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A problem solving approach to mathematics for elementary school teachers

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THEORIES AND PRINCIPLES OF PROBLEM SOLVING IN MATHEMATICS

2023, Article

Doing mathematics means that students are engaged in learning mathematics through reasoning and problem solving (NCTM, 2014). Prospective mathematics teachers need to learn about how to engage students in solving and talking about tasks that can be tackled in different ways by different students. Mathematically, proficient students are able to make sense of a situation, select solution paths, consider alternative strategies and monitor their progress (CCSSO, 2010). Before we can be effective in teaching mathematics, we need to have a good knowledge about what we are supposed to be teaching and how students learn mathematics. We are familiar with why we teach mathematics at the basic and high schools.

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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

89k Accesses

14 Citations

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?

• Mathematical Problem
• Prospective Teacher
• Creative Process
• Digital Technology
• Mathematical Task

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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

Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex, and setting . Buckingham, PA: Open University Press.

Borwein, P., Liljedahl, P., & Zhai, H. (2014). Mathematicians on creativity. Mathematical Association of America.

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Gardner, M. (1982). Aha! gotcha: Paradoxes to puzzle and delight . New York, NY: W. H. Freeman and Company.

Gardner, H. (1993). Creating minds: An anatomy of creativity seen through the lives of Freud, Einstein, Picasso, Stravinsky, Eliot, Graham, and Ghandi . New York, NY: Basic Books.

Glas, E. (2002). Klein’s model of mathematical creativity. Science & Education, 11 (1), 95–104.

Hersh, D. (1997). What is mathematics, really? . New York, NY: Oxford University Press.

Root-Bernstein, R., & Root-Bernstein, M. (1999). Sparks of genius: The thirteen thinking tools of the world’s most creative people . Boston, MA: Houghton Mifflin Company.

Zeitz, P. (2006). The art and craft of problem solving . New York, NY: Willey.

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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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Mathematics for Elementary Teachers

(18 reviews)

Michelle Manes, Honolulu, HI

Copyright Year: 2017

Publisher: University of Hawaii Manoa

Language: English

Formats Available

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Learn more about reviews.

Reviewed by Kevin Voogt, Assistant Professor, Grace College on 4/20/23

There seem to be subjects missing that are typical of the common core mathematics for elementary teachers texts (e.g., Ratios/Proportions, clear Partitive/Measurement division ideas, percentages, certain ideas in Geometry, Measurement). read more

Comprehensiveness rating: 4 see less

There seem to be subjects missing that are typical of the common core mathematics for elementary teachers texts (e.g., Ratios/Proportions, clear Partitive/Measurement division ideas, percentages, certain ideas in Geometry, Measurement).

Content Accuracy rating: 5

I did not find mathematical errors in the text during my review.

Relevance/Longevity rating: 3

I think there is need for quite a few updates to the text in regards to what is covered in elementary mathematics through the common core. The topics listed in my review of the Comprehensiveness above are just a start. I also see a need to add more activities to each section where prospective elementary teachers could do more exploration of the mathematics rather than what seems to be a more traditional approach of having the text explain it followed by problem sets alone.

Clarity rating: 5

The wording was quite clear and had nice explanations throughout.

Consistency rating: 5

It seems consistent throughout - with recurrent use of the same technical terms as needed.

Modularity rating: 4

There were a few issues with being able to assign the texts at different points within the course, as is the case for many math texts, in that many of the sections rely heavily on prior knowledge. If reorganization were to occur, there would be some need to re-structure how certain sections are taught.

Organization/Structure/Flow rating: 4

The text lacks much of the wonderful mathematical connections that could be made between ideas. While some connections are made, they seem a little outdated at times. I also think it would make more sense to have the properties of operations within their corresponding sections on operations rather than after all 4 operations are introduced.

Interface rating: 5

I did not see any issues with the interface. It was pretty user-friendly.

Grammatical Errors rating: 5

I did not notice any errors during my review.

Cultural Relevance rating: 5

I did not see anything insensitive or offensive in the text.

The text is just a small sampling of the many methods that could be used in teaching these mathematical ideas. I would have liked to see more activities for elementary teachers built into the lessons in each chapter as a means for learning and exploring ideas to facilitate more discussion as this text is used. There also are so many more connections that could be made between mathematical ideas that were lost a bit, especially with the general organization. On the whole, it is a nice resource and I could see it as useful for students studying for their certification exams to get some perspectives on the mathematical ideas they might encounter.

Reviewed by Sandra Zirkes, Teaching Professor, Bowling Green State University on 4/14/23

The text covers whole numbers, fractions, decimals, and operations well, and it provides some topics in geometry and algebraic thinking. However, the topics of ratio, proportion, and percent, as well as a more thorough coverage of geometry and... read more

The text covers whole numbers, fractions, decimals, and operations well, and it provides some topics in geometry and algebraic thinking. However, the topics of ratio, proportion, and percent, as well as a more thorough coverage of geometry and measurement are missing.

All information in the text is mathematically accurate and the writing and diagrams are error-free.

Relevance/Longevity rating: 4

While all of the information in the text is accurate and thought-provoking, some specific approaches are outdated with respect to the current standards and pedagogy. Approaching the concept of place value through the "Dots and Boxes" method, without reference to base ten blocks that are overwhelmingly used in the elementary math classroom, limits the coverage of this important topic. Similarly, approaching fractions using the "pies and kids" scenario is not consistent with the standards which emphasize the understanding of all fractions as iterations of unit fractions.

The text is written using clear and understandable prose that is both mathematically accurate and accessible to college level pre-service teachers.

The text has a clear organization and focus and uses consistent approaches and terminology throughout.

Much of the text is easily divisible into smaller subsections for student use. With respect to reorganization and realignment for a particular course, while some topics are revisited at appropriate points in the text, if those original topics were not covered in the course, revisiting the topic may not provide enough basis for the new topic. For example, the understanding of decimals is highly reliant on a student's understanding of the Dots and Boxes approach to place value earlier in the text.

Organization/Structure/Flow rating: 5

The topics in the text are organized in a logical way that is consistent with the structure of a typical mathematics education course.

Interface rating: 4

Navigating the text itself was seamless and intuitive. However, the videos that I viewed had poor visual quality and there was no audio.

The text is well written with no grammatical errors.

There is no apparent cultural insensitivity in the text.

This text has a problem solving focus and emphasizes deep thinking and reasoning about mathematics. Its approaches are clear and understandable. While its approaches are mathematically correct and thought-provoking, it is missing some key topics such as ratio, proportion, percent, and a more thorough coverage of geometry and measurement, as well as some standards-based approaches such as base ten blocks and understanding fractions as iterations of unit fractions.

Reviewed by Fred Coon, Assistant Professor, Anderson University on 2/16/23

The text covers all major points to help develop future teachers. read more

Comprehensiveness rating: 5 see less

The text covers all major points to help develop future teachers.

Text appears to be accurate.

The content is consist with concepts that elementary teachers should know. The methods are small in diversity.

Topics where well explained.

Text appears to use understandable and consistent terms.

Modularity rating: 5

Units appear to be mostly independent and can be used as stand alone units.

The topics are presented in a manner that build on each other but can be rearrange if desired.

Interface was useful and aided in navigating text.

I found no errors.

The text has no culturally insensitive or offensive items that I noticed.

I would like to have seen more diversity in methods discussed.

Reviewed by Perpetual Opoku Agyemang, Professor of Mathematics, Holyoke Community College on 6/17/21

The content in this text is built to help its readers, especially, pre-service elementary education majors learn to think like a mathematician in some very specific ways. The content addresses the subject framework in a complete yet concise... read more

The content in this text is built to help its readers, especially, pre-service elementary education majors learn to think like a mathematician in some very specific ways. The content addresses the subject framework in a complete yet concise manner. Although it does not provide an effective index/or glossary, LCD was not extensively tackled using factor tree, multiples or tables to express it, I still give props to the author since there are a lot of pictorial examples and a question bank for most of the various concepts. Furthermore, Dots and Boxes game on chapter 1 was very engaging and fun.

This text is very accurate and informative using a variety of felicitous examples to suit a diverse student population.

Conventional concepts are presented in a current and applied manner which allows for easier association with similar organized and retained information. This text could use some updated fraction problems and examples involving mixed numbers. Some of the YouTube videos have no sound at all.

Clarity rating: 4

Content material was presented in an easy to understand prose. Introduction of concepts and new terms were usually done by association or relevant previous knowledge. Some of the concepts like Multiplying Fractions, have YouTube videos embedded in the introductions.

Terminologies and framework are consistent throughout the text. The use of different notations were consistent throughout the various chapters and subunits.

This text has easily divisible content as stand alone subunits. However, numbering these chapters and subunits would have gone a long way to help its readers.

The topics in this text are organized from basic to complex concepts in a logical, clear fashion.

This text has an awesome interface (Online, PDF and XML). Moreover, it is untainted by distractions that may confuse its reader. Hyperlinks should have been included in the content.

I did not spot any grammatical errors in this text.

This content material contains no recognizable cultural insensitivity. It could use more examples involving modern affairs that are inclusive of diverse backgrounds.

I truly love the concise format of this text and how many different examples it uses to explain the concepts. The Geometry of Arts and Science and Tangrams were so informative with fun activities. It's easy to tell when one example ends and another begins, although index/or glossary and a system of links from the table of contents would be greatly appreciated. I did not see Points on a Coordinate Plane. Additionally, the number of exercises per section is too small. Of course this can be remedied by adding more. As with any textbook, the reader will need to supplement certain sections and clarify particular terms and concepts to best fit their situation. Pre-service elementary education majors could transition to this book fairly easily and successfully teach K-6 students in the United States in alignment with current Common Core Math Standards.

Reviewed by April Slack, Math Instructor, Aiken Technical College on 5/13/21

This text covers elementary mathematics strands including place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. Measurement and Data and Statistics strands are not included in this particular text. ... read more

This text covers elementary mathematics strands including place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. Measurement and Data and Statistics strands are not included in this particular text. The last chapter supplies the audience with problem-based learning approaches that include some measurement, but not in the detail of previous chapters of the book. It does incorporate problem solving strategies and pedagogical techniques teachers may use in the classroom. Examples with solutions and clarifying notes are provided throughout the text. The text does address Common Core Standards as well as the eight mathematical process standards. The textbook also provide teachers with a conceptual understanding of elementary mathematics along with appropriate mathematical terminology. The text does not offer an index or glossary.

The mathematics content provided in this text is accurate and provides thorough examples of teaching elementary mathematics for pre-service teachers. I found the text to build conceptual understanding and procedural fluency rather than just focus on basic algorithms to solve math problems. This is especially important for pre-service teachers, as they need to truly understand the "why" behind the math tricks that are often taught in early grades. The embedded links throughout the text are all in working order, as well.

The problem-solving approach to mathematics is especially relevant for elementary pre-service teachers; the intended audience. The book does expand beyond elementary mathematics, however, this is deemed extremely useful for all levels of mathematics teachers. Knowing the mathematical concepts beyond elementary strands allows teachers to know where there students are going and the mathematical purpose of content standards at each grade level. Many of the pedagogical techniques presented in the text are aligned with current research and instructional strategies for the elementary classroom.

This text provides explanations and defines mathematical terminology and has accessible prose. Beginning with the problem solving chapter before the specific content strands allows teachers to apply and consider strategies throughout the text. Often times, textbooks save problem solving for the end, but this text addresses strategies upfront and spirals nicely throughout the text. Some of the examples and visual representations are intended for an audience with mathematical background knowledge and strengths. A pre-service teacher may need help with content review prior to understanding the selection of particular problems highlighted in the text.

The text is well-organized and consistent with terminology throughout. The text is also consistent with provided examples that are used by mathematics teachers in everyday classrooms. There are multiple examples throughout each of the content chapters for pre-service teachers to reference and use in their own experiences.

This book is an easy read and may be easily broken up for weekly reading assignments and reflections. It seems as if mathematics teachers had a hand in writing this book. Bulleted and numbered lists are used throughout the text. The text also presents examples in clear, colored blocks. Visual models are clear and concise.

The book is well-organized with headings, subheadings, and the use of italics and boldface make this book extremely student friendly. The topics and content presented in this text are clear and in a logical order. Bulleted and numbered lists are reader friendly and easily understood. I found having the problem solving chapter appear first in the text stresses the importance and relevance of helping students become natural problem solvers. Often times texts and even worksheets save problem solving until the end, which poses a problem with students in the classroom.

This book is very easily navigated. The contents tab and drop down menu allows for the reader to quickly navigate to particular chapters and specific content. The previous and next buttons located at the bottom of the text allows readers to toggle between chapters very quickly. All embedded links work as they should and visual models are clear and understandable. There are no distractors present when trying to navigate the text. There is no index / glossary offered with this text.

The text is free from grammatical errors.

Cultural Relevance rating: 4

This text is not culturally offensive in any way. The final chapter of the text is dedicated to problem based learning and is centered around Voyaging on Hōkūle`a. The text provides embedded links to culturally relevant videos and models that help illustrate the cultural practices of Polynesians.

This textbook has a solid foundation and is well-organized for it's intended audience, the elementary mathematics pre-service teacher. This text will help build conceptual understanding of mathematics that will lead to procedural fluency for teachers. The text also provides clear examples of instructional strategies to be used in today's classrooms. Methods courses for pre-service teachers will find this text extremely useful and easy to incorporate in elementary mathematics methods instruction.

Reviewed by Kane Jessen, Math Instructor, Community College of Aurora on 8/13/20

This textbook is intended to cover the mathematics topics necessary to prepare pre-service elementary education majors to successfully teach K-6 students in the United States in alignment with current Common Core Math Standards. The textbook is a... read more

This textbook is intended to cover the mathematics topics necessary to prepare pre-service elementary education majors to successfully teach K-6 students in the United States in alignment with current Common Core Math Standards. The textbook is a mostly comprehensive collection of K-6 Common Core elementary math topics ranging from non-numerical problem solving through summative PBL assessments incorporating algebra, geometry and authentic problem solving. However, several topics related to K-6 CCSS Standards are not covered or minimally covered. CCSS topics with minimal coverage include set theory, logic, integers, probability, graphing and data analysis. At the beginning of the book, there is an effective and accessible table of contents with links included. However, sections and subsections are labeled only with names and page numbers. The text does not contain an index, glossary or appendices. Chapter summaries and links to previous concepts/problems are not included but would support student learning if included. More visuals and historical explorations would increase comprehensiveness.

Content was found to be accurate, error-free and unbiased.

Relevance/Longevity rating: 5

The language and examples of this text are written with a constructivist and meta-pedagogical voice that is both academic and accessible. The author immediately addresses the importance of CCSS and consistently utilizes the “Exploding Dots” curriculum. The “Exploding Dots” curriculum is a brave and differentiated approach to holistically teaching multi-base mathematics to K-12 students. “Exploding Dots” has been a core focus of K-12 Global Math Project and was pioneered by James Tanton . As future teachers, students can expect to teach “Exploding Dots” or similar CCSS curriculum sometime during their teaching career.

The language of the text is well-written, accessible and clear. Some sections and examples could be expanded for clarity/depth. Prior definitions/review concepts are not consistently linked.

The text is internally consistent in terms of its own terminology, framework and graphics. The “Exploding Dots” infusion helps maintain continuity throughout the text but is not present in all modules.

This text follows the common sequence that many “Mathematics for Elementary Teachers” textbooks commonly follow. The text is organized into eight modules. The text initially builds upon itself without being overly self-referential. The text’s sections, subsections, definitions, axioms and problem banks are all well delineated but lack sections/subsection numbers/identifiers and links to previous concepts/definitions

This textbook has a solid flow and follows a common sequence shared by most for profit “Mathematics for Elementary Teachers” texts. The text is well organized and builds upon itself.

Minimal issues involving interface were observed. Observed interface issues include, one broken video link and unnumbered sections. Definitions and review topics are not linked or referenced with page numbers/sections, however, this creates minimal usability issues. The text contains adequate procedural visuals and also cultural and historical visuals that enhance the student learning experience.

This text is largely free from grammatical errors. Grammatical errors that were observed were minor and non-persistent.

The text is not culturally insensitive or offensive in any way. It consistently uses examples that are inclusive of a variety of races, ethnicities, and backgrounds. Textbook examples often include references to Hawaiin culture. These references are easily understandable and could be readily adapted for students in other places. In an effort to increase relevance, further additions to the text could be made to provide a more equitable and historical focus on women, minorities and problem based learning cross-sectional explorations similar to the Hōkūleʻa section.

This textbook has a solid structure and great flow, I thoroughly enjoyed reviewing this textbook. I am genuinely excited to incorporate Michelle Manes ‘Mathematics for Elementary Teachers’ into my upcoming semester’s curriculum. With subsequent editions and revisions, this textbook will become a wonderful text for students majoring in primary education, especially those who are either lacking in basic math skills or math confidence.

Reviewed by Reina Ojiri, Assistant Professor, Leeward Community College on 7/27/20

The book begins with a reference to the Common Core State Standards (CCSS) for Mathematics and the eight “Mathematical Practices". Though not all states have adopted and/or are currently using the Common Core Standards, with its incorporation at... read more

Comprehensiveness rating: 2 see less

The book begins with a reference to the Common Core State Standards (CCSS) for Mathematics and the eight “Mathematical Practices". Though not all states have adopted and/or are currently using the Common Core Standards, with its incorporation at the beginning of the text I initially thought that the Common Core standards would be revisited consistently throughout the text.

Though the "Think Pair Share" sections are great additions for discussion to the book they do not include common misconceptions or tips for instructors to use to help guide these discussion prompts. The focus on just one type of discussion "Think Pair Share" also does not give future teachers a broader experience with different cooperative learning strategies in the classroom. There are many strategies in addition to “Think Pair Share” that are also great and seeing the same strategy over and over did not provide variation or keep me engaged as I read through the text.

There are a few key concepts that are not included in the text including Measurement & Data and Statistics & Probability.

The text also does not include an effective index and/or glossary. I have found that students do use the index and/or glossary that is typically in the back of the book to help them find information in the text quickly.

Content Accuracy rating: 3

The content is error-free however some of the images included on the PDF version are blurry and hard to read. There does not seem to be consistency between the different readable versions of the text.

There also seems to be a bias for the dots and boxes strategy throughout the text and the content lacks current practices of teaching concepts.

Just like any text, this textbook needs to be updated to match current best practices and research in math education. Since this text is Attribution-ShareAlike which allows “others to remix, adapt, and build upon your work even for commercial purposes, as long as they credit you and license their new creations under the identical terms” it does seem that updates and instructor/course-specific content will be relatively easy to implement as needed.

Clarity rating: 3

This text is written in a way unique way that makes it easier for students to read through and follow. It is very student friendly however might not be as useful as an instructor text since the instructor needs to fill-in-the-blanks on their own.

Consistency rating: 3

The text is written with consistent terminology however the framework for each chapter is not consistent. Some chapters include Explorations and additional sections while others end consistently with a problem bank.

Modularity rating: 3

The text is divided into smaller reading sections however the titles of each section are not easily recognized by students. Though I imagine the titles were meant to be creative for each section, having something more straight forward to make it easier for students to navigate is more important than creativity especially for future teachers who might be teaching these concepts for the first time.

Organization/Structure/Flow rating: 3

It would be good to organize the material consistently throughout the text (e.g.each section should end with a problem bank). The variation in the different sections can be confusing to both the instructor and student when trying to find something in the text.

I also noticed that the online version does not include page numbers while the PDF version does. This is not helpful when referring students to particular sections of the book. The PDF version also has many completely blank pages. I am not sure if this was meant to be on purpose (for printing purposes) but these pages can be very distracting to the reader.

Interface rating: 2

Navigation throughout the text is fine however, there are noticeable differences between the online and PDF versions of the text. The images in the PDF versions are noticeably blurry and lower quality than those in the online version. In some instances, it seems as though images were screenshot and copied and pasted which could account for the image quality.

Some images, in particular, should not have been included at all and are unreadable, for example, the Hokulea on page 441.

I did not notice grammatical errors.

The connection to the Hawaiian culture was a nice touch.

I would use this text as a reference but would not adopt this book as the main text for my class.

Reviewed by Thomas Starmack, Professor, Bloomsburg University of Pennsylvania on 3/26/20

The book is somewhat dated and does not include current research based best practices like concrete, representational, then abstract. Like most authors, they make assumptions that students have the ability to understand abstract and start the... read more

The book is somewhat dated and does not include current research based best practices like concrete, representational, then abstract. Like most authors, they make assumptions that students have the ability to understand abstract and start the lesson there, which is contradictory to how the brain works and what current research says about effective math instruction and learning.

I agree the content is accurate, but in many areas the learner must have a very strong understanding of mathematical concepts, structures, and applications. There lacks current best practice and current NCTM recommendations to approaching the teaching of mathematical content.

Relevance/Longevity rating: 1

Although mathematical concepts at the elementary level remain the same, the approach to engaging students in learning and the methods of instruction have evolved greatly. The book lacks many of the newer approaches and is outdated. The arrangement of the concepts is okay. I would recommend that the big ideas of teaching math are in the beginning and providing an overview of what is mathematics and best approaches to teaching/learning mathematics. Then scaffold the specific concepts. Fractions is one of the most complex and abstract, and this book starts there as a first topic.

Once again, the book is okay in terms of math learning but dated on best practice approaches. The book does not use jargon per say, but does not provide the best approaches for students to learn how to effectively teach mathematics.

Consistency rating: 4

Yes the book is consistent throughout.

The text is divisible, just not relevant to today nor provides current approaches. The order of the content is not in line with a methods of teaching course I would follow.

Organization/Structure/Flow rating: 2

I think the topics are clear but dated and not in the order as described above.

The text provides a variety of interfaces, none of which are confusing for the student who has a very strong math background. The text does mislead students to think starting with abstract is how to instruct elementary students, which is contradictory to brain research and current best practices.

Grammatical Errors rating: 4

I did not notice any grammar errors.

Cultural Relevance rating: 3

I think the text is culturally appropriate. Not certain about the final chapter as it focuses on one population. Having a chapter or theme woven throughout the text that provides students with a stronger understanding that although mathematics is a universal language, there are cultural differences to teaching and learning as evidenced in the 1999 TIMSS report.

The text is outdated. The text is an okay resource but I would not be able to use as the main guide for learning in a college level methods of teaching elementary mathematics course.

Reviewed by Jamie Price, Assistant Professor, East Tennessee State University on 3/20/20

This book introduces the reader to the standards for mathematical practice (SMP) from the Common Core standards in the introduction. I appreciated this as these standards cover all grades and are a unifying theme of the Common Core standards, yet... read more

This book introduces the reader to the standards for mathematical practice (SMP) from the Common Core standards in the introduction. I appreciated this as these standards cover all grades and are a unifying theme of the Common Core standards, yet many times overlooked. In addition, many states, including mine, that are not following Common Core directly have adopted the SMPs. The book does not cover two of the mathematical strands, namely measurement and statistics/data. Among the strands that are covered, however, the author does a thorough job of explaining the content, using a unified theme throughout, such as dots and boxes introduced in place value that appear again in number operations. I particularly liked the final chapter of the book and its connection to Hawaiian culture. The author could easily incorporate ideas related to teaching and learning measurement into this chapter in order to make the book more comprehensive.

The content was very accurate. I did not come across any mathematical errors or biases. The author did a good job of incorporating "think, pair, share" elements throughout each chapter as a model for future teachers. To further guide future teachers, I would have liked to see the author include information in each chapter about common misconceptions students have when learning the related material and ideas on how to address those misconceptions. In my experience, I find that pre-service teachers are unaware of these misconceptions and it is helpful to make them aware of them so that they can anticipate them in their own classrooms.

The content presented in this book is up-to-date and will remain relevant for a long time. Due to the fact that this book focuses more on content rather than methods, I do not foresee a need for many updates moving forward.

The book is written in a very clear and concise way that is approachable to future and current elementary teachers. The author presents key words in bold throughout the book to draw attention to them. I liked the way that the author included videos as well as written explanations of ideas, such as in the Number and Operations chapter, section titled Addition: Dots and Boxes. The author explains, in words, how to use this method to add multi-digit numbers and follows the written example with a video explanation. This helps to reach a variety of learners and learning styles. The author also addresses common "jargon" associated with particular mathematical concepts, such as proper and improper fractions (section titled What is a Fraction?), and discusses how this jargon can be misleading for students.

Each chapter in the book includes an introduction, multiple opportunities for think-pair-share discussions, and several problem sets to practice. I appreciated the consistency in the Dots and Boxes method introduced in the Place Value chapter and then carried into the Number and Operations chapter.

The book uses a modular approach to present the material. Each module contains numerous sections that help to break up the content into smaller chunks so that the content does not seem overwhelming. The modules are set up in an order that makes sense for the mathematics, but a reader could begin reading at any module and still make sense of the content.

The organization of the topics makes sense according to the mathematics presented and is logical.

I did not find anything distracting or confusing in relation to the interface of the text. The book was easy to navigate, with a clearly defined table of contents. I was able to easily click through the various modules and sections within each module. The book uses figures well to provide engagement to the reader as well as to further clarify content. The use of videos embedded within the modules helps to strengthen understanding of the content. It did take me a minute to find the navigation link that allowed me to move to the next section in a module (right arrow at bottom right corner of the page), but once I found it I was able to navigate seamlessly to each subsequent section.

I did not find any grammatical errors in the text.

In my opinion, this was one of the biggest strengths of this text. The author did a nice job of incorporating Hawaiian culture into the text. For example, the author includes an image in the Place Value chapter (Number Systems section) that references the use of tally marks on a sign at Hanakapiai Beach. In addition, a full chapter was devoted to Voyaging on Hōkūle`a. I particularly liked how the author connected this idea to beginning teaching of elementary mathematics and encouraged future teachers to think about ways to see mathematics outside of traditional mathematical settings.

I am glad that I came across this resource. I primarily teach math methods courses for elementary pre-service teachers, but I found many aspects of this text that I can incorporate into my classes to help students think more deeply about the mathematics that they will teach. I appreciated the author's attempt to challenge students in their thinking about elementary mathematics. Initially, I was surprised to find that there was no "answer key" provided for the many problem sets that were included throughout the text. After reading the quote presented on the introductory page to the Problem Solving chapter, I realized that this may have been an intentional decision made by the author to encourage readers to go beyond "a trail someone else has laid." I find that many pre-service elementary teachers want to "just know the answer" when it comes to mathematics; a no answer key approach will encourage discussion and justification, two elements important to ensuring equity in the teaching and learning of mathematics.

Reviewed by Shay Kidd, Assistant Professor- Mathematics Education, University of Montana - Western on 12/30/19

The content that elementary teachers need to have that is not covered in this book is graphing, probability, statistics, exponents, visual displays of data. The coverage of operations is very specific in the examples and does not cover the wide... read more

The content that elementary teachers need to have that is not covered in this book is graphing, probability, statistics, exponents, visual displays of data. The coverage of operations is very specific in the examples and does not cover the wide range that should be presented in this type of text.

Content Accuracy rating: 4

While the core topics presented are correct, the number of problems that are provided without any solutions is alarming. The majority of problems that are provided are meant for the reader to perform but do not provide any type of answer key for checking the work. In this way, the book seems to assume the reader to have a solid knowledge of the topics already and this book discusses a few different approaches to these topics.

The specific content presented is up-to-date and usable.

The book's prose seem to be more of a teaching guide than a textbook. This is nice for the conversational aspect that a reader may want in their learning, but should be explained more or possibly a change of title for the book. Something more like "Exploring the concepts of Elementary Mathematics" would provide a more reading friendly approach the book offers.

The author has a consistent voice of teaching and presenting the material.

Modularity rating: 2

The break-up of the text with boxes is difficult to follow the purpose of each box. While some of the box styles are clear, such as the think, pair, share or problem boxes, others seem to break up the line of discussion. A problem box may be discussed more directly immediately following the box and the presentation of the problem. Most of the problem boxes are not discussed again in the main text. This cased issues for wanting to read with a specific purpose. When the reader wants to understand a problem more, there is generally not more discussion, but unclear about when that would be provided or not. Other times boxes were used without any "box type" provided and these were just to break up the flow of the text.

Place value was a major topic to start the book and had good coverage, then operations and fractions were discussed, then a return to place value with decimals. It would seem that a connection of place value and decimals would work better to follow the other place value discussion.

Interface rating: 3

There are several pages that have large blank parts or are totally blank. This may be due to the PDF version that I chose. When I did use the internet-connected version, there seems to be a dependence on youtube to help do some of the teaching.

There are a few minor issues that would be resolved with a good proofread.

The book does seem to be written with the Hawaiian culture in mind. This may be difficult for other cultures to connect to or understand but does not present any insensitivities.

The book's title suggests a full discussion of the topics that elementary education pre-service teachers would need to know and teach, but this book is very lacking in the topics required for this. I selected this book to review because I teach classes that would use the textbook, but I would not use this textbook as is. There are a few topics that I plan to add to my own instruction, but the book as a whole needs additional help to be able to stand alone. This really appears to be a teaching guide based on the constant think-pair-share setup. This also is a specific teaching and method that seems to require the students to already have much of the content mastered. It does not teach all the content that is required to the level of the discussion had.

Reviewed by Ryan Nivens, Associate Professor, East Tennessee State University on 10/25/19

The book covers all the expected mathematical strands except for measurement and data/stats. There are some obvious connections to the strands of mathematical practice from the Common Core standards. While the abstract specifically lists MP1, MP2,... read more

The book covers all the expected mathematical strands except for measurement and data/stats. There are some obvious connections to the strands of mathematical practice from the Common Core standards. While the abstract specifically lists MP1, MP2, and MP3, the introduction clearly lists all 8. The chapter "Voyaging on Hōkūle`a" contains activities that will require use of measurement and units, but there is no explanation on how measurement topics should be taught or approached. However, this chapter does provide a good project-based learning set of materials, and is an exceptional resource for navigation. The book also includes a chapter on Problem Solving, which is important for those students who must complete the EdTPA and address the 3rd subject specific emphasis area. All embedded links to Youtube videos or Vimeo videos are working and play within the textbook pages.

I find the mathematics to be entirely accurate. There are many teaching strategies, such as "think pair share" that are found throughout the chapters. This is particularly helpful for future teachers.

This book should last a very long time in terms of relevance.

This book is very clear, with mathematical words in bold and proper definitions provided. The text also addresses common math classroom jargon. For an excellent example of this, see the heading "What is a Fraction" in the chapter on Fractions. Toward the bottom is a sub-heading "Jargon: Improper Fractions" that has students consider the usefulness of proper and improper fractions.

This book is consistently laid out, with multiple examples, problems to try, and diagrams to support the transfer of information.

This book is entirely modular. You can pick it up, and easily start in any chapter and not be lost. The heading, subheading, use of italics and boldface make it easy to locate information. As a mathematics education book, this is quite nice.

A mathematician wrote this, the layout is logical without question.

The book is extremely easy to navigate, with a logical structure to the table of contents that you can easily click through. A drop down menu in the upper left corner allows you to view the outline of the book while still viewing a page, and you can collapse/expand chapters within the menu.

The many figures that are present throughout the textbook are perfectly displayed and fit the reading material.

There is nothing I find distracting in the layout and interface.

I could not find any errors.

An entire chapter is dedicated to Voyaging on Hōkūle`a, with exceptional videos and diagrams to illustrate the cultural practices of the early Polynesians.

I was excited to find this book in the Open Educational Resources library. As a professor who frequently teaches methods courses in mathematics for elementary teachers, I feel that this book may be a terrific book to use to replace previous texts that I've adopted. I would like to see a chapter on Measurement to make the Voyaging on Hōkūle`a chapter more useful. It is obvious from the first page you open to that this book was well planned and thought out. I'm impressed.

Reviewed by Monica Rose Gilmore, Graduate Student, CU Boulder on 7/1/19

This textbook goes into depth about different mathematical concepts that are important for elementary school teachers to understand in teaching mathematics. However, the text is missing a focus on statistics and probability, which are key areas of... read more

Comprehensiveness rating: 3 see less

This textbook goes into depth about different mathematical concepts that are important for elementary school teachers to understand in teaching mathematics. However, the text is missing a focus on statistics and probability, which are key areas of focus in elementary math classrooms. The text is also missing an index or glossary but does define new terms as they are introduced.

The content, mathematical diagrams and depictions are accurate and error-free. Each chapter also accurately shows various ways to understand mathematical concepts. However, the diagrams are geared towards an audience that already has some understanding of advanced mathematics.

The content is organized in a way that necessary updates would be straightforward to implement. More specifically, much of the content reflects current mathematical practices and activities endorsed by up-to-date research in mathematics education.

The text is written in accessible prose and provides context for jargon and technical terminology. Additionally, the text clearly separates different terms for different strategies and concepts. For example, in the Problem Solving Strategy section, the interface is divided into different strategies for the reader to explore. This is helpful in keeping new concepts and strategies organized for the reader.

The text is written with consistent terminology. More specifically, the text consistently gives examples of what concepts are called by mathematicians and teachers. This is helpful for pre-service teachers that might be teaching mathematical concepts and strategies for the first time.

The text is easily divided into smaller reading sections. These sections include not only explanations of mathematical concepts, but also theorems, activities and diagrams which can be referenced by the teacher at any point. Also, the text gives teachers ideas for activities and additional problems to try with students.

Though the topics in the texts are presented in a logical, clear fashion, it might be beneficial for pre-service or elementary teachers to see how to specifically scaffold the different concepts within those topics for elementary students at different grade levels. Additionally, the text could also demonstrate how students typically confuse topics so teachers and pre-service teachers are prepared to navigate new concepts for the class.

The interface is easy to navigate since the content clearly outlines chapters and the topics within them. Sections such as notation and vocabulary, think pair shares and theorems are clearly outlined, organized and conceptually scaffolded. However, it might be helpful to have an index so the reader does not have to click within each topic to find the concept they are exploring.

This text is free from grammatical errors.

This text is not culturally insensitive or offensive and includes examples from the Hawaiian culture. Though the text is mainly made up of mathematical explanations, there are a variety of people's names in different problems that could be attributed to a variety of cultures. Additionally, the text reflects Polya's advice (1945) to try adapt the problem until it makes sense. Though the text includes mainly mathematical explanations, it does call for adapting problems which could potentially be applied to a variety of students of different backgrounds.

Reviewed by Glenna Gustafson, Professor, Radford University on 5/22/19

This is book is fairly comprehensive and I feel could be used by most foundational courses in elementary mathematics. The structure and writing provide a good foundation for students learning the "why" behind the mathematics and becoming... read more

This is book is fairly comprehensive and I feel could be used by most foundational courses in elementary mathematics. The structure and writing provide a good foundation for students learning the "why" behind the mathematics and becoming mathematical thinkers. There were some areas that could possibly use more development. In geometry for example there was no discussion of perimeter, area, and volume. Estimation, measurement of weight, time, and probability also appears to be missing. The text is well organized and written so that the chapters do not have to be completed in the order in which they are presented. While there is not index or glossary, the author uses colored text boxes to explain specific content or terms.

The content of the text is accurate and represented in a variety of formats to support learning, Not only does it provide solutions to problems, but also the mathematical thinking behind those solutions.

The text is very relevant for K-6 elementary pre-service teachers. It would be beneficial to know the specific grade levels that the author considers as "elementary" since this does vary by location. The content is "standard" for most elementary math courses and would not need to be updated often and the consistent layout and formation would make changes easy to make.

The text is written in a conversational tone. The simplicity and straight-forwardness of the text should appeal to those students that have sometimes been overwhelmed by writing in more traditional math texts.

The text is organized consistently from chapter to chapter. The table of contents and chunking of content in the chapters is logical and clear, Each chapter includes graphics as well as sections for: Think-Pair-Share; Definitions; Theorems (when appropriate); and, Problems. This consistent structure makes navigation easy.

The table of contents and chunking of content in the chapters is logical and clear. This also makes it easy to not necessary to move sequentially through the text, but to have the option of reviewing or using only needed topics. Subtitles and graphic captioning are appropriate for the content.

The text is easy to read and the organization within each chapters makes navigation easy.

This text is easy to navigate. The inclusion of graphics, charts, photos, and videos support learning. There are several pages where graphics in the Geometry chapter are skewed in the PDF version, but this does not seem to be a problem in the online version, Not all of the video links work within the PDF version.

There were no obvious grammatical errors. Several of the errors that were found were typos and/or word omissions.

The text is culturally inclusive. One thing that should be noted is that it seems male names are over-represented in the Problem sections. A reference to Hawaiian culture and life is evident. The Hōkūle`a voyage found in the last chapter is a good example of problem based learning and the integration of math with other subject areas.

This would be a wonderful text to use as a supplement or compliment to an elementary math methods course. It is not as overwhelming as other math texts, and would provide pre-service teachers with a good foundational review of math concepts, including vocabulary and some pedagogy.

Reviewed by Karise Mace, Mathematics Instructor, Kuztown University on 5/16/19, updated 11/9/20

This book is fairly comprehensive for a one-semester course, although it does not include much detail about several topics. The section on number systems barely touches on Roman numerals and only mentions Mayan and Babylonian counting systems.... read more

This book is fairly comprehensive for a one-semester course, although it does not include much detail about several topics. The section on number systems barely touches on Roman numerals and only mentions Mayan and Babylonian counting systems. The sections on addition, subtraction, and division would be more robust if the author included other algorithms for these operations. The chapter on Geometry does not address perimeter, area, surface area and volume. The book does not include an index or glossary.

While the book is not error free, it is unbiased. Most of the errors seem to be typographical and/or related to web links or LaTeX. In the section on number systems, the author incorrectly explains how one million would be represented using Roman numerals and incorrectly claims that the Mayans did not use a symbol for zero. Further, the Mayan number system was not a true vigesimal system, as the text indicates.

This text uses a constructivist approach to help students build their understanding of the mathematics included in the book. It is well organized and written so that the chapters do not have to be completed in the order in which they are presented. Because of this, the text should be easy to update. When concepts that are presented earlier in the text are used in later chapters, the author includes a brief but thorough review that would allow students to understand the later chapter even if they had not read and completed the problems in the earlier chapter. The "dots and boxes" approach is timely, as it uses the idea of the "exploding dots" that are part of the Global Math Project (https://www.globalmathproject.org).

The textbook is clearly written and enjoyable to read...even for the math-phobic student. The tone is conversational and is even funny at times. The author defines important mathematical terminology in a way that is both mathematically accurate and accessible to students. The chapter on problem solving is fantastic and really gives students insight into how to think and problem solve like a mathematician. The pies per child model for fractions is not the most effective model for helping students understand fractions and this part of the text would be improved if the author replaced this type of modeling with pattern block modeling.

Overall, the text is consistent in its chapter structure and terminology use. However, there is inconsistent notation when using "dots and boxes."

The text is well-organized but can be reorganized in order to suit an instructor's preference. However, it would be best to complete the chapter on problem solving first, as it sets the stage for the rest of the book. Most of the chapters are structured more like an activity book with lots of great problems and thought provoking questions that will help students think deeply about the mathematical concepts being presented. With the exception of the chapter on problem solving, there is not a whole lot of text for students to read.

Although the topics presented could be reorganized to meet student needs, the order in which they are presented is logical and clear.

With only a few exceptions, the images it the text are clear. In the section titled "Careful Use of Language in Mathematics: =" some of the scale images need to be modified so that the items on the scale appear to actually sit on the pans. The same issue occurs in the section titled "Structural and Procedural Algebra." Some of the images in the sections titled "Platonic Solids" and "Symmetry" spill off of the page. The image that appears on page 89 and then again on page 144 would be more clear if a different font was used to label the line segments.

No grammatical errors were noted. However, there were a few typographical errors that could cause confusion for students as on page 219.

The text was culturally sensitive and nothing offensive was noted. As the focus of the text is purely mathematical, there are not many cultural references at all, unless they are references to historical cultures. The author does use names for hypothetical students that are diverse and represent a variety of ethnicities. The last chapter is an integrated unit that focuses on the Hawaiian culture. Unfortunately, the links and web addresses in this chapter do not work and/or are no longer active.

The book includes three sections at the end of the problem solving chapter in which the author articulately explains the language that mathematicians use to succinctly and precisely explain their problem solving and solutions. These sections will help students who may not think of themselves as mathematicians learn to think like mathematicians. So many mathematics textbooks are full of exercises but no true problems. On the other hand, this text is full of wonderful problem solving and critical thinking problems that are embedded in the sections as well as in the problem banks. The author also includes many "Think/Pair/Share" exercises and questions that will facilitate mathematical thinking and conversation among students. The constructivist approach used by the author will help students build deep understanding about the mathematics covered in the text. While there is some room for revision and improvement, this is a very good text to use with elementary education majors, and I definitely plan to use this book the next time I teach them.

Reviewed by Desley Plaisance, Associate Professor, Nicholls State University on 4/29/19

This textbook seems to be appropriate for the first course typically taught for elementary teachers which usually includes topics of problem solving, place value, number and operations. Most books are able to be used for a second course which... read more

This textbook seems to be appropriate for the first course typically taught for elementary teachers which usually includes topics of problem solving, place value, number and operations. Most books are able to be used for a second course which focuses on geometry. This book could not be used for the second course.

Content seems to be accurate.

Topics are somewhat static for a course like this, so the textbook will not become obsolete within a short period of time.

Appears to be clear.

The flow from topic to topic is consistent in presentation.

Divided into clear sections.

Topics are presented logically and in a similar order to most books of this type.

Easy to navigate with clear images and other items such as tables.

Book is written in simple language and appears to be free of grammatical errors.

Appears to be culturally diverse.

This book could definitely be used for a first course of elementary math for teachers with the teacher providing resources. As with many open books, the print and layout is very simple without cluttering pages with unnecessary items.

Reviewed by Lisa Cooper, Assistant Professor, LSUS on 4/26/19

This text covers many concepts appropriately; however, a few concepts are missing, such as; data analysis and statistics. For more than ten years, data-driven instruction has been a major focus in education along with many other uses. This text... read more

This text covers many concepts appropriately; however, a few concepts are missing, such as; data analysis and statistics. For more than ten years, data-driven instruction has been a major focus in education along with many other uses. This text has a table of contents but not an index and/or glossary; however, does define words in chapters when needed.

The content is well organized and accurate. Multiple representations and diverse examples are provided throughout the text which supports an unbiased approach to those entering elementary education.

The text is quite relevant to the classroom today, incorporating such resources as YouTube, varied strategies to promote differentiated instruction, scaffolding between concepts, and problem-solving opportunities. Some states may find issues with Common Core standards being addressed; however, mathematical practices could be interchanged with the "standards."

The text is written free from educational jargon; it is straightforward and easy to understand.

The text is consistent in its structure; color is not distracting, problems, strategies, diagrams, charts, and definitions are provided throughout.

The text is appealing with the page layout; it's not too busy or distracting. Colors are attractive and text is broken down into appropriate amounts.

The text has a well-organized flow with the layout of each topic/chapter.

The text has charts, pictures, diagrams, real-world examples throughout; several different versions of the text are offered too.

No grammatical errors were observed in my review of the text.

The text provides a variety of backgrounds, races, and ethnicities while providing learning experiences and pedagogical approaches to support student engagement and learning.

Reviewed by Demetrice Smith-Mutegi, Instructor/Coordinator, Marian University on 3/6/19

This text covers place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. However, elementary teachers are expected to also know and understand statistics and probability. This text does not address... read more

This text covers place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. However, elementary teachers are expected to also know and understand statistics and probability. This text does not address this mathematical concept.

The text makes non-traditional, yet, accurate representations of mathematical concepts. In some sections, different solutions are presented and explained. This eliminates bias and provides a diverse representation of ideas when solving math problems.

The text is representative of common core problem-solving standards, however, it does require mathematical knowledge beyond elementary school. The problem-solving nature of the text is very relevant to elementary pre-service and in-service teachers (the audience for the book).

The text language is clear and accessible. There is a section on terminology, which is very helpful. Additional diagrams would help to improve the clarity in some cases.

I was expecting to see videos embedded throughout, after seeing them in the first section. It would be great to have a consistent format throughout the text, however, I understand that it is not always feasible to do so. There were other obvious and clear patterns presented, color-coded sections (think/pair/share), problems, examples.

The chapters and subchapters can be easily accessed, breaking the material into smaller sections.

The topics were presented in a logical, clear fashion, however, not all of the chapters would end with a problem bank. In some cases, there were additional sections after the problem bank. It would be great if each section included key objectives or goals of the section.

The text comes in pdf, XML, and an online web version. The search feature on the online version was a valuable addition.

I did not observe any obvious grammatical errors.

Cultural awareness was very obvious in this text. While it was more relevant to Hawaiian culture, it also included cultural awareness of other cultures and backgrounds.

Overall, this text assumes that the student has successfully completed mathematics through basic calculus. There should be more support in this area, as some elementary math students are not prepared to complete problems with this focus.

This a great "discussion" text.

Reviewed by Kandy Noles Stevens, Assistant Professor of Education, Southwest Minnesota State University on 12/28/18

This text covers the areas applicable to elementary mathematics extremely well (with the exception of omitting probability and data analysis) and provides graphically visual boxes within the text to define terms and instructional strategies. ... read more

This text covers the areas applicable to elementary mathematics extremely well (with the exception of omitting probability and data analysis) and provides graphically visual boxes within the text to define terms and instructional strategies. Additionally, the text provides thinking routines that support understanding more than just the concept, but also, the how's and why's of conceptual understanding.

The content is accurate and organized in a way to supports student learning for those training to become elementary teachers of mathematics.

The content of the book is relevant to today's elementary classroom in that it provides future elementary educators with the content knowledge, but also pedagogical approaches that would support student learning. Additionally, the text is organized in a way that is consistent and provides scaffolding support for those who might struggle with any one of the concepts. For Minnesota standards the only item of note is that there is not a section devoted to probability or data analysis, but the latter is touched upon in other chapters. There are three mentions of the Common Core standards in the text. Minnesota is not an adopter of the CC mathematics standards, but the references to the CCSS are in regards to the practices of mathematics and not on standards specifically.

While a great text for training future math teachers, this book does not read as a "typical" mathematics textbook. Students who have struggled in the past with mathematics might find the authors' writing style to be approachable and accessible for all levels of mathematics competence and confidence.

The text is consistent in its terminology and the structure of the framework is uniform throughout, relying on supporting student learning through exercises, think-pair-share activities, and continuous dialog and reflection.

A majority of the chapters begin with a section that introduces the strand of elementary mathematics covered. Not all chapters have this introduction which may pose challenging to interrupt the mathematical progression of some established courses.

The text is very well organized and has an easy-to-read format and flow.

The text is graphically rich with succinct advanced organizers, diagrams, and photos to support learning.

The text is written with professional level writing and is free of grammatical errors.

A variety of races, ethnicities, and backgrounds are present in the exercises used to support student learning throughout. The end of the text involves a Hōkūle`a voyage as a part of a problem-based learning (integrated curriculum) experience. This was something that really made this text stand out in that it gave future elementary teachers an example of using mathematical concepts in authentic (and exciting) learning experiences. This Polynesian voyage would provide many students with an introduction to life culturally different from their own.

I have been a STEM educator for more than two decades and I come from a long line of mathematics educators. While wrapping up my reading of this text, I happened to have my father (a 46 year veteran mathematics educator) here visiting. I shared the text with him and several times I heard him utter, "I like the way this problem is set up". We both found the book to be very knowledgeable for mathematical conceptual understandings, but even more so for introducing ideas for instructional strategies and classroom discourse to help future teachers become equipped with speaking the "language of mathematics" to guide their future students.

Table of Contents

I. Problem Solving

• Introduction
• Problem or Exercise?
• Problem Solving Strategies
• Beware of Patterns!
• Problem Bank
• Careful Use of Language in Mathematics
• Explaning Your Work
• The Last Step

II. Place Value

• Dots and Boxes
• Other Rules
• Binary Numbers
• Other Bases
• Number Systems
• Even Numbers
• Exploration

III. Number and Operations

• Addition: Dots and Boxes
• Subtration: Dots and Boxes
• Multiplication: Dots and Boxes
• Division: Dots and Boxes
• Number Line Model
• Area Model for Multiplication
• Properties of Operations
• Division Explorations

IV. Fractions

• What is a Fraction?
• The Key Fraction Rule
• Adding and Subtracting Fractions
• What is a Fraction? Revisited
• Multiplying Fractions
• Dividing Fractions: Meaning
• Dividing Fractions: Invert and Multiply
• Dividing Fractions: Problems
• Fractions involving zero
• Egyptian Fractions
• Algebra Connections
• What is a Fraction? Part 3

V. Patterns and Algebraic Thinking

• Borders on a Square
• Careful Use of Language in Mathematics: =
• Growing Patterns
• Matching Game
• Structural and Procedural Algebra

VI. Place Value and Decimals

• Review of Dots & Boxes Model
• Division and Decimals
• More x -mals
• Terminating or Repeating?
• Operations on Decimals
• Orders of Magnitude

VII. Geometry

• Triangles and Quadrilaterals
• Platonic Solids
• Painted Cubes
• Geometry in Art and Science

VIII. Voyaging on Hokule?a

• Worldwide Voyage

Ancillary Material

About the book.

This book will help you to understand elementary mathematics more deeply, gain facility with creating and using mathematical notation, develop a habit of looking for reasons and creating mathematical explanations, and become more comfortable exploring unfamiliar mathematical situations. The primary goal of this book is to help you learn to think like a mathematician in some very specific ways. You will: • Make sense of problems and persevere in solving them. You will develop and demonstrate this skill by working on difficult problems, making incremental progress, and revising solutions to problems as you learn more. • Reason abstractly and quantitatively. You will demonstrate this skill by learning to represent situations using mathematical notation (abstraction) as well as creating and testing examples (making situations more concrete). • Construct viable arguments and critique the reasoning of others. You will be expected to create both written and verbal explanations for your solutions to problems. The most important questions in this class are “Why?” and “How do you know you're right?” Practice asking these questions of yourself, of your professor, and of your fellow students. Throughout the book, you will learn how to learn mathematics on you own by reading, working on problems, and making sense of new ideas on your own and in collaboration with other students in the class.

About the Contributors

Michelle Manes, Associate Professor, Department of Mathematics, University of Hawaii

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Mathematics > Optimization and Control

Title: solving the waste bin location problem with uncertain waste generation rate: a bi-objective robust optimization approach.

Abstract: An efficient Municipal solid waste (MSW) system is critical to modern cities in order to enhance sustainability and livability of urban life. With this aim, the planning phase of the MSW system should be carefully addressed by decision makers. However, planning success is dependent on many sources of uncertainty that can affect key parameters of the system, e.g., the waste generation rate in an urban area. With this in mind, this paper contributes with a robust optimization model to design the network of collection points (i.e., location and storage capacity), which are the first points of contact with the MSW system. A central feature of the model is a bi-objective function that aims at simultaneously minimizing the network costs of collection points and the required collection frequency to gather the accumulated waste (as a proxy of the collection cost). The value of the model is demonstrated by comparing its solutions with those obtained from its deterministic counterpart over a set of realistic instances considering different scenarios defined by different waste generation rates. The results show that the robust model finds competitive solutions in almost all cases investigated. An additional benefit of the model is that it allows the user to explore trade-offs between the two objectives.

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