implications of problem solving to mathematics teaching and learning

• Open access
• Published: 19 December 2019

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

• Yeping Li 1 &
• Alan H. Schoenfeld 2  

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

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

Introduction

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

How the ways that mathematics is often taught cause concerns.

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

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

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

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

How to change?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Connecting changes in mathematics and STEM education

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

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

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

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

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

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

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

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

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

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

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

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

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

Extending to STEM education

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

The domain-general version of the TRU framework

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

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

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

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

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

Making student thinking central to classroom discourse.

Ensuring that classroom discourse is respectful and inviting.

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

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

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

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

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

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Original research article, mathematical problem-solving through cooperative learning—the importance of peer acceptance and friendships.

• 1 Department of Education, Uppsala University, Uppsala, Sweden
• 2 Department of Education, Culture and Communication, Malardalen University, Vasteras, Sweden
• 3 School of Natural Sciences, Technology and Environmental Studies, Sodertorn University, Huddinge, Sweden
• 4 Faculty of Education, Gothenburg University, Gothenburg, Sweden

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional approaches supporting student learning in this area. This study aims to contribute to previous research by studying the effects of an instructional approach of cooperative learning on students’ mathematical problem-solving in heterogeneous classrooms in grade five, in which students with special needs are educated alongside with their peers. The intervention combined a cooperative learning approach with instruction in problem-solving strategies including mathematical models of multiplication/division, proportionality, and geometry. The teachers in the experimental group received training in cooperative learning and mathematical problem-solving, and implemented the intervention for 15 weeks. The teachers in the control group received training in mathematical problem-solving and provided instruction as they would usually. Students (269 in the intervention and 312 in the control group) participated in tests of mathematical problem-solving in the areas of multiplication/division, proportionality, and geometry before and after the intervention. The results revealed significant effects of the intervention on student performance in overall problem-solving and problem-solving in geometry. The students who received higher scores on social acceptance and friendships for the pre-test also received higher scores on the selected tests of mathematical problem-solving. Thus, the cooperative learning approach may lead to gains in mathematical problem-solving in heterogeneous classrooms, but social acceptance and friendships may also greatly impact students’ results.

Introduction

The research on instruction in mathematical problem-solving has progressed considerably during recent decades. Yet, there is still a need to advance our knowledge on how teachers can support their students in carrying out this complex activity ( Lester and Cai, 2016 ). Results from the Program for International Student Assessment (PISA) show that only 53% of students from the participating countries could solve problems requiring more than direct inference and using representations from different information sources ( OECD, 2019 ). In addition, OECD (2019) reported a large variation in achievement with regard to students’ diverse backgrounds. Thus, there is a need for instructional approaches to promote students’ problem-solving in mathematics, especially in heterogeneous classrooms in which students with diverse backgrounds and needs are educated together. Small group instructional approaches have been suggested as important to promote learning of low-achieving students and students with special needs ( Kunsch et al., 2007 ). One such approach is cooperative learning (CL), which involves structured collaboration in heterogeneous groups, guided by five principles to enhance group cohesion ( Johnson et al., 1993 ; Johnson et al., 2009 ; Gillies, 2016 ). While CL has been well-researched in whole classroom approaches ( Capar and Tarim, 2015 ), few studies of the approach exist with regard to students with special educational needs (SEN; McMaster and Fuchs, 2002 ). This study contributes to previous research by studying the effects of the CL approach on students’ mathematical problem-solving in heterogeneous classrooms, in which students with special needs are educated alongside with their peers.

Group collaboration through the CL approach is structured in accordance with five principles of collaboration: positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing ( Johnson et al., 1993 ). First, the group tasks need to be structured so that all group members feel dependent on each other in the completion of the task, thus promoting positive interdependence. Second, for individual accountability, the teacher needs to assure that each group member feels responsible for his or her share of work, by providing opportunities for individual reports or evaluations. Third, the students need explicit instruction in social skills that are necessary for collaboration. Fourth, the tasks and seat arrangements should be designed to promote interaction among group members. Fifth, time needs to be allocated to group processing, through which group members can evaluate their collaborative work to plan future actions. Using these principles for cooperation leads to gains in mathematics, according to Capar and Tarim (2015) , who conducted a meta-analysis on studies of cooperative learning and mathematics, and found an increase of .59 on students’ mathematics achievement scores in general. However, the number of reviewed studies was limited, and researchers suggested a need for more research. In the current study, we focused on the effect of CL approach in a specific area of mathematics: problem-solving.

Mathematical problem-solving is a central area of mathematics instruction, constituting an important part of preparing students to function in modern society ( Gravemeijer et al., 2017 ). In fact, problem-solving instruction creates opportunities for students to apply their knowledge of mathematical concepts, integrate and connect isolated pieces of mathematical knowledge, and attain a deeper conceptual understanding of mathematics as a subject ( Lester and Cai, 2016 ). Some researchers suggest that mathematics itself is a science of problem-solving and of developing theories and methods for problem-solving ( Hamilton, 2007 ; Davydov, 2008 ).

Problem-solving processes have been studied from different perspectives ( Lesh and Zawojewski, 2007 ). Problem-solving heuristics Pólya, (1948) has largely influenced our perceptions of problem-solving, including four principles: understanding the problem, devising a plan, carrying out the plan, and looking back and reflecting upon the suggested solution. Schoenfield, (2016) suggested the use of specific problem-solving strategies for different types of problems, which take into consideration metacognitive processes and students’ beliefs about problem-solving. Further, models and modelling perspectives on mathematics ( Lesh and Doerr, 2003 ; Lesh and Zawojewski, 2007 ) emphasize the importance of engaging students in model-eliciting activities in which problem situations are interpreted mathematically, as students make connections between problem information and knowledge of mathematical operations, patterns, and rules ( Mousoulides et al., 2010 ; Stohlmann and Albarracín, 2016 ).

Not all students, however, find it easy to solve complex mathematical problems. Students may experience difficulties in identifying solution-relevant elements in a problem or visualizing appropriate solution to a problem situation. Furthermore, students may need help recognizing the underlying model in problems. For example, in two studies by Degrande et al. (2016) , students in grades four to six were presented with mathematical problems in the context of proportional reasoning. The authors found that the students, when presented with a word problem, could not identify an underlying model, but rather focused on superficial characteristics of the problem. Although the students in the study showed more success when presented with a problem formulated in symbols, the authors pointed out a need for activities that help students distinguish between different proportional problem types. Furthermore, students exhibiting specific learning difficulties may need additional support in both general problem-solving strategies ( Lein et al., 2020 ; Montague et al., 2014 ) and specific strategies pertaining to underlying models in problems. The CL intervention in the present study focused on supporting students in problem-solving, through instruction in problem-solving principles ( Pólya, 1948 ), specifically applied to three models of mathematical problem-solving—multiplication/division, geometry, and proportionality.

Students’ problem-solving may be enhanced through participation in small group discussions. In a small group setting, all the students have the opportunity to explain their solutions, clarify their thinking, and enhance understanding of a problem at hand ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ). In fact, small group instruction promotes students’ learning in mathematics by providing students with opportunities to use language for reasoning and conceptual understanding ( Mercer and Sams, 2006 ), to exchange different representations of the problem at hand ( Fujita et al., 2019 ), and to become aware of and understand groupmates’ perspectives in thinking ( Kazak et al., 2015 ). These opportunities for learning are created through dialogic spaces characterized by openness to each other’s perspectives and solutions to mathematical problems ( Wegerif, 2011 ).

However, group collaboration is not only associated with positive experiences. In fact, studies show that some students may not be given equal opportunities to voice their opinions, due to academic status differences ( Langer-Osuna, 2016 ). Indeed, problem-solvers struggling with complex tasks may experience negative emotions, leading to uncertainty of not knowing the definite answer, which places demands on peer support ( Jordan and McDaniel, 2014 ; Hannula, 2015 ). Thus, especially in heterogeneous groups, students may need additional support to promote group interaction. Therefore, in this study, we used a cooperative learning approach, which, in contrast to collaborative learning approaches, puts greater focus on supporting group cohesion through instruction in social skills and time for reflection on group work ( Davidson and Major, 2014 ).

Although cooperative learning approach is intended to promote cohesion and peer acceptance in heterogeneous groups ( Rzoska and Ward, 1991 ), previous studies indicate that challenges in group dynamics may lead to unequal participation ( Mulryan, 1992 ; Cohen, 1994 ). Peer-learning behaviours may impact students’ problem-solving ( Hwang and Hu, 2013 ) and working in groups with peers who are seen as friends may enhance students’ motivation to learn mathematics ( Deacon and Edwards, 2012 ). With the importance of peer support in mind, this study set out to investigate whether the results of the intervention using the CL approach are associated with students’ peer acceptance and friendships.

The Present Study

In previous research, the CL approach has shown to be a promising approach in teaching and learning mathematics ( Capar and Tarim, 2015 ), but fewer studies have been conducted in whole-class approaches in general and students with SEN in particular ( McMaster and Fuchs, 2002 ). This study aims to contribute to previous research by investigating the effect of CL intervention on students’ mathematical problem-solving in grade 5. With regard to the complexity of mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach in this study was combined with problem-solving principles pertaining to three underlying models of problem-solving—multiplication/division, geometry, and proportionality. Furthermore, considering the importance of peer support in problem-solving in small groups ( Mulryan, 1992 ; Cohen, 1994 ; Hwang and Hu, 2013 ), the study investigated how peer acceptance and friendships were associated with the effect of the CL approach on students’ problem-solving abilities. The study aimed to find answers to the following research questions:

a) What is the effect of CL approach on students’ problem-solving in mathematics?

b) Are social acceptance and friendship associated with the effect of CL on students’ problem-solving in mathematics?

Participants

The participants were 958 students in grade 5 and their teachers. According to power analyses prior to the start of the study, 1,020 students and 51 classes were required, with an expected effect size of 0.30 and power of 80%, provided that there are 20 students per class and intraclass correlation is 0.10. An invitation to participate in the project was sent to teachers in five municipalities via e-mail. Furthermore, the information was posted on the website of Uppsala university and distributed via Facebook interest groups. As shown in Figure 1 , teachers of 1,165 students agreed to participate in the study, but informed consent was obtained only for 958 students (463 in the intervention and 495 in the control group). Further attrition occurred at pre- and post-measurement, resulting in 581 students’ tests as a basis for analyses (269 in the intervention and 312 in the control group). Fewer students (n = 493) were finally included in the analyses of the association of students’ social acceptance and friendships and the effect of CL on students’ mathematical problem-solving (219 in the intervention and 274 in the control group). The reasons for attrition included teacher drop out due to sick leave or personal circumstances (two teachers in the control group and five teachers in the intervention group). Furthermore, some students were sick on the day of data collection and some teachers did not send the test results to the researchers.

FIGURE 1 . Flow chart for participants included in data collection and data analysis.

As seen in Table 1 , classes in both intervention and control groups included 27 students on average. For 75% of the classes, there were 33–36% of students with SEN. In Sweden, no formal medical diagnosis is required for the identification of students with SEN. It is teachers and school welfare teams who decide students’ need for extra adaptations or special support ( Swedish National Educational Agency, 2014 ). The information on individual students’ type of SEN could not be obtained due to regulations on the protection of information about individuals ( SFS 2009 ). Therefore, the information on the number of students with SEN on class level was obtained through teacher reports.

TABLE 1 . Background characteristics of classes and teachers in intervention and control groups.

Intervention

The intervention using the CL approach lasted for 15 weeks and the teachers worked with the CL approach three to four lessons per week. First, the teachers participated in two-days training on the CL approach, using an especially elaborated CL manual ( Klang et al., 2018 ). The training focused on the five principles of the CL approach (positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing). Following the training, the teachers introduced the CL approach in their classes and focused on group-building activities for 7 weeks. Then, 2 days of training were provided to teachers, in which the CL approach was embedded in activities in mathematical problem-solving and reading comprehension. Educational materials containing mathematical problems in the areas of multiplication and division, geometry, and proportionality were distributed to the teachers ( Karlsson and Kilborn, 2018a ). In addition to the specific problems, adapted for the CL approach, the educational materials contained guidance for the teachers, in which problem-solving principles ( Pólya, 1948 ) were presented as steps in problem-solving. Following the training, the teachers applied the CL approach in mathematical problem-solving lessons for 8 weeks.

Solving a problem is a matter of goal-oriented reasoning, starting from the understanding of the problem to devising its solution by using known mathematical models. This presupposes that the current problem is chosen from a known context ( Stillman et al., 2008 ; Zawojewski, 2010 ). This differs from the problem-solving of the textbooks, which is based on an aim to train already known formulas and procedures ( Hamilton, 2007 ). Moreover, it is important that students learn modelling according to their current abilities and conditions ( Russel, 1991 ).

In order to create similar conditions in the experiment group and the control group, the teachers were supposed to use the same educational material ( Karlsson and Kilborn, 2018a ; Karlsson and Kilborn, 2018b ), written in light of the specified view of problem-solving. The educational material is divided into three areas—multiplication/division, geometry, and proportionality—and begins with a short teachers’ guide, where a view of problem solving is presented, which is based on the work of Polya (1948) and Lester and Cai (2016) . The tasks are constructed in such a way that conceptual knowledge was in focus, not formulas and procedural knowledge.

Implementation of the Intervention

To ensure the implementation of the intervention, the researchers visited each teachers’ classroom twice during the two phases of the intervention period, as described above. During each visit, the researchers observed the lesson, using a checklist comprising the five principles of the CL approach. After the lesson, the researchers gave written and oral feedback to each teacher. As seen in Table 1 , in 18 of the 23 classes, the teachers implemented the intervention in accordance with the principles of CL. In addition, the teachers were asked to report on the use of the CL approach in their teaching and the use of problem-solving activities embedding CL during the intervention period. As shown in Table 1 , teachers in only 11 of 23 classes reported using the CL approach and problem-solving activities embedded in the CL approach at least once a week.

Control Group

The teachers in the control group received 2 days of instruction in enhancing students’ problem-solving and reading comprehension. The teachers were also supported with educational materials including mathematical problems Karlsson and Kilborn (2018b) and problem-solving principles ( Pólya, 1948 ). However, none of the activities during training or in educational materials included the CL approach. As seen in Table 1 , only 10 of 25 teachers reported devoting at least one lesson per week to mathematical problem-solving.

Tests of Mathematical Problem-Solving

Tests of mathematical problem-solving were administered before and after the intervention, which lasted for 15 weeks. The tests were focused on the models of multiplication/division, geometry, and proportionality. The three models were chosen based on the syllabus of the subject of mathematics in grades 4 to 6 in the Swedish National Curriculum ( Swedish National Educational Agency, 2018 ). In addition, the intention was to create a variation of types of problems to solve. For each of these three models, there were two tests, a pre-test and a post-test. Each test contained three tasks with increasing difficulty ( Supplementary Appendix SA ).

The tests of multiplication and division (Ma1) were chosen from different contexts and began with a one-step problem, while the following two tasks were multi-step problems. Concerning multiplication, many students in grade 5 still understand multiplication as repeated addition, causing significant problems, as this conception is not applicable to multiplication beyond natural numbers ( Verschaffel et al., 2007 ). This might be a hindrance in developing multiplicative reasoning ( Barmby et al., 2009 ). The multi-step problems in this study were constructed to support the students in multiplicative reasoning.

Concerning the geometry tests (Ma2), it was important to consider a paradigm shift concerning geometry in education that occurred in the mid-20th century, when strict Euclidean geometry gave way to other aspects of geometry like symmetry, transformation, and patterns. van Hiele (1986) prepared a new taxonomy for geometry in five steps, from a visual to a logical level. Therefore, in the tests there was a focus on properties of quadrangles and triangles, and how to determine areas by reorganising figures into new patterns. This means that structure was more important than formulas.

The construction of tests of proportionality (M3) was more complicated. Firstly, tasks on proportionality can be found in many different contexts, such as prescriptions, scales, speeds, discounts, interest, etc. Secondly, the mathematical model is complex and requires good knowledge of rational numbers and ratios ( Lesh et al., 1988 ). It also requires a developed view of multiplication, useful in operations with real numbers, not only as repeated addition, an operation limited to natural numbers ( Lybeck, 1981 ; Degrande et al., 2016 ). A linear structure of multiplication as repeated addition leads to limitations in terms of generalization and development of the concept of multiplication. This became evident in a study carried out in a Swedish context ( Karlsson and Kilborn, 2018c ). Proportionality can be expressed as a/b = c/d or as a/b = k. The latter can also be expressed as a = b∙k, where k is a constant that determines the relationship between a and b. Common examples of k are speed (km/h), scale, and interest (%). An important pre-knowledge in order to deal with proportions is to master fractions as equivalence classes like 1/3 = 2/6 = 3/9 = 4/12 = 5/15 = 6/18 = 7/21 = 8/24 … ( Karlsson and Kilborn, 2020 ). It was important to take all these aspects into account when constructing and assessing the solutions of the tasks.

The tests were graded by an experienced teacher of mathematics (4 th author) and two students in their final year of teacher training. Prior to grading, acceptable levels of inter-rater reliability were achieved by independent rating of students’ solutions and discussions in which differences between the graders were resolved. Each student response was to be assigned one point when it contained a correct answer and two points when the student provided argumentation for the correct answer and elaborated on explanation of his or her solution. The assessment was thus based on quality aspects with a focus on conceptual knowledge. As each subtest contained three questions, it generated three student solutions. So, scores for each subtest ranged from 0 to 6 points and for the total scores from 0 to 18 points. To ascertain that pre- and post-tests were equivalent in degree of difficulty, the tests were administered to an additional sample of 169 students in grade 5. Test for each model was conducted separately, as students participated in pre- and post-test for each model during the same lesson. The order of tests was switched for half of the students in order to avoid the effect of the order in which the pre- and post-tests were presented. Correlation between students’ performance on pre- and post-test was .39 ( p < 0.000) for tests of multiplication/division; .48 ( p < 0.000) for tests of geometry; and .56 ( p < 0.000) for tests of proportionality. Thus, the degree of difficulty may have differed between pre- and post-test.

Measures of Peer Acceptance and Friendships

To investigate students’ peer acceptance and friendships, peer nominations rated pre- and post-intervention were used. Students were asked to nominate peers who they preferred to work in groups with and who they preferred to be friends with. Negative peer nominations were avoided due to ethical considerations raised by teachers and parents ( Child and Nind, 2013 ). Unlimited nominations were used, as these are considered to have high ecological validity ( Cillessen and Marks, 2017 ). Peer nominations were used as a measure of social acceptance, and reciprocated nominations were used as a measure of friendship. The number of nominations for each student were aggregated and divided by the number of nominators to create a proportion of nominations for each student ( Velásquez et al., 2013 ).

Statistical Analyses

Multilevel regression analyses were conducted in R, lme4 package Bates et al. (2015) to account for nestedness in the data. Students’ classroom belonging was considered as a level 2 variable. First, we used a model in which students’ results on tests of problem-solving were studied as a function of time (pre- and post) and group belonging (intervention and control group). Second, the same model was applied to subgroups of students who performed above and below median at pre-test, to explore whether the CL intervention had a differential effect on student performance. In this second model, the results for subgroups of students could not be obtained for geometry tests for subgroup below median and for tests of proportionality for subgroup above median. A possible reason for this must have been the skewed distribution of the students in these subgroups. Therefore, another model was applied that investigated students’ performances in math at both pre- and post-test as a function of group belonging. Third, the students’ scores on social acceptance and friendships were added as an interaction term to the first model. In our previous study, students’ social acceptance changed as a result of the same CL intervention ( Klang et al., 2020 ).

The assumptions for the multilevel regression were assured during the analyses ( Snijders and Bosker, 2012 ). The assumption of normality of residuals were met, as controlled by visual inspection of quantile-quantile plots. For subgroups, however, the plotted residuals deviated somewhat from the straight line. The number of outliers, which had a studentized residual value greater than ±3, varied from 0 to 5, but none of the outliers had a Cook’s distance value larger than 1. The assumption of multicollinearity was met, as the variance inflation factors (VIF) did not exceed a value of 10. Before the analyses, the cases with missing data were deleted listwise.

What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

As seen in the regression coefficients in Table 2 , the CL intervention had a significant effect on students’ mathematical problem-solving total scores and students’ scores in problem solving in geometry (Ma2). Judging by mean values, students in the intervention group appeared to have low scores on problem-solving in geometry but reached the levels of problem-solving of the control group by the end of the intervention. The intervention did not have a significant effect on students’ performance in problem-solving related to models of multiplication/division and proportionality.

TABLE 2 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving.

The question is, however, whether CL intervention affected students with different pre-test scores differently. Table 2 includes the regression coefficients for subgroups of students who performed below and above median at pre-test. As seen in the table, the CL approach did not have a significant effect on students’ problem-solving, when the sample was divided into these subgroups. A small negative effect was found for intervention group in comparison to control group, but confidence intervals (CI) for the effect indicate that it was not significant.

Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

As seen in Table 3 , students’ peer acceptance and friendship at pre-test were significantly associated with the effect of the CL approach on students’ mathematical problem-solving scores. Changes in students’ peer acceptance and friendships were not significantly associated with the effect of the CL approach on students’ mathematical problem-solving. Consequently, it can be concluded that being nominated by one’s peers and having friends at the start of the intervention may be an important factor when participation in group work, structured in accordance with the CL approach, leads to gains in mathematical problem-solving.

TABLE 3 . Mean scores (standard deviation in parentheses) and unstandardized multilevel regression estimates for tests of mathematical problem-solving, including scores of social acceptance and friendship in the model.

In light of the limited number of studies on the effects of CL on students’ problem-solving in whole classrooms ( Capar and Tarim, 2015 ), and for students with SEN in particular ( McMaster and Fuchs, 2002 ), this study sought to investigate whether the CL approach embedded in problem-solving activities has an effect on students’ problem-solving in heterogeneous classrooms. The need for the study was justified by the challenge of providing equitable mathematics instruction to heterogeneous student populations ( OECD, 2019 ). Small group instructional approaches as CL are considered as promising approaches in this regard ( Kunsch et al., 2007 ). The results showed a significant effect of the CL approach on students’ problem-solving in geometry and total problem-solving scores. In addition, with regard to the importance of peer support in problem-solving ( Deacon and Edwards, 2012 ; Hwang and Hu, 2013 ), the study explored whether the effect of CL on students’ problem-solving was associated with students’ social acceptance and friendships. The results showed that students’ peer acceptance and friendships at pre-test were significantly associated with the effect of the CL approach, while change in students’ peer acceptance and friendships from pre- to post-test was not.

The results of the study confirm previous research on the effect of the CL approach on students’ mathematical achievement ( Capar and Tarim, 2015 ). The specific contribution of the study is that it was conducted in classrooms, 75% of which were composed of 33–36% of students with SEN. Thus, while a previous review revealed inconclusive findings on the effects of CL on student achievement ( McMaster and Fuchs, 2002 ), the current study adds to the evidence of the effect of the CL approach in heterogeneous classrooms, in which students with special needs are educated alongside with their peers. In a small group setting, the students have opportunities to discuss their ideas of solutions to the problem at hand, providing explanations and clarifications, thus enhancing their understanding of problem-solving ( Yackel et al., 1991 ; Webb and Mastergeorge, 2003 ).

In this study, in accordance with previous research on mathematical problem-solving ( Lesh and Zawojewski, 2007 ; Degrande et al., 2016 ; Stohlmann and Albarracín, 2016 ), the CL approach was combined with training in problem-solving principles Pólya (1948) and educational materials, providing support in instruction in underlying mathematical models. The intention of the study was to provide evidence for the effectiveness of the CL approach above instruction in problem-solving, as problem-solving materials were accessible to teachers of both the intervention and control groups. However, due to implementation challenges, not all teachers in the intervention and control groups reported using educational materials and training as expected. Thus, it is not possible to draw conclusions of the effectiveness of the CL approach alone. However, in everyday classroom instruction it may be difficult to separate the content of instruction from the activities that are used to mediate this content ( Doerr and Tripp, 1999 ; Gravemeijer, 1999 ).

Furthermore, for successful instruction in mathematical problem-solving, scaffolding for content needs to be combined with scaffolding for dialogue ( Kazak et al., 2015 ). From a dialogical perspective ( Wegerif, 2011 ), students may need scaffolding in new ways of thinking, involving questioning their understandings and providing arguments for their solutions, in order to create dialogic spaces in which different solutions are voiced and negotiated. In this study, small group instruction through CL approach aimed to support discussions in small groups, but the study relies solely on quantitative measures of students’ mathematical performance. Video-recordings of students’ discussions may have yielded important insights into the dialogic relationships that arose in group discussions.

Despite the positive findings of the CL approach on students’ problem-solving, it is important to note that the intervention did not have an effect on students’ problem-solving pertaining to models of multiplication/division and proportionality. Although CL is assumed to be a promising instructional approach, the number of studies on its effect on students’ mathematical achievement is still limited ( Capar and Tarim, 2015 ). Thus, further research is needed on how CL intervention can be designed to promote students’ problem-solving in other areas of mathematics.

The results of this study show that the effect of the CL intervention on students’ problem-solving was associated with students’ initial scores of social acceptance and friendships. Thus, it is possible to assume that students who were popular among their classmates and had friends at the start of the intervention also made greater gains in mathematical problem-solving as a result of the CL intervention. This finding is in line with Deacon and Edwards’ study of the importance of friendships for students’ motivation to learn mathematics in small groups ( Deacon and Edwards, 2012 ). However, the effect of the CL intervention was not associated with change in students’ social acceptance and friendship scores. These results indicate that students who were nominated by a greater number of students and who received a greater number of friends did not benefit to a great extent from the CL intervention. With regard to previously reported inequalities in cooperation in heterogeneous groups ( Cohen, 1994 ; Mulryan, 1992 ; Langer Osuna, 2016 ) and the importance of peer behaviours for problem-solving ( Hwang and Hu, 2013 ), teachers should consider creating inclusive norms and supportive peer relationships when using the CL approach. The demands of solving complex problems may create negative emotions and uncertainty ( Hannula, 2015 ; Jordan and McDaniel, 2014 ), and peer support may be essential in such situations.

Limitations

The conclusions from the study must be interpreted with caution, due to a number of limitations. First, due to the regulation of protection of individuals ( SFS 2009 ), the researchers could not get information on type of SEN for individual students, which limited the possibilities of the study for investigating the effects of the CL approach for these students. Second, not all teachers in the intervention group implemented the CL approach embedded in problem-solving activities and not all teachers in the control group reported using educational materials on problem-solving. The insufficient levels of implementation pose a significant challenge to the internal validity of the study. Third, the additional investigation to explore the equivalence in difficulty between pre- and post-test, including 169 students, revealed weak to moderate correlation in students’ performance scores, which may indicate challenges to the internal validity of the study.

Implications

The results of the study have some implications for practice. Based on the results of the significant effect of the CL intervention on students’ problem-solving, the CL approach appears to be a promising instructional approach in promoting students’ problem-solving. However, as the results of the CL approach were not significant for all subtests of problem-solving, and due to insufficient levels of implementation, it is not possible to conclude on the importance of the CL intervention for students’ problem-solving. Furthermore, it appears to be important to create opportunities for peer contacts and friendships when the CL approach is used in mathematical problem-solving activities.

Data Availability Statement

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

Ethics Statement

The studies involving human participants were reviewed and approved by the Uppsala Ethical Regional Committee, Dnr. 2017/372. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

Author Contributions

NiK was responsible for the project, and participated in data collection and data analyses. NaK and WK were responsible for intervention with special focus on the educational materials and tests in mathematical problem-solving. PE participated in the planning of the study and the data analyses, including coordinating analyses of students’ tests. MK participated in the designing and planning the study as well as data collection and data analyses.

The project was funded by the Swedish Research Council under Grant 2016-04,679.

Conflict of Interest

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

Publisher’s Note

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

Acknowledgments

We would like to express our gratitude to teachers who participated in the project.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2021.710296/full#supplementary-material

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Keywords: cooperative learning, mathematical problem-solving, intervention, heterogeneous classrooms, hierarchical linear regression analysis

Citation: Klang N, Karlsson N, Kilborn W, Eriksson P and Karlberg M (2021) Mathematical Problem-Solving Through Cooperative Learning—The Importance of Peer Acceptance and Friendships. Front. Educ. 6:710296. doi: 10.3389/feduc.2021.710296

Received: 15 May 2021; Accepted: 09 August 2021; Published: 24 August 2021.

Reviewed by:

Copyright © 2021 Klang, Karlsson, Kilborn, Eriksson and Karlberg. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Nina Klang, [email protected]

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Softening the sharp edges in mathematics.

For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start — to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics — a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

“Mathematics should be flexible, conceptual, a place where we play with ideas and make connections," says Professor Jo Boaler. (Photo: Robert Houser Photography)

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced “How to Learn Math,” the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users. 

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer. 

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 — but the most common answer 13-year-olds gave was 19. The second most common was 21. 

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important? 

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mind-set. I think we all need a little more ish in our lives. 

You also argue that mathematics should be taught in more visual ways. What do you mean by that? 

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things — all of that contributes to our understanding of how it works. 

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking. 

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, ‘I’m in geometry class now, and I still remember that  sugar cube, what it looked like and felt like.’ His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing — elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

Click to enlarge: A depiction of various ways to calculate 38 x 5, numerically and visually. (Image: Courtesy of Jo Boaler)

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them. 

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

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A model for problem creation: implications for teacher training

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• Published: 16 January 2024

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• María Burgos   ORCID: orcid.org/0000-0002-4598-7684 1 ,
• Nicolás Tizón-Escamilla   ORCID: orcid.org/0000-0003-3221-7928 1 &
• Jorhan Chaverri   ORCID: orcid.org/0000-0003-3504-5308 2  

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The invention of problems is a fundamental competence that enhances the didactic-mathematical knowledge of mathematics teachers and therefore should be an objective in teacher training plans. In this paper, we revise different proposals for categorizing problem-creation activities and propose a theoretical model for problem posing that, based on the assumptions of the Onto-Semiotic Approach, considers both the elements that characterize a problem and a categorization of different types of problem-posing tasks. In addition, the model proposes a description of the mathematical processes that occur during the sequence of actions carried out when a new problem is created. The model is illustrated by its application to analyze the practices developed by pre-service teachers in three problem-posing tasks aimed at specific didactic-mathematical purposes (mobilizing certain mathematical knowledge or reasoning, contributing to achieving learning goals, or addressing students’ difficulties). We conclude discussing the potential of our model to analyze the mathematical processes involved in problem creation from the perspective of teacher education.

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Introduction

Although mathematical problem posing has been at the forefront of discussion for the past few decades (Cai et al., 2022 ), the recent intensification in research activity is reflected in the wide range of topics studied in relation to problem posing (Baumanns & Rott, 2021a ; Cai et al., 2022 ; Cai & Huang, 2020 ; Cai & Leikin, 2020 ; Liljedahl & Cai, 2021 , among other synthesis works). According to Cai and Leikin, problem posing in Mathematics Education can be considered in step with activity theory (Leont'ev, 1978 ), which “draws connections between goals, actions, conditions, and tools in any human activity” (Cai & Leikin, 2020 , p. 287). The authors suggest that literature in this field can be categorized according to how problem posing is viewed.

Research viewing problem posing as a tool for teaching mathematics considers this activity as a tool which can be used by teachers to help their students learn mathematics. Research in this area examines how students’ learning of mathematics (measured in terms of cognitive and non-cognitive aspects) can be improved by involving them in problem-posing activities (Cai & Hwang, 2022 ; Chen & Cai, 2020 ; Koichu, 2020 ; Silber & Cai, 2021 ; Xu et al., 2020 ).

The literature considering problem posing as a goal of mathematics teaching focuses on how the ability to pose good problems develops. This includes the study of creating new problems as part of other types of mathematical activities such as problem-solving, proof, and inquiry (Guo et al., 2021 ; Leikin & Elgrably, 2020 ; Ponte & Henriques, 2013 ).

Problem posing appears as a research tool in works focusing on other aspects of learning, thinking (Erdogan, 2020 ), reasoning, and creativity of students (Elgrably & Leikin, 2021 ; Leikin & Elgrably, 2020 ; Singer & Voica, 2015 ) or on the effects of its inclusion in curricula (Cai et al., 2013 ). It is emphasized that problem posing could be seen as a complementary proposal to problem-solving (Pino-Fan et al., 2020 ; Silver, 2013 ) as it improves students’ perception of mathematics, enhances their reasoning and problem-solving skills, improves their attitudes and confidence in mathematics, and fosters a broader understanding of mathematical concepts, properties, and procedures (Ayllón et al., 2016 ; Christou et al., 2005 ; Fernández & Carrillo, 2020 ).

Finally, studies that consider problem creation as an object of research focus on understanding the nature of problem posing itself, including the analysis and evaluation of the typology, variety, and quality of posed problems (Ellerton, 2013 ; Guo et al., 2021 ), as well as the competences, strategies, and other factors enable effective problem formulation (Leikin & Elgrably, 2020 ).

Regardless of how problem invention is perceived, a large number of researchers in Mathematics Education have emphasized the importance of incorporating problem posing in teacher education programs (Grundmeier, 2015 ; Malaspina et al., 2019 ; Singer et al., 2013 ). Without specific training, teachers pose predictable, undemanding, poorly formulated, or unsolvable problems, or those more focused on procedural exercise than on reasoning and conceptual understanding, so it is important to broaden the knowledge of teachers in problem posing (Crespo, 2003 ; Crespo & Sinclair, 2008 ; Lee et al., 2018 ; Stein et al., 2000 ; Xie & Masingila, 2017 ). On one hand, it is stressed that specific teacher training is the only way to firmly integrate problem posing into mathematics curricula and classroom practices (Ellerton, 2013 ). On the other hand, problem posing is considered an appropriate way to introduce pre-service teachers to mathematics teaching, which allows them to explore mathematical content in depth and become aware of their possible deficiencies (Tichá & Hošpesová, 2013 ; Yao et al., 2021 ). To this regard, research on the creation of mathematics problems for educational purposes explicitly mentions their close connection with teaching competencies (Ellerton, 2013 ; Malaspina et al., 2015 ; Mallart et al., 2018 ; Silver, 2013 ), particularly with the capacity for didactic analysis (Malaspina et al., 2019 ). Consequently, problem posing in teacher education should be seen:

both as a means of instruction (meant to engage students in genuine learning activities that produce deep understanding of mathematics concepts and procedures) and as an object of instruction (focused on developing students' proficiency in identifying and formulating problems from unstructured situations). (Singer et al., 2013 , p. 5)

While criteria for categorizing problem-posing activities have largely depended on individual preferences of researchers (Lee et al., 2018 ), traditionally, studies on problem invention have primarily focused on posed problems as a product (Baumanns & Root, 2022 ; Bicer et al., 2020 ; Van Harpen & Presmeg, 2013 ). However, analyzing the processes of problem posing is equally relevant because “it is in the processes that problem posers come up with ideas for new problems, evaluate those ideas, and develop or reject them” (Cai et al., 2022 , p. 123). This has led to an increasing consideration of problem creation as a process in recent studies (Baumanns & Rott, 2022 ; Cai & Leikin, 2020 ; Christou et al., 2005 ; Crespo & Harper, 2020 ; Headrick et al., 2020 ; Koichu & Kontorovich, 2013 ; Patáková, 2014 ; Ponte & Henriques, 2013 ). Nevertheless, these works mainly focus on the cognitive processes that students (Christou et al., 2005 ) or pre-service and in-service teachers (Cruz, 2006 ; Kontorovich & Koichu, 2016 ; Pelczer & Gamboa, 2009 ) activate in problem creation (Kontorovich, 2023 ). In other words, these studies analyzed problem posing from the point of view of the mathematical thinking that accompanies it, seeking to understand “the complex relationship between problem posing and problem solving related to the cognitive processes of problem posing” (Zhang et al., 2022 , p. 498).

The processes involved in problem posing have their own (epistemic) nature independently of the individual learner’s thinking, so that studies from a purely cognitive point of view do not allow us to address the entirety of the educational issue. In fact, many of the difficulties observed in the teaching and learning of mathematics, not only in problem creation or resolution, are related to the fact that institutional mathematical objects exhibit a complexity of inherently mathematical character (Font et al., 2013 ; Godino et al., 2007 ).

From this perspective, we believe that the theoretical and methodological tools developed by the Onto-Semiotic Approach (OSA) to mathematical knowledge and instruction (Godino et al., 2007 ) enable us to tackle the inherent difficulty in researching problem posing by analyzing it in onto-semiotic terms, that is, by means of the mathematical entities and processes involved and their relations. Additionally, to analyze the professional knowledge and competencies required by teachers to create problems with an educational purpose, we adopt the teacher’s Didactic-Mathematical Knowledge and Competencies (DMKC) model (Breda et al., 2017 ; Godino et al., 2017 ), based on the analytical tools of mathematical and didactic activity developed within the core of OSA (Pino-Fan et al., 2015 ).

To address the study of problem-posing processes beyond the cognitive dimension, in this manuscript, we develop a theoretical model for problem posing which, by articulating mathematical and didactic dimensions, focuses the analysis on the involved mathematical (epistemic) processes. To support this model, we first provide a review of proposals for categorizing problem creation activities, particularly those aimed at teacher education, identifying their common characteristics and limitations. Next, we introduce the essential elements of OSA and DMKC frameworks on which our proposal is based. Subsequently, we describe the model and illustrate it by applying it to analyze practices of pre-service teachers in three problem creation tasks aimed at specific didactic-mathematical goals (mobilizing certain mathematical knowledge or reasoning, contributing to learning objectives, or addressing student difficulties). It is intended to show how the analysis based on the proposed model catches the complexity of problem posing through a microscopic analysis of the mathematical activity involved. The article concludes with a discussion of the purpose and potential of our model.

State of the art: problem creation as a product and as a process

Although different authors have given different names to the activity of creating problems, in essence, it involves both formulating new situations and reformulating given problems (English, 1998 ; Silver, 1994 ; Silver & Cai, 1996 ). In this section, we summarize some of the multiple categories and methodological proposals on problem creation found in previous literature.

For Silver ( 1994 ), problem posing is an important component of problem-solving, which leads him to assume a classification of problem creation in terms of the moment in the problem-solving process: before (pre-solution), during (within the solution), and after (post-solution). The goal of problem creation before its resolution is not to solve the problem but to pose a mathematical problem based on a situation or experience previously given to the student. In this case, problems are generated from a concrete stimulus, such as a story, an image, and a representation. When problems are created within the problem-solving process, they are reformulated, for example, by changing objectives and conditions to facilitate their understanding and resolution. In problem creation after resolution, new related problems are generated once properties or characteristics of the initial situation have been analyzed, modifying conditions, objectives, or questions. In this latter category, experiences from the problem-solving context are applied to new situations (Christou et al., 2005 ).

Stoyanova and Ellerton ( 1996 ) assume that problem creation processes can take place under three categories: free, semi-structured, or structured situations. In a free situation , students create problems without any kind of restriction, based on their experiences inside and outside of school (Stoyanova, 1998 ). For example, “create a difficult problem,” “create a money problem,” or “there are 10 girls and 10 boys standing in a line. Invent as many problems as you can that use this information in some way” (Van Harpen & Presmeg, 2013 ).

In the semi-structured category , students are given an open situation and are requested to explore its structure and complete it by applying knowledge, skills, and relationships derived from their previous mathematical experiences. Visual representations in which an image, a graph, or a table is presented; open verbal stories, among others, are frequently used in studies of semi-structured problem posing (English, 1998 ; Silver & Cai, 2005 ). Thus, within the category of semi-structured problem creation, it is possible to consider subcategories according to the starting situation (Akay & Boz, 2010 ): mathematical, open, modelling, and unknown data semi-structured situation. In the mathematical semi-structured situation , an appropriate problem related to the mathematical situation is required to be posed (for example, students must create a problem related to a graph in which the region bounded by the graphical representation of two functions is observed). In an open semi-structured situation , there are no restrictions on the mathematical components involved in a real-life scenario, and students can pose problems according to the structure and conditions they prefer (for example, posing an integral problem to calculate the volume of a solid figure that is not a surface of revolution). A modelling semi-structured situation leads students to concretize and anticipate real-life situations that can be solved through mathematics (for example, posing a problem about the design and cost calculation of a ring). Finally, in an unknown data semi-structured situation , the problem is created by adding the missing data to a situation in which some structures conflict.

In a structured situation , the objective can be determined by all the given elements and relationships. The problem is elaborated based on a previous one, reformulating the given situation or changing its conditions or questions (Van Harpen & Presmeg, 2013 ). In this process, students establish problems taking into account strategies and situations that may be limited by their teachers (English & Watson, 2015 ).

Authors such as Baumanns and Rott ( 2021a ) encountered difficulties in distinguishing between free and semi-structured situations, leading them to differentiate between unstructured and structured situations based on the degree of given information (Baumanns & Rott, 2021a , b , 2022 ). The former form a spectrum of situations without an initial problem, where information ranges from almost none (“create a problem for a math olympiad”) to open situations rich in information where exploring the structure using mathematical knowledge is necessary. In structured situations, new problems based on a specific problem need to be posed, for example, by varying its conditions.

Models such as Silver’s ( 1994 ) which categorize problem-posing activities by their role in problem-solving, or those of Stoyanova and Ellerton ( 1996 ) or derivatives, which classify problem-posing tasks according to the degree of information or restrictions enabling the elaboration of a new problem, focus essentially on the product of problem creation. However, Christou et al. ( 2005 ) consider that these categories allow for the classification of students’ thinking processes in problem posing (studying the product to discover the process). Thus, Christou et al. ( 2005 ) classified problem creation through four different processes: editing, selection, comprehension, and translation of quantitative information. The editing processes refer to tasks requiring students to pose a problem without any restrictions regarding provided information, stories, or instructions. The process of selecting quantitative information is associated with tasks that aim to have students pose appropriate problems or questions for specific, predetermined answers. The known response functions as a restriction, requiring students to focus on the structural context and relationships between the provided information, making selection more difficult than editing. In the processes of understanding and organizing quantitative information , students engage in problem-posing activities based on given equations or mathematical calculations. Understanding requires “understanding the meaning of the operations and students usually follow an algorithmic process focusing on the operational and not the semantic structure of the problems” (Christou et al., 2005 , p. 151). Finally, the translation of quantitative information occurs when students pose problems or questions appropriate to graphs, diagrams, or tables. According Christou et al. ( 2005 ), translation is more demanding than understanding, as it requires interpreting different representations of mathematical relationships.

To approach the procedural study of problem posing, several authors (Baumann & Rott, 2022 ; Cruz, 2006 ; Koichu & Kontorovich, 2013 ; Pelczer & Gamboa, 2009 ) develop descriptive models (showing how subjects actually create problems) of phases (actions) from the numerous processes involved.

Cruz ( 2006 ) model, contextualized in a professional development program, aims to guide teachers in the stages of the “macrostructure of the process of elaborating new problems” (p. 83) in teaching and learning situations. It considers that elaboration, formulation, and posing are differentiated activities in the “metaproblem”:

Problem elaboration will be referred to a complex cognitive activity that the teacher acts (at a macro level); problem formulation constitutes a substructure of such activity, which is at the same time made of several different actions (at a meso level), while the problem statement will be related to a final operation of the formulation (at a micro level). (Cruz, 2006 , p. 83)

For Cruz ( 2006 ), the process of formulating a problem begins with the selection of the mathematical object, which may be conditioned by didactic needs and purposes. Once selected, a classification (analysis) of the components of the mathematical object follows, in which information is obtained and organized according to various criteria. This analysis may result in transforming the object into a generalization of itself or using analogies. After transforming or not transforming the object, the subsequent action involves associating the elements obtained during the classification that are abstracted to relate to mathematical concepts, through their properties or relationships. The search for dependence between the selected concepts allows for making necessary decisions to pose the questions of the problem.

Pelczer and Gamboa ( 2009 ) developed a descriptive model for problem posing, which distinguishes five actions: configuration (reflection on the context of the given situation and the necessary knowledge to understand it), transformation (analysis of the problem's conditions and identification and assessment of possible modifications to execute them), formulation (exploration of problem formulations and possible alterations), evaluation (to verify if it satisfies the initial conditions or if it needs modifications), and final assessment (reflection on the entire process).

Koichu and Kontorovich ( 2013 ) also developed a descriptive model in which, based on two activities of future mathematics teachers called “success stories,” they identified four phases of problem posing. In the first phase, warming up , the given task is associated with certain prototypical or familiar types of problems, and spontaneous ideas are discussed. In the next phase, searching for an interesting mathematical phenomenon , the initial problems serve as material to be critically considered, based on mathematical aspects that will underlie the next interesting problem. Then, problem creators hide the problem-posing process in the formulation of the problems, with the intention that the processes they go through in posing and solving the intermediate problems are not visible to solvers. In the final revision phase, the potential and relevance of the posed problems are evaluated and tested with peers.

For Baumann and Rott ( 2022 ), there is a conceptual and empirical need for a more generally applicable model for problem-posing research than previous ones. In this regard, they propose a new descriptive phase model based on the general structure, i.e., the sequence of distinguishable actions of the observed problems. The authors consider that problem posing for a structured situation follows the following sequence: (1) analysis of the situation, in which simple or multiple conditions of the initial task are collected, identifying which ones and to what extent they are suitable for creating a new task by variation or generation; (2) variation (some of the individual or multiple conditions of the initial task are changed or omitted and the new task is formulated or written); (3) generation of the new task by elaborating one or several new conditions, which are then formulated and written; (4) solution of the posed problem; and (5) evaluation , in which the posed problem is assessed as to whether it is solvable, well-defined, similar to the initial task, appropriate or interesting for a specific objective or group of students. Based on this evaluation, the created problem is accepted or rejected.

The four models presented above describe, through phases, the cognitive actions that unfold in the process of problem creation. These proposals consider that problem posing is part of a complex system in which the teacher acts, for example, to select the mathematical content according to the didactic need or purpose (Cruz, 2006 ) or to evaluate the relevance of the problems (Baumann & Rott, 2022 ; Koichu & Kontorovich, 2013 ; Pelczer & Gamboa, 2009 ). However, in the description of problem creation processes as a sequence of actions, the boundary between mathematical and didactical aspects is not always clear, which hinders a micro-level analysis of the activity.

Kontorovich et al. ( 2012 ) introduce an exploratory framework to address the complexity of mathematical problem posing, moving away from a purely cognitive approach. The framework integrates the following facets based on previous research: (1) task organization (compound of both the specific problem-posing task and the didactic specifications taken by the teacher when planning the activity), (2) knowledge (and competences) base (concepts, properties, routine procedures, system of prototypical problems (Kontorovich & Koichu, 2016 ); competence for mathematical discourse and writing), (3) problem-posing heuristics and schemes (examples of problem-posing strategies) and (4) dynamics and social interactions. Furthermore, they introduce a new facet, (5) aptness (suitability or appropriateness) to oneself, to the potential evaluators of the product or to its potential solvers. The authors consider that to better understanding the problem-posing product, the problem-posing process must be analyzed through the modular structure offered by these facets.

In models like the ones proposed by Contreras ( 2007 ) or Grundmeier ( 2015 ), designed to assist future mathematics teachers in generating mathematical problems, actions indicate on which attributes of a base problem one acts and how to create a new problem. According to Contreras ( 2007 ), some problems can be modified to generate new problems by applying the following fundamental mathematical processes: proof , inversion , particularization , generalization , and extension . By applying these processes, the following types of problems are generated: proof problems (which ask for the proof of a certain property included among the attributes of the base problem), inverse problems (where a known attribute of the base problem is replaced by an unknown one and vice versa, adding additional conditions or constraints if necessary), particular problems (obtained by substituting a mathematical object from the base problem for a particular example or case of the original mathematical object), general problems (obtained by substituting a mathematical object from the initial problem for another one for which the original is an example), and extended problems (obtained by substituting a mathematical object from the base problem for another one that is similar or analogous, without being a particular case of the previous one). In any case, the didactic purpose of generating new problems is to achieve a deeper understanding of the properties of the mathematical objects involved (Contreras, 2007 ). As we can see, Contreras model clearly distinguishes between the process of modifying the original problem and the product (resulting problem) in problem creation.

Grundmeier’s ( 2015 ) study incorporates problem posing in a mathematics content course from two perspectives: problem reformulation and problem generation. In the problem reformulation process , which focuses on a problem that has already been or is being solved, the following techniques are considered: (a) exchanging the given and required information (the same context as the original problem with the given and required information exchanged), (b) changing the context (same structure, but with the context changed), (c) changing the given information (same context and structure of the problem, but with the given information changed), (d) changing the required information (same context and structure of the problem, but changing what the question asks), (e) extending the given problem to more general situations, f) adding information (same context and structure of the problem with added information), and (g) modifying the wording of the problem. In the process of generating a problem from certain given information, additional information can be included, but it must be related to the original set of information (Grundmeier, 2015 ). For this author, adding information, changing what is given, changing what is required, or modifying the wording of the problem are “superficial techniques,” as they do not require a change in the structure of the problem, as opposed to creating a problem by exchanging the given (information) and the required (requirement), modifying the context or extending it. Reformulating the structure involves more creativity and a better understanding of the mathematical content by the problem creator.

To determine what is involved in creating a problem, Malaspina ( 2013 ) begins by specifying the elements characterizing a mathematical problem, namely: information, requirement, context and mathematical environment. Information refers to the quantitative or relational data given in the problem. The requirement is what is asked to be found, examined or concluded, which can be quantitative or qualitative, including graphs and demonstrations. The context can be intra-mathematical or extra-mathematical determining the environment or setting that gives rise to the mathematical activity. The mathematical environment is the global framework in which the mathematical concepts that intervene or may intervene to solve the problem, their properties and relationships are located, that is, the mathematical structure (Grundmeier, 2015 ). For Malaspina ( 2013 ), problem creation is a process by which a new problem is obtained. He distinguishes between variation (of a problem) and elaboration. In a variation , a new problem is constructed by modifying one or more of the four elements (information, requirement, context, environment) of an initial problem. In the process of elaboration of a problem, a new problem is constructed in an open form, starting from a situation (given or configured by the author), or with a specific requirement or purpose, which may have a mathematical or didactic emphasis. In the elaboration of a problem from a situation, the context originates in the situation, the information is obtained by selection or by modification of that which is perceived in such situation, the requirement is a consequence of the relationships between the elements of the information implicit in the statement, and the mathematical environment can be determined by the author or by the ways of solving the problem. In this model, problem-posing tasks are linked, on the one hand, to the modifiable attributes in problems (thus overcoming the unclear boundaries between different problem-posing categories) and, on the other hand, to the presence or absence of an educational purpose that motivates them.

Theoretical framework

Problem solving in the construction of meaning.

From the anthropological and pragmatic perspective of mathematics adopted by OSA (Godino et al., 2007 ), the activity of individuals in problem-solving occupies a central place in the construction of mathematical knowledge. Mathematical practice is considered any action or expression carried out by someone to solve mathematical problems, communicate their solution, validate it, or generalize it to other situations (Godino et al., 2007 ).

OSA assumes that the solution of mathematics education epistemological, cognitive, and instructional problems previously requires addressing the ontological problem, i.e., clarifying the nature and types of mathematical objects that appear in mathematics teaching and learning activities such as problem creation (Font et al., 2013 ; Godino et al., 2007 ).

The term object is used in a broad sense in the OSA to refer to any entity that is involved in some way in mathematical practice and that can be separated or individualized. A typology of primary mathematical objects is proposed: problem-situations (exercises and problems understood as tasks that induce mathematical activity), languages (mathematical terms and expressions), concepts (mathematical entities that can be introduced by description), propositions (statements about concepts, properties), procedures (calculation techniques, operations, algorithms), and arguments (statements required to demonstrate propositions or explain procedures). The consideration of an object as primary is not an absolute matter, as it pertains to functional entities relative to language games (institutional frameworks, communities of practice, and usage contexts) in which they participate. Primary objects involved in mathematical practices and their emergents, according to the language game they participate in, can be considered from five dual facets that dialectically complement each other (Fig.  1 ). These facets are regarded as attributes that, when applied to different primary objects, result in the following typology of secondary objects: ostensive (public, material, perceptible)— non-ostensive (abstract, ideal, immaterial); extensive (particular)— intensive (general); personal (relative to individual subjects)— institutional (shared in an institution or community of practices); significant (expression)— meaning (content); unitary (objects considered globally as a previously known whole)— systemic (objects as systems formed by structured components). These objects and their relationships (configurations), both in the personal and institutional facet, are constituted over time through mathematical processes.

Source: Godino et al., ( 2020 , p. 7)

Onto-semiotic configuration of practice, objects, and processes.

Although OSA does not attempt to give a definition of “process” upfront, since there are many different types of processes (such as processes as series of practices, cognitive processes, metacognitive processes, instructional processes), an effort is made to establish certain characteristics of what a mathematical process is (Font et al., 2008 ; Godino et al., 2009 ). A mathematical process is considered to be any sequence of actions activated or developed over a certain time to achieve an objective, usually the response to a proposed task subject to mathematical or metamathematical rules.

Both primary and secondary objects (derived from the application of dualities) can be considered from the process–product perspective, which provides criteria for distinguishing types of primary and secondary mathematical processes (Fig.  1 ). On the one hand, primary objects emerge from systems of practices through the primary processes of problematization, communication, definition, enunciation, algorithmization-description, and argumentation, respectively. On the other hand, dualities give rise to the following secondary processes: institutionalization-personalization; materialization-idealization; representation-signification (or interpretation); particularization-generalization; unitarization (or reification)-decomposition (or splitting), from which secondary objects emerge (Font et al., 2013 ; Godino et al., 2007 ). The distinction between particularization and generalization processes with respect to idealization and materialization processes (Font & Contreras, 2008 ) and between these and unitarization and decomposition processes allows for a more detailed analysis of each of these processes and their combined presence in mathematical activity.

Problem-solving or modelling is considered in the OSA framework as hyper or mega-processes (Godino et al., 2009 ) in the sense that they involve some or several of the aforementioned processes.

From the OSA perspective, the (epistemic) analysis of mathematical practices, objects, and processes constitutes a first level of didactic analysis of any mathematical activity, particularly that which revolves around the creation of mathematical problems (Font et al., 2013 ; Godino et al., 2007 ). When problem creation takes place under the guidance of a teacher and in interaction with other students, didactic analysis must progress from the study of the epistemic configuration to the analysis of the didactic configuration, which also considers, in addition to the system of mathematical practices, the sequence of actions and means used by the teacher for this activity.

Mathematics teacher knowledge and competencies model

The DMKC model articulates categories of didactic-mathematical knowledge and competences of mathematics teachers, through the facets that condition instructional processes (Godino et al., 2017 ). In that way, it is accepted that the teacher must have mathematical knowledge per se, that is, common and relative to the educational level where they teach, as well as expanded knowledge allowing them to articulate it with higher levels. In addition, the teacher must have specialized or didactic-mathematical knowledge of the different facets that affect the educational process: epistemic (didactic-mathematical knowledge about the content itself, institutional reference meanings), ecological (relationships between mathematical contents and other disciplines, curricular factors and socio-professional factors that condition mathematical instruction processes), cognitive (how students reason and understand mathematics, and how they progress in their learning), affective (knowledge of the affective, emotional, attitudinal and belief aspects of students regarding mathematical objects and their study), mediational (technological, material, and temporal resources), and interactional (knowledge about mathematics teaching, selection and organization of tasks, resolution of student difficulties and classroom interaction management). These dimensions interact in every process aimed at teaching and learning mathematical content.

In addition to having this knowledge, the DMKC model proposes that the teacher should be competent in addressing the basic didactic problems present in instructional processes. The general competence of didactic analysis and intervention allows the teacher to describe, explain, and judge what has happened in the study process and make improvement proposals for future implementations (Godino et al., 2017 ). The theoretical and methodological tools of OSA allow the development of this competence, which encompasses, among other sub-competences, the identification and description of operational and discursive practices implied in mathematical activity ( analysis of global meanings ) and the recognition of the configuration of objects and processes emerging from mathematical practices ( ontosemiotic analysis ). Both skills are fundamental in the creation of problems to respond to certain requirements. But, reciprocally, the creation of problems with didactic purposes serves as a means to develop them, since this process requires: reflecting on the overall structure of the problem (what it seeks and whether the information provided is sufficient to solve it); analyzing both the possible ways of solving it and the mathematical objects and processes involved and how they are related, and recognizing possible difficulties that students may encounter and how to address them in the approach to new situations.

The literature review reveals the existence of (1) models considering the attributes or elements characterizing a problem (Contreras, 2007 ; Grundmeier, 2015 ; Malaspina, 2013 ), (2) models categorizing different tasks in problem creation (Akay & Boz, 2010 ; Stoyanova & Ellerton, 1996 ; Van Harpen & Presmeg, 2013 ), and (3) models describing the problem creation process (Baumanns & Rott, 2022 ; Christou et al., 2005 ; Cruz, 2006 ; Koichu & Kontorovich, 2013 ; Pelczer & Gamboa, 2009 ). Footnote 1 However, these three aspects are not articulated within a framework for problem posing. Moreover, when these models describe processes observed during problem creation, they tend to focus on cognitive processes that occur afterward. In our model, we start from the OSA categorization of the types of mathematical processes (primary and secondary) involved in and modelling mathematical practice and identify those that are present in problem creation.

Hence, the aim of our research is to present a theoretical framework that integrates these three aspects, bridging the mathematical dimension (epistemic analysis of processes involved in the mega-process of problem posing) with the didactic dimension (categorization of problem-posing tasks based on problem components and facets of didactic-mathematical knowledge involved).

A framework for problem creation

Based on Malaspina’s ( 2013 ) elements of a problem (Fig.  2 ) and considering previous models, we propose the following categories for problem creation:

Free or unstructured elaboration . There is no starting (structured) problem-situation, and no indications, guidelines, or restrictions are included on any of the components (context, environment, information, and requirement) of the new problem. For example, “create an easy problem.” The context-information-requirement field is completely empty and thus is the environment.

Semi-structured elaboration . There is no base problem situation, but information or restrictions on elements of the new problem are included. That is, some of the context-information-requirement components are given. For example, it may contain context and information (Van Harper & Presmeg, 2013 ) that must be completed with the objective, or data (information) must be completed to answer a given question (requirement) (Cai & Jiang, 2017 ; Espinoza et al., 2018 ; Grundmeier, 2015 ). On other occasions, the restriction may be given as part of the mathematical environment, often as a strategy or certain calculation that must lead to the solution (Christou et al., 2005 ).

Structured elaboration or variation . A problem is posed based on a previous one (Stoyanova & Ellerton, 1996 ; Van Harpen & Presmeg, 2013 ), so that the different elements (context, information, requirement and mathematical environment) are known. The variation may be partial if some of these components are modified but not all of them, for example, quantitative or relational data (information) or questions (requirement) (Van Harpen & Presmeg, 2013 ), or complete (if the four components are modified). The exchange between information and requirement, adding new information or asking new questions (Cai & Jiang, 2017 ), is also considered within the category of structured elaboration.

Components of a structured problem-situation

We consider that the elaboration of a problem to respond to a mathematical purpose or demand, such as employing a certain strategy or arithmetic calculations (English & Watson, 2015 ), or to a didactic-mathematical one (Malaspina, 2016 ; Malaspina & Vallejo, 2014 ), in the case of teachers, such as involving certain concepts or properties (epistemic dimension), or meeting a certain level of cognitive demand or diagnosing certain expected difficulties (cognitive dimension), can occur both in the semi-structured and structured (variation) case.

In the same way that problem-solving or modelling, OSA considers problem posing as a mega-process in that it may involve several primary (communication, enunciation, problematization) and secondary (representation-signification, materialization-idealization, particularization-generalization, decomposition-unitarization) processes. The elementary processes involved depend on the creative task that motivates them, i.e., what is the product expected to be obtained; reciprocally, certain processes determine different problem-products.

Let us take as an example a semi-structured elaboration situation in which some of the components of context-information-requirement are given (shown in grey in Fig.  3 ). To create the problem, it is necessary to determine the remaining elements. First, this involves separating into semiotic units (decomposition process) the given components, giving meaning (signification process) to each of the linguistic objects (in their various registers) in the situation (upper center part of Fig.  3 ). Next, information and requirement must be linked through the mathematical structure. Mathematical objects are abstract entities, that is, immaterial (non-ostensive) and general (intensive), so idealization and generalization processes occur in their articulation with the mathematical environment, i.e., a configuration of concepts, properties, and procedures, is established that enables giving meaning to the requirement based on the information or vice versa. In this way, all the elements that determine the problem are available (upper right part of Fig.  3 ). Analysis of mathematical objects involves classification, association, and search for relationships (Cruz, 2006 ). The previously identified semiotic units must be composed to acquire unitary entity (unitarization), so abstract objects that make up the mathematical structure of the problem (the environment) must be particularized, materialized, and represented in the enunciation of the new problem (central right part of Fig.  3 ). Additionally, as part of the problem creation process, it is possible to consider the (mega-process) of solving it (involving other primary processes such as representation, algorithmization, and argumentation of the solution), as occurs in the models of Baumanns and Rott ( 2022 ), Cruz ( 2006 ), and Pelczer and Gamboa ( 2009 ), within the evaluation phase. If the solution evaluation is not satisfactory, it may be necessary to resignify the components of the problem and reanalyze the mathematical objects to obtain new problem formulations (lower left part of Fig.  3 ). When problem posing serves a didactic purpose, it is not only the solution to this intermediate problem that is evaluated; the aptness of the provisional product is also assessed (Kontorovich et al., 2012 ), meaning its quality or the extent to which it aligns with the intended educational objectives.

The mega-process of problem posing. The case of a semi-structured elaboration

As depicted in Fig.  4 , we consider a dual dimension in problem creation: mathematical and didactic. In the mathematical dimension, problem creation is an activity shared by students and teachers, modelled in terms of mathematical processes and their emergent objects (see Fig.  3 ). In the didactic dimension, problem creation involves professional knowledge in various facets some of which are related to task organization, problem-posing heuristics, and schemes suggested by Kontorovich et al. ( 2012 ). Indeed, when creating a problem, the teacher should be aware of and employ a variety of problem-solving strategies that are appropriate from an instructional point of view (interactional facet) recognizing the material resources that could be helpful and the temporal requirements (mediational facet). Teacher should also identify the knowledge involved in its solution (epistemic facet) and evaluate its adequacy to the educational level and curriculum (ecological facet). This analysis will allow them to be aware of the complexity of the situations proposed to students, responding to their knowledge and (potential or encountered) difficulties (cognitive facet) and catering to their interests (affective facet). The deployment of these knowledge is determined by the didactic purpose of the problem-posing task, which can in turn have an epistemic, ecological, cognitive, etc. character.

Facets of didactic-mathematical knowledge in problem posing

Illustrating the model’s implementation

In this section, we aim to illustrate and operationalize the problem-posing framework proposed in the previous section. To do so, we describe some tasks implemented in a primary education teacher training course aimed at developing the competence to create problems with a didactic purpose. Footnote 2 This represents a particular case of the formative experiences we are conducting in the specific field of proportional reasoning focused on the epistemic and cognitive facets of didactic-mathematical knowledge (Burgos, et al., 2018 ; Burgos & Chaverri, 2022 , 2023 ; Burgos & Godino, 2022a , b ). In this experience, prospective teachers receive specialized training on the notions of practices and pragmatic meaning, the nature and typology of mathematical objects and processes (Godino et al., 2007 ), the elements that characterize a mathematical problem (Malaspina, 2013 ), as well as specific instruction on proportional reasoning and the difficulties it poses for students (Hilton & Hilton, 2019 ).

The responses of the trainee teachers (chosen, in each case, to be representative of the productions of the rest of the participants Footnote 3 ) are analyzed from the point of view of the mathematical processes involved and the emergent objects in the mega-process of problem posing, as shown in Fig.  3 . These responses include both the new problem and the (mathematical and didactical) practices on which they are based to address the intended educational purpose.

Problem creation with an epistemic didactic-mathematical purpose

In the first part of the situation shown in Fig.  5 , a semi-structured elaboration task with a didactic-mathematical purpose of epistemic nature is proposed. The context of the problem to be elaborated, the information, and the mathematical environment in which it should be framed are provided. The teacher in training must recognize from the information provided which magnitudes can be related in an additive way, which in a proportional way, complete it and establish the requirement so that solving the problem requires differentiating these relationships (epistemic facet of didactic-mathematical knowledge).

Source: Own elaboration

Problem creation according to a didactic-mathematical purpose.

Figure  6 shows the problem created by a trainee teacher, Laura (a fictitious name), and how she justifies the relevance of her proposal.

Problem created by Laura to solve the task given in Fig.  5

Laura decomposes the context and information given in the proposed situation into semiotic units. Considering the stated objective, Laura selects the magnitudes that can potentially be related in an additive or multiplicative way, based on the data and their links. The process of idealization and generalization leads to identify the need to add regularity conditions and relations between the magnitudes in the situation so that they fit the models y  =  x  +  k (additive relation) and y  =  kx (proportional relation). Finally, Laura particularizes these relations (“Juan is faster than Raquel,” “Raquel and Daniel plant at the same speed,” “Daniel started before his classmates”) and provides a value of the magnitude “number of flowers planted by Raquel” to ask for the corresponding values of the magnitudes “number of flowers planted by Juan” and “number of flowers planted by Daniel.” Laura solves the problem, giving as final her formulation of the problem (Fig.  7 ).

Mega-process of problem posing (semi-structured elaboration) followed by Laura

The second part of the task (Fig.  5 ) entails a structured elaboration, as the teacher-in-training must vary her problem with an epistemic purpose: not all the ratios involved should be integers. This implies idealizing and generalizing the information (data and relations) in the created problem and connecting with the mathematical environment provided by the didactic objective: multiplicative relation or not between antecedent and consequent of a ratio. Then, the particularization and materialization allows to decide which elements of the ratios given in the problem are changed to formulate the problem with the established purpose.

As shown in Fig.  8 , Laura interprets the didactic pursue provided in the second part of the task to decide on which elements of the problem she can act on and in what manner. Thus, she keeps the requirement, but modifies the information by changing the number of flowers planted by Juan and Daniel, so that, although the ratio in the number of flowers planted by Raquel, 4 to 20, is still integer, the ratio between the number of flowers planted by Raquel and Juan, 4 to 21, is not.

Problem created by variation of problem 1 (Fig.  6 ) by Laura

Problem creation with a cognitive didactic-mathematical purpose

Figure  9 depicts a (fictional) mathematics classroom situation in which a problem is presented, and a conversation takes place between two students and the teacher during its resolution. In this case, it is intended that teachers in training analyze the educational objective behind the base problem and that they can vary (structured elaboration) it when the result of its implementation is not the intended one from the point of view of expectations and learning achievements (cognitive facet of the didactic-mathematical knowledge).

Structured elaboration task with a cognitive purpose.

In this case, the task organization (Kontorovich et al., 2012 ) does not only consider the didactic specifications or the situation itself but also the students’ discursive mathematical practices regarding its solution. The trainee teacher must analyze these practices to decide on which elements of the base problem to act upon. That is, the processes of decomposition, signification, idealization, and generalization stem not only from the problem itself but also from the mathematical activity of the students.

Figure  10 shows the answer given by a trainee teacher. Daniela (fictitious name) identifies and interprets units of meaning (decomposition, signification) in the pupils’ answers: “Lola is the one who stopped more, Alex who stopped less,” “David saved almost all of them,” “from 2 to 5 there are less than from 11 to 20.” She recognizes the incorrect additive strategy in the answer and situates (idealization, generalisation) the mathematical environment in the proportionality relationship (highlighting the role of the comparison of fractions). Although the objectives she specifies are more curricular than didactic in nature, Daniela describes objects (concepts, procedures, properties) expected in the solution to the mathematical problem. In the variation, she modifies the requirement to make it clearer to the student the need to compare proportionally, preserving context, information, and mathematical environment. In addition, she considers that the graphical representation (without specifying the type) helps in the resolution, so she also includes it as a requirement in the problem. Daniela idealizes and generalizes the information (penalties saved by each goalkeeper and total penalties taken by each one) and the requirement (question about who saved the penalties better) of the initial problem to articulate the objects of the mathematical environment of the problem, identified as proportional relations and comparison of fractions. After that, she particularizes these abstract objects in the given context (penalty kicks) and represents them through natural language to modify the requirement of the problem (“in proportion to the total number of balls taken”).

Analysis and creation of problem by variation of the problem proposed in the episode (Fig.  9 ) developed by Daniela

Creating problems to develop algebraic reasoning

From research, it is highlighted that creating problems enables the development of mathematical reasoning and fosters a deeper understanding of concepts, properties, and procedures. This not only occurs with students (Ayllón et al., 2016 ) but also with prospective teachers (Malaspina et al., 2019 ). Moreover, it is emphasized that prospective teachers should identify the potential of tasks that are not intentionally algebraic and to modify them so that objects and processes of an algebraic nature come into play during their solution (Burgos & Godino, 2022a ; Godino et al., 2014 ). Problem-posing tasks, like the one we illustrate below (Fig.  11 ), aim to encourage prospective teachers to create problems that articulate different types of reasoning, including proportional, probabilistic, and algebraic reasoning. Footnote 4

Creation of problems to promote algebraic and proportional reasoning in the probabilistic context.

In the first part of the task shown in Fig.  11 (semi-structured situation), the trainee teachers have to create a problem in the probabilistic context, taking into account that proportional reasoning must be used (epistemic purpose). The result of this process is the Pre problem. Therefore, the context (urns), the information (number of balls in each urn, by means of graphic/pictorial language), and the environment (proportional reasoning) are provided, leaving the requirement open. Figure  12 shows the problem created by David (fictitious name) from the situation presented.

Pre problem created by David in the environment of comparison of probabilities in urns

In his problem-posing process (Fig.  13 ), David gives meaning (signification) to the image in the probabilistic context (unrs), considering two boxes (A and B) with different numbers of black and white balls. As it must involve proportional reasoning, it is placed in the comparison of probabilities (ideal, abstract mathematical structure), through the correspondence with the comparison of proportions between the number of balls, delimiting the mathematical environment (idealization and generalization). The context of chance is clearly materialized in the Pre problem, specifying that the drawing is done blind.

Problem-posing process followed by David

The didactical goal pursue by the second part of the task (Fig.  11 ) involves specific didactical-mathematical knowledge about algebraic reasoning. The trainee teacher must make a complex microscopic analysis of the mathematical structure involved in the solution of the Pre problem, in order to decide which variation would imply a higher level of algebraic reasoning, i.e., which elements of the Pre problem should be modified so that objects with a higher degree of intension and generalisation and new algebraic processes are involved in its solution (Godino et al., 2014 ). David appreciates that there is no algebraic activity in the solution to the Pre problem, “because particular numbers are used, and no unknowns are reflected.” In addition, he identifies the process of “particularization of the proportionality relation” to the case of the composition of the boxes. This reflection leads him to propose the Pos problem included in Fig.  14 .

Pos Problem created by David to increase the algebraic reasoning required

In this case, David maintains the context, information, and mathematical environment of the problem (comparison of probabilities) but modifies the requirement making use of a third box whose composition must be determined. This requirement mobilizes higher level objects and processes (relations and properties of rational numbers, parameters; generalisation, symbolic representation) in its resolution. The idealization process carried out by David to create the Pos problem is observed when he justifies what makes the level of algebraic reasoning higher in this case. He considers the implication of “properties of equivalent fractions [to] be able to know if it is the same proportion despite having different numbers, thus finding a generalization by knowing that \(a/b=(na)/(nb)\) ,” the use of the “relational meaning of equality as equivalence, and employing literal symbols as unknowns, but without operating with them.”

Final reflection

The creation of problems, not only in the design and planning of lessons but whenever necessary during their implementation, to encourage students to use specific mathematical knowledge, employ certain competencies, overcome difficulties encountered with a prior problem, or achieve specific learning outcomes, is one of the features of a teacher’s competence in analysis and didactic intervention. Reciprocally, the creation of problems for didactic purposes serves as a means to enhance and integrate other didactic-mathematical competencies and knowledge of the teacher: it requires reflection on the elements characterizing the problem, different ways it can be solved, analyzing the mathematical practices, objects, and processes involved, and identifying potential difficulties that students may encounter (Burgos & Chaverri, 2023 ; Calle et al., 2023 ; Godino et al., 2017 ). This justifies the interest in incorporating problem posing into teacher education (Grundmeier, 2015 ; Singer et al., 2013 ) and developing theoretical and methodological tools to guide them in a complex task they may not be familiar with (Ellerton, 2013 ; Mallart et al., 2018 ). With this commitment, this work has presented (founded and illustrated) a theoretical model for problem posing integrating the epistemic analysis of the involved processes, categorization of problem creation tasks based on the components of a problem, and the types of didactic-mathematical knowledge they involve.

Firstly, our model allows us to address the complexity of problem posing from the perspective of the mathematical activity it entails. In this sense, our framework provides a description of the mathematical processes and objects (as detailed in Fig.  1 ) involved in the sequence of actions carried out when posing a problem. This point of view complements previous problem-posing models focusing on describing cognitive processes that intervene when posing a problem (Baumann & Rott, 2022 ; Cruz, 2006 ; Koichu & Kontorovich, 2013 ; Pelczer & Gamboa, 2009 ), analyzing the mathematical actions performed to create a problem (Contreras, 2007 ; Grundmeier, 2015 ), or characterizing the facets involved in problem posing (Kontorovich et al., 2012 ). Secondly, based on Malaspina’s ( 2013 ) description of the elements of a problem, our model structures a problem (Fig.  2 ) as an entity determined by context, information, requirements, and mathematical environment. Thirdly, this characterization of problem’s attributes leads us to establish three categories of problem-posing tasks which groups previous categories of problem creation (Akay & Boz, 2010 ; Baumann & Rott, 2021a ; Cai & Jiang, 2017 ; Stoyanova & Ellerton, 1996 ; Van Harper & Presmeg, 2013 ) as didactic decisions made by the teacher (Kontorovich et al., 2012 ): free or unstructured, semi-structured, and structured elaborations. Once the type of problem-posing task is determined based on the known components in the situation and the didactic instructions that determine the rest, the application of the model reveals different sequences of emerging processes and objects (Fig.  3 ). Up to our knowledge, we presented for the first time a model which articulates these three aspects of problem posing (components of a problem, categorization of problem-posing tasks, and description of mathematical processes involved when posing a problem) into a single framework.

The range of applicability of our model encompass two different dimensions. When focusing on teacher training, using the model allows teacher educators to design a priori problem-posing activities taking into account the mathematical practices, objects, and processes involved, as well as the actions (didactic practices) coming into play, their sequencing (didactic processes), and the emerging entities of such actions (didactic objects) determining the didactic dimension of problem creation (Fig.  4 ). Afterward, it enables the assessment of the achieved competence and the identification of processes (mathematical or didactic) that may have caused difficulties in some cases, guiding instruction towards the development of these capabilities. Problem creation in this case is part of a macrostructure (Kontorovich et al., 2012 ) organized around the different facets of the teacher’s specialized knowledge (Godino et al., 2017 ), which broadens the perspective of previous research (Baumann & Rott, 2022 ; Contreras, 2007 ; Cruz, 2006 ; Grundmeier, 2015 ; Koichu & Kontorovich, 2013 ; Pelczer & Gamboa, 2009 ). When the problem-posing task is proposed to students, the model allows the researcher or teacher to reflect a priori on the adequacy of the information provided to the student for creating the problem, as well as to become aware of the intrinsic complexity of the mathematical processes leading to the problem as the final product. Afterward, it enables a microscopic analysis of the mathematical activity that led to this result.

In this work, we illustrated the applicability of our model by studying the responses of trainee teachers to different problem-posing tasks. In future research, we will apply our framework to analyze and categorize the mathematical processes in which prospective teachers encounter difficulties when creating problems with a didactic-mathematical purpose, not only in epistemic and cognitive but also affective, ecological, interactional, or mediational facets. Additionally, we will investigate how prospective teachers utilize the model to analyze students’ productions in response to various problem-posing tasks, interpreting their difficulties in terms of the underlying processes. Finally, our model could be extended to encompass the didactic-mathematical knowledge and competencies of teacher educators concerning the design of problem-posing tasks.

The comparison and articulation of the models described in the “ State of the art: problem creation as a product and as a process ” section can be significant given that, as we have seen, they employ similar or at least compatible terms, concepts, and principles.

The aim of this section is not to describe the results of these empirical studies per se, but rather to exemplify our model through its implementation in a real teacher training context.

The formative experience from which the examples are taken was carried out with a group of 66 students for teacher from the third year of the Primary Education degree at a Spanish university.

Proportional reasoning is an integral part of the components of the sample space analysis and quantification of probabilities (Bryant & Nunes, 2012 ). It is also an essential component of early algebraic thinking (Van Dooren et al., 2018 ).

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Funding for open access publishing: Universidad de Granada/CBUA. Research conducted as part of the research project, PID2019-105601GB-I00/AEI/10.13039/501100011033 (Ministry of Science and Innovation) and Project PID2022-139748NB-100 funded by MCIN/AEI/10.13039/501100011033/ and by FEDER (una forma de hacer Europa), with support from Research Group FQM-126 (Junta de Andalucía, Spain).

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Burgos, M., Tizón-Escamilla, N. & Chaverri, J. A model for problem creation: implications for teacher training. Math Ed Res J (2024). https://doi.org/10.1007/s13394-023-00482-w

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How to Create Custom AI Chatbots That Enrich Your Classroom

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• Artificial Intelligence
• Classroom Management
• Digital Learning

G iven the breakneck speed at which AI tools and models are developing, and the many implications of this technology on learning, work, and culture, it’s easy to feel a little lost. This is particularly true for educators who are trying to understand generative AI and determine its role in their classrooms and the broader educational landscape.

At Boston College, my colleagues and I have been finding our way by drawing on the university’s mission to help students not only develop academically, but also to find purpose and meaning in their lives.

This focus on purpose was on our minds as we began experimenting with custom chatbots to help educators learn about AI and determine whether they have a role in helping students learn. What we discovered surprised us: It wasn’t just teachers and students who were feeling lost in an AI world; chatbots were struggling to find their purpose, too.

Without our guidance, they didn’t know how to help students learn, and they had no sense of our institution’s context, our values, or what our students needed. They did their best, but they were floundering. They needed us to help them to be productive partners in our teaching.

“What we discovered surprised us: It wasn’t just teachers and students who were feeling lost in an AI world; chatbots were struggling to find their purpose, too.”

What follows are examples of chatbots we created at Boston College, the purposes they serve for our students, and some of the strategies we found useful in doing this work. Everything here is preliminary and provisional, of course, since our approaches to AI will continue to evolve and the tools for creating chatbots will change. But we hope our initial efforts will help you find a path for exploring whether AI could have a purposeful role in your teaching as well.

How to build a custom chatbot: The Stuckbot story

When OpenAI released GPTs late in 2023, my colleagues and I began looking beyond general-purpose chatbots such as ChatGPT , Claude , and Gemini and began exploring how we could create chatbots with customized instructions and access to our own content. This meant that, if our students started a conversation with a chatbot, it would have a built-in sense of direction and context that aligned with our specific goals as teachers. When chatbots can do this, they’re supporting what has always been a central role of teachers—to give educational experiences context so students know not just what they’re learning, but why they’re learning and who they are doing it with.

To avoid the costs associated with OpenAI’s first version of GPTs, I developed something similar using simple WordPress sites and the plugin AI Engine Pro (along with an OpenAI API key). This relatively inexpensive and non-technical method was meant to be an interim solution until other more scalable and secure options became available. It gave us a sandbox environment for small-scale experimentation before we needed to make larger decisions about AI infrastructure at the university.

My colleague, professor George Wyner , was excited to jump on this new opportunity, not just because he teaches data analytics, but because he had spent the semester exploring AI with a cohort of colleagues and staff facilitated by the Center for Digital Innovation in Learning (CDIL). He already planned to have his undergraduate students try using AI in his Data Management for Analytics and Applications course, but now he could experiment with a chatbot that was better informed about his course and how he wanted his students to learn.

“When chatbots have the same teacher as the students they’re assisting, it’s more likely they will contribute something valuable to the classroom.”

He created a specific assignment called “ The Programmer’s Journal ” that provided students with context for how they should use what we affectionately dubbed the “Stuckbot.” The premise for the assignment relied on the idea that having students take notes on their problem-solving process and then reflect on where they get stuck is beneficial for learning, with or without AI. Involving AI in that process added another reflective layer by helping them practice metacognition and providing them chances to discuss the roles AI might play in learning data analytics.

The assignment description aimed to make these layers clear:

A challenge when writing code (in R, SQL, or any other programming language) is that sometimes you get stuck. In this assignment, we will explore strategies for getting unstuck using Stuckbot, an AI chatbot crafted for this purpose. Our goal is to get better at getting unstuck when problem-solving and, at the same time, to explore how AI can help with this and with the learning process in general. In this assignment, you will create a kind of programmer’s journal, in which you add two journal entries that each chronicle a moment when you got stuck and then your efforts to get unstuck. In addition, you will add a brief reflection on this process and the role that AI played in it.

I began working with Wyner to craft instructions for the chatbot that would guide it in how to work with students. We began by giving it a role:

You are a friendly, supportive chatbot designed to help undergraduates talk through the experience of getting stuck when learning programming and data science. Not only are you an expert in applied mathematics, data science, and programming, but you are particularly skilled in supporting student learning by helping them work through getting stuck and reflecting on what they learn from the experience.

We let the chatbot know that it would be working with students in the Data Management for Analytics and Applications course and provided specifics on which tools and packages the students would be using.

Next, we assigned it some general rules to follow:

Don’t give students answers to problems outright. Ask them questions to get them to the next step. Ask only one question at a time and go one step at a time. Share summaries of reflections with the user to help confirm what they are saying. If ever unclear, ask the user how they want to proceed.

Source: Boston College, 2024

A screenshot of Wyner’s Stuckbot.

The remainder of our work on the prompt involved providing instructions for the two different roles the chatbot would need to play in supporting the “Programmer’s Journal” assignment. First, its job was to serve as an assignment tutor to help students when they got stuck on problems. Second, it could help students pause and reflect on how they got unstuck before moving on. (These are two of several roles that AI chatbots can play for your students; more on that in a moment.)

As we tested the chatbot with colleagues and former students, we were pleasantly surprised to see how well it performed in both roles once we gave it instructions for each and the context it needed to understand exactly what students were learning in the course. In a focus group at the end of the course, students reported that the Stuckbot aided their learning by guiding them toward solutions rather than giving them an answer outright, and they said they would have used it more had it been available earlier in the course.

Their feedback suggests that the constraints put on the chatbot were what mattered for students (not just what it could do for them), and these limitations enabled the chatbot to feel like an extension of their teacher’s intentions for the course and his care for them as learners.

Access the full Stuckbot prompt and assignment for “The Programmer’s Journal” here .

Four purposeful roles for chatbots in the classroom

When Wyner shared how he designed his chatbot and how it aligned with very specific learning goals, it inspired others at Boston College to imagine other roles they could experiment with beyond what they first thought was possible.

Over the course of the following spring semester, we developed 10 chatbots with different purposes. The results varied widely, but we learned as much from the failures as from the successes, and this hands-on experimentation allowed us to begin identifying some patterns that could inform and inspire future experiments.

These four chatbot examples represent a first attempt at capturing what we’ve learned so far. We think these delineations make it easier to see how chatbots can align with specific aspects of the learning process, even if they might have more than one role at a time (such as with the Stuckbot, acting as both an assignment tutor and reflective guide, for example).

We hope they will provide you with a starting point as you imagine roles for AI that reflect your particular contexts and purposes, and we look forward to learning from what you discover in the process.

Four purposeful roles for course chatbots.

1. Course assistant: Make courses easier to navigate

A course assistant chatbot specializes in helping students understand how a course works and making it easier for them to navigate through course materials and assignments. It gives students the ability to spend more effort on learning and less on understanding the logistics of the course and finding course content.

By helping to make the expectations and requirements of the course as transparent as possible, this chatbot can be particularly helpful for first-generation college students and those who need more support to make the “ hidden curriculum ” more visible, or in online courses where course websites can be extensive and there is less real-time interaction between faculty and students. It also can be useful in face-to-face classrooms where students often balance multiple courses and other non-academic demands and need key information to be reinforced in multiple ways.

“A course assistant chatbot specializes in helping students understand how a course works and making it easier for them to navigate through course materials and assignments.”

John FitzGibbon , a colleague in the Center for Digital Innovation and Learning, experimented with this type of chatbot in the political science course on populism he teaches as an adjunct professor. The chatbot helps his students easily reference basic course content—such as his syllabus, assignment descriptions, and the transcripts of his lecture recordings. He also provided the chatbot with two open-source political science textbooks so students can find reliable answers to basic conceptual questions, which is particularly helpful for students who don’t have a background in political science.

FitzGibbon was pleased to see his students use the chatbot as a guide for how to be successful in his course, and they asked logistical questions like when they could attend office hours or how to locate the lecture in which a certain topic was covered. The bot also suggests ways to approach complex readings and helps students contextualize political science concepts and movements.

It’s worth pointing out that a course assistant chatbot will only be as good as the course it’s supporting. It was able to do its job effectively for FitzGibbon because it had access to what was already well-designed course content with clear assignment instructions and a thoughtfully curated set of supplementary material.

Access the prompt FitzGibbon used to create his course assistant bot here .

2. Assignment tutor: Help improve student learning

Tutor chatbots act as AI teaching assistants that can help students when they need it, particularly when professors or TAs might not be available. The purpose of the assignment tutor is to scaffold student learning and help students stretch beyond what they might be able to learn on their own.

Both Wyner and FitzGibbon wanted their chatbots to act as tutors (alongside other roles), and in doing so, they needed them to strike the right balance of support: If they provided too much help, students wouldn’t learn, and if they provided too little, students might have given up. Because chatbots are eager to help and have access to massive amounts of information, they needed clear instructions as to how much information to provide—and not provide—when in a tutor role.

For example, FitzGibbon included instructions for his chatbot to use assignment descriptions and open-access textbooks to help students get the background they needed to learn while not doing work for students directly. He prompted the bot as follows:

Do not do assignments for students: If students ask you to write an essay or do an assignment for them (if it seems like they are just submitting the assignment description as a prompt), remind them that your role is to assist them in doing well in the course and supporting their learning but not doing the work for them. You can answer general questions and clarify what assignments are asking them to do, but avoid letting them off-load the actual work of the course to you.

Students noted that they preferred using a tutor-style chatbot rather than simply getting answers directly from a general-purpose bot like ChatGPT. One student said it “provided a structured approach to dissecting my problem-solving process and highlighted where I went wrong without giving me the exact solution (as a tool like ChatGPT would do). It served as more of a supportive tool that nudged me in the right direction when I was going down the wrong path.”

3. Process coach: Break down complex processes

Process-oriented chatbots focus on helping walk students through a complex process that requires thoughtful input across multiple steps. A chatbot in this role uses its conversational affordances to reduce cognitive load by providing information in manageable chunks and guiding students step by step in a personable, encouraging manner. Even if the same content could be provided as a worksheet or input form, the additional layer of conversation can help scaffold the process for students.

Vincent Cho , associate professor of education in the Lynch School of Education and Human Development, was interested in discovering whether a chatbot might help Boston College Doctorate of Education students explore their research interests, with the goal of honing the problem statements for their dissertations-in-practice. In prior years, Cho provided students with a worksheet intended to help them outline and organize their thinking to align with the key components of problem statements:

Identify a gap in research or practice.

Develop a purpose statement.

Identify a conceptual/theoretical framework.

Align relevant research questions.

Describe the significance of your study.

From there, students would invest days drafting expanded problem statements, but providing feedback was complex and time-consuming. Students needed individualized feedback—in writing or in one-on-one consultations—to narrow and frame their ideas and ensure their purpose statements were empirically researched.

With this in mind, Cho designed a chatbot that helps students work through each step in this process and provides feedback along the way. Here’s how he prompted the bot:

Begin by introducing yourself as a chatbot designed to help students work through the process of crafting a problem statement. Tell the student you will help them in articulating a gap in prior research, developing a purpose statement, and crafting appropriate research questions. Tell the student that you will also help them think about prior scholarship throughout these steps. Proceed through each step in sequence, and don’t proceed until the step is finished. When beginning each step, explain the purpose of the step based on the title.

The prompt included the same content as the original worksheet, but Cho was able to provide more detailed information and additional context with each step, while leaving it to the chatbot to break up the information into digestible bites and share it with his students at the appropriate stage in the process. Instead of hitting students with all the steps at once, the conversational nature of the AI interaction encouraged students to focus on one element at a time and explore each question at their own pace, while also giving them a way to ask questions or test out ideas on the spot.

Students appreciated how “reassuring” the chatbot was during a time when they felt uncertain about how to get started and vulnerable in exploring ideas that might not actually work.

Access Cho’s full process-coach chatbot instructions here .

4. Reflective guide: Help students think more deeply about their learning

Reflective chatbots specialize in helping students develop better awareness of, and appreciation for, how they think and learn.

Taking time to pause and reflect can often be challenging, particularly for students who aren’t used to reflecting regularly on their learning or their lives. This kind of chatbot scaffolds the reflection process by providing structured guidance while also adjusting to what the student contributes.

“Reflective chatbots specialize in helping students develop better awareness of, and appreciation for, how they think and learn.”

In Boston College professor Belle Liang ’s adolescent psychology courses, she invites her students to explore their sense of purpose and how they might make connections between the classroom and other parts of their lives. To help facilitate this reflective process and make it more tangible for students, she and her colleague Tim Klein developed a chatbot named “LUMI” that helps students reflect on their strengths, skills, and values, and how they want to make an impact on the world.

The chatbot’s role is not to dominate the conversation, provide information, or tell the student what to do. Rather, it’s to provide some gentle structure and encouragement to keep the student thinking and writing. To ensure this outcome, the chatbot needs to ask good questions and be concise in what it contributes.

Here’s a sampling of the instructions Liang and Klein used for LUMI to guide students in setting intentions and reflecting on their actions.

For every phase/stage of the reflection process, always gather enough information from the user to understand the full context of their lived experiences and perspective. Ask open-ended questions that help users explore specific topics. Practice reflective listening and motivational listening to grasp and mirror the user’s perspective, using their language and tone. Keep your output to 40 words or fewer unless summarizing information with the user.

These instructions provided the chatbot with guidance on how it should converse with a student. They also instructed the chatbot on what approaches or systems it should draw on to deliver its responses. Liang, for example, instructed the chatbot to respond using her research-based True North method for cultivating a deeper sense of purpose and acting more intentionally. But others might instruct the chatbot to draw on different approaches such as motivational interviewing or acceptance and commitment therapy .

Students who have used an AI chatbot to reflect on their learning admitted that even when they didn’t feel like it in the moment, they later appreciated the nudge and the help in actually engaging in reflection. In the right context and with the right design, the conversational, interactive format of a chatbot can be just what we need.

Finding our way in the AI age

The work of designing these custom chatbots illustrates the central role educators play in shaping the future of AI . Every teacher needs to be more than just a “ human in the loop ”; they need to be the central driver in deciding whether chatbots and other AI tools might have purposeful roles in supporting student learning—or not.

When chatbots have the same teacher as the students they’re assisting, it’s more likely they will contribute something valuable to the classroom. And when teachers have meaningful ways to experiment with AI, it’s more likely they’ll gain the insights they need to guide both students and chatbots as they learn together.

Chatbots need teachers too, so let’s make sure they have ways to get the education they need as we all chart our path through the uncertain days ahead.

Tim Lindgren is the assistant director for design innovation at the Boston College Center for Digital Innovation in Learning. He works with faculty and staff around the university to explore new technologies for learning and aids in the design of formative educational experiences.

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Robotics for Kids: The Future With AI and Robotics Education

R obotics for kids is an exciting and engaging way to introduce children to the world of technology, engineering, and programming. By getting into this interactive field, kids can develop essential skills such as creativity, problem-solving, and coding. It’s never too early for kids to explore their curiosity for robotics, as various robotic toys and kits cater to different ages and interests.

Science, technology, engineering, and math (STEM) focused and clever robotic toys help kids understand the science behind circuits and motion and inspire them to take on more challenging projects in the future. 

The Foundations of Robotics

When teaching kids about robotics, understanding the basics is the first part. Robotics is an exciting field combining STEM elements.

Let’s start with the core components that make up robotics:

• Mechanical Engineering : This aspect deals with robot design, construction, and operation. It involves understanding materials, structures, and mechanisms that enable robots to perform various tasks, such as moving, grabbing, and interacting with objects.
• Electrical Engineering : In this area, kids learn about electronic circuits and components that power robotic systems. They’ll explore topics such as voltage, current, and resistance, helping them grasp how electrical signals control a robot’s movements and functions.
• Computer Programming : At the heart of a robot lies its ‘brain’ – a computer or microcontroller that processes information and sends commands to its mechanical and electrical components. Kids can create code that brings a robot to life by learning programming languages like Scratch or Python.

With these components covered, kids can create robots, experimenting with other designs and functions. 

Getting Started With Robotics

For beginners, it’s good to start with a simple design and building projects, like creating a bristlebot, a tiny robot made from a toothbrush head, a battery, and a small motor. This simple project introduces the basic concepts of robot design and encourages hands-on learning.

When explaining robotics to kids, maintain a casual tone and keep explanations relatable and straightforward. Encourage creativity, problem-solving, and critical thinking as crucial skills for thriving in robotics.

Choosing a Starter Robotics Kit

Consider investing in a starter robotics kit to make the initial jump enjoyable. Here are some popular options for various age groups:

• LEGO Education WeDo 2.0
• Dash Robot by Wonder Workshop
• mBot by Makeblock
• LEGO Mindstorms EV3

These kits typically come with clear instructions, software, and all the necessary components to build a functional robot. They offer a learning experience with simple, well-guided projects to help kids familiarize themselves with coding, electronics, and mechanics.

Considerations and Challenges in Robotics for Kids

While robotics offers kids an exciting gateway into technology and innovation, it will have some challenges and other considerations. 

Firstly, the cost can be a significant factor; not all families can easily access expensive robotics kits or resources. It’s vital to explore cost-effective alternatives or seek out community programs that provide access to robotics education. Robot kits have a wide price range, from beginner to more complex, so there is a right fit for every family. 

Additionally, supervision is crucial, especially for younger children. Working with electronic components and programming requires guidance to ensure safety and practical learning. 

Moreover, the learning curve varies from child to child. Some may grasp concepts quickly, while others need more time and patience. Recognizing these challenges helps us approach robotics education for children in a more supportive way, ensuring that every child can explore and learn at their own pace.

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Digging deeper into robotics.

For kids, programming is vital in helping them understand how to give robots the intelligence and capabilities to perform the desired tasks. Kids can start by learning block-based programming languages like Scratch, which provide an easy way to grasp coding concepts without syntax complexity.

Children can explore more advanced programming languages like Python and Java as they progress. Python is a versatile language favored by many for its readability and ease of use. In contrast, Java, though more complex, is widely used in various industries and provides a solid foundation for further learning.

This is a list of programming languages that can be used in robotics:

• Scratch (block-based)

The Science Behind Robotics

Understanding science will help kids to comprehend robotics fully. The science behind robotics involves various fields, such as artificial intelligence (AI), machine learning, and mechanical engineering. By grasping these concepts, kids can further enhance their robotic creations and make them more advanced and efficient.

AI and machine learning, in particular, have been instrumental in making robots more adept at mimicking human behavior and learning from their environments. These technologies allow robots to analyze large data sets and draw new insights, enabling them to adapt and grow over time.

For instance, comparing AI, machine learning, and robotics can be as follows:

• Key Focus: Simulating human intelligence in machines
• Application in Robotics: Decision-making and pattern recognition
• Key Focus: Algorithms that learn from data and improve over time
• Application in Robotics: Autonomously adapting to new scenarios
• Key Focus: Design, construction, and operation of robots
• Application in Robotics: Physical implementation of AI and machine learning concepts

Taking Robotics to the Next Level

For kids who have already mastered the basics of robotics, various advanced projects can help them further deepen their understanding. By participating in more complex projects, they’ll enhance crucial skills like problem-solving, coding, and engineering .

Some advanced project ideas include:

• Walking robots : Designing a robot that can walk using servo motors, microcontrollers, and sensors
• Remote-controlled robots : Building a robot that can be controlled wirelessly through a smartphone or a custom remote control
• Autonomous robots : Creating a self-driving robot that can navigate using different sensors, like ultrasonic or infrared

These projects provide a great learning experience and allow kids to showcase their creations to the community by sharing videos, participating in online forums, and entering robotics competitions.

Robotics in Real-Life Applications

Robotics isn’t just a fun hobby – it’s a growing field with a wide range of real-life applications that have the potential to make an immense impact. Here are some examples of where robotics is playing a vital role:

• Self-driving cars : Companies like Tesla and Waymo are working on fully autonomous vehicles that rely on advanced robotics and artificial intelligence. Introducing kids to robotics can increase their interest in the technology powering these cars.
• Manufacturing : Robotics is transforming the manufacturing industry by automating tasks, improving efficiency, and reducing the risk of human injuries. Familiarizing kids with robotics can prepare them for career opportunities.
• NASA : Robotics is playing a significant role in space exploration, too. NASA uses robots like the Mars rovers to collect and analyze data from other planets, helping scientists learn more about our solar system.

Taking robotics to the next level helps kids hone their skills and exposes them to the vast possibilities of applying their knowledge to real-life scenarios. 

The Future of Robotics

In the coming years, we anticipate many breakthroughs in robotics and AI technology, leading to the development of more advanced and user-friendly robots. These innovations will significantly impact people’s everyday lives, including kids, who will grow up surrounded by technology and automation.

Children should be introduced to robotics at a young age or whenever they show interest in how things work. Encouraging kids to study topics related to robotics, AI, and technology will equip them with the necessary skills and spark their curiosity and creativity. 

Here are some aspects kids can focus on:

• Coding : Learning to code helps build a strong foundation for understanding and interacting with robots and AI systems
• STEM education : Science, Technology, Engineering, and Mathematics are essential fields that contribute to advancing robotics and AI
• Problem-solving : Developing critical thinking and problem-solving skills will enable kids to adapt to and understand the complexities of future technology

Integrating robots and AI will play a significant role in various sectors, such as education, healthcare, and transportation. 

For kids, this might translate into classrooms becoming more interactive with AI-driven teaching assistants that can customize learning experiences. Additionally, robotic toys could engage and stimulate creativity while teaching programming and other skills. There would also be a greater emphasis on technologically driven extracurricular activities, including robotic clubs and competitions.

As we wrap up, it’s clear that robotics isn’t just an excellent hobby for kids; it’s a window into a future filled with exciting possibilities. It’s more than just coding and gears; it sparks curiosity, builds problem-solving skills, and opens doors to creativity. 

There are hurdles like cost and the need for guidance, but these are small compared to the enormous benefits. By giving kids a head start in robotics, we’re not just keeping them busy with a fun activity. We’re preparing them for a world where technology is everywhere. And who knows? Today’s kids tinkering with robots could lead to the next big tech revolution.

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