essence for teaching mathematics and mathematics problem solving skills

Problem Solving in Mathematics Education pp 1–39 Cite as

Problem Solving in Mathematics Education

• Peter Liljedahl 6 ,
• Manuel Santos-Trigo 7 ,
• Uldarico Malaspina 8 &
• Regina Bruder 9  
• Open Access
• First Online: 28 June 2016

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Part of the book series: ICME-13 Topical Surveys ((ICME13TS))

Problem solving in mathematics education has been a prominent research field that aims at understanding and relating the processes involved in solving problems to students’ development of mathematical knowledge and problem solving competencies. The accumulated knowledge and field developments include conceptual frameworks to characterize learners’ success in problem solving activities, cognitive, metacognitive, social and affective analysis, curriculum proposals, and ways to foster problem solving approaches. In the survey, four interrelated areas are reviewed: (i) the relevance of heuristics in problem solving approaches—why are they important and what research tells us about their use? (ii) the need to characterize and foster creative problem solving approaches—what type of heuristics helps learners think of and practice creative solutions? (iii) the importance for learners to formulate and pursue their own problems; and (iv) the role played by the use of both multiple purpose and ad hoc mathematical action types of technologies in problem solving activities—what ways of reasoning do learners construct when they rely on the use of digital technologies and how technology and technology approaches can be reconciled?

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Mathematical problem solving has long been seen as an important aspect of mathematics, the teaching of mathematics, and the learning of mathematics. It has infused mathematics curricula around the world with calls for the teaching of problem solving as well as the teaching of mathematics through problem solving. And as such, it has been of interest to mathematics education researchers for as long as our field has existed. More relevant, mathematical problem solving has played a part in every ICME conference, from 1969 until the forthcoming meeting in Hamburg, wherein mathematical problem solving will reside most centrally within the work of Topic Study 19: Problem Solving in Mathematics Education. This booklet is being published on the occasion of this Topic Study Group.

To this end, we have assembled four summaries looking at four distinct, yet inter-related, dimensions of mathematical problem solving. The first summary, by Regina Bruder, is a nuanced look at heuristics for problem solving. This notion of heuristics is carried into Peter Liljedahl’s summary, which looks specifically at a progression of heuristics leading towards more and more creative aspects of problem solving. This is followed by Luz Manuel Santos Trigo’s summary introducing us to problem solving in and with digital technologies. The last summary, by Uldarico Malaspina Jurado, documents the rise of problem posing within the field of mathematics education in general and the problem solving literature in particular.

Each of these summaries references in some critical and central fashion the works of George Pólya or Alan Schoenfeld. To the initiated researchers, this is no surprise. The seminal work of these researchers lie at the roots of mathematical problem solving. What is interesting, though, is the diverse ways in which each of the four aforementioned contributions draw on, and position, these works so as to fit into the larger scheme of their respective summaries. This speaks to not only the depth and breadth of these influential works, but also the diversity with which they can be interpreted and utilized in extending our thinking about problem solving.

Taken together, what follows is a topical survey of ideas representing the diversity of views and tensions inherent in a field of research that is both a means to an end and an end onto itself and is unanimously seen as central to the activities of mathematics.

1 Survey on the State-of-the-Art

1.1 role of heuristics for problem solving—regina bruder.

The origin of the word heuristic dates back to the time of Archimedes and is said to have come out of one of the famous stories told about this great mathematician and inventor. The King of Syracuse asked Archimedes to check whether his new wreath was really made of pure gold. Archimedes struggled with this task and it was not until he was at the bathhouse that he came up with the solution. As he entered the tub he noticed that he had displaced a certain amount of water. Brilliant as he was, he transferred this insight to the issue with the wreath and knew he had solved the problem. According to the legend, he jumped out of the tub and ran from the bathhouse naked screaming, “Eureka, eureka!”. Eureka and heuristic have the same root in the ancient Greek language and so it has been claimed that this is how the academic discipline of “heuristics” dealing with effective approaches to problem solving (so-called heurisms) was given its name. Pólya ( 1964 ) describes this discipline as follows:

Heuristics deals with solving tasks. Its specific goals include highlighting in general terms the reasons for selecting those moments in a problem the examination of which could help us find a solution. (p. 5)

This discipline has grown, in part, from examining the approaches to certain problems more in detail and comparing them with each other in order to abstract similarities in approach, or so-called heurisms. Pólya ( 1949 ), but also, inter alia, Engel ( 1998 ), König ( 1984 ) and Sewerin ( 1979 ) have formulated such heurisms for mathematical problem tasks. The problem tasks examined by the authors mentioned are predominantly found in the area of talent programmes, that is, they often go back to mathematics competitions.

In 1983 Zimmermann provided an overview of heuristic approaches and tools in American literature which also offered suggestions for mathematics classes. In the German-speaking countries, an approach has established itself, going back to Sewerin ( 1979 ) and König ( 1984 ), which divides school-relevant heuristic procedures into heuristic tools, strategies and principles, see also Bruder and Collet ( 2011 ).

Below is a review of the conceptual background of heuristics, followed by a description of the effect mechanisms of heurisms in problem-solving processes.

1.1.1 Research Review on the Promotion of Problem Solving

In the 20th century, there has been an advancement of research on mathematical problem solving and findings about possibilities to promote problem solving with varying priorities (c.f. Pehkonen 1991 ). Based on a model by Pólya ( 1949 ), in a first phase of research on problem solving, particularly in the 1960s and the 1970s, a series of studies on problem-solving processes placing emphasis on the importance of heuristic strategies (heurisms) in problem solving has been carried out. It was assumed that teaching and learning heuristic strategies, principles and tools would provide students with an orientation in problem situations and that this could thus improve students’ problem-solving abilities (c.f. for instance, Schoenfeld 1979 ). This approach, mostly researched within the scope of talent programmes for problem solving, was rather successful (c.f. for instance, Sewerin 1979 ). In the 1980s, requests for promotional opportunities in everyday teaching were given more and more consideration: “ problem solving must be the focus of school mathematics in the 1980s ” (NCTM 1980 ). For the teaching and learning of problem solving in regular mathematics classes, the current view according to which cognitive, heuristic aspects were paramount, was expanded by certain student-specific aspects, such as attitudes, emotions and self-regulated behaviour (c.f. Kretschmer 1983 ; Schoenfeld 1985 , 1987 , 1992 ). Kilpatrick ( 1985 ) divided the promotional approaches described in the literature into five methods which can also be combined with each other.

Osmosis : action-oriented and implicit imparting of problem-solving techniques in a beneficial learning environment

Memorisation : formation of special techniques for particular types of problem and of the relevant questioning when problem solving

Imitation : acquisition of problem-solving abilities through imitation of an expert

Cooperation : cooperative learning of problem-solving abilities in small groups

Reflection : problem-solving abilities are acquired in an action-oriented manner and through reflection on approaches to problem solving.

Kilpatrick ( 1985 ) views as success when heuristic approaches are explained to students, clarified by means of examples and trained through the presentation of problems. The need of making students aware of heuristic approaches is by now largely accepted in didactic discussions. Differences in varying approaches to promoting problem-solving abilities rather refer to deciding which problem-solving strategies or heuristics are to imparted to students and in which way, and not whether these should be imparted at all or not.

1.1.2 Heurisms as an Expression of Mental Agility

The activity theory, particularly in its advancement by Lompscher ( 1975 , 1985 ), offers a well-suited and manageable model to describe learning activities and differences between learners with regard to processes and outcomes in problem solving (c.f. Perels et al. 2005 ). Mental activity starts with a goal and the motive of a person to perform such activity. Lompscher divides actual mental activity into content and process. Whilst the content in mathematical problem-solving consists of certain concepts, connections and procedures, the process describes the psychological processes that occur when solving a problem. This course of action is described in Lompscher by various qualities, such as systematic planning, independence, accuracy, activity and agility. Along with differences in motivation and the availability of expertise, it appears that intuitive problem solvers possess a particularly high mental agility, at least with regard to certain contents areas.

According to Lompscher, “flexibility of thought” expresses itself

… by the capacity to change more or less easily from one aspect of viewing to another one or to embed one circumstance or component into different correlations, to understand the relativity of circumstances and statements. It allows to reverse relations, to more or less easily or quickly attune to new conditions of mental activity or to simultaneously mind several objects or aspects of a given activity (Lompscher 1975 , p. 36).

These typical manifestations of mental agility can be focused on in problem solving by mathematical means and can be related to the heurisms known from the analyses of approaches by Pólya et al. (c.f. also Bruder 2000 ):

Reduction : Successful problem solvers will intuitively reduce a problem to its essentials in a sensible manner. To achieve such abstraction, they often use visualisation and structuring aids, such as informative figures, tables, solution graphs or even terms. These heuristic tools are also very well suited to document in retrospect the approach adopted by the intuitive problem solvers in a way that is comprehensible for all.

Reversibility : Successful problem solvers are able to reverse trains of thought or reproduce these in reverse. They will do this in appropriate situations automatically, for instance, when looking for a key they have mislaid. A corresponding general heuristic strategy is working in reverse.

Minding of aspects : Successful problem solvers will mind several aspects of a given problem at the same time or easily recognise any dependence on things and vary them in a targeted manner. Sometimes, this is also a matter of removing barriers in favour of an idea that appears to be sustainable, that is, by simply “hanging on” to a certain train of thought even against resistance. Corresponding heurisms are, for instance, the principle of invariance, the principle of symmetry (Engel 1998 ), the breaking down or complementing of geometric figures to calculate surface areas, or certain terms used in binomial formulas.

Change of aspects : Successful problem solvers will possibly change their assumptions, criteria or aspects minded in order to find a solution. Various aspects of a given problem will be considered intuitively or the problem be viewed from a different perspective, which will prevent “getting stuck” and allow for new insights and approaches. For instance, many elementary geometric propositions can also be proved in an elegant vectorial manner.

Transferring : Successful problem solvers will be able more easily than others to transfer a well-known procedure to another, sometimes even very different context. They recognise more easily the “framework” or pattern of a given task. Here, this is about own constructions of analogies and continual tracing back from the unknown to the known.

Intuitive, that is, untrained good problem solvers, are, however, often unable to access these flexibility qualities consciously. This is why they are also often unable to explain how they actually solved a given problem.

To be able to solve problems successfully, a certain mental agility is thus required. If this is less well pronounced in a certain area, learning how to solve problems means compensating by acquiring heurisms. In this case, insufficient mental agility is partly “offset” through the application of knowledge acquired by means of heurisms. Mathematical problem-solving competences are thus acquired through the promotion of manifestations of mental agility (reduction, reversibility, minding of aspects and change of aspects). This can be achieved by designing sub-actions of problem solving in connection with a (temporarily) conscious application of suitable heurisms. Empirical evidence for the success of the active principle of heurisms has been provided by Collet ( 2009 ).

Against such background, learning how to solve problems can be established as a long-term teaching and learning process which basically encompasses four phases (Bruder and Collet 2011 ):

Intuitive familiarisation with heuristic methods and techniques.

Making aware of special heurisms by means of prominent examples (explicit strategy acquisition).

Short conscious practice phase to use the newly acquired heurisms with differentiated task difficulties.

Expanding the context of the strategies applied.

In the first phase, students are familiarised with heurisms intuitively by means of targeted aid impulses and questions (what helped us solve this problem?) which in the following phase are substantiated on the basis of model tasks, are given names and are thus made aware of their existence. The third phase serves the purpose of a certain familiarisation with the new heurisms and the experience of competence through individualised practising at different requirement levels, including in the form of homework over longer periods. A fourth and delayed fourth phase aims at more flexibility through the transfer to other contents and contexts and the increasingly intuitive use of the newly acquired heurisms, so that students can enrich their own problem-solving models in a gradual manner. The second and third phases build upon each other in close chronological order, whilst the first phase should be used in class at all times.

All heurisms can basically be described in an action-oriented manner by means of asking the right questions. The way of asking questions can thus also establish a certain kind of personal relation. Even if the teacher presents and suggests the line of basic questions with a prototypical wording each time, students should always be given the opportunity to find “their” wording for the respective heurism and take a note of it for themselves. A possible key question for the use of a heuristic tool would be: How to illustrate and structure the problem or how to present it in a different way?

Unfortunately, for many students, applying heuristic approaches to problem solving will not ensue automatically but will require appropriate early and long-term promoting. The results of current studies, where promotion approaches to problem solving are connected with self-regulation and metacognitive aspects, demonstrate certain positive effects of such combination on students. This field of research includes, for instance, studies by Lester et al. ( 1989 ), Verschaffel et al. ( 1999 ), the studies on teaching method IMPROVE by Mevarech and Kramarski ( 1997 , 2003 ) and also the evaluation of a teaching concept on learning how to solve problems by the gradual conscious acquisition of heurisms by Collet and Bruder ( 2008 ).

1.2 Creative Problem Solving—Peter Liljedahl

There is a tension between the aforementioned story of Archimedes and the heuristics presented in the previous section. Archimedes, when submersing himself in the tub and suddenly seeing the solution to his problem, wasn’t relying on osmosis, memorisation, imitation, cooperation, or reflection (Kilpatrick 1985 ). He wasn’t drawing on reduction, reversibility, minding of aspects, change of aspect, or transfer (Bruder 2000 ). Archimedes was stuck and it was only, in fact, through insight and sudden illumination that he managed to solve his problem. In short, Archimedes was faced with a problem that the aforementioned heuristics, and their kind, would not help him to solve.

According to some, such a scenario is the definition of a problem. For example, Resnick and Glaser ( 1976 ) define a problem as being something that you do not have the experience to solve. Mathematicians, in general, agree with this (Liljedahl 2008 ).

Any problem in which you can see how to attack it by deliberate effort, is a routine problem, and cannot be an important discover. You must try and fail by deliberate efforts, and then rely on a sudden inspiration or intuition or if you prefer to call it luck. (Dan Kleitman, participant cited in Liljedahl 2008 , p. 19).

Problems, then, are tasks that cannot be solved by direct effort and will require some creative insight to solve (Liljedahl 2008 ; Mason et al. 1982 ; Pólya 1965 ).

1.2.1 A History of Creativity in Mathematics Education

In 1902, the first half of what eventually came to be a 30 question survey was published in the pages of L’Enseignement Mathématique , the journal of the French Mathematical Society. The authors, Édouard Claparède and Théodore Flournoy, were two Swiss psychologists who were deeply interested in the topics of mathematical discovery, creativity and invention. Their hope was that a widespread appeal to mathematicians at large would incite enough responses for them to begin to formulate some theories about this topic. The first half of the survey centered on the reasons for becoming a mathematician (family history, educational influences, social environment, etc.), attitudes about everyday life, and hobbies. This was eventually followed, in 1904, by the publication of the second half of the survey pertaining, in particular, to mental images during periods of creative work. The responses were sorted according to nationality and published in 1908.

During this same period Henri Poincaré (1854–1912), one of the most noteworthy mathematicians of the time, had already laid much of the groundwork for his own pursuit of this same topic and in 1908 gave a presentation to the French Psychological Society in Paris entitled L’Invention mathématique —often mistranslated to Mathematical Creativity Footnote 1 (c.f. Poincaré 1952 ). At the time of the presentation Poincaré stated that he was aware of Claparède and Flournoy’s work, as well as their results, but expressed that they would only confirm his own findings. Poincaré’s presentation, as well as the essay it spawned, stands to this day as one of the most insightful, and thorough treatments of the topic of mathematical discovery, creativity, and invention.

Just at this time, I left Caen, where I was living, to go on a geological excursion under the auspices of the School of Mines. The incident of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuschian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had the time, as, upon taking my seat in the omnibus, I went on with the conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the results at my leisure. (Poincaré 1952 , p. 53)

So powerful was his presentation, and so deep were his insights into his acts of invention and discovery that it could be said that he not so much described the characteristics of mathematical creativity, as defined them. From that point forth mathematical creativity, or even creativity in general, has not been discussed seriously without mention of Poincaré’s name.

Inspired by this presentation, Jacques Hadamard (1865–1963), a contemporary and a friend of Poincaré’s, began his own empirical investigation into this fascinating phenomenon. Hadamard had been critical of Claparède and Flournoy’s work in that they had not adequately treated the topic on two fronts. As exhaustive as the survey appeared to be, Hadamard felt that it failed to ask some key questions—the most important of which was with regard to the reason for failures in the creation of mathematics. This seemingly innocuous oversight, however, led directly to his second and “most important criticism” (Hadamard 1945 ). He felt that only “first-rate men would dare to speak of” (p. 10) such failures. So, inspired by his good friend Poincaré’s treatment of the subject Hadamard retooled the survey and gave it to friends of his for consideration—mathematicians such as Henri Poincaré and Albert Einstein, whose prominence were beyond reproach. Ironically, the new survey did not contain any questions that explicitly dealt with failure. In 1943 Hadamard gave a series of lectures on mathematical invention at the École Libre des Hautes Études in New York City. These talks were subsequently published as The Psychology of Mathematical Invention in the Mathematical Field (Hadameard 1945 ).

Hadamard’s classic work treats the subject of invention at the crossroads of mathematics and psychology. It provides not only an entertaining look at the eccentric nature of mathematicians and their rituals, but also outlines the beliefs of mid twentieth-century mathematicians about the means by which they arrive at new mathematics. It is an extensive exploration and extended argument for the existence of unconscious mental processes. In essence, Hadamard took the ideas that Poincaré had posed and, borrowing a conceptual framework for the characterization of the creative process from the Gestaltists of the time (Wallas 1926 ), turned them into a stage theory. This theory still stands as the most viable and reasonable description of the process of mathematical creativity.

1.2.2 Defining Mathematical Creativity

The phenomena of mathematical creativity, although marked by sudden illumination, actually consist of four separate stages stretched out over time, of which illumination is but one stage. These stages are initiation, incubation, illumination, and verification (Hadamard 1945 ). The first of these stages, the initiation phase, consists of deliberate and conscious work. This would constitute a person’s voluntary, and seemingly fruitless, engagement with a problem and be characterized by an attempt to solve the problem by trolling through a repertoire of past experiences. This is an important part of the inventive process because it creates the tension of unresolved effort that sets up the conditions necessary for the ensuing emotional release at the moment of illumination (Hadamard 1945 ; Poincaré 1952 ).

Following the initiation stage the solver, unable to come up with a solution stops working on the problem at a conscious level and begins to work on it at an unconscious level (Hadamard 1945 ; Poincaré 1952 ). This is referred to as the incubation stage of the inventive process and can last anywhere from several minutes to several years. After the period of incubation a rapid coming to mind of a solution, referred to as illumination , may occur. This is accompanied by a feeling of certainty and positive emotions (Poincaré 1952 ). Although the processes of incubation and illumination are shrouded behind the veil of the unconscious there are a number of things that can be deduced about them. First and foremost is the fact that unconscious work does, indeed, occur. Poincaré ( 1952 ), as well as Hadamard ( 1945 ), use the very real experience of illumination, a phenomenon that cannot be denied, as evidence of unconscious work, the fruits of which appear in the flash of illumination. No other theory seems viable in explaining the sudden appearance of solution during a walk, a shower, a conversation, upon waking, or at the instance of turning the conscious mind back to the problem after a period of rest (Poincaré 1952 ). Also deducible is that unconscious work is inextricably linked to the conscious and intentional effort that precedes it.

There is another remark to be made about the conditions of this unconscious work: it is possible, and of a certainty it is only fruitful, if it is on the one hand preceded and on the other hand followed by a period of conscious work. These sudden inspirations never happen except after some days of voluntary effort which has appeared absolutely fruitless and whence nothing good seems to have come … (Poincaré 1952 , p. 56)

Hence, the fruitless efforts of the initiation phase are only seemingly so. They not only set up the aforementioned tension responsible for the emotional release at the time of illumination, but also create the conditions necessary for the process to enter into the incubation phase.

Illumination is the manifestation of a bridging that occurs between the unconscious mind and the conscious mind (Poincaré 1952 ), a coming to (conscious) mind of an idea or solution. What brings the idea forward to consciousness is unclear, however. There are theories of the aesthetic qualities of the idea, effective surprise/shock of recognition, fluency of processing, or breaking functional fixedness. For reasons of brevity I will only expand on the first of these.

Poincaré proposed that ideas that were stimulated during initiation remained stimulated during incubation. However, freed from the constraints of conscious thought and deliberate calculation, these ideas would begin to come together in rapid and random unions so that “their mutual impacts may produce new combinations” (Poincaré 1952 ). These new combinations, or ideas, would then be evaluated for viability using an aesthetic sieve, which allows through to the conscious mind only the “right combinations” (Poincaré 1952 ). It is important to note, however, that good or aesthetic does not necessarily mean correct. Correctness is evaluated during the verification stage.

The purpose of verification is not only to check for correctness. It is also a method by which the solver re-engages with the problem at the level of details. That is, during the unconscious work the problem is engaged with at the level of ideas and concepts. During verification the solver can examine these ideas in closer details. Poincaré succinctly describes both of these purposes.

As for the calculations, themselves, they must be made in the second period of conscious work, that which follows the inspiration, that in which one verifies the results of this inspiration and deduces their consequences. (Poincaré 1952 , p. 62)

Aside from presenting this aforementioned theory on invention, Hadamard also engaged in a far-reaching discussion on a number of interesting, and sometimes quirky, aspects of invention and discovery that he had culled from the results of his empirical study, as well as from pertinent literature. This discussion was nicely summarized by Newman ( 2000 ) in his commentary on the elusiveness of invention.

The celebrated phrenologist Gall said mathematical ability showed itself in a bump on the head, the location of which he specified. The psychologist Souriau, we are told, maintained that invention occurs by “pure chance”, a valuable theory. It is often suggested that creative ideas are conjured up in “mathematical dreams”, but this attractive hypothesis has not been verified. Hadamard reports that mathematicians were asked whether “noises” or “meteorological circumstances” helped or hindered research [..] Claude Bernard, the great physiologist, said that in order to invent “one must think aside”. Hadamard says this is a profound insight; he also considers whether scientific invention may perhaps be improved by standing or sitting or by taking two baths in a row. Helmholtz and Poincaré worked sitting at a table; Hadamard’s practice is to pace the room (“Legs are the wheels of thought”, said Emile Angier); the chemist J. Teeple was the two-bath man. (p. 2039)

1.2.3 Discourses on Creativity

Creativity is a term that can be used both loosely and precisely. That is, while there exists a common usage of the term there also exists a tradition of academic discourse on the subject. A common usage of creative refers to a process or a person whose products are original, novel, unusual, or even abnormal (Csíkszentmihályi 1996 ). In such a usage, creativity is assessed on the basis of the external and observable products of the process, the process by which the product comes to be, or on the character traits of the person doing the ‘creating’. Each of these usages—product, process, person—is the roots of the discourses (Liljedahl and Allan 2014 ) that I summarize here, the first of which concerns products.

Consider a mother who states that her daughter is creative because she drew an original picture. The basis of such a statement can lie either in the fact that the picture is unlike any the mother has ever seen or unlike any her daughter has ever drawn before. This mother is assessing creativity on the basis of what her daughter has produced. However, the standards that form the basis of her assessment are neither consistent nor stringent. There does not exist a universal agreement as to what she is comparing the picture to (pictures by other children or other pictures by the same child). Likewise, there is no standard by which the actual quality of the picture is measured. The academic discourse that concerns assessment of products, on the other hand, is both consistent and stringent (Csíkszentmihályi 1996 ). This discourse concerns itself more with a fifth, and as yet unmentioned, stage of the creative process; elaboration . Elaboration is where inspiration becomes perspiration (Csíkszentmihályi 1996 ). It is the act of turning a good idea into a finished product, and the finished product is ultimately what determines the creativity of the process that spawned it—that is, it cannot be a creative process if nothing is created. In particular, this discourse demands that the product be assessed against other products within its field, by the members of that field, to determine if it is original AND useful (Csíkszentmihályi 1996 ; Bailin 1994 ). If it is, then the product is deemed to be creative. Note that such a use of assessment of end product pays very little attention to the actual process that brings this product forth.

The second discourse concerns the creative process. The literature pertaining to this can be separated into two categories—a prescriptive discussion of the creativity process and a descriptive discussion of the creativity process. Although both of these discussions have their roots in the four stages that Wallas ( 1926 ) proposed makes up the creative process, they make use of these stages in very different ways. The prescriptive discussion of the creative process is primarily focused on the first of the four stages, initiation , and is best summarized as a cause - and - effect discussion of creativity, where the thinking processes during the initiation stage are the cause and the creative outcome are the effects (Ghiselin 1952 ). Some of the literature claims that the seeds of creativity lie in being able to think about a problem or situation analogically. Other literature claims that utilizing specific thinking tools such as imagination, empathy, and embodiment will lead to creative products. In all of these cases, the underlying theory is that the eventual presentation of a creative idea will be precipitated by the conscious and deliberate efforts during the initiation stage. On the other hand, the literature pertaining to a descriptive discussion of the creative process is inclusive of all four stages (Kneller 1965 ; Koestler 1964 ). For example, Csíkszentmihályi ( 1996 ), in his work on flow attends to each of the stages, with much attention paid to the fluid area between conscious and unconscious work, or initiation and incubation. His claim is that the creative process is intimately connected to the enjoyment that exists during times of sincere and consuming engagement with a situation, the conditions of which he describes in great detail.

The third, and final, discourse on creativity pertains to the person. This discourse is space dominated by two distinct characteristics, habit and genius. Habit has to do with the personal habits as well as the habits of mind of people that have been deemed to be creative. However, creative people are most easily identified through their reputation for genius. Consequently, this discourse is often dominated by the analyses of the habits of geniuses as is seen in the work of Ghiselin ( 1952 ), Koestler ( 1964 ), and Kneller ( 1965 ) who draw on historical personalities such as Albert Einstein, Henri Poincaré, Vincent Van Gogh, D.H. Lawrence, Samuel Taylor Coleridge, Igor Stravinsky, and Wolfgang Amadeus Mozart to name a few. The result of this sort of treatment is that creative acts are viewed as rare mental feats, which are produced by extraordinary individuals who use extraordinary thought processes.

These different discourses on creativity can be summed up in a tension between absolutist and relativist perspectives on creativity (Liljedahl and Sriraman 2006 ). An absolutist perspective assumes that creative processes are the domain of genius and are present only as precursors to the creation of remarkably useful and universally novel products. The relativist perspective, on the other hand, allows for every individual to have moments of creativity that may, or may not, result in the creation of a product that may, or may not, be either useful or novel.

Between the work of a student who tries to solve a problem in geometry or algebra and a work of invention, one can say there is only a difference of degree. (Hadamard 1945 , p. 104).

Regardless of discourse, however, creativity is not “part of the theories of logical forms” (Dewey 1938 ). That is, creativity is not representative of the lock-step logic and deductive reasoning that mathematical problem solving is often presumed to embody (Bibby 2002 ; Burton 1999 ). Couple this with the aforementioned demanding constraints as to what constitutes a problem, where then does that leave problem solving heuristics? More specifically, are there creative problem solving heuristics that will allow us to resolve problems that require illumination to solve? The short answer to this question is yes—there does exist such problem solving heuristics. To understand these, however, we must first understand the routine problem solving heuristics they are built upon. In what follows, I walk through the work of key authors and researchers whose work offers us insights into progressively more creative problem solving heuristics for solving true problems.

1.2.4 Problem Solving by Design

In a general sense, design is defined as the algorithmic and deductive approach to solving a problem (Rusbult 2000 ). This process begins with a clearly defined goal or objective after which there is a great reliance on relevant past experience, referred to as repertoire (Bruner 1964 ; Schön 1987 ), to produce possible options that will lead towards a solution of the problem (Poincaré 1952 ). These options are then examined through a process of conscious evaluations (Dewey 1933 ) to determine their suitability for advancing the problem towards the final goal. In very simple terms, problem solving by design is the process of deducing the solution from that which is already known.

Mayer ( 1982 ), Schoenfeld ( 1982 ), and Silver ( 1982 ) state that prior knowledge is a key element in the problem solving process. Prior knowledge influences the problem solver’s understanding of the problem as well as the choice of strategies that will be called upon in trying to solve the problem. In fact, prior knowledge and prior experiences is all that a solver has to draw on when first attacking a problem. As a result, all problem solving heuristics incorporate this resource of past experiences and prior knowledge into their initial attack on a problem. Some heuristics refine these ideas, and some heuristics extend them (c.f. Kilpatrick 1985 ; Bruder 2000 ). Of the heuristics that refine, none is more influential than the one created by George Pólya (1887–1985).

1.2.5 George Pólya: How to Solve It

In his book How to Solve It (1949) Pólya lays out a problem solving heuristic that relies heavily on a repertoire of past experience. He summarizes the four-step process of his heuristic as follows:

Understanding the Problem

First. You have to understand the problem.

What is the unknown? What are the data? What is the condition?

Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?

Draw a figure. Introduce suitable notation.

Separate the various parts of the condition. Can you write them down?

Devising a Plan

Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

Have you seen it before? Or have you seen the same problem in a slightly different form?

Do you know a related problem? Do you know a theorem that could be useful?

Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?

Could you restate the problem? Could you restate it still differently? Go back to definitions.

If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?

Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

Carrying Out the Plan

Third. Carry out your plan.

Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

Looking Back

Fourth. Examine the solution obtained.

Can you check the result? Can you check the argument?

Can you derive the solution differently? Can you see it at a glance?

Can you use the result, or the method, for some other problem?

The emphasis on auxiliary problems, related problems, and analogous problems that are, in themselves, also familiar problems is an explicit manifestation of relying on a repertoire of past experience. This use of familiar problems also requires an ability to deduce from these related problems a recognizable and relevant attribute that will transfer to the problem at hand. The mechanism that allows for this transfer of knowledge between analogous problems is known as analogical reasoning (English 1997 , 1998 ; Novick 1988 , 1990 , 1995 ; Novick and Holyoak 1991 ) and has been shown to be an effective, but not always accessible, thinking strategy.

Step four in Pólya’s heuristic, looking back, is also a manifestation of utilizing prior knowledge to solve problems, albeit an implicit one. Looking back makes connections “in memory to previously acquired knowledge [..] and further establishes knowledge in long-term memory that may be elaborated in later problem-solving encounters” (Silver 1982 , p. 20). That is, looking back is a forward-looking investment into future problem solving encounters, it sets up connections that may later be needed.

Pólya’s heuristic is a refinement on the principles of problem solving by design. It not only makes explicit the focus on past experiences and prior knowledge, but also presents these ideas in a very succinct, digestible, and teachable manner. This heuristic has become a popular, if not the most popular, mechanism by which problem solving is taught and learned.

1.2.6 Alan Schoenfeld: Mathematical Problem Solving

The work of Alan Schoenfeld is also a refinement on the principles of problem solving by design. However, unlike Pólya ( 1949 ) who refined these principles at a theoretical level, Schoenfeld has refined them at a practical and empirical level. In addition to studying taught problem solving strategies he has also managed to identify and classify a variety of strategies, mostly ineffectual, that students invoke naturally (Schoenfeld 1985 , 1992 ). In so doing, he has created a better understanding of how students solve problems, as well as a better understanding of how problems should be solved and how problem solving should be taught.

For Schoenfeld, the problem solving process is ultimately a dialogue between the problem solver’s prior knowledge, his attempts, and his thoughts along the way (Schoenfeld 1982 ). As such, the solution path of a problem is an emerging and contextually dependent process. This is a departure from the predefined and contextually independent processes of Pólya’s ( 1949 ) heuristics. This can be seen in Schoenfeld’s ( 1982 ) description of a good problem solver.

To examine what accounts for expertise in problem solving, you would have to give the expert a problem for which he does not have access to a solution schema. His behavior in such circumstances is radically different from what you would see when he works on routine or familiar “non-routine” problems. On the surface his performance is no longer proficient; it may even seem clumsy. Without access to a solution schema, he has no clear indication of how to start. He may not fully understand the problem, and may simply “explore it for a while until he feels comfortable with it. He will probably try to “match” it to familiar problems, in the hope it can be transformed into a (nearly) schema-driven solution. He will bring up a variety of plausible things: related facts, related problems, tentative approaches, etc. All of these will have to be juggled and balanced. He may make an attempt solving it in a particular way, and then back off. He may try two or three things for a couple of minutes and then decide which to pursue. In the midst of pursuing one direction he may go back and say “that’s harder than it should be” and try something else. Or, after the comment, he may continue in the same direction. With luck, after some aborted attempts, he will solve the problem. (p. 32-33)

Aside from demonstrating the emergent nature of the problem solving process, this passage also brings forth two consequences of Schoenfeld’s work. The first of these is the existence of problems for which the solver does not have “access to a solution schema”. Unlike Pólya ( 1949 ), who’s heuristic is a ‘one size fits all (problems)’ heuristic, Schoenfeld acknowledges that problem solving heuristics are, in fact, personal entities that are dependent on the solver’s prior knowledge as well as their understanding of the problem at hand. Hence, the problems that a person can solve through his or her personal heuristic are finite and limited.

The second consequence that emerges from the above passage is that if a person lacks the solution schema to solve a given problem s/he may still solve the problem with the help of luck . This is an acknowledgement, if only indirectly so, of the difference between problem solving in an intentional and mechanical fashion verses problem solving in a more creative fashion, which is neither intentional nor mechanical (Pehkonen 1997 ).

1.2.7 David Perkins: Breakthrough Thinking

As mentioned, many consider a problem that can be solved by intentional and mechanical means to not be worthy of the title ‘problem’. As such, a repertoire of past experiences sufficient for dealing with such a ‘problem’ would disqualify it from the ranks of ‘problems’ and relegate it to that of ‘exercises’. For a problem to be classified as a ‘problem’, then, it must be ‘problematic’. Although such an argument is circular it is also effective in expressing the ontology of mathematical ‘problems’.

Perkins ( 2000 ) also requires problems to be problematic. His book Archimedes’ Bathtub: The Art and Logic of Breakthrough Thinking (2000) deals with situations in which the solver has gotten stuck and no amount of intentional or mechanical adherence to the principles of past experience and prior knowledge is going to get them unstuck. That is, he deals with problems that, by definition, cannot be solved through a process of design [or through the heuristics proposed by Pólya ( 1949 ) and Schoenfeld ( 1985 )]. Instead, the solver must rely on the extra-logical process of what Perkins ( 2000 ) calls breakthrough thinking .

Perkins ( 2000 ) begins by distinguishing between reasonable and unreasonable problems. Although both are solvable, only reasonable problems are solvable through reasoning. Unreasonable problems require a breakthrough in order to solve them. The problem, however, is itself inert. It is neither reasonable nor unreasonable. That quality is brought to the problem by the solver. That is, if a student cannot solve a problem by direct effort then that problem is deemed to be unreasonable for that student. Perkins ( 2000 ) also acknowledges that what is an unreasonable problem for one person is a perfectly reasonable problem for another person; reasonableness is dependent on the person.

This is not to say that, once found, the solution cannot be seen as accessible through reason. During the actual process of solving, however, direct and deductive reasoning does not work. Perkins ( 2000 ) uses several classic examples to demonstrate this, the most famous being the problem of connecting nine dots in a 3 × 3 array with four straight lines without removing pencil from paper, the solution to which is presented in Fig.  1 .

Nine dots—four lines problem and solution

To solve this problem, Perkins ( 2000 ) claims that the solver must recognize that the constraint of staying within the square created by the 3 × 3 array is a self-imposed constraint. He further claims that until this is recognized no amount of reasoning is going to solve the problem. That is, at this point in the problem solving process the problem is unreasonable. However, once this self-imposed constraint is recognized the problem, and the solution, are perfectly reasonable. Thus, the solution of an, initially, unreasonable problem is reasonable.

The problem solving heuristic that Perkins ( 2000 ) has constructed to deal with solvable, but unreasonable, problems revolves around the idea of breakthrough thinking and what he calls breakthrough problems . A breakthrough problem is a solvable problem in which the solver has gotten stuck and will require an AHA! to get unstuck and solve the problem. Perkins ( 2000 ) poses that there are only four types of solvable unreasonable problems, which he has named wilderness of possibilities , the clueless plateau , narrow canyon of exploration , and oasis of false promise . The names for the first three of these types of problems are related to the Klondike gold rush in Alaska, a time and place in which gold was found more by luck than by direct and systematic searching.

The wilderness of possibilities is a term given to a problem that has many tempting directions but few actual solutions. This is akin to a prospector searching for gold in the Klondike. There is a great wilderness in which to search, but very little gold to be found. The clueless plateau is given to problems that present the solver with few, if any, clues as to how to solve it. The narrow canyon of exploration is used to describe a problem that has become constrained in such a way that no solution now exists. The nine-dot problem presented above is such a problem. The imposed constraint that the lines must lie within the square created by the array makes a solution impossible. This is identical to the metaphor of a prospector searching for gold within a canyon where no gold exists. The final type of problem gets its name from the desert. An oasis of false promise is a problem that allows the solver to quickly get a solution that is close to the desired outcome; thereby tempting them to remain fixed on the strategy that they used to get this almost-answer. The problem is, that like the canyon, the solution does not exist at the oasis; the solution strategy that produced an almost-answer is incapable of producing a complete answer. Likewise, a desert oasis is a false promise in that it is only a reprieve from the desolation of the dessert and not a final destination.

Believing that there are only four ways to get stuck, Perkins ( 2000 ) has designed a problem solving heuristic that will “up the chances” of getting unstuck. This heuristic is based on what he refers to as “the logic of lucking out” (p. 44) and is built on the idea of introspection. By first recognizing that they are stuck, and then recognizing that the reason they are stuck can only be attributed to one of four reasons, the solver can access four strategies for getting unstuck, one each for the type of problem they are dealing with. If the reason they are stuck is because they are faced with a wilderness of possibilities they are to begin roaming far, wide, and systematically in the hope of reducing the possible solution space to one that is more manageable. If they find themselves on a clueless plateau they are to begin looking for clues, often in the wording of the problem. When stuck in a narrow canyon of possibilities they need to re-examine the problem and see if they have imposed any constraints. Finally, when in an oasis of false promise they need to re-attack the problem in such a way that they stay away from the oasis.

Of course, there are nuances and details associated with each of these types of problems and the strategies for dealing with them. However, nowhere within these details is there mention of the main difficulty inherent in introspection; that it is much easier for the solver to get stuck than it is for them to recognize that they are stuck. Once recognized, however, the details of Perkins’ ( 2000 ) heuristic offer the solver some ways for recognizing why they are stuck.

1.2.8 John Mason, Leone Burton, and Kaye Stacey: Thinking Mathematically

The work of Mason et al. in their book Thinking Mathematically ( 1982 ) also recognizes the fact that for each individual there exists problems that will not yield to their intentional and mechanical attack. The heuristic that they present for dealing with this has two main processes with a number of smaller phases, rubrics, and states. The main processes are what they refer to as specializing and generalizing. Specializing is the process of getting to know the problem and how it behaves through the examination of special instances of the problem. This process is synonymous with problem solving by design and involves the repeated oscillation between the entry and attack phases of Mason et al. ( 1982 ) heuristic. The entry phase is comprised of ‘getting started’ and ‘getting involved’ with the problem by using what is immediately known about it. Attacking the problem involves conjecturing and testing a number of hypotheses in an attempt to gain greater understanding of the problem and to move towards a solution.

At some point within this process of oscillating between entry and attack the solver will get stuck, which Mason et al. ( 1982 ) refer to as “an honourable and positive state, from which much can be learned” (p. 55). The authors dedicate an entire chapter to this state in which they acknowledge that getting stuck occurs long before an awareness of being stuck develops. They proposes that the first step to dealing with being stuck is the simple act of writing STUCK!

The act of expressing my feelings helps to distance me from my state of being stuck. It frees me from incapacitating emotions and reminds me of actions that I can take. (p. 56)

The next step is to reengage the problem by examining the details of what is known, what is wanted, what can be introduced into the problem, and what has been introduced into the problem (imposed assumptions). This process is engaged in until an AHA!, which advances the problem towards a solution, is encountered. If, at this point, the problem is not completely solved the oscillation is then resumed.

At some point in this process an attack on the problem will yield a solution and generalizing can begin. Generalizing is the process by which the specifics of a solution are examined and questions as to why it worked are investigated. This process is synonymous with the verification and elaboration stages of invention and creativity. Generalization may also include a phase of review that is similar to Pólya’s ( 1949 ) looking back.

1.2.9 Gestalt: The Psychology of Problem Solving

The Gestalt psychology of learning believes that all learning is based on insights (Koestler 1964 ). This psychology emerged as a response to behaviourism, which claimed that all learning was a response to external stimuli. Gestalt psychologists, on the other hand, believed that there was a cognitive process involved in learning as well. With regards to problem solving, the Gestalt school stands firm on the belief that problem solving, like learning, is a product of insight and as such, cannot be taught. In fact, the theory is that not only can problem solving not be taught, but also that attempting to adhere to any sort of heuristic will impede the working out of a correct solution (Krutestkii 1976 ). Thus, there exists no Gestalt problem solving heuristic. Instead, the practice is to focus on the problem and the solution rather than on the process of coming up with a solution. Problems are solved by turning them over and over in the mind until an insight, a viable avenue of attack, presents itself. At the same time, however, there is a great reliance on prior knowledge and past experiences. The Gestalt method of problem solving, then, is at the same time very different and very similar to the process of design.

Gestalt psychology has not fared well during the evolution of cognitive psychology. Although it honours the work of the unconscious mind it does so at the expense of practicality. If learning is, indeed, entirely based on insight then there is little point in continuing to study learning. “When one begins by assuming that the most important cognitive phenomena are inaccessible, there really is not much left to talk about” (Schoenfeld 1985 , p. 273). However, of interest here is the Gestalt psychologists’ claim that focus on problem solving methods creates functional fixedness (Ashcraft 1989 ). Mason et al. ( 1982 ), as well as Perkins ( 2000 ) deal with this in their work on getting unstuck.

1.2.10 Final Comments

Mathematics has often been characterized as the most precise of all sciences. Lost in such a misconception is the fact that mathematics often has its roots in the fires of creativity, being born of the extra-logical processes of illumination and intuition. Problem solving heuristics that are based solely on the processes of logical and deductive reasoning distort the true nature of problem solving. Certainly, there are problems in which logical deductive reasoning is sufficient for finding a solution. But these are not true problems. True problems need the extra-logical processes of creativity, insight, and illumination, in order to produce solutions.

Fortunately, as elusive as such processes are, there does exist problem solving heuristics that incorporate them into their strategies. Heuristics such as those by Perkins ( 2000 ) and Mason et al. ( 1982 ) have found a way of combining the intentional and mechanical processes of problem solving by design with the extra-logical processes of creativity, illumination, and the AHA!. Furthermore, they have managed to do so without having to fully comprehend the inner workings of this mysterious process.

1.3 Digital Technologies and Mathematical Problem Solving—Luz Manuel Santos-Trigo

Mathematical problem solving is a field of research that focuses on analysing the extent to which problem solving activities play a crucial role in learners’ understanding and use of mathematical knowledge. Mathematical problems are central in mathematical practice to develop the discipline and to foster students learning (Pólya 1945 ; Halmos 1994 ). Mason and Johnston-Wilder ( 2006 ) pointed out that “The purpose of a task is to initiate mathematically fruitful activity that leads to a transformation in what learners are sensitized to notice and competent to carry out” (p. 25). Tasks are essential for learners to elicit their ideas and to engage them in mathematical thinking. In a problem solving approach, what matters is the learners’ goals and ways to interact with the tasks. That is, even routine tasks can be a departure point for learners to extend initial conditions and transform them into some challenging activities.

Thus, analysing and characterizing ways in which mathematical problems are formulated (Singer et al. 2015 ) and the process involved in pursuing and solving those problems generate important information to frame and structure learning environments to guide and foster learners’ construction of mathematical concepts and problem solving competences (Santos-Trigo 2014 ). Furthermore, mathematicians or discipline practitioners have often been interested in unveiling and sharing their own experience while developing the discipline. As a results, they have provided valuable information to characterize mathematical practices and their relations to what learning processes of the discipline entails. It is recognized that the work of Pólya ( 1945 ) offered not only bases to launch several research programs in problem solving (Schoenfeld 1992 ; Mason et al. 1982 ); but also it became an essential resource for teachers to orient and structure their mathematical lessons (Krulik and Reys 1980 ).

1.3.1 Research Agenda

A salient feature of a problem solving approach to learn mathematics is that teachers and students develop and apply an enquiry or inquisitive method to delve into mathematical concepts and tasks. How are mathematical problems or concepts formulated? What types of problems are important for teachers/learners to discuss and engage in mathematical reasoning? What mathematical processes and ways of reasoning are involved in understanding mathematical concepts and solving problems? What are the features that distinguish an instructional environment that fosters problem-solving activities? How can learners’ problem solving competencies be assessed? How can learners’ problem solving competencies be characterized and explained? How can learners use digital technologies to understand mathematics and to develop problem-solving competencies? What ways of reasoning do learners construct when they use digital technologies in problem solving approaches? These types of questions have been important in the problem solving research agenda and delving into them has led researchers to generate information and results to support and frame curriculum proposals and learning scenarios. The purpose of this section is to present and discuss important themes that emerged in problem solving approaches that rely on the systematic use of several digital technologies.

In the last 40 years, the accumulated knowledge in the problem solving field has shed lights on both a characterization of what mathematical thinking involves and how learners can construct a robust knowledge in problem solving environments (Schoenfeld 1992 ). In this process, the field has contributed to identify what types of transformations traditional learning scenarios might consider when teachers and students incorporate the use of digital technologies in mathematical classrooms. In this context, it is important to briefly review what main themes and developments the field has addressed and achieved during the last 40 years.

1.3.2 Problem Solving Developments

There are traces of mathematical problems and solutions throughout the history of civilization that explain the humankind interest for identifying and exploring mathematical relations (Kline 1972 ). Pólya ( 1945 ) reflects on his own practice as a mathematician to characterize the process of solving mathematical problems through four main phases: Understanding the problem, devising a plan, carrying out the plan, and looking back. Likewise, Pólya ( 1945 ) presents and discusses the role played by heuristic methods throughout all problem solving phases. Schoenfeld ( 1985 ) presents a problem solving research program based on Pólya’s ( 1945 ) ideas to investigate the extent to which problem solving heuristics help university students to solve mathematical problems and to develop a way of thinking that shows consistently features of mathematical practices. As a result, he explains the learners’ success or failure in problem solving activities can be characterized in terms their mathematical resources and ways to access them, cognitive and metacognitive strategies used to represent and explore mathematical tasks, and systems of beliefs about mathematics and solving problems. In addition, Schoenfeld ( 1992 ) documented that heuristics methods as illustrated in Pólya’s ( 1945 ) book are ample and general and do not include clear information and directions about how learners could assimilate, learn, and use them in their problem solving experiences. He suggested that students need to discuss what it means, for example, to think of and examining special cases (one important heuristic) in finding a closed formula for series or sequences, analysing relationships of roots of polynomials, or focusing on regular polygons or equilateral/right triangles to find general relations about these figures. That is, learners need to work on examples that lead them to recognize that the use of a particular heuristic often involves thinking of different type of cases depending on the domain or content involved. Lester and Kehle ( 2003 ) summarize themes and methodological shifts in problem solving research up to 1995. Themes include what makes a problem difficult for students and what it means to be successful problem solvers; studying and contrasting experts and novices’ problem solving approaches; learners’ metacognitive, beliefs systems and the influence of affective behaviours; and the role of context; and social interactions in problem solving environments. Research methods in problem solving studies have gone from emphasizing quantitative or statistical design to the use of cases studies and ethnographic methods (Krutestkii ( 1976 ). Teaching strategies also evolved from being centred on teachers to the active students’ engagement and collaboration approaches (NCTM 2000 ). Lesh and Zawojewski ( 2007 ) propose to extend problem solving approaches beyond class setting and they introduce the construct “model eliciting activities” to delve into the learners’ ideas and thinking as a way to engage them in the development of problem solving experiences. To this end, learners develop and constantly refine problem-solving competencies as a part of a learning community that promotes and values modelling construction activities. Recently, English and Gainsburg ( 2016 ) have discussed the importance of modeling eliciting activities to prepare and develop students’ problem solving experiences for 21st Century challenges and demands.

Törner et al. ( 2007 ) invited mathematics educators worldwide to elaborate on the influence and developments of problem solving in their countries. Their contributions show a close relationship between countries mathematical education traditions and ways to frame and implement problem solving approaches. In Chinese classrooms, for example, three instructional strategies are used to structure problem solving lessons: one problem multiple solutions , multiple problems one solution , and one problem multiple changes . In the Netherlands, the realistic mathematical approach permeates the students’ development of problem solving competencies; while in France, problem solving activities are structured in terms of two influential frameworks: The theory of didactical situations and anthropological theory of didactics.

In general, problem solving frameworks and instructional approaches came from analysing students’ problem solving experiences that involve or rely mainly on the use of paper and pencil work. Thus, there is a need to re-examined principles and frameworks to explain what learners develop in learning environments that incorporate systematically the coordinated use of digital technologies (Hoyles and Lagrange 2010 ). In this perspective, it becomes important to briefly describe and identify what both multiple purpose and ad hoc technologies can offer to the students in terms of extending learning environments and representing and exploring mathematical tasks. Specifically, a task is used to identify features of mathematical reasoning that emerge through the use digital technologies that include both mathematical action and multiple purpose types of technologies.

1.3.3 Background

Digital technologies are omnipresent and their use permeates and shapes several social and academic events. Mobile devices such as tablets or smart phones are transforming the way people communicate, interact and carry out daily activities. Churchill et al. ( 2016 ) pointed out that mobile technologies provide a set of tools and affordances to structure and support learning environments in which learners continuously interact to construct knowledge and solve problems. The tools include resources or online materials, efficient connectivity to collaborate and discuss problems, ways to represent, explore and store information, and analytical and administration tools to management learning activities. Schmidt and Cohen ( 2013 ) stated that nowadays it is difficult to imagine a life without mobile devices, and communication technologies are playing a crucial role in generating both cultural and technical breakthroughs. In education, the use of mobile artefacts and computers offers learners the possibility of continuing and extending peers and groups’ mathematical discussions beyond formal settings. In this process, learners can also consult online materials and interact with experts, peers or more experienced students while working on mathematical tasks. In addition, dynamic geometry systems (GeoGebra) provide learners a set of affordances to represent and explore dynamically mathematical problems. Leung and Bolite-Frant ( 2015 ) pointed out that tools help activate an interactive environment in which teachers and students’ mathematical experiences get enriched. Thus, the digital age brings new challenges to the mathematics education community related to the changes that technologies produce to curriculum, learning scenarios, and ways to represent, explore mathematical situations. In particular, it is important to characterize the type of reasoning that learners can develop as a result of using digital technologies in their process of learning concepts and solving mathematical problems.

1.3.4 A Focus on Mathematical Tasks

Mathematical tasks are essential elements for engaging learners in mathematical reasoning which involves representing objects, identifying and exploring their properties in order to detect invariants or relationships and ways to support them. Watson and Ohtani ( 2015 ) stated that task design involves discussions about mathematical content and students’ learning (cognitive perspective), about the students’ experiences to understand the nature of mathematical activities; and about the role that tasks played in teaching practices. In this context, tasks are the vehicle to present and discuss theoretical frameworks for supporting the use of digital technology, to analyse the importance of using digital technologies in extending learners’ mathematical discussions beyond formal settings, and to design ways to foster and assess the use of technologies in learners’ problem solving environments. In addition, it is important to discuss contents, concepts, representations and strategies involved in the process of using digital technologies in approaching the tasks. Similarly, it becomes essential to discuss what types of activities students will do to learn and solve the problems in an environment where the use of technologies fosters and values the participation and collaboration of all students. What digital technologies are important to incorporate in problem solving approaches? Dynamic Geometry Systems can be considered as a milestone in the development of digital technologies. Objects or mathematical situations can be represented dynamically through the use of a Dynamic Geometry System and learners or problem solvers can identify and examine mathematical relations that emerge from moving objects within the dynamic model (Moreno-Armella and Santos-Trigo 2016 ).

Leung and Bolite-Frant ( 2015 ) stated that “dynamic geometry software can be used in task design to cover a large epistemic spectrum from drawing precise robust geometrical figures to exploration of new geometric theorems and development of argumentation discourse” (p. 195). As a result, learners not only need to develop skills and strategies to construct dynamic configuration of problems; but also ways of relying on the tool’s affordances (quantifying parameters or objects attributes, generating loci, graphing objects behaviours, using sliders, or dragging particular elements within the configuration) in order to identify and support mathematical relations. What does it mean to represent and explore an object or mathematical situation dynamically?

A simple task that involves a rhombus and its inscribed circle is used to illustrate how a dynamic representation of these objects and embedded elements can lead learners to identify and examine mathematical properties of those objects in the construction of the configuration. To this end, learners are encouraged to pose and pursue questions to explain the behaviours of parameters or attributes of the family of objects that is generated as a result of moving a particular element within the configuration.

1.3.5 A Task: A Dynamic Rhombus

Figure  2 represents a rhombus APDB and its inscribed circle (O is intersection of diagonals AD and BP and the radius of the inscribed circle is the perpendicular segment from any side of the rhombus to point O), vertex P lies on a circle c centred at point A. Circle c is only a heuristic to generate a family of rhombuses. Thus, point P can be moved along circle c to generate a family of rhombuses. Indeed, based on the symmetry of the circle it is sufficient to move P on the semicircle B’CA to draw such a family of rhombuses.

A dynamic construction of a rhombus

1.3.6 Posing Questions

A goal in constructing a dynamic model or configuration of problems is always to identify and explore mathematical properties and relations that might result from moving objects within the model. How do the areas of both the rhombus and the inscribed circle behave when point P is moved along the arc B’CB? At what position of point P does the area of the rhombus or inscribed circle reach the maximum value? The coordinates of points S and Q (Fig.  3 ) are the x -value of point P and as y -value the corresponding area values of rhombus ABDP and the inscribed circle respectively. Figure  2 shows the loci of points S and Q when point P is moved along arc B’CB. Here, finding the locus via the use of GeoGebra is another heuristic to graph the area behaviour without making explicit the algebraic model of the area.

Graphic representation of the area variation of the family of rhombuses and inscribed circles generated when P is moved through arc B’CB

The area graphs provide information to visualize that in that family of generated rhombuses the maximum area value of the inscribed circle and rhombus is reached when the rhombus becomes a square (Fig.  4 ). That is, the controlled movement of particular objects is an important strategy to analyse the area variation of the family of rhombuses and their inscribed circles.

Visualizing the rhombus and the inscribed circle with maximum area

It is important to observe the identification of points P and Q in terms of the position of point P and the corresponding areas and the movement of point P was sufficient to generate both area loci. That is, the graph representation of the areas is achieved without having an explicit algebraic expression of the area variation. Clearly, the graphic representations provide information regarding the increasing or decreasing interval of both areas; it is also important to explore what properties both graphic representations hold. The goal is to argue that the area variation of the rhombus represents an ellipse and the area of the inscribed circle represents a parabola. An initial argument might involve selecting five points on each locus and using the tool to draw the corresponding conic section (Fig.  5 ). In this case, the tool affordances play an important role in generating the graphic representation of the areas’ behaviours and in identifying properties of those representations. In this context, the use of the tool can offer learners the opportunity to problematize (Santos-Trigo 2007 ) a simple mathematical object (rhombus) as a means to search for mathematical relations and ways to support them.

Drawing the conic section that passes through five points

1.3.7 Looking for Different Solutions Methods

Another line of exploration might involve asking for ways to construct a rhombus and its inscribed circle: Suppose that the side of the rhombus and the circle are given, how can you construct the rhombus that has that circle inscribed? Figure  6 shows the given data, segment A 1 B 1 and circle centred at O and radius OD. The initial goal is to draw the circle tangent to the given segment. To this end, segment AB is congruent to segment A 1 B 1 and on this segment a point P is chosen and a perpendicular to segment AB that passes through point P is drawn. Point C is on this perpendicular and the centre of a circle with radius OD and h is the perpendicular to line PC that passes through point C. Angle ACB changes when point P is moved along segment AB and point E and F are the intersection of line h and the circle with centre M the midpoint of AB and radius MA (Fig.  6 ).

Drawing segment AB tangent to the given circle

Figure  7 a shows the right triangle AFB as the base to construct the rhombus and the inscribed circle and Fig.  7 b shows the second solution based on triangle AEB.

a Drawing the rhombus and the inscribed circle. b Drawing the second solution

Another approach might involve drawing the given circle centred at the origin and the segment as EF with point E on the y-axis. Line OC is perpendicular to segment EF and the locus of point C when point E moves along the y-axis intersects the given circle (Fig.  8 a, b). Both figures show two solutions to draw the rhombus that circumscribe the given circle.

a and b Another solution that involves finding a locus of point C

In this example, the GeoGebra affordances not only are important to construct a dynamic model of the task; but also offer learners and opportunity to explore relations that emerge from moving objects within the model. As a result, learners can rely on different concepts and strategies to solve the tasks. The idea in presenting this rhombus task is to illustrate that the use of a Dynamic Geometry System provides affordances for learners to construct dynamic representation of mathematical objects or problems, to move elements within the representation to pose questions or conjectures to explain invariants or patterns among involved parameters; to search for arguments to support emerging conjectures, and to develop a proper language to communicate results.

1.3.8 Looking Back

Conceptual frameworks used to explain learners’ construction of mathematical knowledge need to capture or take into account the different ways of reasoning that students might develop as a result of using a set of tools during the learning experiences. Figure  9 show some digital technologies that learners can use for specific purpose at the different stages of problem solving activities.

The coordinated use of digital tools to engage learners in problem solving experiences

The use of a dynamic system (GeoGebra) provides a set of affordances for learners to conceptualize and represent mathematical objects and tasks dynamically. In this process, affordances such as moving objects orderly (dragging), finding loci of objects, quantifying objects attributes (lengths, areas, angles, etc.), using sliders to vary parameters, and examining family of objects became important to look for invariance or objects relationships. Likewise, analysing the parameters or objects behaviours within the configuration might lead learners to identify properties to support emerging mathematical relations. Thus, with the use of the tool, learners might conceptualize mathematical tasks as an opportunity for them to engage in mathematical activities that include constructing dynamic models of tasks, formulating conjectures, and always looking for different arguments to support them. Similarly, learners can use an online platform to share their ideas, problem solutions or questions in a digital wall and others students can also share ideas or solution methods and engaged in mathematical discussions that extend mathematical classroom activities.

1.4 Problem Posing: An Overview for Further Progress—Uldarico Malaspina Jurado

Problem posing and problem solving are two essential aspects of the mathematical activity; however, researchers in mathematics education have not emphasized their attention on problem posing as much as problem solving. In that sense, due to its importance in the development of mathematical thinking in students since the first grades, we agree with Ellerton’s statement ( 2013 ): “for too long, successful problem solving has been lauded as the goal; the time has come for problem posing to be given a prominent but natural place in mathematics curricula and classrooms” (pp. 100–101); and due to its importance in teacher training, with Abu-Elwan’s statement ( 1999 ):

While teacher educators generally recognize that prospective teachers require guidance in mastering the ability to confront and solve problems, what is often overlooked is the critical fact that, as teachers, they must be able to go beyond the role as problem solvers. That is, in order to promote a classroom situation where creative problem solving is the central focus, the practitioner must become skillful in discovering and correctly posing problems that need solutions. (p. 1)

Scientists like Einstein and Infeld ( 1938 ), recognized not only for their notable contributions in the fields they worked, but also for their reflections on the scientific activity, pointed out the importance of problem posing; thus it is worthwhile to highlight their statement once again:

The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old questions from a new angle, requires creative imagination and marks real advance in science. (p. 92)

Certainly, it is also relevant to remember mathematician Halmos’s statement ( 1980 ): “I do believe that problems are the heart of mathematics, and I hope that as teachers (…) we will train our students to be better problem posers and problem solvers than we are” (p. 524).

An important number of researchers in mathematics education has focused on the importance of problem posing, and we currently have numerous, very important publications that deal with different aspects of problem posing related to the mathematics education of students in all educational levels and to teacher training.

1.4.1 A Retrospective Look

Kilpatrick ( 1987 ) marked a historical milestone in research related to problem posing and points out that “problem formulating should be viewed not only as a goal of instruction but also as a means of instruction” (Kilpatrick 1987 , p. 123); and he also emphasizes that, as part of students’ education, all of them should be given opportunities to live the experience of discovering and posing their own problems. Drawing attention to the few systematic studies on problem posing performed until then, Kilpatrick contributes defining some aspects that required studying and investigating as steps prior to a theoretical building, though he warns, “attempts to teach problem-formulating skills, of course, need not await a theory” (p. 124).

Kilpatrick refers to the “Source of problems” and points out how virtually all problems students solve have been posed by another person; however, in real life “many problems, if not most, must be created or discovered by the solver, who gives the problem an initial formulation” (p. 124). He also points out that problems are reformulated as they are being solved, and he relates this to investigation, reminding us what Davis ( 1985 ) states that, “what typically happens in a prolonged investigation is that problem formulation and problem solution go hand in hand, each eliciting the other as the investigation progresses” (p. 23). He also relates it to the experiences of software designers, who formulate an appropriate sequence of sub-problems to solve a problem. He poses that a subject to be examined by teachers and researchers “is whether, by drawing students’ attention to the reformulating process and given them practice in it, we can improve their problem solving performance” (p. 130). He also points out that problems may be a mathematical formulation as a result of exploring a situation and, in that sense, “school exercises in constructing mathematical models of a situation presented by the teacher are intended to provide students with experiences in formulating problems.” (p. 131).

Another important section of Kilpatrick’s work ( 1987 ) is Processes of Problem Formulating , in which he considers association, analogy, generalization and contradiction. He believes the use of concept maps to represent concept organization, as cognitive scientists Novak and Gowin suggest, might help to comprehend such concepts, stimulate creative thinking about them, and complement the ideas Brown and Walter ( 1983 ) give for problem posing by association. Further, in the section “Understanding and developing problem formulating abilities”, he poses several questions, which have not been completely answered yet, like “Perhaps the central issue from the point of view of cognitive science is what happens when someone formulates the problem? (…) What is the relation between problem formulating, problem solving and structured knowledge base? How rich a knowledge base is needed for problem formulating? (…) How does experience in problem formulating add to knowledge base? (…) What metacognitive processes are needed for problem formulating?”

It is interesting to realize that some of these questions are among the unanswered questions proposed and analyzed by Cai et al. ( 2015 ) in Chap. 1 of the book Mathematical Problem Posing (Singer et al. 2015 ). It is worth stressing the emphasis on the need to know the cognitive processes in problem posing, an aspect that Kilpatrick had already posed in 1987, as we just saw.

1.4.2 Researches and Didactic Experiences

Currently, there are a great number of publications related to problem posing, many of which are research and didactic experiences that gather the questions posed by Kilpatrick, which we just commented. Others came up naturally as reflections raised in the framework of problem solving, facing the natural requirement of having appropriate problems to use results and suggestions of researches on problem solving, or as a response to a thoughtful attitude not to resign to solving and asking students to solve problems that are always created by others. Why not learn and teach mathematics posing one’s own problems?

1.4.3 New Directions of Research

Singer et al. ( 2013 ) provides a broad view about problem posing that links problem posing experiences to general mathematics education; to the development of abilities, attitudes and creativity; and also to its interrelation with problem solving, and studies on when and how problem-solving sessions should take place. Likewise, it provides information about research done regarding ways to pose new problems and about the need for teachers to develop abilities to handle complex situations in problem posing contexts.

Singer et al. ( 2013 ) identify new directions in problem posing research that go from problem-posing task design to the development of problem-posing frameworks to structure and guide teachers and students’ problem posing experiences. In a chapter of this book, Leikin refers to three different types of problem posing activities, associated with school mathematics research: (a) problem posing through proving; (b) problem posing for investigation; and (c) problem posing through investigation. This classification becomes evident in the problems posed in a course for prospective secondary school mathematics teachers by using a dynamic geometry environment. Prospective teachers posed over 25 new problems, several of which are discussed in the article. The author considers that, by developing this type of problem posing activities, prospective mathematics teachers may pose different problems related to a geometric object, prepare more interesting lessons for their students, and thus gradually develop their mathematical competence and their creativity.

1.4.4 Final Comments

This overview, though incomplete, allows us to see a part of what problem posing experiences involve and the importance of this area in students mathematical learning. An important task is to continue reflecting on the questions posed by Kilpatrick ( 1987 ), as well as on the ones that come up in the different researches aforementioned. To continue progressing in research on problem posing and contribute to a greater consolidation of this research line, it will be really important that all mathematics educators pay more attention to problem posing, seek to integrate approaches and results, and promote joint and interdisciplinary works. As Singer et al. ( 2013 ) say, going back to Kilpatrick’s proposal ( 1987 ),

Problem posing is an old issue. What is new is the awareness that problem posing needs to pervade the education systems around the world, both as a means of instruction (…) and as an object of instruction (…) with important targets in real-life situations. (p. 5)

Although it can be argued that there is a difference between creativity, discovery, and invention (see Liljedahl and Allan 2014 ) for the purposes of this book these will be assumed to be interchangeable.

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Further Reading

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Liljedahl, P., Santos-Trigo, M., Malaspina, U., Bruder, R. (2016). Problem Solving in Mathematics Education. In: Problem Solving in Mathematics Education. ICME-13 Topical Surveys. Springer, Cham. https://doi.org/10.1007/978-3-319-40730-2_1

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Students' critical mathematical thinking abilities through flip-problem based learning model based on LMS-google classroom

R Ramadhani 1 , N S Bina 1 , S F Sihotang 1 , S D Narpila 1 and M R Mazaly 1

Published under licence by IOP Publishing Ltd Journal of Physics: Conference Series , Volume 1657 , The 2nd International Seminar on Applied Mathematics and Mathematics Education (2nd ISAMME) 2020 5 August 2020, Cimahi, Indonesia Citation R Ramadhani et al 2020 J. Phys.: Conf. Ser. 1657 012025 DOI 10.1088/1742-6596/1657/1/012025

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1 Universitas Potensi Utama, Jl. K.L.Yos Sudarso KM. 6,5 No. 3-A Medan, Indonesia

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This research aimed to calculate the increase of students' mathematics critical thinking ability in Senior High School who taught with Flip-Problem Based Learning based on LMS-Google Classroom. This research is quasi-experiment research used pre-test post-test control group design research. Sample in this research is 60 students divided into two research groups, 30 students in the experiment group (taught with Flip-Problem Based Learning) and 30 students in the control group (taught with Problem-Based Learning). The homogeneity test and normality data test used Levenes' test and Kolmogorov Smirnov test. The result of the homogeneity test and normality data test is homogenous and distributed normally. Based on the result of the homogeneity test and normality data test, hence the hypothesis research test used Independent Sample T-Test. The test result showed that mathematical critical thinking ability for students' Senior High School who taught with Flip-Problem Based Learning based on LMS-Google Classroom. The result proved that the mathematics learning process of Flip-Problem Based Learning has a positive and significant effect on students' mathematics critical thinking ability. Based on this research result, Flip-Problem Based Learning was recommended to be applied in mathematics learning. LMS-Google Classroom can also be one of the solutions in mathematics learning based on digital.

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ORIGINAL RESEARCH article

Challenges of teachers when teaching sentence-based mathematics problem-solving skills.

• Faculty of Education, Universiti Kebangsaan Malaysia, Bangi, Malaysia

Sentence-based mathematics problem-solving skills are essential as the skills can improve the ability to deal with various mathematical problems in daily life, increase the imagination, develop creativity, and develop an individual’s comprehension skills. However, mastery of these skills among students is still unsatisfactory because students often find it difficult to understand mathematical problems in verse, are weak at planning the correct solution strategy, and often make mistakes in their calculations. This study was conducted to identify the challenges that mathematics teachers face when teaching sentence-based mathematics problem-solving skills and the approaches used to address these challenges. This study was conducted qualitatively in the form of a case study. The data were collected through observations and interviews with two respondents who teach mathematics to year four students in a Chinese national primary school in Kuala Lumpur. This study shows that the teachers have faced three challenges, specifically low mastery skills among the students, insufficient teaching time, and a lack of ICT infrastructure. The teachers addressed these challenges with creativity and enthusiasm to diversify the teaching approaches to face the challenges and develop interest and skills as part of solving sentence-based mathematics problems among year four students. These findings allow mathematics teachers to understand the challenges faced while teaching sentence-based mathematics problem solving in depth as part of delivering quality education for every student. Nevertheless, further studies involving many respondents are needed to understand the problems and challenges of different situations and approaches that can be used when teaching sentence-based mathematics problem-solving skills.

1. Introduction

To keep track of the development of the current world, education has changed over time to create a more robust and effective system for producing a competent and competitive generation ( Hashim and Wan, 2020 ). The education system of a country is a significant determinant of the growth and development of the said country ( Ministry of Education Malaysia, 2013 ). In the Malaysian context, the education system has undergone repeated changes alongside the latest curriculum, namely the revised Primary School Standard Curriculum (KSSR) and the revised Secondary School Standard Curriculum (KSSM). These changes have been implemented to ensure that Malaysian education is improving continually so then the students can guide the country to compete globally ( Adam and Halim, 2019 ). However, Malaysian students have shown limited skills in international assessments such as Trends in International Mathematics and Science Study (TIMSS) and the Program for International Student Assessment (PISA).

According to the PISA 2018 results, the students’ performance in mathematics is still below the average level of the Organization for Economic Co-operation and Development (OECD; Avvisati et al., 2019 ). The results show that almost half of the students in Malaysia have still not mastered mathematical skills fully. Meanwhile, the TIMSS results in 2019 have shown there to be a descent in the achievements of Malaysian students compared to the results in 2015 ( Ministry of Education Malaysia, 2020b ). This situation is worrying as most students from other countries such as China, Singapore, Korea, Japan, and others have a higher level of mathematical skills than Malaysian students. According to Mullis et al. (2016) , these two international assessments have in common that both assessments test the level of the students’ skills when solving real-world problems. In short, PISA and TIMSS have proven that Malaysian students are still weak when it comes to solving sentence-based mathematics problems.

According to Hassan et al. (2019) , teachers must emphasize the mastery of sentence-based mathematics problem-solving skills and apply it in mathematics teaching in primary school. Sentence-based mathematics problem-solving skills can improve the students’ skills when dealing with various mathematical problems in daily life ( Gurat, 2018 ), increase the students’ imagination ( Wibowo et al., 2017 ), develop the students’ creativity ( Suastika, 2017 ), and develop the students’ comprehension skills ( Mulyati et al., 2017 ). The importance of sentence-based mathematic problem-solving skills is also supported by Ismail et al. (2021) . They stated that mathematics problem-solving skills are similar to high-level thinking skills when it comes to guiding students with how to deal with problems creatively and critically. Moreover, problem-solving skills are also an activity that requires an individual to select an appropriate strategy to be performed by the individual to ensure that movement occurs between the current state to the expected state ( Sudarmo and Mariyati, 2017 ). There are various strategies that can be used by teachers to guide students when developing their problem-solving skills such as problem-solving strategies based on Polya’s Problem-Solving Model (1957). Various research studies have used problem-solving models to solve specific problems to improve the students’ mathematical skills. Polya (1957) , Lester (1980) , Gick (1986) , and DeMuth (2007) are examples. One of the oldest problem-solving models is the George Polya model (1957). The model is divided into four major stages: (i) understanding the problem; (ii) devising a plan that will lead to the solution; (iii) Carrying out the plan; and (iv) looking back. In contrast to traditional mathematics classroom environments, Polya’s Problem-Solving Process allows the students to practice adapting and changing strategies to match new scenarios. As a result, the teachers must assist the students to help them recognize whether the strategy is appropriate, including where and how to apply the technique.

In addition, problem-solving skills are one of the 21st-century skills that need to be mastered by students through education now so then they are prepared to face the challenges of daily life ( Khoiriyah and Husamah., 2018 ). This statement is also supported by Widodo et al. (2018) who put forward four main reasons why students need to master problem-solving skills through mathematics learning. One reason is that sentence-based mathematics problem-solving skills are closely related to daily life ( Wong, 2015 ). Such skills can be used to formulate concepts and develop mathematical ideas, a skill that needs to be conveyed according to the school’s content standards. The younger generation is expected to develop critical, logical, systematic, accurate, and efficient thinking when solving a problem. Accordingly, problem solving has become an element that current employers emphasize when looking to acquire new energy sources ( Zainuddin et al., 2018 ). This clearly shows that problem-solving skills are essential skills that must be mastered by students and taken care of by mathematics teachers in primary school.

In the context of mathematics learning in Malaysia, students are required to solve sentence-based mathematics problems by applying mathematical concepts learned at the end of each topic. Two types of sentence-based mathematics problems are presented when teaching mathematics: routine and non-routine ( Wong and Matore, 2020 ). According to Nurkaeti (2018) , routine sentence-based math problems are questions that require the students to solve problems using algorithmic calculations to obtain answers. For non-routine sentence-based math problems, thinking skills and the ability to apply more than one method or solution step are needed by the student to solve the problem ( Shawan et al., 2021 ). According to Rohmah and Sutiarso (2018) , problem-solving skills when solving a non-routine sentence-based mathematical problem is a high-level intellectual skill where the students need to use logical thinking and reasoning. This statement also aligns with Wilson's (1997) opinion that solving non-routine sentence-based mathematics always involves high-order thinking skills (HOTS). To solve non-routine and HOTS fundamental sentence-based math problems, a student is required to know various problem-solving strategies for solving the problems ( Wong and Matore, 2020 ). This situation has indirectly made the mastery of sentence-based mathematics problem-solving skills among students more challenging ( Mahmud, 2019 ).

According to Alkhawaldeh and Khasawneh’s findings (2021) , the failure of students stems from the teachers’ inability to perform their role effectively in the classroom. This statement is also supported by Abdullah (2020) . He argues that the failure of students in mastering non-routine sentence-based mathematics problem-solving skills is due to the teachers rarely supplying these types of questions during the process of learning mathematics in class. A mathematics teacher should consider this issue because the quality of their teaching will affect the students’ mastery level of sentence-based mathematics problem-solving skills.

In addition, the teachers’ efforts to encourage the students to engage in social interactions with the teachers ( Jatisunda, 2017 ) and the teachers’ method of teaching and assessing the level of sentence-based mathematics problem-solving skills ( Buschman, 2004 ) are also challenges that the teachers must face. Strategies that are not appropriate for the students will affect the quality of delivery of the sentence-based mathematics problem-solving skills as well as cause one-way interactions to exist in the classroom. According to Rusdin and Ali (2019) , a practical teaching approach plays a vital role in developing the students’ skills when mastering specific knowledge. However, based on previous studies, the main challenges that mathematics teachers face when teaching sentence-based mathematics problem solving are due to the students. These challenges include the students having difficulty understanding sentence-based math problems, lacking knowledge about basic mathematical concepts, not calculating accurately, and not transforming the sentence-based mathematics problems into an operational form ( Yoong and Nasri, 2021 ). This also means that they cannot transform the sentence-based math problems into an operational form ( Yoong and Nasri, 2021 ). As a result, the teacher should diversify his or her teaching strategy by emphasizing understanding the mathematical concepts rather than procedural teaching to reinforce basic mathematical concepts, to encourage the students to work on any practice problems assigned by the teacher before completing any assignments to help them do the calculation correctly, and engaging in the use of effective oral questioning to stimulate student thinking related to the operational need when problem solving. All of these strategies actually help the teachers facilitate and lessen the students’ difficulty understanding sentence-based math problems ( Subramaniam et al., 2022 ).

Meanwhile, Dirgantoro et al. (2019) stated several challenges that the students posed while solving the sentence-based problem. For example, students do not read the questions carefully, the students lack mastery of mathematical concepts, the students solve problems in a hurry due to poor time management, the students are not used to making hypotheses and conclusions, as well as the students, being less skilled at using a scientific calculator. These factors have caused the students to have difficulty mastering sentence-based mathematics problem-solving skills, which goes on to become an inevitable challenge in maths classes. Therefore, teachers need to study these challenges to self-reflect so then their self-professionalism can be further developed ( Dirgantoro et al., 2019 ).

As for the school factor, challenges such as limited teaching resources, a lack of infrastructure facilities, and a large number of students in a class ( Rusdin and Ali, 2019 ) have meant that a conducive learning environment for learning sentence-based mathematics problem-solving skills cannot be created. According to Ersoy (2016) , problem-solving skills can be learned if an appropriate learning environment is provided for the students to help them undergo a continuous and systematic problem-solving process.

To develop sentence-based mathematics problem-solving skills among students, various models, pedagogies, activities, etc. have been introduced to assist mathematics teachers in delivering sentence-based mathematics problem-solving skills more effectively ( Gurat, 2018 ; Khoiriyah and Husamah., 2018 ; Özreçberoğlu and Çağanağa, 2018 ; Hasibuan et al., 2019 ). However, students nowadays still face difficulties when trying to master sentence-based mathematics problem-solving skills. This situation occurs due to the lack of studies examining the challenges faced by these mathematics teachers and how teachers use teaching approaches to overcome said challenges. This has led to various issues during the teaching and facilitation of sentence-based mathematics problem-solving skills in mathematics classes. According to Rusdin and Ali (2019) , these issues need to be addressed by a teacher wisely so then the quality of teaching can reach the best level. Therefore, mathematics teachers must understand and address these challenges to improve their teaching.

However, so far, not much is known about how primary school mathematics teachers face the challenges encountered when teaching sentence-based mathematics problem-solving skills and what approaches are used to address the challenges in the context of education in Malaysia. Therefore, this study needs to be carried out to help understand the teaching of sentence-based mathematics problem-solving skills in primary schools ( Pazin et al., 2022 ). Due to the challenges when teaching mathematics as stipulated in the Mathematics Curriculum and Assessment Standard Document ( Ministry of Education Malaysia, 2020a ) which emphasizes mathematical problem-solving skills as one of the main skills that students need to master in comprehensive mathematics learning, this study focuses on identifying the challenges faced by mathematics teachers when teaching sentence-based mathematics problem-solving skills and the approaches that mathematics teachers have used to overcome those challenges. The results of this study can provide information to mathematics teachers to help them understand the challenges when teaching sentence-based mathematics problem-solving skills and the approaches that can be applied to overcome the challenges faced. Therefore, it is very important for this study to be carried out so then all the visions set within the framework of the Malaysian National Mathematics Curriculum can be successfully achieved.

2. Conceptual framework

The issue of students lacking mastery of sentence-based mathematics problem-solving skills is closely related to the challenges that teachers face and the teaching approach used. Based on the overall findings of the previous studies, the factors that pose a challenge to teachers when delivering sentence-based mathematics problem-solving skills include challenges from the teacher ( Buschman, 2004 ; Jatisunda, 2017 ; Abdullah, 2020 ), challenges from the pupils ( Dirgantoro et al., 2019 ), and challenges from the school ( Rusdin and Ali, 2019 ). As for the teaching approach, previous studies have suggested teaching approaches such as mastery learning, contextual learning, project-based learning, problem-based learning, simulation, discovery inquiry, the modular approach, the STEM approach ( Curriculum Development Division, 2019 ), game-based teaching which uses digital games ( Muhamad et al., 2018 ), and where a combination of the modular approach especially the flipped classroom is applied alongside the problem-based learning approach when teaching sentence-based mathematics problem solving ( Alias et al., 2020 ). This is as well as the constructivism approach ( Jatisunda, 2017 ). The conceptual framework in Figure 1 illustrates that the teachers will face various challenges during the ongoing teaching and facilitation of sentence-based mathematics problem-solving skills.

Figure 1 . Conceptual framework of the study.

3. Methodology

The objective of this study was to determine the challenges that teachers face while teaching sentence-based mathematics problem-solving skills and the approaches used when teaching those skills. Therefore, a qualitative research approach in the form of a case study was used to collect data from the participants in a Chinese national type of school (SJKC) in Bangsar and Pudu, in the Federal Territory of Kuala Lumpur. The school, SJKC, in the districts of Bangsar and Pudu, was chosen as the location of this study because the school is implementing the School Transformation Program 2025 (TS25). One of the main objectives of the TS25 program is to apply the best teaching concepts and practices so then the quality of the learning and teaching in the classes is improved. Thus, schools that go through the program are believed to be able to diversify their teachers’ teaching and supply more of the data needed to answer the questions of this study. This is because case studies can develop an in-depth description and analysis of the case to be studied ( Creswell and Poth, 2018 ). All data collected through the observations, interviews, audio-visual materials, documents, and reports can be reported on in terms of both depth and detail based on the theme of the case. Therefore, this study collected data related to the challenges and approaches of SJKC mathematics teachers through observations, interviews, and document analysis.

Two primary school mathematics teachers who teach year four mathematics were selected to be the participants of this research using the purposive sampling technique to identify the challenges faced and the approaches used to overcome those challenges. The number of research participants in this study was sufficient enough to allow the researcher to explore the real picture of the challenges found when teaching sentence-based mathematics problem-solving skills and the approaches that can be applied when teaching to overcome the challenges faced. According to Creswell and Creswell (2018) , the small number of study participants is sufficient when considering that the main purpose of the study is to obtain findings that can give a holistic and meaningful picture of the teaching and learning process in the classroom. However, based on the data analysis for both study participants, the researcher considered repeated information until it reached a saturation point. The characteristics of the study participants required when they were supplying the information for this study were as follows:

i. New or experienced teachers.

ii. Year four math teacher.

iii. Teachers teach in primary schools.

iv. The teacher teaches the topic of sentence-based mathematics problem-solving skills.

The types of instruments used in the study were the observation protocol, field notes, interview protocol, and participants’ documents. In this study, the researcher used participatory type observations to observe the teaching style of the teachers when engaged in sentence-based mathematics problem-solving skills lessons. Before conducting the study, the researcher obtained consent to conduct the study from the school as well as informed consent from the study participants to observe their teaching. During the observation, the teacher’s teaching process was recorded and transcribed using the field notes provided. Then, the study participants submitted and validated the field notes to avoid biased data. After that, the field notes were analyzed based on the observation protocol to identify the teachers’ challenges and teaching approaches in relation to sentence-based mathematics problem-solving skills. Throughout the observation process of this study, the researcher observed the teaching of mathematics teachers online at least four times during the 2 months of the data collection at the research location.

Semi-structured interviews were used to identify the teachers’ perspectives and views on teaching sentence-based mathematics problem-solving skills in terms of the challenges faced when teaching sentence-based mathematics problem-solving skills and the approaches used by the teachers to overcome those challenges. To ensure that the interview data collected could answer the research questions, an interview protocol was prepared so then the required data could be collected from the study participants ( Cohen et al., 2007 ). Two experts validated the interview protocol, and a pilot study was conducted to ensure that the questions were easy to understand and would obtain the necessary data. Before the interview sessions began, the participants were informed of their rights and of the related research ethics. Throughout the interview sessions, the participants were asked two questions, namely:

1. What are the challenges faced when teaching mathematical problem-solving skills earlier?

2. What teaching approaches are used by teachers when facing these challenges? Why?

Semi-structured interviews were used to interview the study participants for 30 min every interview session. The timing ensured sufficient time for both parties to complete the question-and-answer process. Finally, the entire interview process was recorded in audio form. The audio recordings were then transcribed into text form and verified by the study participants.

The types of document collected in this study included informal documents, namely the daily lesson plan documents of the study participants, the work of the students of the study participants, and any teaching aids used. All of the documents were analyzed and used to ensure that the triangulation of the data occurred between the data collected from observations, interviews, and document analysis.

All data collected through the observations, interviews, and documentary analyses were entered into the NVIVO 11 software to ensure that the coding process took place simultaneously. The data in this study were analyzed using the constant comparative analysis method including open coding, axial coding, and selective coding to obtain the themes and subthemes related to the focus of the study ( Kolb, 2012 ). The NVIVO 11 software was also used to manage the data stack obtained from the interviews, observations, and document analysis during the data analysis process itself. In order to ensure that the themes generated from all of the data were accurate, the researcher carried out a repetitive reading process. The process of theme development involved numerous steps. First, the researcher examined the verbatim instruction data several times while looking for statements or paragraphs that could summarize a theme in a nutshell. This process had already been completed during the verbatim formation process of the teaching, while preparing the transcription. Second, the researcher kept reading (either from the same or different data), and if the researcher found a sentence that painted a similar picture to the theme that had been developed, the sentence was added to the same theme. This process is called “pattern matching” because the coding of the sentences refers to the existing categories ( Yin, 2003 ). Third, if the identified sentence was incompatible with an existing theme, a new theme was created. Fourth, this coding procedure continued throughout each data set’s theme analysis. The repeated reading process was used to select sentences able to explain the theme or help establish a new one. In short, the researcher conducted the data analysis process based on the data analysis steps proposed by Creswell and Creswell (2018) , as shown in Figure 2 .

Figure 2 . Data analysis steps ( Creswell and Creswell, 2018 ).

4. Findings

The findings of this study are presented based on the objective of the study, which was to identify the challenges faced by teachers and the approaches used to addressing those challenges when imparting sentence-based mathematics problem-solving skills to students in year four. Several themes were formed based on the analysis of the field notes, interview transcripts, and daily lesson plans of the study participants. This study found that teachers will face challenges that stem from the readiness of students to master sentence-based mathematics problem-solving skills, the teachers’ teaching style, and the equipment used for delivering sentence-based mathematics problem-solving skills. Due to facing these challenges, teachers have diversified their teaching approaches ( Figure 3 ; Table 1 ).

Figure 3 . Sentence-based mathematics problem-solving teaching approaches.

Table 1 . Teacher challenges when teaching sentence-based mathematics problem-solving skills.

5. Discussion

5.1. challenges for teachers when imparting sentence-based mathematics problem-solving skills.

A mathematics teacher will face three challenges when teaching sentence-based mathematics problem-solving skills. The first challenge stems from the low mastery skills held by a student. Pupils can fail to solve sentence-based mathematics problems because they have poor reading skills, there is a poor medium of instruction used, or they have a poor mastery of mathematical concepts ( Johari et al., 2022 ). This indicates that students who are not ready or reach a minimum level of proficiency in a language, comprehension, mathematical concepts, and calculations will result in them not being able to solve sentence-based mathematics problems smoothly.

These findings are consistent with the findings of the studies by Raifana et al. (2016) and Dirgantoro et al. (2019) who showed that students who are unprepared in terms of language skills, comprehension, mathematical concepts, and calculations are likely to make mistakes when solving sentence-based mathematics problems. If these challenges are not faced well, the students will become passive and not interact when learning sentence-based mathematics problem-solving skills. This situation occurs because students who frequently make mistakes will incur low self-confidence in mathematics ( Jailani et al., 2017 ). This situation should be avoided by teachers and social interaction should be encouraged during the learning process because the interaction between students and teachers can ensure that the learning outcomes are achieved by the students optimally ( Jatisunda, 2017 ).

The next challenge stems from the teacher-teaching factor. This study found that how teachers convey problem-solving skills has been challenging in terms of ensuring that their students master sentence-based mathematics problem-solving skills ( Nang et al., 2022 ). The mastery teaching approach has caused the teaching time spent on mathematical content to be insufficient. Based on the findings of this study, the allocation of time spent ensuring that the students master the skills of solving sentence-based mathematics problems through a mastery approach has caused the teaching process not to follow the rate set in the annual lesson plan.

In this study, the participants spent a long time correcting the students’ mathematical concepts and allowing students to apply the skills learned. The actions of the participants of this study are in line with the statement of Adam and Halim (2019) that teachers need more time to arouse their students’ curiosity and ensure that students understand the correct ideas and concepts before doing more challenging activities. However, this approach has indirectly posed challenges regarding time allocation and ensuring that the students master the skills of sentence-based mathematics problem-solving. Aside from ensuring that the students’ master problem-solving skills, the participants must also complete the syllabus set in the annual lesson plan.

Finally, teachers also face challenges in terms of the lack of information and communication technology (ICT) infrastructure when implementing the teaching and facilitation of sentence-based mathematics problem-solving skills. In this study, mathematics teachers were found to face challenges caused by an unstable internet connection such as the problem of their students dropping out of class activities and whiteboard links not working. These problems have caused one mathematics class to run poorly ( Mahmud and Law, 2022 ). Throughout the implementation of teaching and its facilitation, ICT infrastructure equipment in terms of hardware, software, and internet services has become an element that will affect the effectiveness of virtual teaching ( Saifudin and Hamzah, 2021 ). In this regard, a mathematics teacher must be wise when selecting a teaching approach and diversifying the learning activities to implement a suitable mathematics class for students such as systematically using tables, charts, or lists, creating digital simulations, using analogies, working back over the work, involving reasoning activities and logic, and using various new applications such as Geogebra and Kahoot to help enable their students’ understanding.

5.2. Teaching sentence-based mathematics problem-solving skills—Approaches

In this study, various approaches have been used by the teachers facing challenges while imparting sentence-based mathematics problem-solving skills. Among the approaches that the mathematics teachers have used when teaching problem-solving skills are the oral questioning approach, mastery learning approach, contextual learning approach, game approach, and modular approach. This situation has shown that mathematics teachers have diversified their teaching approaches when facing the challenges associated with teaching sentence-based mathematics problem-solving skills. This action is also in line with the excellent teaching and facilitation of mathematics proposal in the Curriculum and Assessment Standards Document revised KSSR Mathematics Year 4 ( Curriculum Development Division, 2019 ), stating that teaching activities should be carefully planned by the teachers and combine a variety of approaches that allow the students not only to understand the content in depth but also to think at a higher level. Therefore, a teacher needs to ensure that this teaching approach is applied when teaching sentence-based mathematics problem-solving skills so then the students can learn sentence-based mathematics problem-solving teaching skills in a more fun, meaningful, and challenging environment ( Mahmud et al., 2022 ).

Through the findings of this study, the teaching approach used by mathematics teachers was found to have a specific purpose, namely facing the challenges associated with teaching sentence-based mathematics problem-solving skills in the classroom. First of all, the oral questioning approach has been used by teachers facing the challenge of students having a poor understanding of the medium of instruction. The participants stated that questioning the students in stages can guide them to understanding the question and helping them plan appropriate problem-solving strategies. This opinion is also supported by Maat (2015) who stated that low-level oral questions could help the students achieve a minimum level of understanding, in particular remembering, and strengthening abstract mathematical concepts. The teacher’s action of guiding the students when solving sentence-based mathematics problems through oral questioning has ensured that the learning takes place in a student-centered manner, providing opportunities for the students to think and solve problems independently ( Mahmud and Yunus, 2018 ). This action is highly encouraged because teaching mathematics through the conventional approach is only effective for a short period, as the students can lack an understanding or fail to remember the mathematical concepts presented by the teacher ( Ali et al., 2021 ).

In addition, this study also found that the participants used the mastery approach to overcome the challenges of poor reading skills and poor mastery of mathematical concepts among the students. The mastery approach was used because it can provide more opportunities and time for the students to improve their reading skills and mastery of mathematical concepts ( Shawan et al., 2021 ). This approach has ensured that all students achieve the teaching objectives and that the teachers have time to provide enrichment and rehabilitation to the students as part of mastering the basic skills needed to solve sentence-based mathematics problems. This approach is very effective at adapting students to solving sentence-based mathematics problems according to the solution steps of the Polya model as well as the mathematical concepts learned in relation to a particular topic. The finding is in line with Ranggoana et al. (2018) and Mahmud (2019) study, which has shown that teaching through a mastery approach can enhance the student’s learning activities. This situation clearly shows that the mastery approach has ensured that the students have time to learn at their own pace, where they often try to emulate the solution shown by the teacher to solve a sentence-based mathematics problem.

Besides that, this study also found that mathematics teachers apply contextual learning approaches when teaching and facilitating sentence-based mathematics problem-solving skills. In this study, mathematics teachers have linked non-routine problems with examples from everyday life to guide the students with poor language literacy to help them understand non-routine problems and plan appropriate solution strategies. Such relationships can help the students process non-routine problems or mathematical concepts in a more meaningful context where the problem is relevant to real situations ( Siew et al., 2016 ). This situation can develop the students’ skill of solving sentence-based math problems where they can choose the right solution strategy to solve a non-routine problem. This finding is consistent with the results of Afni and Hartono (2020) . They showed that the contextual approach applied in learning could guide the students in determining appropriate strategies for solving sentence-based math problems. These findings are also supported by Seliaman and Dollah (2018) who stated that the practice of teachers giving examples that exist around the students and in real situations could make teaching and the subject facilitation easier to understand and fun.

Furthermore, the game approach was also used by the participants when imparting sentence-based mathematics problem-solving skills. According to Sari et al. (2018) , the game approach to teaching mathematics can improve the student learning outcomes because the game approach facilitates the learning process and provides a more enjoyable learning environment for achieving the learning objectives. In this study, the game approach was used by the teachers to overcome the challenge of mastering the concept of unit conversion, which was not strong among the students ( Tobias et al., 2015 ; Hui and Mahmud, 2022 ). The participants used the game approach to teach induction sets that guided the students in recalling mathematical concepts. The action provided a fun learning environment and attracted the students to learning mathematical concepts, especially in the beginning of the class. This situation is consistent with the findings of Muhamad et al. (2018) . They showed that the game approach improved the students’ problem-solving skills, interests, and motivation to find a solution to the problem.

Regarding the challenge of insufficient teaching time and a lack of ICT infrastructure, modular approaches such as flipped classrooms have been used to encourage students to learn in a situation that focuses on self-development ( UNESCO, 2020 ). In this study, the participants used instructional videos with related content, clear instructions, and worksheets as part of the Google classroom learning platform. The students can follow the instructions to engage in revision or self-paced learning in their spare time. This modular approach has ensured that teachers can deliver mathematical content and increase the effectiveness of learning a skill ( Alias et al., 2020 ). For students with unstable internet connections, the participants have used a modular approach to ensure that the students continue learning and send work through other channels such as WhatsApp, by email, or as a hand-in hardcopy. In short, an appropriate teaching approach needs to be planned and implemented by the mathematics teachers to help students master sentence-based mathematics problem-solving skills.

6. Conclusion

Overall, this study has expanded the literature related to the challenges when teaching sentence-based mathematics problem-solving skills and the approaches that can be applied while teaching to overcome the challenges faced. This study has shown that students have difficulty mastering sentence-based mathematics problem-solving skills because they do not achieve the minimum mastery of factual knowledge, procedural skills, conceptual understanding, and the ability to choose appropriate strategies ( Collins and Stevens, 1983 ). This situation needs to be taken into account because sentence-based mathematics problem-solving skills train the students to always be prepared to deal with problems that they will be faced with in their daily life. Through this study, teachers were found to play an essential role in overcoming the challenges faced by choosing the most appropriate teaching approach ( Baul and Mahmud, 2021 ). An appropriate teaching approach can improve the students’ sentence-based mathematics problem-solving skills ( Wulandari et al., 2020 ). Teachers need to work hard to equip themselves with varied knowledge and skills to ensure that sentence-based mathematics problem-solving skills can be delivered to the students more effectively. Finally, the findings of this study were part of obtaining extensive data regarding the challenges that mathematics teachers face when teaching sentence-based mathematics problem-solving skills and the approaches used to address those challenges in the process of teaching mathematics. It is suggested that a quantitative study be conducted to find out whether the findings obtained can be generalized to other populations. This is because this study is a qualitative one, and the findings of this study cannot be generalized to other populations.

The findings of this study can be used as a reference to develop the professionalism of mathematics teachers when teaching mathematical problem-solving skills. However, the study’s findings, due to being formulated from a small sample size, cannot be generalized to all mathematics teachers in Malaysia. Further studies are proposed to involve more respondents to better understand the different challenges and approaches used when teaching sentence-based mathematics problem-solving skills.

Data availability statement

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

Ethics statement

This study was reviewed and approved by The Malaysian Ministry of Education. The participants provided their written informed consent to participate in this study.

Author contributions

AL conceived and designed the study, collected and organized the database, and performed the analysis. AL and MM co-wrote the manuscript and contributed to manuscript revision. All authors read and approved the final submitted version.

The publication of this article is fully sponsored by the Faculty of Education Universiti Kebangsaan Malaysia and University Research Grant: GUP-2022-030, GGPM-2021-014, and GG-2022-022.

Acknowledgments

The authors appreciate the commitment from the respondent. Thank you to the Faculty of Education, Universiti Kebangsaan Malaysia, and University Research Grant: GUP-2022-030, GGPM-2021-014, and GG-2022-022 for sponsoring the publication of this article. Thanks also to all parties directly involved in helping the publication of this article to success.

Conflict of interest

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

Publisher’s note

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

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Keywords: mathematics, problem solving, teaching challenges, teaching approaches, primary school

Citation: Ling ANB and Mahmud MS (2023) Challenges of teachers when teaching sentence-based mathematics problem-solving skills. Front. Psychol . 13:1074202. doi: 10.3389/fpsyg.2022.1074202

Received: 19 October 2022; Accepted: 21 December 2022; Published: 01 February 2023.

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Copyright © 2023 Ling and Mahmud. 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: Muhammad Sofwan Mahmud, ✉ [email protected]

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• Open access
• Published: 04 November 2023

Exploring the multifaceted roles of mathematics learning in predicting students' computational thinking competency

• Silvia Wen-Yu Lee   ORCID: orcid.org/0000-0001-6111-2055 1 , 4 ,
• Hsing-Ying Tu 1 ,
• Guang-Lin Chen 2 &
• Hung-Ming Lin 3  

International Journal of STEM Education volume  10 , Article number:  64 ( 2023 ) Cite this article

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There exist shared competencies between computational thinking (CT) and mathematics, and these two domains also mutually benefit from various teaching approaches. However, the linkages between mathematics and computational thinking lack robust empirical support, particularly from student-centered learning perspectives. Our study aimed to enhance our understanding of the connections between students' mathematics learning and computational thinking. To assess students' mathematics learning, we measured their beliefs about mathematics learning and their level of mathematical literacy (ML). Our hypothesis posited that students' beliefs concerning mathematics learning, encompassing their views on the nature of mathematics and their attitude towards the subject, can both directly and indirectly influence their CT, with ML serving as a mediating factor. Our data were gathered through surveys and tests administered to eighth- and ninth-grade students. Data were analyzed using partial least squares–structural equation modeling (PLS–SEM).

The evaluation of the measurement model indicated strong internal consistency for each construct. Both convergent and discriminant validity were also established. Upon assessing the structural model, it was found that beliefs about the nature of mathematics positively predicted attitudes towards mathematics, and this belief also indirectly predicted ML through positive attitudes towards mathematics. In addition, ML directly and positively predicted both CT subscales. Notably, a comprehensive mediating effect of ML on beliefs about mathematics learning and CT was identified in the analysis.

Conclusions

This study advances the understanding of the relationships between mathematics learning and CT. We have further confirmed the importance of mathematical literacy in predicting CT and its mediating role between beliefs about mathematics learning and CT. It is suggested that teachers could promote students’ CT competence by enhancing their mathematical literacy or integrating mathematics and CT into the same learning activities. Finally, we propose that upcoming investigations treat CT assessments as formative constructs, diverging from their reflective counterparts.

Introduction

Computational thinking (CT) is a widely applicable form of literacy in the twenty-first century that is necessary for solving problems in various domains (Çoban & Korkmaz, 2021 ; Grover & Pea, 2018 ; Peel et al., 2022 ). The importance of providing students with CT-integrated education has been gaining global attention, and the presence of CT in K-12 classrooms has also been increasing over the last decade (Hurt et al., 2023 ; Lee & Lee, 2021 ; Shute et al., 2017 ; Weintrop et al., 2021 ; Ye et al., 2023 ). CT is regarded as a series of thinking abilities that use fundamental concepts of computer science to solve problems (Wing, 2006 ). It integrates mathematical thinking, engineering thinking and scientific thinking in solving problems, designing and evaluating systems, and understanding intelligence and human behavior (Wing, 2008 ). Researchers have also claimed that CT should be considered as a domain-general ability, that is, the ability to solve complex daily-life problems (Li et al., 2020b ; Yadav et al., 2016 ). In other words, CT involves solving problems using the abilities to order logically, analyze data, and create solutions by following sequences and rules. It also allows students to apply what they have learned in real-life situations (Yadav et al., 2016 ).

Recent literature provides valuable insights into the intricate interplay between CT and mathematics education, as well as the connection between CT and mathematical thinking (MT). From a competency perspective, CT and mathematics share common competencies, such as problem solving, modeling, analyzing and interpreting data, statistics and probability (Sneider et al., 2014 ). Mathematics thinking and computational thinking are similar in terms of abstract problem solving, and they may support each other (Rycroft-Smith & Connolly, 2019 ). Researchers have also claimed that the process of mathematical problem solving involves components of CT (Denning, 2017 ; Nurhayati & Lutfianto, 2020 ).

From a teaching perspective, mathematical activities can be used as a starting point for CT (Kallia et al., 2021 ), and CT and mathematics learning objectives can be integrated into the same curriculum (Chan et al., 2023 ; Pei et al., 2018 ). Through a systemic literature review, Khoo et al. ( 2022 ) found that CT was implemented into mathematics education through two common approaches, namely, using software tools as a medium for CT in the mathematics curriculum, and teaching CT as a way of thinking. The former approach was also commonly used in engineering and STEM, where computing skills and computer programming were taught using CT in relation to mathematics (Ersozlu et al., 2023 ). In another study, Wu and Yang ( 2022 ) reviewed how CT and MT have been integrated into mathematics education. They concluded three types of relationships between CT and MT, namely: (1) contribution of CT to MT, usually when using software and programming; (2) contribution of MT to CT in problem solving, and (3) reciprocal relationships between CT and MT through embedding CT into mathematics education.

Although the aforementioned studies have classified the relationships between mathematics and CT from teaching perspectives, empirical evidence from learners' perspectives and competence in this area remains limited. Moreover, while past studies have been using qualitative methods to explore the relationships between mathematics and CT teaching, there is a lack of statistical models that accurately depict the direct and indirect relationships among constructs related to mathematics learning and CT (Lv et al., 2023 ). Thus, our study aimed to gain insights into the associations between students’ mathematical learning and CT through quantitative modeling. In our model, we specifically include beliefs about learning mathematics as a crucial component. Learners' beliefs about, attitude towards, and perceptions of mathematics can significantly influence their engagement in, motivation for, and approach to mathematical learning (Gjicali & Lipnevich, 2021 ; Metzger et al., 2019 ). To adopt a competency-focused perspective and to emphasize the real-life applicability of mathematics and computational thinking, we chose to measure two key constructs: mathematical literacy and non-programming CT competence.

The Organization for Economic Cooperation and Development (OECD, ) defines mathematical literacy as the ability to engage with and use mathematics in various contexts, effectively applying mathematical concepts, reasoning, and problem-solving skills to solve real-life problems. It encompasses the capacity to understand, interpret, and critically evaluate mathematical information. Non-programming CT competence refers to the ability to think critically and algorithmically, solve problems, analyze data, and make informed decisions using computational thinking strategies that do not require programming (Selby & Woollard, 2013 ; Shute et al., 2017 ). Further details about our hypothesized model will be provided later, offering a more comprehensive explanation.

Computational thinking (CT)

Researchers have provided various definitions of CT. On one hand, CT has been described as being related to computer programming and computer concepts. For instance, CT has been defined as identifying computational aspects of the world and applying computer science tools and methodologies to comprehend the functioning of natural and man-made systems (Royal Society, 2012 ). Brennan and Resnick ( 2012 ) have specifically defined three key dimensions of CT: computational concepts, computational practices, and computational perspectives. Looking through a data modeling lens, Weintrop et al. ( 2016 ) classified CT into four categories, namely, data practices, modeling and simulation, computational problem solving, and systems thinking. In sum, the programming-related CT definitions are diverse and focus on different aspects of computing or programming.

On the other hand, CT has been recognized as a fundamental competence for general problem solving and as a cognitive process rooted in logical reasoning (Csizmadia et al., 2015 ; Tang et al., 2020 ). Many definitions encompass algorithmic thinking, decomposition, pattern generalization, abstractions, and evaluation, which can be effectively utilized for curriculum development and assessment (Csizmadia et al., 2015 ; Selby & Woollard, 2013 ; Shute et al., 2017 ). According to Csizmadia et al.'s ( 2015 ) suggested CT definitions, algorithmic thinking refers to the capability to discern sequences and rules to address issues or understand scenarios. Decomposition involves the ability to break down problems into components, while generalization is the ability for solving problems rapidly based on previous solutions. In addition, abstraction denotes the skill to eliminate unnecessary details for making problem solving more feasible and effective. Finally, evaluation pertains to the capacity to ascertain if a solution, whether an algorithm, system, or process, is suitable for its intended purpose. In our study, we adopted the CT definition aligned with the problem-solving perspective and assessed students' thinking skills in real-life situations (Selby & Woollard, 2013 ; Shute et al., 2017 ).

Scholars have developed different assessment tools based on the aforementioned definitions of CT, and the assessment tools for CT are categorized into self-reported CT scales and CT tests (Cutumisu et al., 2019 ). An instance of a CT scale is the Computational Thinking Scale (CTS) devised by Korkmaz et al. ( 2017 ), comprising subscales, such as algorithmic thinking, creativity, cooperativity, critical thinking, and problem solving. CT tests could be further categorized into two subtypes: domain-specific CT assessment, such as programming-based or computer concept-based CT assessment, and domain-general CT assessment. An illustration of a domain-specific assessment is the computational thinking test (CTT), a programming-based CT assessment introduced by Román-González et al. ( 2018a ). It was designed to assess middle school students’ CT skills through computer science concepts, such as sequences and loops. A sample question asks students which code makes the correct path that takes Pac-Man to the ghost. An example of domain-general CT assessment is the Bebras® Computing Challenge, an international competition for informatics and computational thinking ( https://www.bebras.org/ ). Bebras aims to foster students’ CT interest and competence by solving problems based on real-life situations. Devoid of any programming involvement, the Bebras challenge aims to evaluate participants' abilities, whether directly or indirectly linked to CT (Dagiene & Stupuriene, 2016 ). In this study, the CT measurement was derived from Bebras to assess students' proficiency in applying their CT skills to diverse problems and situations (del Olmo-Muñoz et al., 2020 ).

• Mathematical literacy

Researchers have also uncovered the interplay between ML and various psychological factors, including enhanced mathematics self-efficacy, motivation, mathematics intentions, and perseverance (Kitsantas et al., 2021 ; Ozgen, 2013 ; Rahmi et al., 2017 ; Skaalvik et al., 2015 ). Mathematical Literacy (ML) encompasses a spectrum of attributes, spanning from adept mathematical reasoning and comprehensive content knowledge to a positive mathematical disposition. It further encapsulates an awareness of mathematics' practical utility and an appreciation of the essence and nature of mathematics (Edge, 2009 ).

Beyond mere proficiency in mathematical knowledge and skills, mathematical literacy (ML) encompasses the adeptness and confidence to apply mathematical insights within real-life contexts (Ojose, 2011 ). In other words, the significance of mathematical literacy lies in its potential to inspire students to address practical problems rooted in mathematical concepts (Consortium for Mathematics and Its Applications [COMAP], 2015 ). This imperative is reflected in the assessment of ML within the Programme for International Student Assessment (PISA) test, where it emphasizes the application of mathematics in authentic scenarios. Here, students' capacity for mathematical reasoning and depicting relationships holds more weight than their proficiency in tackling conventional textbook queries, serving as a robust indicator of their mathematical competencies (Lin & Tai, 2015 ).

In this study, we adopted the ML definitions coined by the Organization for Economic Co-operation and Development (OECD) by which ML is as "an individual's capacity to formulate, employ, and interpret mathematics across a spectrum of situations" (OECD, 2013 , p. 25). The term employ refers to the application of accumulated mathematical knowledge, which encompasses principles, methodologies, facts, and tools, to address mathematical challenges. Formulate signifies the ability to identify opportunities for applying mathematics and to transform problems into patterns that align with mathematical structural representations. Interpret refers to the ability to recognize and explain the relationship between mathematical solutions or responses and the contexts of real-world scenarios (OECD, 2019 ).

Beliefs related to mathematics learning

In the field of cognitive psychology, there has been increasing interest in studying students’ beliefs about mathematics learning (Jin et al., 2010 ; Leder et al., 2006 ). Mathematical beliefs refer to the personal opinions, attitude, or values that individuals hold about mathematics and the nature of mathematics (Ernest, 1989 ; Underhill, 1988 ). Other researchers have also included students’ perceptions of mathematics usefulness, mathematics learning, and mathematics teaching as constructs of beliefs about mathematics learning (Lazim, 2004 ; McLeod, 1992 ). Researchers have found empirical evidence of a relationship between students’ beliefs about the nature of mathematics and their mathematics learning. Students with positive beliefs about and attitude towards mathematics tend to perform better (Papanastasiou, 2000 ). Studies have shown that beliefs about mathematics learning could positively predict mathematics outcomes, such as performance on mathematics tests (Bonne & Johnston, 2016 ; House & Telese, 2008 ; Suthar & Tarmizi, 2010 ; Suthar et al., 2010 ). Scholars have suggested that future research could further investigate beliefs about the nature of mathematics and about students’ mathematical capability due to their relationships with students’ mathematics self-efficacy and their mathematics achievements (Yin et al., 2020 ).

Hypothesized model

In this study, we hypothesized that students’ beliefs about the nature of mathematics and their attitude towards mathematics could both directly and indirectly predict their CT through the mediation of mathematics performance. First, we hypothesized that beliefs about mathematics learning are predictors of ML and CT. These hypotheses have been based on previous studies which found that students’ beliefs may contribute to higher levels of thinking or competencies. As mentioned earlier, studies have shown that beliefs about mathematics learning can positively predict mathematics outcomes (Bonne & Johnston, 2016 ; House & Telese, 2008 ; Suthar & Tarmizi, 2010 ; Suthar et al., 2010 ). Sanico’s ( 2019 ) study demonstrated that mathematics belief was positively correlated with mathematics problem-solving performance, but could only indirectly predict mathematical problem-solving performance. Schommer-Aikins et al. ( 2005 ) also discovered that mathematical problem-solving beliefs and epistemological beliefs could positively predict students’ mathematics performance. To date, few studies have explored the relationships between beliefs about mathematics learning and CT. Nevertheless, other beliefs have been found to be predictors of CT. For instance, research has shown that students’ beliefs about computer programming can predict their computational thinking (Lee et al., 2023 ).

Second, we hypothesized that ML predicts students’ CT. Since our study focused on non-programming CT, we adopted the perspective of the “contribution of mathematical thinking to CT,” as previously mentioned (Wu & Yang, 2022 ). Studies integrating CT into mathematics education have shown positive results regarding the relationship between CT and mathematics learning outcomes (Sung & Black, 2020 ; Sung et al., 2017 ; Suters & Suters, 2020 ). For instance, Özgür ( 2020 ) conducted research on the relationship between CT and students’ previous mathematics academic achievement through structural equation modeling (SEM). The results demonstrated that students’ previous mathematics academic achievement was not only positively correlated with CT, but could also positively predict CT, and the same result was found by Durak and Saritepeci ( 2018 ) and Finke et al. ( 2022 ). Guggemos ( 2021 ) measured CT and mathematical skills at three different timepoints to see the relationships between these two types of skills. The results showed that CT at any timepoint was positively correlated with students’ mathematical skills, and CT was positively predicted by mathematical skills. What is worth mentioning here is that most of the aforementioned studies studied mathematics achievement. Mathematics achievement measures an individual's performance within the confines of a specific curriculum or learning content within a certain period of time (i.e., a semester), while mathematical literacy, as the focus of the current study, goes beyond this and assesses the practical application of mathematics knowledge in real-world contexts. Nevertheless, both concepts are important for overall mathematics proficiency, and are interconnected.

Finally, we hypothesized relationships between the sub-scales of beliefs about mathematics learning and the sub-scales of CT. Our primary interest was to explore potential hierarchical relationships between these two sets of sub-scales. Furthermore, we chose to use first-order rather than second-order constructs as they offer a more straightforward representation of relationships (Hair et al., 2021 ).

Specifically, we hypothesized that beliefs about nature of mathematics would predict attitude towards mathematics. However, we also acknowledged the possibility of a reciprocal relationship, where the nature of mathematics could also predict attitude towards mathematics. Beliefs about nature of mathematics represents a learner's philosophical orientation towards the nature of knowledge, while attitude towards mathematics reflects self-evaluations of emotions, interest, motivation, and perceived usefulness of the subject. We assume that an epistemic and philosophical construct can serve as a predictor of an affective construct.

In a previous study, Tsai et al. ( 2022 ) found that CT dispositions were divided into lower level and higher level based on the level of cognitive complexity and the information-processing theory of human problem solving. Tsai et al. ( 2022 ) showed that lower level CT components, such as abstraction and decomposition, significantly predicted higher level CT, including algorithmic thinking, evaluation, and generalization. We aimed to investigate whether a similar pattern exists in the context of CT competence.

In this study, the measurement of CT, the computational thinking test for Junior High Students (CTT–JH) (Lee et al., 2023 ), was adapted and modified from the Bebras Challenges and was used to determine how well students could apply their CT skills to real-life problems and situations (del Olmo-Muñoz et al., 2020 ). This instrument focused on a generous process of problem solving which is independent of the type of programming language. In addition, this instrument has been well-validated and has good reliability overall, as well as in all five dimensions based on empirical data (Lee et al., 2023 ).

In sum, the hypothesized model is shown in Fig.  1 . The purposes of this study were (1) to investigate the roles of beliefs about mathematics learning and mathematical literacy in predicting CT competence, and (2) to investigate the mediating role of ML in the proposed model.

Hypothesized model showing the relationships between beliefs about the nature of mathematics, attitude towards mathematics, mathematical literacy, and computational thinking (lower level and higher level)

Research instruments

Computational thinking test.

We adapted items from the measurement of CT, the computational thinking test for Junior High Students (CTT–JH) (Lee et al., 2023 ). The test was developed based on Bebras Challenge tasks for assessing five dimensions of CT, namely, abstraction, decomposition, algorithmic thinking, evaluation, and generalization. These five dimensions were defined as the competence of abstracting essential information, breaking down complicated problems into manageable parts, thinking procedurally as a sequence of steps to reach a solution, deciding the most appropriate solution to the problem, and adapting and transferring solutions to other problems (Lee et al., 2023 ). Each item was designed to assess one or more CT dimensions simultaneously, and collectively CTT–JH is a multi-dimensional research instrument. The assessment comprises multiple-choice questions, with each correct answer earning 1 point. To further categorize the items into lower and higher levels of CT, the first author and two other experts in computing education and CT research reviewed the items. The group of experts examined the CT dimensions assessed by each item and, as a result, four items assessing lower level CT and four assessing higher level CT were selected. Lower level CT primarily assesses abstraction and decomposition, while higher level CT assesses algorithmic thinking, evaluation, and generalization (Tsai et al., 2022 ). A sample question is shown in Appendix 1 . The corresponding CT dimensions for each item are shown in Appendix 2 .

Mathematical literacy test

The ML measurement was adapted from the mathematical reasoning items of PISA 2012. A total of nine items were included in the test in this study, with three questions for each of the employ, formulate and interpret constructs. Each item in the assessment consisted of a multiple-choice question, with 1 point awarded for a correct answer. Students’ ability to work out solutions to mathematical problems through applying mathematical concepts ( employ ), transform problems into mathematical structures ( formulate ), and determine and explain the connection between mathematical solutions and real-life situations ( interpret ) were assessed. To further ensure the content validity of the items, the first author and two other experts in mathematics education reviewed and selected the items. Sample questions are shown in Appendix 3 .

The beliefs about the nature of mathematics scale and the attitude towards mathematics scale

The items for both the Beliefs about the Nature of Mathematics Scale and the Attitude towards Mathematics Scale were adapted from a questionnaire about beliefs in mathematics (Lazim et al., 2004 ; Su, 2018 ). Beliefs about the nature of mathematics refers to students' perspectives on mathematics and its significance in their lives. Attitude towards mathematics encompasses students' holistic evaluation, perception, and emotional stance towards mathematics, encompassing their preferences, confidence, interest, and perceived utility or relevance of mathematics in their daily lives. We made the decision to rename the original “self-evaluation” scale to “attitude towards mathematics” to better capture the essence of the items included in this scale.

The initial questionnaire consisted of five items each for beliefs about the nature of mathematics (NM) and attitude towards mathematics (AM). All items were rated on a 5-point Likert scale, ranging from 1 ( strongly disagree ) to 5 ( strongly agree ). For example, an NM sample item is “Mathematics enables people to understand the world better,” while an AM sample item is “I have been performing well in mathematics exams.” During the PLS–SEM analysis, one item from the original set of questions for the NM scale was excluded due to its low factor loading. For a comprehensive list of the remaining items in both scales, please refer to Table 1 in the Results section.

Data collection and data analysis

Participants were recruited from eighth- and ninth-grade students attending four junior high schools in central Taiwan. Data collection took place in both urban and rural areas, with two schools selected from each setting. This approach ensured the inclusion of a more diverse range of socio-economic backgrounds. Out of the 265 students who initially participated in the study, 247 valid samples were retained, comprising 131 females and 116 males. Data were collected using face-to-face paper-and-pencil questionnaires and tests. Prior to administering the survey, the research team obtained oral consent from each participant. This process involved explaining the study's purpose and nature, highlighting voluntary participation, and assuring the confidentiality and anonymity of responses. Participants were assured that their test results would not impact their school grades. Their rights were fully disclosed, and they had the option to decline participation or withdraw at any time without any consequences. The research team also ensured that participants had the opportunity to ask questions and seek clarification before giving their oral consent. Subsequently, the questionnaires and tests were administered consecutively, with the entire process typically taking no more than 90 min to complete.

The decision to employ partial least squares–structural equation modeling (PLS–SEM) instead of covariance-based structural equation modeling (CB–SEM) in this study was driven by several considerations. First, due to the relatively small amount of data, PLS–SEM was deemed more appropriate. Second, PLS–SEM is recognized for its strong predictive capabilities, as it focuses on minimizing residual variance and enabling accurate predictions based on the model (Lin et al., 2020 ). This aligns well with the study's focus on variables related to mathematics learning that potentially predict CT.

Third, PLS–SEM offers the advantage of accommodating both formative and reflective constructs within the model (Hair et al., 2021 ). Reflective constructs represent latent variables measured by observed indicators, while formative constructs are composed of observed variables that collectively define the latent variable. Formative constructs are particularly useful when the construct being measured is seen as a combination of different dimensions or factors, with each observed variable contributing uniquely to the construct (Hair et al., 2021 ). In this study, while beliefs about the nature of mathematics and attitude towards mathematics were treated as reflective constructs, the CT test and ML test were considered as formative constructs. In sum, by utilizing PLS–SEM, this study addressed the limitations posed by the small amount of data, leveraged the predictive capabilities of the model, and accommodated both formative and reflective constructs.

Two-stage evaluation was employed using PLS–SEM: evaluation of the measurement model and evaluation of the structural model. The evaluation criteria for the measurement model (i.e., the first stage evaluation) have to be met before further analyzing the structural model (i.e., the second-stage evaluation). For the measurement model, the reflective constructs were evaluated by assessing the internal consistency reliability, convergent validity, and discriminant validity. The formative constructs were evaluated by assessing variance inflation factor values (VIF), and outer weights. The structural model was then evaluated using the PLS–SEM algorithm and bootstrapping resampling to test the statistical significance of the path coefficients. The path weighting scheme was selected as the weighting method, with a threshold value of 1·10 –5 (i.e., stop criterion), and a value of at least 300 for the maximum number of iterations (Hair et al., 2021 ). This study applied the bootstrapping procedure with 247 cases and 5000 samples to test the contribution of the formative indicators to their associated constructs and the structural path significance.

To assess the predictive power of the model, several measures were calculated, including the effect size ( f 2 ), coefficient of determination ( R 2 value), and Q 2 value. The R 2 value quantifies the extent to which the exogenous constructs linked to an endogenous construct explain its variance (Sarstedt et al., 2016 ). The effect size f 2 , on the other hand, quantifies the magnitude of change in R 2 when a construct is omitted from the model, offering insights into the influence of one construct on another (Hair et al., 2021 ). In addition, the Q 2 value (Geisser, 1974 ) serves as another measure to assess the internal model, specifically its predictive relevance for a given endogenous construct.

Measurement model

Construct reliabilities and construct validities were tested to examine the quality of the measurement model. Reliability for verifying internal consistency of the indicators of each construct was tested by the composite reliability (CR) values and the Cronbach’s alpha values (Fornell & Larcker, 1981 ). As shown in Table 1 , the CR values of the constructs were 0.82 and 0.93, which were above the suggested value of 0.70. The Cronbach’s alpha values of the constructs were 0.71 and 0.90, which also met the requirement of being greater than 0.70. These results show that the internal consistency of the indicators for each construct was good, and the measurement model had sufficient reliability.

To verify whether the measurements effectively reflected the corresponding measured constructs, convergent validity and discriminant validity were assessed. Convergent validity of the measurements was validated by factor loadings of indicators and the average variance extracted (AVE) of constructs (Hair et al., 2019 ). The factor loadings of the individual items were all above 0.6 (see Table 1 ), which is acceptable according to Hair et al. ( 2021 ). The AVE values of the constructs were 0.53 and 0.72, which were higher than the suggested value of 0.5. In accordance with Fornell and Larcker ( 1981 ), these results suggested adequate convergent validity.

Discriminant validity, which refers to the degree to which each construct in the resulting model is distinct from the others, was measured through the Fornell–Larcker criterion (Fornell & Larcker, 1981 ) and cross loadings. The cross loadings of measurement variables should be higher than the related latent variable (Chin, 1998 ). The correlation matrix and the square root of the AVE value of Nature of Mathematics and Attitude towards Mathematics are shown in Table 2 . The results showed that the two variables were significantly correlated with each other ( r  = 0.56, p  < 0.001) and the square root of the AVE value of each variable was higher than 0.5 (0.73, 0.85) and larger than the Pearson’s correlation coefficient between the two variables. As shown in Table 3 , all cross loadings were higher than each related latent variable. Therefore, the discriminant validity of the variables was confirmed (Hair et al., 2021 ).

In our analysis, the collinearity, significance, and relevance of formative indicators or items within the measurement models were assessed by variance inflation factor (VIF), outer weights, and outer loadings for this analysis. In this research, both CT and ML were considered as formative constructs. Moreover, CT was categorized into lower level (LLCT) and higher level CT (HLCT). The presence of acceptable collinearity and adequate construct validity were signified by VIF values less than 5. This indicated that an item's contribution to the primary latent construct was unique, as noted by Hair et al. ( 2021 ). As shown in Table 4 , the VIF values for all items in this study ranged from 1.02 to 1.25.

According to Hair et al. ( 2021 ), an item's relative importance in formative constructs is determined by its outer weight, while its absolute importance to the construct is determined by its outer loading. An item was kept in the measurement model if it had a significant outer weight ( p  < 0.05), or if its outer loading was higher than 0.5. Items that did not meet these criteria were further evaluated based on the significance of their outer loading. If an item's outer loading was lower than 0.5 and not significant, indicating no absolute importance to the construct, it was ultimately removed from the model. As per these criteria, the items that were retained are shown in Table 4 .

The structural relationships

We used PLS–SEM to test our proposed theoretical hypotheses, which included the relationships between NM, AM, ML, and the higher and lower levels of CT. The paths with statistical significance ( p  < 0.05) are shown in Fig.  2 . The results indicated that beliefs about the nature of mathematics positively predicted attitude towards mathematics ( β  = 0.56, t  = 11.22, p  < 0.001); attitude towards mathematics positively predicted mathematical literacy ( β  = 0.48, t  = 7.06, p  < 0.001); mathematical literacy positively predicted lower level computational thinking ( β  = 0.42, t  = 6.40, p  < 0.001) and higher level computational thinking ( β  = 0.38, t  = 5.77); and lower level computational thinking positively predicted higher level computational thinking ( β  = 0.34, t  = 5.61, p  < 0.001).

Structural model results (only significant paths are shown). NM: beliefs about the nature of mathematics; AM: attitude towards mathematics; ML: mathematical literacy; LLCT: lower level CT; HLCT: higher level CT. *** p  < 0.001

In addition, as shown in Fig.  2 , the f 2 effect sizes for significant paths were all larger than 0.15, showing a moderate or large effect according to Cohen’s ( 1988 ) suggestion that 0.02, 0.15 and 0.35 indicate small, moderate, and large effects, respectively. It is worth noting that the f 2 value of beliefs about the nature of mathematics predicting attitude towards mathematics was 0.45, which showed a large effect on attitude towards mathematics.

The predictive validities of the model were indicated by the coefficient of determination (R 2 ) and predictive relevance ( Q 2 ) (see Fig.  2 ). The endogenous constructs including attitude towards mathematics ( R 2  = 30.95%), mathematical literacy ( R 2  = 18.33%), lower level CT ( R 2  = 20.09%) and higher level CT ( R 2  = 40.06%) present acceptable levels of explained variance (Chin, 1998 ; Tenenhaus et al., 2005 ). Moreover, the predictive relevance is evaluated through Stone–Geisser’s Q 2 test. Q 2 greater than 0 implies that the model has predictive relevance, whereas a Q 2 less than 0 suggests that the model lacks predictive relevance (Stone, 1974 ). Our model shows positive Q 2 values of the endogenous constructs (see Fig.  2 ; attitude towards mathematics Q 2  = 0.216, mathematical literacy Q 2  = 0.049, lower level CT Q 2  = 0.053 and higher level CT Q 2  = 0.136) which suggests that the model has predictive validity.

As shown in Table 5 , attitude towards mathematics, mathematical literacy and lower level computational thinking played significant mediating roles. Attitude towards mathematics mediated the relationship between beliefs about the nature of mathematics and mathematical literacy ( β  = 0.269, p  < 0.001), while mathematical literacy mediated the relationship between attitude towards mathematics and lower level computational thinking ( β  = 0.203, p  < 0.001), and between attitude towards mathematics and higher level computational thinking ( β  = 0.181, p  < 0.001). In addition, lower level computational thinking mediated the relationship between mathematical literacy and higher level computational thinking ( β  = 0.142, p  < 0.001).

Analysis showed a serial mediation effect between beliefs about the nature of mathematics and lower level computational thinking through attitude towards mathematics and mathematical literacy ( β  = 0.113, p  < 0.001), and also between beliefs about the nature of mathematics and higher level computational thinking ( β  = 0.101, p  < 0.001). Moreover, there was a serial mediation effect between beliefs about the nature of mathematics and higher level computational thinking through attitude towards mathematics, mathematical literacy, and lower level computational thinking ( β  = 0.038, p  = 0.002). Finally, a serial mediation effect between attitude towards mathematics and higher level computational thinking through mathematical literacy and lower level computational thinking was also shown by the analysis ( β  = 0.069, p  = 0.001).

As there were no significant direct relationships between beliefs about the nature of mathematics and mathematical literacy, the indirect effect for the mediation by attitude towards mathematics was full mediation. Similarly, the indirect effect for the mediation by mathematical literacy was full mediation as well. Because mathematical literacy directly predicted higher level computational thinking, the indirect effect for the mediation by lower level computational thinking was partial mediation.

Discussion and implications

This study proposed a hypothesized model to assess the relationships among beliefs about mathematics learning (beliefs about the nature of mathematics, and attitude towards mathematics), ML, and CT (lower level CT, higher level CT). According to the model analysis, it was confirmed that beliefs about mathematics learning could positively predict ML, and ML could, in turn, positively predict CT. Although the direct relationship between beliefs about mathematics learning and CT was not found in the model, beliefs about mathematics learning could indirectly predict CT through ML, which confirms the mediating role of ML in the proposed model. In addition to the above findings, beliefs about the nature of mathematics were found to positively predict attitude towards mathematics, and lower level CT was also confirmed to positively predict higher level CT. In the following, we will discuss our major findings.

First, it was found that CT was directly predicted by ML, indicating that an increase in students’ ML corresponded to an enhancement in their CT skills. These results resonate with the findings of Durak and Saritepeci ( 2018 ) and Özgür ( 2020 ). Wing’s ( 2008 ) comments indicating that CT is a concept that integrates mathematical thinking into solving a problem, designing and evaluating a system, and understanding intelligence and human behavior support this finding in this research. Similarly, Alyahya and Alotaibi's ( 2019 ) discovery of a significantly positive correlation between CT and TIMSS mathematics achievements aligns with our findings. In addition, the outcomes from Román-González et al. ( 2018b ) affirm that CT serves as a crucial indicator for estimating mathematics achievement, further reinforcing our study's outcomes.

Moreover, the findings indicated that students’ beliefs about mathematics could positively predict ML. Besides the positive correlation between beliefs about mathematics and ML, the increase in students’ beliefs about mathematics could positively increase their ML. Moreover, the results showed that beliefs about the nature of mathematics as part of beliefs about mathematics positively predicted the other subscale— attitude towards mathematics , which indicates that the stronger students’ thoughts about the value of mathematics, the higher their confidence in learning mathematics. A past study found that beliefs about the nature of mathematics was positively correlated with attitude towards mathematics (Kaldo & Hannula, 2014 ), and this study provides further indication of the direction of the predictive relationship. Nevertheless, beliefs about the nature of mathematics only indirectly predicted ML through attitude towards mathematics . This outcome might be attributed to the abstract nature of beliefs about the nature of mathematic s for junior high school students. In this stage of learning, students’ attitude towards mathematics plays a pivotal role in predicting their mathematical literacy. Future studies could explore whether older students, such as high school or post-secondary students, possess a better grasp of these abstract concepts.

Finally, our findings revealed that beliefs about mathematics positively predicted CT, albeit through the intermediary of ML. In essence, enhancing students' perceptions of mathematics and its significance in life not only boosts their confidence in learning mathematics but also augments their real-life problem-solving abilities. While previous studies primarily focused on the relationship between pairs of constructs, our results extend this examination to encompass the relationships among all three constructs.

There are two implications of teaching mathematics and CT. This study demonstrated the importance of students’ beliefs about mathematics and ML in predicting CT competence. Therefore, to promote CT competencies, in addition to providing more learning opportunities for CT, improving students’ understanding of mathematics may be beneficial. Specifically, researchers and teachers should pay more attention to students’ beliefs about mathematics in addition to promoting their mathematical literacy. Some researchers have suggested that students’ beliefs about mathematics depend on their mathematics learning experiences and their view of mathematics (Lazim et al, 2004 ). One approach to enhancing students’ mathematical literacy and providing meaningful mathematics learning experiences is through integrating real-life situations into mathematics learning (Barcelos et al., 2018 ; Cui et al., 2023 ). Designing a mathematics curriculum based on topics relevant to students’ daily life experiences may help students develop mathematical literacy and views of mathematics (Amirali, 2010 ), which, in turn, can help them develop CT competencies.

Second, we suggest integrating the teaching of CT with mathematics, such as designing STEM (science, technology, engineering, and mathematics) -oriented activities or curricula (Zhang et al., 2023 ). Recently, the trend of incorporating CT into interdisciplinary education, particularly within STEM domains, has been noted (Lee et al., 2020 ; Li et al., 2020a ). In addition to gaining knowledge and skills, students can develop multi-dimensional and multidisciplinary competence through integration (Dolgopolovas & Dagienė, 2021 ). Studies have also shown the benefit of STEM content for CT learning (Bortz et al., 2020 ). Studies have implemented problem-based instruction, games, or robots in the STEM curriculum for students of various ages and for pre-service teachers as well (Wang et al., 2021 ; Wawan et al., 2022 ). Examples include using Scratch (Rodríguez-Martínez et al., 2020 ) or the integrated 6E Learning by DeSIGN™ Instructional Model (i.e., engage, explore, explain, engineer, enrich, evaluate) with LEGO robots to improve students’ CT (Chiang et al., 2022 ), and designing an engineering-based activity at a STEM camp (Shang et al., 2023 ). There are also studies that have implemented innovative teaching methods in the curriculum, such as integrating CT into mathematics education through programming role-playing or embodiment (Sung et al., 2017 ).

Finally, the findings of this study also have some methodological implications. First, while most of the studies treated the CT scale or CT assessment as a reflective construct (e.g., Korkmaz et al., 2017 ), in this study, CT was treated as a formative construct. Researchers have also suggested the importance of carefully considering the nature of the items and making informed decisions regarding whether to treat the measurement as formative or reflective to prevent model mis-specification (Lin et al., 2020 ). Second, based on Tsai et al. ( 2022 ), this study confirmed that CT could be divided into lower level (abstraction, decomposition) and higher level (algorithmic thinking, evaluation, and generalization), where lower level CT could positively predict higher level CT. The results of categorization in the previous study were based on CT disposition, and this study used assessment data to further verify the two-level structure of the constructs. Future studies should consider CT as a two-level rather than a one-level construct.

While this study contributes insights into the relationships between mathematics learning and computational thinking, it has some limitations that should be acknowledged. One limitation is the sample size and the age of the students. Due to the time needed to complete the survey and tests, we were only able to recruit a limited number of participants, which may not fully represent the diverse range of learners. In addition, our sample was limited to students at the junior high school level. The relationships between mathematics learning and computational thinking may change due to, for instance, students’ mastery of abstract thinking. Another limitation of the educational study is the exclusion of heuristic and meta-cognitive aspects of mathematics learning, such as the monitoring and control of problem-solving strategies (Schoenfeld, 1992 ). Furthermore, this study failed to account for personal attributes, such as self-efficacy, persistence, and creativity (Rozgonjuk et al., 2020 ). Neglecting these factors may hinder a comprehensive understanding of the multifaceted nature of mathematics learning. Incorporating these aspects into future research could provide a more holistic perspective on the relationships between mathematics learning and CT.

In this study, we proposed a model suggesting the relationships between students’ understanding of mathematics and CT. The model highlights the roles of mathematical literacy (ML) in both directly predicting CT competence and mediating the relationships between beliefs about mathematics and CT. Therefore, we suggest enhancing students’ mathematical literacy or integrating mathematics and CT into the same learning activities to promote CT competence. Another interesting finding is that attitude towards mathematics directly predicted students’ ML and also mediated the relationships between beliefs about the nature of mathematics and ML. The results indicate that students’ attitude towards mathematics may be a more important predictor than their beliefs about the nature of mathematics, at least for students at this stage. Finally, we recommend that future studies consider CT as a two-level construct and treat CT tests as a formative rather than a reflective construct.

Availability of data and materials

The data sets analyzed during the current study are not publicly available but are available from the corresponding author on reasonable request.

Abbreviations

Attitude towards mathematics

Average variance extracted

Composite reliability

• Computational thinking

Higher level computational thinking

Lower level computational thinking

Mathematical thinking

Beliefs about the nature of mathematics

Partial least square–structural equation modeling

Science, technology, engineering, and mathematics

Variance inflation factor values

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This work was supported by the National Science and Technology Council in Taiwan [Grant Number MOST 109-2511-H-003-052-MY3].

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SWL designed and supervised the research. She analyzed and interpreted the data and provided suggestions for the structure of the manuscript. She contributed to the writing and revisions. HYT analyzed the data and contributed to the writing and revisions. GLC contributed to data collection. HML provided suggestions regarding statistical methods.

This research is funded by the National Science and Technology Council in Taiwan. It was also supported by the “Institute for Research Excellence in Learning Sciences” of NTNU sponsored by the Ministry of Education (MOE) in Taiwan.

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Lee, S.WY., Tu, HY., Chen, GL. et al. Exploring the multifaceted roles of mathematics learning in predicting students' computational thinking competency. IJ STEM Ed 10 , 64 (2023). https://doi.org/10.1186/s40594-023-00455-2

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Challenges of teachers when teaching sentence-based mathematics problem-solving skills

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Sentence-based mathematics problem-solving skills are essential as the skills can improve the ability to deal with various mathematical problems in daily life, increase the imagination, develop creativity, and develop an individual’s comprehension skills. However, mastery of these skills among students is still unsatisfactory because students often find it difficult to understand mathematical problems in verse, are weak at planning the correct solution strategy, and often make mistakes in their calculations. This study was conducted to identify the challenges that mathematics teachers face when teaching sentence-based mathematics problem-solving skills and the approaches used to address these challenges. This study was conducted qualitatively in the form of a case study. The data were collected through observations and interviews with two respondents who teach mathematics to year four students in a Chinese national primary school in Kuala Lumpur. This study shows that the teachers have faced three challenges, specifically low mastery skills among the students, insufficient teaching time, and a lack of ICT infrastructure. The teachers addressed these challenges with creativity and enthusiasm to diversify the teaching approaches to face the challenges and develop interest and skills as part of solving sentence-based mathematics problems among year four students. These findings allow mathematics teachers to understand the challenges faced while teaching sentence-based mathematics problem solving in depth as part of delivering quality education for every student. Nevertheless, further studies involving many respondents are needed to understand the problems and challenges of different situations and approaches that can be used when teaching sentence-based mathematics problem-solving skills.

1. Introduction

To keep track of the development of the current world, education has changed over time to create a more robust and effective system for producing a competent and competitive generation ( Hashim and Wan, 2020 ). The education system of a country is a significant determinant of the growth and development of the said country ( Ministry of Education Malaysia, 2013 ). In the Malaysian context, the education system has undergone repeated changes alongside the latest curriculum, namely the revised Primary School Standard Curriculum (KSSR) and the revised Secondary School Standard Curriculum (KSSM). These changes have been implemented to ensure that Malaysian education is improving continually so then the students can guide the country to compete globally ( Adam and Halim, 2019 ). However, Malaysian students have shown limited skills in international assessments such as Trends in International Mathematics and Science Study (TIMSS) and the Program for International Student Assessment (PISA).

According to the PISA 2018 results, the students’ performance in mathematics is still below the average level of the Organization for Economic Co-operation and Development (OECD; Avvisati et al., 2019 ). The results show that almost half of the students in Malaysia have still not mastered mathematical skills fully. Meanwhile, the TIMSS results in 2019 have shown there to be a descent in the achievements of Malaysian students compared to the results in 2015 ( Ministry of Education Malaysia, 2020b ). This situation is worrying as most students from other countries such as China, Singapore, Korea, Japan, and others have a higher level of mathematical skills than Malaysian students. According to Mullis et al. (2016) , these two international assessments have in common that both assessments test the level of the students’ skills when solving real-world problems. In short, PISA and TIMSS have proven that Malaysian students are still weak when it comes to solving sentence-based mathematics problems.

According to Hassan et al. (2019) , teachers must emphasize the mastery of sentence-based mathematics problem-solving skills and apply it in mathematics teaching in primary school. Sentence-based mathematics problem-solving skills can improve the students’ skills when dealing with various mathematical problems in daily life ( Gurat, 2018 ), increase the students’ imagination ( Wibowo et al., 2017 ), develop the students’ creativity ( Suastika, 2017 ), and develop the students’ comprehension skills ( Mulyati et al., 2017 ). The importance of sentence-based mathematic problem-solving skills is also supported by Ismail et al. (2021) . They stated that mathematics problem-solving skills are similar to high-level thinking skills when it comes to guiding students with how to deal with problems creatively and critically. Moreover, problem-solving skills are also an activity that requires an individual to select an appropriate strategy to be performed by the individual to ensure that movement occurs between the current state to the expected state ( Sudarmo and Mariyati, 2017 ). There are various strategies that can be used by teachers to guide students when developing their problem-solving skills such as problem-solving strategies based on Polya’s Problem-Solving Model (1957). Various research studies have used problem-solving models to solve specific problems to improve the students’ mathematical skills. Polya (1957) , Lester (1980) , Gick (1986) , and DeMuth (2007) are examples. One of the oldest problem-solving models is the George Polya model (1957). The model is divided into four major stages: (i) understanding the problem; (ii) devising a plan that will lead to the solution; (iii) Carrying out the plan; and (iv) looking back. In contrast to traditional mathematics classroom environments, Polya’s Problem-Solving Process allows the students to practice adapting and changing strategies to match new scenarios. As a result, the teachers must assist the students to help them recognize whether the strategy is appropriate, including where and how to apply the technique.

In addition, problem-solving skills are one of the 21st-century skills that need to be mastered by students through education now so then they are prepared to face the challenges of daily life ( Khoiriyah and Husamah., 2018 ). This statement is also supported by Widodo et al. (2018) who put forward four main reasons why students need to master problem-solving skills through mathematics learning. One reason is that sentence-based mathematics problem-solving skills are closely related to daily life ( Wong, 2015 ). Such skills can be used to formulate concepts and develop mathematical ideas, a skill that needs to be conveyed according to the school’s content standards. The younger generation is expected to develop critical, logical, systematic, accurate, and efficient thinking when solving a problem. Accordingly, problem solving has become an element that current employers emphasize when looking to acquire new energy sources ( Zainuddin et al., 2018 ). This clearly shows that problem-solving skills are essential skills that must be mastered by students and taken care of by mathematics teachers in primary school.

In the context of mathematics learning in Malaysia, students are required to solve sentence-based mathematics problems by applying mathematical concepts learned at the end of each topic. Two types of sentence-based mathematics problems are presented when teaching mathematics: routine and non-routine ( Wong and Matore, 2020 ). According to Nurkaeti (2018) , routine sentence-based math problems are questions that require the students to solve problems using algorithmic calculations to obtain answers. For non-routine sentence-based math problems, thinking skills and the ability to apply more than one method or solution step are needed by the student to solve the problem ( Shawan et al., 2021 ). According to Rohmah and Sutiarso (2018) , problem-solving skills when solving a non-routine sentence-based mathematical problem is a high-level intellectual skill where the students need to use logical thinking and reasoning. This statement also aligns with Wilson's (1997) opinion that solving non-routine sentence-based mathematics always involves high-order thinking skills (HOTS). To solve non-routine and HOTS fundamental sentence-based math problems, a student is required to know various problem-solving strategies for solving the problems ( Wong and Matore, 2020 ). This situation has indirectly made the mastery of sentence-based mathematics problem-solving skills among students more challenging ( Mahmud, 2019 ).

According to Alkhawaldeh and Khasawneh’s findings (2021) , the failure of students stems from the teachers’ inability to perform their role effectively in the classroom. This statement is also supported by Abdullah (2020) . He argues that the failure of students in mastering non-routine sentence-based mathematics problem-solving skills is due to the teachers rarely supplying these types of questions during the process of learning mathematics in class. A mathematics teacher should consider this issue because the quality of their teaching will affect the students’ mastery level of sentence-based mathematics problem-solving skills.

In addition, the teachers’ efforts to encourage the students to engage in social interactions with the teachers ( Jatisunda, 2017 ) and the teachers’ method of teaching and assessing the level of sentence-based mathematics problem-solving skills ( Buschman, 2004 ) are also challenges that the teachers must face. Strategies that are not appropriate for the students will affect the quality of delivery of the sentence-based mathematics problem-solving skills as well as cause one-way interactions to exist in the classroom. According to Rusdin and Ali (2019) , a practical teaching approach plays a vital role in developing the students’ skills when mastering specific knowledge. However, based on previous studies, the main challenges that mathematics teachers face when teaching sentence-based mathematics problem solving are due to the students. These challenges include the students having difficulty understanding sentence-based math problems, lacking knowledge about basic mathematical concepts, not calculating accurately, and not transforming the sentence-based mathematics problems into an operational form ( Yoong and Nasri, 2021 ). This also means that they cannot transform the sentence-based math problems into an operational form ( Yoong and Nasri, 2021 ). As a result, the teacher should diversify his or her teaching strategy by emphasizing understanding the mathematical concepts rather than procedural teaching to reinforce basic mathematical concepts, to encourage the students to work on any practice problems assigned by the teacher before completing any assignments to help them do the calculation correctly, and engaging in the use of effective oral questioning to stimulate student thinking related to the operational need when problem solving. All of these strategies actually help the teachers facilitate and lessen the students’ difficulty understanding sentence-based math problems ( Subramaniam et al., 2022 ).

Meanwhile, Dirgantoro et al. (2019) stated several challenges that the students posed while solving the sentence-based problem. For example, students do not read the questions carefully, the students lack mastery of mathematical concepts, the students solve problems in a hurry due to poor time management, the students are not used to making hypotheses and conclusions, as well as the students, being less skilled at using a scientific calculator. These factors have caused the students to have difficulty mastering sentence-based mathematics problem-solving skills, which goes on to become an inevitable challenge in maths classes. Therefore, teachers need to study these challenges to self-reflect so then their self-professionalism can be further developed ( Dirgantoro et al., 2019 ).

As for the school factor, challenges such as limited teaching resources, a lack of infrastructure facilities, and a large number of students in a class ( Rusdin and Ali, 2019 ) have meant that a conducive learning environment for learning sentence-based mathematics problem-solving skills cannot be created. According to Ersoy (2016) , problem-solving skills can be learned if an appropriate learning environment is provided for the students to help them undergo a continuous and systematic problem-solving process.

To develop sentence-based mathematics problem-solving skills among students, various models, pedagogies, activities, etc. have been introduced to assist mathematics teachers in delivering sentence-based mathematics problem-solving skills more effectively ( Gurat, 2018 ; Khoiriyah and Husamah., 2018 ; Özreçberoğlu and Çağanağa, 2018 ; Hasibuan et al., 2019 ). However, students nowadays still face difficulties when trying to master sentence-based mathematics problem-solving skills. This situation occurs due to the lack of studies examining the challenges faced by these mathematics teachers and how teachers use teaching approaches to overcome said challenges. This has led to various issues during the teaching and facilitation of sentence-based mathematics problem-solving skills in mathematics classes. According to Rusdin and Ali (2019) , these issues need to be addressed by a teacher wisely so then the quality of teaching can reach the best level. Therefore, mathematics teachers must understand and address these challenges to improve their teaching.

However, so far, not much is known about how primary school mathematics teachers face the challenges encountered when teaching sentence-based mathematics problem-solving skills and what approaches are used to address the challenges in the context of education in Malaysia. Therefore, this study needs to be carried out to help understand the teaching of sentence-based mathematics problem-solving skills in primary schools ( Pazin et al., 2022 ). Due to the challenges when teaching mathematics as stipulated in the Mathematics Curriculum and Assessment Standard Document ( Ministry of Education Malaysia, 2020a ) which emphasizes mathematical problem-solving skills as one of the main skills that students need to master in comprehensive mathematics learning, this study focuses on identifying the challenges faced by mathematics teachers when teaching sentence-based mathematics problem-solving skills and the approaches that mathematics teachers have used to overcome those challenges. The results of this study can provide information to mathematics teachers to help them understand the challenges when teaching sentence-based mathematics problem-solving skills and the approaches that can be applied to overcome the challenges faced. Therefore, it is very important for this study to be carried out so then all the visions set within the framework of the Malaysian National Mathematics Curriculum can be successfully achieved.

2. Conceptual framework

The issue of students lacking mastery of sentence-based mathematics problem-solving skills is closely related to the challenges that teachers face and the teaching approach used. Based on the overall findings of the previous studies, the factors that pose a challenge to teachers when delivering sentence-based mathematics problem-solving skills include challenges from the teacher ( Buschman, 2004 ; Jatisunda, 2017 ; Abdullah, 2020 ), challenges from the pupils ( Dirgantoro et al., 2019 ), and challenges from the school ( Rusdin and Ali, 2019 ). As for the teaching approach, previous studies have suggested teaching approaches such as mastery learning, contextual learning, project-based learning, problem-based learning, simulation, discovery inquiry, the modular approach, the STEM approach ( Curriculum Development Division, 2019 ), game-based teaching which uses digital games ( Muhamad et al., 2018 ), and where a combination of the modular approach especially the flipped classroom is applied alongside the problem-based learning approach when teaching sentence-based mathematics problem solving ( Alias et al., 2020 ). This is as well as the constructivism approach ( Jatisunda, 2017 ). The conceptual framework in Figure 1 illustrates that the teachers will face various challenges during the ongoing teaching and facilitation of sentence-based mathematics problem-solving skills.

Conceptual framework of the study.

3. Methodology

The objective of this study was to determine the challenges that teachers face while teaching sentence-based mathematics problem-solving skills and the approaches used when teaching those skills. Therefore, a qualitative research approach in the form of a case study was used to collect data from the participants in a Chinese national type of school (SJKC) in Bangsar and Pudu, in the Federal Territory of Kuala Lumpur. The school, SJKC, in the districts of Bangsar and Pudu, was chosen as the location of this study because the school is implementing the School Transformation Program 2025 (TS25). One of the main objectives of the TS25 program is to apply the best teaching concepts and practices so then the quality of the learning and teaching in the classes is improved. Thus, schools that go through the program are believed to be able to diversify their teachers’ teaching and supply more of the data needed to answer the questions of this study. This is because case studies can develop an in-depth description and analysis of the case to be studied ( Creswell and Poth, 2018 ). All data collected through the observations, interviews, audio-visual materials, documents, and reports can be reported on in terms of both depth and detail based on the theme of the case. Therefore, this study collected data related to the challenges and approaches of SJKC mathematics teachers through observations, interviews, and document analysis.

Two primary school mathematics teachers who teach year four mathematics were selected to be the participants of this research using the purposive sampling technique to identify the challenges faced and the approaches used to overcome those challenges. The number of research participants in this study was sufficient enough to allow the researcher to explore the real picture of the challenges found when teaching sentence-based mathematics problem-solving skills and the approaches that can be applied when teaching to overcome the challenges faced. According to Creswell and Creswell (2018) , the small number of study participants is sufficient when considering that the main purpose of the study is to obtain findings that can give a holistic and meaningful picture of the teaching and learning process in the classroom. However, based on the data analysis for both study participants, the researcher considered repeated information until it reached a saturation point. The characteristics of the study participants required when they were supplying the information for this study were as follows:

• New or experienced teachers.
• Year four math teacher.
• Teachers teach in primary schools.
• The teacher teaches the topic of sentence-based mathematics problem-solving skills.

The types of instruments used in the study were the observation protocol, field notes, interview protocol, and participants’ documents. In this study, the researcher used participatory type observations to observe the teaching style of the teachers when engaged in sentence-based mathematics problem-solving skills lessons. Before conducting the study, the researcher obtained consent to conduct the study from the school as well as informed consent from the study participants to observe their teaching. During the observation, the teacher’s teaching process was recorded and transcribed using the field notes provided. Then, the study participants submitted and validated the field notes to avoid biased data. After that, the field notes were analyzed based on the observation protocol to identify the teachers’ challenges and teaching approaches in relation to sentence-based mathematics problem-solving skills. Throughout the observation process of this study, the researcher observed the teaching of mathematics teachers online at least four times during the 2 months of the data collection at the research location.

Semi-structured interviews were used to identify the teachers’ perspectives and views on teaching sentence-based mathematics problem-solving skills in terms of the challenges faced when teaching sentence-based mathematics problem-solving skills and the approaches used by the teachers to overcome those challenges. To ensure that the interview data collected could answer the research questions, an interview protocol was prepared so then the required data could be collected from the study participants ( Cohen et al., 2007 ). Two experts validated the interview protocol, and a pilot study was conducted to ensure that the questions were easy to understand and would obtain the necessary data. Before the interview sessions began, the participants were informed of their rights and of the related research ethics. Throughout the interview sessions, the participants were asked two questions, namely:

• What are the challenges faced when teaching mathematical problem-solving skills earlier?
• What teaching approaches are used by teachers when facing these challenges? Why?

Semi-structured interviews were used to interview the study participants for 30 min every interview session. The timing ensured sufficient time for both parties to complete the question-and-answer process. Finally, the entire interview process was recorded in audio form. The audio recordings were then transcribed into text form and verified by the study participants.

The types of document collected in this study included informal documents, namely the daily lesson plan documents of the study participants, the work of the students of the study participants, and any teaching aids used. All of the documents were analyzed and used to ensure that the triangulation of the data occurred between the data collected from observations, interviews, and document analysis.

All data collected through the observations, interviews, and documentary analyses were entered into the NVIVO 11 software to ensure that the coding process took place simultaneously. The data in this study were analyzed using the constant comparative analysis method including open coding, axial coding, and selective coding to obtain the themes and subthemes related to the focus of the study ( Kolb, 2012 ). The NVIVO 11 software was also used to manage the data stack obtained from the interviews, observations, and document analysis during the data analysis process itself. In order to ensure that the themes generated from all of the data were accurate, the researcher carried out a repetitive reading process. The process of theme development involved numerous steps. First, the researcher examined the verbatim instruction data several times while looking for statements or paragraphs that could summarize a theme in a nutshell. This process had already been completed during the verbatim formation process of the teaching, while preparing the transcription. Second, the researcher kept reading (either from the same or different data), and if the researcher found a sentence that painted a similar picture to the theme that had been developed, the sentence was added to the same theme. This process is called “pattern matching” because the coding of the sentences refers to the existing categories ( Yin, 2003 ). Third, if the identified sentence was incompatible with an existing theme, a new theme was created. Fourth, this coding procedure continued throughout each data set’s theme analysis. The repeated reading process was used to select sentences able to explain the theme or help establish a new one. In short, the researcher conducted the data analysis process based on the data analysis steps proposed by Creswell and Creswell (2018) , as shown in Figure 2 .

Data analysis steps ( Creswell and Creswell, 2018 ).

4. Findings

The findings of this study are presented based on the objective of the study, which was to identify the challenges faced by teachers and the approaches used to addressing those challenges when imparting sentence-based mathematics problem-solving skills to students in year four. Several themes were formed based on the analysis of the field notes, interview transcripts, and daily lesson plans of the study participants. This study found that teachers will face challenges that stem from the readiness of students to master sentence-based mathematics problem-solving skills, the teachers’ teaching style, and the equipment used for delivering sentence-based mathematics problem-solving skills. Due to facing these challenges, teachers have diversified their teaching approaches ( Figure 3 ; Table 1 ).

Sentence-based mathematics problem-solving teaching approaches.

Teacher challenges when teaching sentence-based mathematics problem-solving skills.

5. Discussion

5.1. challenges for teachers when imparting sentence-based mathematics problem-solving skills.

A mathematics teacher will face three challenges when teaching sentence-based mathematics problem-solving skills. The first challenge stems from the low mastery skills held by a student. Pupils can fail to solve sentence-based mathematics problems because they have poor reading skills, there is a poor medium of instruction used, or they have a poor mastery of mathematical concepts ( Johari et al., 2022 ). This indicates that students who are not ready or reach a minimum level of proficiency in a language, comprehension, mathematical concepts, and calculations will result in them not being able to solve sentence-based mathematics problems smoothly.

These findings are consistent with the findings of the studies by Raifana et al. (2016) and Dirgantoro et al. (2019) who showed that students who are unprepared in terms of language skills, comprehension, mathematical concepts, and calculations are likely to make mistakes when solving sentence-based mathematics problems. If these challenges are not faced well, the students will become passive and not interact when learning sentence-based mathematics problem-solving skills. This situation occurs because students who frequently make mistakes will incur low self-confidence in mathematics ( Jailani et al., 2017 ). This situation should be avoided by teachers and social interaction should be encouraged during the learning process because the interaction between students and teachers can ensure that the learning outcomes are achieved by the students optimally ( Jatisunda, 2017 ).

The next challenge stems from the teacher-teaching factor. This study found that how teachers convey problem-solving skills has been challenging in terms of ensuring that their students master sentence-based mathematics problem-solving skills ( Nang et al., 2022 ). The mastery teaching approach has caused the teaching time spent on mathematical content to be insufficient. Based on the findings of this study, the allocation of time spent ensuring that the students master the skills of solving sentence-based mathematics problems through a mastery approach has caused the teaching process not to follow the rate set in the annual lesson plan.

In this study, the participants spent a long time correcting the students’ mathematical concepts and allowing students to apply the skills learned. The actions of the participants of this study are in line with the statement of Adam and Halim (2019) that teachers need more time to arouse their students’ curiosity and ensure that students understand the correct ideas and concepts before doing more challenging activities. However, this approach has indirectly posed challenges regarding time allocation and ensuring that the students master the skills of sentence-based mathematics problem-solving. Aside from ensuring that the students’ master problem-solving skills, the participants must also complete the syllabus set in the annual lesson plan.

Finally, teachers also face challenges in terms of the lack of information and communication technology (ICT) infrastructure when implementing the teaching and facilitation of sentence-based mathematics problem-solving skills. In this study, mathematics teachers were found to face challenges caused by an unstable internet connection such as the problem of their students dropping out of class activities and whiteboard links not working. These problems have caused one mathematics class to run poorly ( Mahmud and Law, 2022 ). Throughout the implementation of teaching and its facilitation, ICT infrastructure equipment in terms of hardware, software, and internet services has become an element that will affect the effectiveness of virtual teaching ( Saifudin and Hamzah, 2021 ). In this regard, a mathematics teacher must be wise when selecting a teaching approach and diversifying the learning activities to implement a suitable mathematics class for students such as systematically using tables, charts, or lists, creating digital simulations, using analogies, working back over the work, involving reasoning activities and logic, and using various new applications such as Geogebra and Kahoot to help enable their students’ understanding.

5.2. Teaching sentence-based mathematics problem-solving skills—Approaches

In this study, various approaches have been used by the teachers facing challenges while imparting sentence-based mathematics problem-solving skills. Among the approaches that the mathematics teachers have used when teaching problem-solving skills are the oral questioning approach, mastery learning approach, contextual learning approach, game approach, and modular approach. This situation has shown that mathematics teachers have diversified their teaching approaches when facing the challenges associated with teaching sentence-based mathematics problem-solving skills. This action is also in line with the excellent teaching and facilitation of mathematics proposal in the Curriculum and Assessment Standards Document revised KSSR Mathematics Year 4 ( Curriculum Development Division, 2019 ), stating that teaching activities should be carefully planned by the teachers and combine a variety of approaches that allow the students not only to understand the content in depth but also to think at a higher level. Therefore, a teacher needs to ensure that this teaching approach is applied when teaching sentence-based mathematics problem-solving skills so then the students can learn sentence-based mathematics problem-solving teaching skills in a more fun, meaningful, and challenging environment ( Mahmud et al., 2022 ).

Through the findings of this study, the teaching approach used by mathematics teachers was found to have a specific purpose, namely facing the challenges associated with teaching sentence-based mathematics problem-solving skills in the classroom. First of all, the oral questioning approach has been used by teachers facing the challenge of students having a poor understanding of the medium of instruction. The participants stated that questioning the students in stages can guide them to understanding the question and helping them plan appropriate problem-solving strategies. This opinion is also supported by Maat (2015) who stated that low-level oral questions could help the students achieve a minimum level of understanding, in particular remembering, and strengthening abstract mathematical concepts. The teacher’s action of guiding the students when solving sentence-based mathematics problems through oral questioning has ensured that the learning takes place in a student-centered manner, providing opportunities for the students to think and solve problems independently ( Mahmud and Yunus, 2018 ). This action is highly encouraged because teaching mathematics through the conventional approach is only effective for a short period, as the students can lack an understanding or fail to remember the mathematical concepts presented by the teacher ( Ali et al., 2021 ).

In addition, this study also found that the participants used the mastery approach to overcome the challenges of poor reading skills and poor mastery of mathematical concepts among the students. The mastery approach was used because it can provide more opportunities and time for the students to improve their reading skills and mastery of mathematical concepts ( Shawan et al., 2021 ). This approach has ensured that all students achieve the teaching objectives and that the teachers have time to provide enrichment and rehabilitation to the students as part of mastering the basic skills needed to solve sentence-based mathematics problems. This approach is very effective at adapting students to solving sentence-based mathematics problems according to the solution steps of the Polya model as well as the mathematical concepts learned in relation to a particular topic. The finding is in line with Ranggoana et al. (2018) and Mahmud (2019) study, which has shown that teaching through a mastery approach can enhance the student’s learning activities. This situation clearly shows that the mastery approach has ensured that the students have time to learn at their own pace, where they often try to emulate the solution shown by the teacher to solve a sentence-based mathematics problem.

Besides that, this study also found that mathematics teachers apply contextual learning approaches when teaching and facilitating sentence-based mathematics problem-solving skills. In this study, mathematics teachers have linked non-routine problems with examples from everyday life to guide the students with poor language literacy to help them understand non-routine problems and plan appropriate solution strategies. Such relationships can help the students process non-routine problems or mathematical concepts in a more meaningful context where the problem is relevant to real situations ( Siew et al., 2016 ). This situation can develop the students’ skill of solving sentence-based math problems where they can choose the right solution strategy to solve a non-routine problem. This finding is consistent with the results of Afni and Hartono (2020) . They showed that the contextual approach applied in learning could guide the students in determining appropriate strategies for solving sentence-based math problems. These findings are also supported by Seliaman and Dollah (2018) who stated that the practice of teachers giving examples that exist around the students and in real situations could make teaching and the subject facilitation easier to understand and fun.

Furthermore, the game approach was also used by the participants when imparting sentence-based mathematics problem-solving skills. According to Sari et al. (2018) , the game approach to teaching mathematics can improve the student learning outcomes because the game approach facilitates the learning process and provides a more enjoyable learning environment for achieving the learning objectives. In this study, the game approach was used by the teachers to overcome the challenge of mastering the concept of unit conversion, which was not strong among the students ( Tobias et al., 2015 ; Hui and Mahmud, 2022 ). The participants used the game approach to teach induction sets that guided the students in recalling mathematical concepts. The action provided a fun learning environment and attracted the students to learning mathematical concepts, especially in the beginning of the class. This situation is consistent with the findings of Muhamad et al. (2018) . They showed that the game approach improved the students’ problem-solving skills, interests, and motivation to find a solution to the problem.

Regarding the challenge of insufficient teaching time and a lack of ICT infrastructure, modular approaches such as flipped classrooms have been used to encourage students to learn in a situation that focuses on self-development ( UNESCO, 2020 ). In this study, the participants used instructional videos with related content, clear instructions, and worksheets as part of the Google classroom learning platform. The students can follow the instructions to engage in revision or self-paced learning in their spare time. This modular approach has ensured that teachers can deliver mathematical content and increase the effectiveness of learning a skill ( Alias et al., 2020 ). For students with unstable internet connections, the participants have used a modular approach to ensure that the students continue learning and send work through other channels such as WhatsApp, by email, or as a hand-in hardcopy. In short, an appropriate teaching approach needs to be planned and implemented by the mathematics teachers to help students master sentence-based mathematics problem-solving skills.

6. Conclusion

Overall, this study has expanded the literature related to the challenges when teaching sentence-based mathematics problem-solving skills and the approaches that can be applied while teaching to overcome the challenges faced. This study has shown that students have difficulty mastering sentence-based mathematics problem-solving skills because they do not achieve the minimum mastery of factual knowledge, procedural skills, conceptual understanding, and the ability to choose appropriate strategies ( Collins and Stevens, 1983 ). This situation needs to be taken into account because sentence-based mathematics problem-solving skills train the students to always be prepared to deal with problems that they will be faced with in their daily life. Through this study, teachers were found to play an essential role in overcoming the challenges faced by choosing the most appropriate teaching approach ( Baul and Mahmud, 2021 ). An appropriate teaching approach can improve the students’ sentence-based mathematics problem-solving skills ( Wulandari et al., 2020 ). Teachers need to work hard to equip themselves with varied knowledge and skills to ensure that sentence-based mathematics problem-solving skills can be delivered to the students more effectively. Finally, the findings of this study were part of obtaining extensive data regarding the challenges that mathematics teachers face when teaching sentence-based mathematics problem-solving skills and the approaches used to address those challenges in the process of teaching mathematics. It is suggested that a quantitative study be conducted to find out whether the findings obtained can be generalized to other populations. This is because this study is a qualitative one, and the findings of this study cannot be generalized to other populations.

The findings of this study can be used as a reference to develop the professionalism of mathematics teachers when teaching mathematical problem-solving skills. However, the study’s findings, due to being formulated from a small sample size, cannot be generalized to all mathematics teachers in Malaysia. Further studies are proposed to involve more respondents to better understand the different challenges and approaches used when teaching sentence-based mathematics problem-solving skills.

Data availability statement

Ethics statement.

This study was reviewed and approved by The Malaysian Ministry of Education. The participants provided their written informed consent to participate in this study.

Author contributions

AL conceived and designed the study, collected and organized the database, and performed the analysis. AL and MM co-wrote the manuscript and contributed to manuscript revision. All authors read and approved the final submitted version.

The publication of this article is fully sponsored by the Faculty of Education Universiti Kebangsaan Malaysia and University Research Grant: GUP-2022-030, GGPM-2021-014, and GG-2022-022.

Conflict of interest

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

Publisher’s note

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

Acknowledgments

The authors appreciate the commitment from the respondent. Thank you to the Faculty of Education, Universiti Kebangsaan Malaysia, and University Research Grant: GUP-2022-030, GGPM-2021-014, and GG-2022-022 for sponsoring the publication of this article. Thanks also to all parties directly involved in helping the publication of this article to success.

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Challenges of teachers when teaching sentence-based mathematics problem-solving skills

Affiliation.

• 1 Faculty of Education, Universiti Kebangsaan Malaysia, Bangi, Malaysia.
• PMID: 36817370
• PMCID: PMC9928858
• DOI: 10.3389/fpsyg.2022.1074202

Sentence-based mathematics problem-solving skills are essential as the skills can improve the ability to deal with various mathematical problems in daily life, increase the imagination, develop creativity, and develop an individual's comprehension skills. However, mastery of these skills among students is still unsatisfactory because students often find it difficult to understand mathematical problems in verse, are weak at planning the correct solution strategy, and often make mistakes in their calculations. This study was conducted to identify the challenges that mathematics teachers face when teaching sentence-based mathematics problem-solving skills and the approaches used to address these challenges. This study was conducted qualitatively in the form of a case study. The data were collected through observations and interviews with two respondents who teach mathematics to year four students in a Chinese national primary school in Kuala Lumpur. This study shows that the teachers have faced three challenges, specifically low mastery skills among the students, insufficient teaching time, and a lack of ICT infrastructure. The teachers addressed these challenges with creativity and enthusiasm to diversify the teaching approaches to face the challenges and develop interest and skills as part of solving sentence-based mathematics problems among year four students. These findings allow mathematics teachers to understand the challenges faced while teaching sentence-based mathematics problem solving in depth as part of delivering quality education for every student. Nevertheless, further studies involving many respondents are needed to understand the problems and challenges of different situations and approaches that can be used when teaching sentence-based mathematics problem-solving skills.

Keywords: mathematics; primary school; problem solving; teaching approaches; teaching challenges.

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