and problem-solving strategies to formulate and pursue complex problems such as those involving climate changes, wealth distribution, immigration, pollution, mobility, connectivity, etc. To formulate and approach these problems, students need to develop a multidisciplinary thinking and rely on different tools to represent, explore, and share and continuously report partial solutions. To this end, they are encouraged to work with peers and groups as a part of learning community that fosters and values collective problem solutions. Finding multiple paths to solve problems becomes important for students to develop creative and innovative problem solutions (Leikin & Guberman, 2023 ). In this perspective, learning environments should provide conditions for students to transform digital applications in problem-solving tools to work on problematic situations. Online students’ assignments become an important component to structure and organize students and teachers’ face-to-face interactions. Likewise, the use of technology can also provide a tool for students to register and monitor their work and learning experiences. A digital wall or a problem-solving digital notebook (Santos-Trigo et al., 2022 ) could be introduced for students to register and monitor their learning experiences. Here, Students are asked to record on a weekly basis their work, questions, comments, and ideas that include: Questions they pose to understand concepts and problem statements; online resources and platforms they consult to contextualize problems and review and extend their understanding of involved concepts; concepts and strategies used to solve problems through different approaches; the Identification of other problems that can be solved with the methods that were used to solve the problem; digital technologies and online resources used to work on and solve the problem; dynamic models used to solve the problem and strategies used to identify and explore mathematical relations (dragging objects, measuring object attributes, tracing loci, using sliders, etc.; the formulation of new related problems including possible extensions for the initial problem; discussion of solutions of some new problems; and short recorded video presentation of their work and problem solutions. That is, the digital wall becomes an space for learners to share their work and to contrast and reflect on their peers work including extending their problem-solving approaches based on their teachers feedback and peers’ ideas or solutions.

The term Dynamic Geometry System is used, instead of Dynamic Geometry Environment or Dynamic Geometry Software, to emphasize that the app or tool interface encompasses a system of affordances that combines the construction of dynamic models, the use of Computer Algebra Systems and the use spreadsheet programs.

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STUDY OF VARIOUS PROBLEMS FACED BY THE STUDENTS AND TEACHERS IN LEARNING & TEACHING MATHEMATICS AND THEIR SUGGESTIVE MEASURES

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