Isomorphism and homomorphism are topics central to abstract algebra, but research on mathematicians’ views of these topics, especially with respect to sameness, remains limited. This study examines open response survey data from 197 mathematicians on how sameness could be helpful or harmful when studying isomorphism and homomorphism. Using thematic analysis, we examined whether sameness was viewed as helpful or harmful for isomorphism and homomorphism before examining rationales for those views. Making use of values of the mathematical community, we note that mathematicians saw conceptual and pedagogical benefits to connecting isomorphism and sameness, which connects to leveraging intuition and valuing ways of increasing understanding. Mathematicians’ concerns around using sameness largely revolved around the violation of the mathematical community’s idealized value of expressing a priori truth via a-contextual justifications. However, these concerns can be addressed through only targeted usage of sameness and explicit discussions around the utility and relevance of sameness. Implications include the importance of considering how interventions proposed by mathematics educators align with or provoke tension between values held by the mathematical community in order to mitigate or lean into those tensions and encourage fruitful dialogue between the mathematics and mathematics education communities.
An essential task for mathematics teachers is posing problems. Selecting mathematics problems that develop mathematical proficiency and engage students in desirable mathematical practices is a critical decision-making process. We present the Framework for Posing Elementary Mathematics Problems (F-PosE) developed to focus prospective teacher noticing on desirable features of mathematics problems and inform decision-making processes around the selection of problems for use in elementary classrooms. Development of the framework was informed by a three-phase design research process consisting of an extensive review of the literature, document content analysis and successive testing of mathematics problems in elementary classrooms in partnership with teachers and children. Consequently, it draws from emergent practice informed by the collective endeavour of a community of educators. The framework consists of eight indicators: use of a motivating and engaging context, clarity in language and cultural context, curriculum coherence, attention to cognitive demand, an appropriate number of solution steps to support reasoning, a variety of solution strategies, facilitating multiple solutions and opportunity for success. This F-PosE provides a critical focusing lens for prospective teachers when creating and selecting mathematics problems specifically for use in elementary classrooms.
In mathematics education research, proofs are often conceptualized as sequences of mathematical assertions. We argue that this ignores proofs that contain instructions to perform mathematical actions, often in the form of imperatives, which are common both in mathematical practice and in undergraduate mathematics textbooks. We consider in detail a specific type of proof which we call recipe proofs, which are comprised of sequence of instructions that direct the reader to produce mathematical objects with desirable properties. We present a model of what it means to understand a recipe proof, and use this model in conjunction with process-object theories from mathematics education research, to explain why recipe proofs are inherently difficult for students to understand.
In this article, we report on a typology of the demands of statistical and mathematical products (StaMPs) embedded in media items related to the COVID-19 (coronavirus) pandemic. The typology emerged from a content analysis of a large purposive sample of diverse media items selected from digital news sources based in four countries. The findings encompass nine categories of StaMPs: (1) descriptive quantitative information, (2) models, predictions, causality and risk, (3) representations and displays, (4) data quality and strength of evidence, (5) demographics and comparative thinking, (6) heterogeneity and contextual factors, (7) literacy and language demands, (8) multiple information sources, and (9) critical demands. We illustrate these categories via selected media items, substantiate them through relevant research literature, and point to categories that encompass new or enhanced types of demands. Our findings offer insights into the rich set of capabilities that citizens (including both young people and adults) must possess in order to engage these mass media demands, critically analyze statistical and mathematical information in the media, evaluate the meaning and credibility of news reports, understand public policies, and make evidenced-informed judgments. Our conclusions point to the need to revise current curricular frameworks and conceptual models (e.g., regarding statistical and probability literacy, adult numeracy), to better incorporate notions such as blended knowledge, vagueness, risk, strength of evidence, and criticality. Furthermore, more attention is needed to the literacy and language demands of media items involving statistical and mathematical information. Implications for further research and educational practice are discussed.
Drawing bar diagrams has been shown to improve performance on mathematical word problems wherein the relational keyword is consistent with the required arithmetic operation. This study extends this by testing the effectiveness of bar diagram drawing for word problems with an inconsistent keyword-arithmetic operation mapping. Seventy-five fifth graders solved consistent and inconsistent word problems while encouraged to draw bar diagrams. For each word problem, we assessed problem type (consistent/inconsistent), performance (correct/incorrect), and bar diagrams (accurate/inaccurate/no drawing). Overall, bar diagram drawing was associated with increased performance on both consistent and inconsistent word problems, but the strongest benefits of drawing were found for inconsistent word problems. For inconsistent word problems, bar diagram accuracy was more clearly related to performance (accurate bar diagrams related to correct answers, but inaccurate ones to incorrect answers) than for consistent word problems. We conclude that bar diagram drawing provides an effective graphical support for solving inconsistent word problems.
Variance and invariance are two powerful mathematical ideas to support geometrical and spatial thinking, yet there is limited research about teachers’ knowledge of variance and invariance. In this paper, we examined how high school teachers deal with the task of looking for invariant properties in a dynamic geometry environment (DGE) setting. Specifically, we investigated if they even attend to invariant properties; what invariant properties they discern and discuss; and how DGE can support such discernment. Our analysis found that teachers tend to discern and discuss invariant properties mainly when they were probed to consider invariance. We also found four categories of invariant properties that seem to be important for a robust and rich understanding of geometric objects in the context of invariance and DGE. The use of DGE allowed teachers to see and interact with invariant properties, thus suggesting that accessing geometry dynamically may have structural affordances especially when exploring invariance. Teachers were able to enact different DGE movements to discern and discuss invariant properties, as well as to reason with and about them. We also saw that teachers’ backgrounds and past experiences can play an important role in their descriptions of invariant properties. Possible future research directions and implications to teacher education are discussed.
Students’ issues dealing with reflection tasks, especially with inclined mirror lines, are widely known. Previous research conducted with primary school students mainly focussed on task difficulty and students’ outcomes, especially their errors of reflection. Little is known about which not obviously sustainable understanding of reflection leads to what kind of errors. To address this issue, the purpose of this study is to explore which approaches students use in their solution finding processes when dealing with reflection tasks. We also investigated whether there was a relationship between the students’ approaches and correct or incorrect solution products. This article discusses a task-based interview study focussing on approaches used in the students’ solution process of four reflection tasks. To gather data, 42 interviews with students at the end of primary school in Germany were conducted. For the analysis, a category system with the categories “properties of reflection,” “holistic approaches,” “gestures and actions illustrating the process of reflection,” and “mental images illustrating the process of reflection,” was developed and validated. We observed that there is a relation between the approaches used by the students during their process of solving reflection tasks and their solution products. The observed relationship between the approaches applied during the solution processes and the solutions themselves leads to the assumption that the approaches the students used reveal varying degrees of mathematical understanding of reflection. As emphasized in the discussion of the results, it is important to take a differentiated look on the students’ solution products and the importance of adequate instruction methods.
Within educational research, dyslexia and other disabilities are typically conceptualized as deficits. The theory of neurodiversity encourages researchers to conceptualize cognitive differences as natural forms of human diversity with unique sets of challenges and strengths. Using neurodiversity as our theoretical framework, we analyze the experiences of five research mathematicians with dyslexia as told through personal narratives to find common strengths and challenges for dyslexic thinkers at the highest level of mathematics. We report on 4 themes: (1) highly visual and intuitive ways of mathematical thinking, (2) issues with language and translation between forms, (3) issues with memorization of mathematical facts and procedures, and (4) resilience as a strength of dyslexia that matters in mathematics. We use our participants’ insights to explore how neurodiversity, a theory of cognitive disability developed by and for neurodiverse people, could expand opportunities for research. We call for mathematics educators to consider the strengths and challenges of dyslexic learners, as well as valuing expertise from insider perspectives.
Understanding the secondary-tertiary transition is not only an educational matter, but also an inclusion and equality matter as mathematics qualifications lead to better employment and better earning for students. Research on what causes the crisis associated to this transition has so far focused on the impact of cognitive and epistemological changes in university mathematics. Recently attention has been given to sociocultural and affective issues, which indicate the impact of diverse contexts on such event in the students' life. Here we present a study investigating the experiences of first-year mathematics students in three European countries. The study adopted the framework of transition as a rite of passage and to detect the changes in attitudes towards mathematics which originate the crisis the Three-Dimensional Model for Attitude toward mathematics (TMA) was used. Data consist in one questionnaire translated in three languages and administered to students in the first year of study. Most of the questions were open-ended and the data were analysed first qualitatively and then quantitized. Results show that while the transition experiences of the students may seem on the surface uniform, some significant differences emerge for motivation to join mathematics degrees, motives for changes in perceived competence and impact of university mathematics formalism. We hypothesise a link between these changes and the differences in educational environments in the three countries. We conclude highlighting the need for more research of this kind to understand the secondary-tertiary transition in the educational context in which it happens.
One typical challenge in algebra education is that many students justify the equivalence of expressions only by referring to transformation rules that they perceive as arbitrary without being able to justify these rules. A good algebraic understanding involves connecting the transformation rules to other characterizations of equivalence of expressions (e.g., description equivalence that both expressions describe the same situation or figure). In order to overcome this disconnection even before variables are introduced, a design research study was conducted in Grade 5 to design and investigate an early algebra learning environment to establish stronger connections between different mental models and representations of equivalence of expressions. The qualitative analysis of design experiments with 14 fifth graders revealed deep insights into complexities of connecting representations. It confirmed that many students first relate the representations in ways that are too superficial without establishing deep connections. Analyzing successful students’ processes helped to identify an additional characterization that can support students in bridging the connection between other characterizations, which we call restructuring equivalence. By including learning opportunities for restructuring equivalence, students can be supported to compare expressions in graphical and symbolic representation simultaneously and dynamically. The design research study disentangles the complex requirements for realizing the design principle of connecting multiple representations, which should be of relevance beyond the specific concept of equivalence and applicable to other mathematical topics.
Within the commognitive perspective, ritual and explorative routines are used in a very particular way to distinguish students’ routines according to whether they are driven by social reward or by generating a substantiated narrative. Explorative routines in this theorisation may refer not to inquiry-based activity but to the result of a student’s routine moving from being process-oriented to becoming outcome-oriented, a deritualisation. Choice of tasks as well as a teacher’s moves offer students different opportunities to engage in rituals, explorative routines and deritualisations. Through nuancing the space spanned by opportunities to engage in rituals and explorative routines respectively, we describe and contrast classroom practices in three lessons from three contexts. The lessons share a commonality in encouraging explorative routines as a starting point, yet being adapted towards ritual activity through decreased openings for student agentivity, fewer invitations for students’ own substantiations or both. We argue that such adaptations are driven by the teachers’ commitment to reach mathematical closure in a lesson, to balance considerations of the classroom community and individual students and to meet curricular requirements. Our model helps interrogate the nature and relevance of hybrids of explorative routines and rituals.
Mathematical objects are the outcomes of human discourse and come to life through the process of objectification. Primary and concrete discursive objects (d-objects) play an important role in this process, forming different realizations through the objectification process. In the present study, we analyze the written discourses about the derivative at a point in fourteen Year 11 and Year 12 Iranian calculus textbooks over 42 years (1979–2020) in terms of primary objects, concrete d-objects, and the realizations used in them, here based on Sfard’s commognitive theory. The research method is a qualitative content analysis in which the analysis units were texts, examples, activities, and exercises in the textbooks. The results show that out of the 14 textbooks, only five included primary and concrete d-objects. The other textbooks only used abstract objects in their written discourse about the derivative. The pictures (visual mediators) that were used as primary objects gradually changed from schematic to real colored photos, and locally developed technologies and cultural elements were included in the new textbook editions. Additionally, the number of concrete d-objects increased in the new editions. In terms of the realizations, the number of roots also increased over time; however, none of the textbooks had all fve main realizations (i.e., symbolic, graphical, verbal, numerical, and physical) for the derivative at a point. The realization tree developed as part of this study can be used as an analytical framework to analyze calculus textbooks in other contexts and can be used by teachers and lecturers to help students have explorative participation in discourse on the derivative.
Our research aimed to investigate the potential learning benefits to young children of implementing digital interactive multimodal technologies that provide both visual and haptic experiences in elementary mathematics classrooms. We studied the ways in which fourth-grade students collaboratively create collective strategies for solving mathematical problems utilizing dynamic geometry software with multi-touch interfaces, a combination we call a multi-touch Dynamic Geometry Environment. We examine in-depth two case studies each illustrating how mathematical strategies, collaboration, and socially mediated metacognition emerge in the small groups of children while working on an activity using the Geometer’s Sketchpad® on the iPad to make sense of an intuitive idea of covariation. We found that children’s interactions with their peers, the interviewer, and the mDGE favored the emergence of varied collaborative behaviors and socially mediated metacognitive processes that fostered the co-construction and development of mathematical strategies over a short period of time.
Due to the COVID-19 pandemic in Shanghai, China, all school classes were delivered through an online environment from February 24 to May 22, 2020. To support this transition, the Shanghai Education Commission led expert teachers and specialists to develop a series of online video lessons based on the Shanghai unified curriculum, and suggested students watch the online video lessons individually from home, followed by an online synchronous lesson supported by class teachers. This study investigated what primary mathematics teachers learned from addressing these challenges through a case study. By following two purposefully selected teachers over 2 weeks during the transition, multiple data sets including online video lessons, online synchronous lessons, daily reflections, and post-online teacher interviews were collected. A fine-grained analysis of the data from the lens of the documentational approach to didactics found that teachers adaptively used online video lessons as important resources for their online synchronous lessons and virtual Teaching Research Groups as a teachers' collaboration mechanism supported them to develop online video lessons and address various technological constraints. Finally, implications of this case study for mathematics education globally are discussed.
The online version contains supplementary material available at 10.1007/s10649-022-10172-2.
In this article, we document the support provided by highly recognised journals in Mathematics Education in response to the challenges faced by English non-dominant language (EnDL) authors when they attempt to publish in English. To address this issue, we first conducted a synthesis of research literature related to those influences that direct EnDL authors’ publication efforts towards English language journals and the associated challenges that result. Second, we gathered survey and interview data from participant editors-in-chief of leading journals in Mathematics Education about the support, enacted and planned, provided by their journal for EnDL authors and associated challenges. Finally, we discuss the findings of the study from the perspective of heteroglossia. Findings indicate that while a range of initiatives have been employed to support EnDL authors, current journal policies lag somewhat behind the plans of editors-in-chief, which have been limited, to date, by available resources.
Thirty-seven third graders and thirty-two first graders engaged in solving a tangram puzzle in the shape of a fox. They had five minutes to solve the puzzle, and after this time, they received guidance on the particular piece they had difficulties with. Through the lenses of navigating flexible abstraction, reinterpretation, combinations, and borrowing structure to expand upon the existing 2D shape composition and decomposition learning trajectory, we examined ways in which students’ puzzle-solving processes and their challenges related to the fox puzzle’s features. Students’ initial shape placements suggest that parts of the fox puzzle primed the use of particular pieces, which reduced the abstraction of the puzzle. The most challenging part of the puzzle for students was navigating reinterpretation to place the square and two small triangles on the fox’s head in nonstandard orientations. Even though students faced challenges at different steps, they overcame them similarly by trying new combinations and by borrowing structure. Some students did not complete the puzzle even though they used flips and turns (reinterpretation) strategically. The results suggest potential modifications of the current learning trajectory to account for differences between tangram and pattern block puzzles and differences due to tangram puzzles’ features. Because the puzzle’s features played a role in students’ challenges, future work needs to focus on the interaction between students’ puzzle-solving processes and puzzles’ features for a variety of tangram puzzles.
Offering explanations is a central part of teaching mathematics, and understanding those explanations is a vital activity for learners. Given this, it is natural to ask what makes a good mathematical explanation. This question has received surprisingly little attention in the mathematics education literature, perhaps because the field has no agreed method by which explanation quality can be reliably assessed. In this paper, we explore this issue by asking whether mathematicians and undergraduates agree with each other about explanation quality. A corpus of 10 explanations produced by 10 mathematicians was used. Using a comparative judgement method, we analysed 320 paired comparisons from 16 mathematicians and 320 from 32 undergraduate students. We found that both mathematicians and undergraduates were able to reliably assess the quality of a set of mathematical explanations. Furthermore, the assessments were largely consistent across the two groups. Implications for theories of mathematical explanation are discussed. We conclude by arguing that comparative judgement is a promising technique for exploring explanation quality.
Mathematicians frequently attend their peers' lectures to learn new mathematical content. The goal of this paper is to investigate what mathematicians learned from the lectures. Our research took place at a two-week workshop on inner model theory, a topic of set theory, which was largely comprised of a series of lectures. We asked the six workshop organizers and seven conference attendees what could be learned from the lectures in the workshop, and from mathematics lectures in general. A key finding was that participants felt the motivation and road maps that were provided by the lecturers could facilitate the attendees' future individual studying of the material. We conclude by discussing how our findings inform the development of theory on how individuals can learn from lectures and suggest interesting directions for future research.
Teachers’ questioning plays an essential role in shaping collective argumentative discourse. This paper demonstrated that rationality dimensions in teacher questions can be assessed by adapting Habermas’ three components of rationality. By coordinating Habermas’ construct with Toulmin’s model for argumentation, this paper investigated how two secondary mathematics teachers used rational questioning to support student participation in collective argumentation. This paper identified various ways in which two participating teachers used rational questioning to support student participation in argumentation via contributions of argument components. The results establish a theoretical connection between the use of rational questions and students’ contributions of components of arguments. The results indicated that not all rational questions were associated with a component of argument, and rational questions may additionally support argumentation in general for the development of a culture of rationality. The study has implications in terms of theory and professional development of teachers.
Self-efficacy in mathematics is related to engagement, persistence, and academic performance. Prior research focused mostly on examining changes to students’ self-efficacy across large time intervals (months or years), and paid less attention to changes at the level of lesson sequences. Knowledge of how self-efficacy changes during a sequence of lessons is important as it can help teachers better support students’ self-efficacy in their everyday work. In this paper, we expanded previous studies by investigating changes in students’ self-efficacy across a sequence of 3–4 lessons when students were learning a new topic in mathematics (n Students = 170, n Time-points = 596). Nine classes of Norwegian grade 6 ( n = 77) and grade 10 students ( n = 93) reported their self-efficacy for easy, medium difficulty, and hard tasks. Using multilevel models for change, we found (a) change of students’ self-efficacy across lesson sequences, (b) differences in the starting point and change of students’ self-efficacy according to perceived task difficulty and grade, (c) more individual variation of self-efficacy starting point and change in association with harder tasks, and (d) students in classes who were taught a new topic in geometry had stronger self-efficacy at the beginning of the first lesson as compared to those who were taught a new topic in algebra (grade 10), and students in classes who were taught a new topic in fractions had steeper growth across the lesson sequence as compared to those who were taught a new topic in measurement (grade 6). Implications for both research and practice on how new mathematics topics are introduced to students are discussed.
The success or failure of education systems in promoting the problem-solving skills of students depends on attitudinal, political, and pedagogical variables. Among these variables, the design of mathematics textbooks is thought to partially explain why students from high-achieving countries show better problem-solving ability in international assessments. In the current study, we delved into this question and compared the frequency and characteristics of arithmetic word problems (AWPs) contained in primary math textbooks of two countries with different levels of performance in international assessments—Singapore and Spain. In our analyses, we focused on: 1) the quantity of arithmetic word problems; 2) a variety of problems in terms of their additive or multiplicative structures and semantic-mathematical substructures; and 3) the quantity and nature of illustrations that were presented together with arithmetic word problems. Although a larger proportion of AWP activities was found in Singaporean textbooks, the results showed a similar variety of AWPs in math textbooks of Singapore and Spain. Furthermore, in both countries, math textbooks emphasized (additive) combine 1 and multiplication-simple rate problems. Notably, Singaporean textbooks contained a larger percentage of illustrations that reflected the semantic-mathematical structures of the problems and helped students learn how to solve AWPs (e.g., bar models). The findings are discussed in light of theories that posit that textbooks constitute a fundamental part of the teaching-learning process in the classroom.
Public media both reflects and shapes societal perceptions and attitudes. Teachers and others around students in mathematics classrooms have expectations for the students, projected with what appears in these media. We are most concerned about the expectations placed on students who are identified with minoritized groups—particularly students who are Indigenous or migrated to Norway. We investigate how minoritized group contexts and mathematics education appear together in Norwegian news media texts. Our analysis uses the notion of storylines to describe the expectations about minoritized groups that news media project. We found seven entangled storylines: “the majority language and culture are keys to learning and knowing mathematics,” “mathematics is language- and culture-neutral,” “minoritized groups’ mathematics achievements are linked to culture and gender,” “extraordinary measures are needed to teach students from minoritized groups mathematics,” “students from minoritized groups underachieve,” “students from minoritized groups put in extraordinary effort and time to learn mathematics,” and “minoritized mathematics students are motivated by gratitude.”
In this article, we direct attention to what becomes critical in teaching activities for toddlers (1–3-year-olds) to learn the meaning of numbers. One activity we thoroughly explore is interactive book reading, based on previous research indicating positive learning outcomes from this type of mathematical activity, as it has shown to simultaneously embrace the child’s perspective and encourage interaction and ‘number talk.’ A specially designed picture book presenting small quantities was developed, and variation theory principles were embedded in both the book design and the teaching acts. Through qualitative analyses, we aim to identify what is critical in the interactive book reading sessions for toddlers to discern essential aspects of numbers, with a specific focus on the conditions for making modes of representations into resources for learning. Preschool teachers frequently read the book to 27 toddlers over the course of a year. Video documentation of their reading sessions was analyzed, and exposed the significance of addressing the child’s perspective when choosing what representation to emphasize and in what ways connections within and between representations can be made. Thus, the study contributes knowledge on the teaching of numbers with toddlers, and problematizes as well as extends the potential of interactive book reading as a quality-enhancing educational tool.
This study aimed to investigate seventh-grade students’ visuospatial thinking processes in an art studio environment, where students were engaged with geometrically rich artworks. The students were asked to observe minimalist artworks, then create and critique their own and others’ artworks based on the Studio Thinking Framework. Data were collected through interviews conducted with students, video recordings in the studio, and students’ documents (sketches, artworks, and notes). The data were analyzed based on previous studies on spatial thinking and emergent data. The study’s findings indicate that the Studio Thinking-based environment has the potential to elicit students’ visuospatial thinking processes, mainly in recognizing shapes, decomposing and composing shapes, patterning, and transforming shapes rigidly and non-rigidly (scaling). The present study, which includes accounts of three studio works, suggests an emergent framework for the characterization of visuospatial thinking within a particular art-math-related environment. The findings of the study shed light on other studies on visual arts and mathematics education and on mathematical thinking and learning in informal learning settings.
Several studies have shown that children do not only erroneously use additive reasoning in proportional word problems, but also erroneously use proportional reasoning in additive word problems. Traditionally, these errors were contributed to a lack of calculation and discrimination skills. Recent research evidence puts forward an additional explanation, namely, children's relational preference (i.e., in tasks where both, additive and multiplicative reasoning, are appropriate, some children have a preference for additive relations, while others have a preference for multiplicative relations). Children's relational preference offers a unique explanation for erroneous word-problem solving, after taking into account computation and discrimination skills in 8-to 12-year-olds. However, it is still unclear whether relational preference is also associated with word-problem solving at an earlier age, before the start of formal instruction in word-problem solving. A task measuring children's relational preference as well as three additive and three proportional word problems was administered to a large group (n = 343) of 6-to 7-year-olds. Results show that relational preference is also associated with word-problem solving behavior at this young age: an additive preference is related with better performance on additive word problems but also with more erroneous additive reasoning in proportional word problems. Similarly, a multiplicative preference is related with better performance on proportional word problems but not yet with more erroneous proportional reasoning in additive word problems. The latter is possibly due to the low number of proportional errors that were made in the additive word problems at this young age. The implications of these findings for further research and educational practice are discussed.
Emotions play an essential role in pre-service teachers' competence development, particularly in mathematics. However, the emotion of shame in mathematics has been largely neglected so far. This article deals with shameful experiences of pre-service primary school teachers during their mathematical education at school and the various effects of shame on their decision to study mathematics as a subject at university. The research consists of a qualitative and a quantitative study with 311 prospective primary school teachers who responded to a survey about their experiences of shame in mathematics at school when they were students. Results of the qualitative study emphasize the different experiences in mathematics during the school years and reveal the characteristics of these situations, for example, social exposure or competition games. In the quantitative study, pre-service primary teachers' subject choice was analyzed in relation to their experienced shame in mathematics at school. Results reveal that shame experienced at school has effects on the initial choice in favor of mathematics at university. Implications for primary teacher education are finally discussed. Keywords Shame · Mathematics · Pre-service primary teachers · Teacher education Research on pre-service teachers' emotions has received rising interest over the last decades (Coppola et al., 2013; Hannula, 2012). Particularly, studies have revealed that emotions can influence both pre-service teachers' competence development throughout teacher
In mathematical whole-class discussions, teachers can build on various student ideas and develop these ideas toward mathematical goals. This requires teachers to make sense of their students’ mathematical thinking, which evidently involves mathematical thinking on the teacher’s part. Teacher sense-making of student mathematical thinking has been studied and conceptualized as an aspect of teacher noticing and has also been conceptualized as a mathematical activity. We combine these perspectives to explore the role of teacher mathematical thinking in making sense of student mathematical thinking. In this study, we investigated that role using video-based teacher discussions in a teacher researcher collaboration in which five Dutch high school mathematics teachers and one researcher developed discourse based lessons in cycles of design, enactment, and evaluation. In video-based discussions, they collaboratively reflected on whole-class discussions from the teachers’ own lessons. We analyzed these discussions to explore the mathematical thinking that teachers articulated during sense-making of students’ mathematical thinking and how teachers’ mathematical thinking affected their sense-making. We found five categories concerning the role of teacher mathematical thinking in their sense-making: flexibility, preoccupation, incomprehension, exemplification, and projection. These categories show how both the content and the process of teacher mathematical thinking can support or impede their sense-making. In addition, we found that the teachers often did not articulate explicit mathematical thinking. Our findings suggest that sense-making of students’ mathematical thinking requires teachers to (re-)engage in reflective thinking with regard to the mathematical content as well as the process of their own mathematical thinking.
Affect (e.g., beliefs, attitudes, emotions) plays a crucial role in mathematics learning, but reliance on verbal and written responses (from surveys, interviews, etc.) limits students’ expression of their affective states. As a complement to existing methods that rely on verbal reports, we explore how graphing can be used to study affect during mathematical experiences. We analyze three studies that used graphing to represent, stimulate recall, and reflect on affect. In each, students were asked to draw their perception of an affective construct, such as confidence or intensity of emotion, against time. The studies differed in participant populations, target affect, timescales of participant experience, and structural features of the graphs. The affordances of graphing include reduced dependence on verbal data, temporal ordering of participants’ recollections, explicit representation of change over time, and the creation of objects (the graph) for discussion. These studies as examples show that well-structured graphing can productively supplement existing methods for studying affect in mathematics education, as a different medium through which students can communicate their experience.
Although there is much research exploring students’ covariational reasoning, there is less research exploring the ways students can leverage such reasoning to coordinate more than two quantities. In this paper, we describe a system of covariational relationships as a comprehensive image of how two varying quantities, having the same attribute across different objects, each covary with respect to a third quantity and in relation to each other. We first describe relevant theoretical constructs, including reasoning covariationally to construct relationships between quantities and reasoning covariationally to compare quantities. We then present a conceptual analysis entailing three interrelated activities we conjectured could support students in reasoning covariationally to conceive of a system of covariational relationships and represent the system graphically. We provide results from two teaching experiments with four middle school students engaging in tasks designed with our conceptual analysis in mind. We highlight two different ways students reasoned covariationally compatible with our conceptual analysis. We discuss the implications of our results and provide areas for future research.
In this study, the authors examined a group of related literatures that used the same large-scale nationally representative dataset with algebra achievement as the outcome to explore why results from the same dataset may differ across studies. More specially, the authors synthesized the extant research literature that utilized data from the High School Longitudinal Study of 2009 (HSLS:09) to investigate the student-, teacher-, school-, and parent-level characteristics associated with algebra achievement and whether these relationships were consistent across students’ backgrounds (i.e., race/ethnicity, gender, socio-economic status, and prior achievement). The authors have summarized the results of 21 studies across 2 outcome variables (ninth- and eleventh-grade algebra achievement) using optimal resource theory as a framework to summarize evidence-based practices for improving student outcomes.
In activities based on Proof Without Words (PWW), we developed, students are given a PWW– a diagram that alludes to the proof of a mathematical theorem. The students work collaboratively to construct a proof alluded by the PWW, and then each student writes and submits a proof attempt. In a three-year design-based study, we investigate and develop PWW-based activities for advancing upper secondary students' proficiency in constructing proofs. We use the concept of gap-filling as a theoretical framework. In a nutshell, gap-filling is an action of adding information absent in a text that a reader does for sense-making. We inquire whether secondary school students independently construct a proof when a PWW is at their disposal, what characterizes those gaps that students identify and fill, and how PWW design principles influence students' gap-filling. We identified four categories of gaps: key idea, generality, constructional, and figure property justifications. We find that students mainly fill key idea gaps but not generality gaps and that PWWs that presented the construction procedure led more students to fill the constructional gaps. However, PWWs that did not explicitly present some figures' property led more students to fill those justification gaps. We identify five PWW design principles that can enhance secondary students’ gap-filling and conclude that a meticulous design paves the way to implement PWW-based activities for fostering mathematical proving.
Mathematical modelling is endorsed as both a means and an end to learning mathematics. Despite its utility and inclusion as a curricular objective, one of many questions remaining about learners’ modelling regards how modelers choose relevant situational attributes and express mathematical relationships in terms of them. Research on quantitative reasoning has informed the field on how individuals quantify attributes and conceive of covariational relationships among them. However, this research has not often attended to modelers’ mathematization in open modelling tasks, an endeavor that invites further attention to theoretical and methodological details. To this end, we offer a synthesis of existing theories to present a cognitive account of mathematical model construction through a quantity-oriented lens. Second, we use empirical data to illustrate why it is productive for theories of modelling to attend to and account for students’ quantitative reasoning while modelling. Finally, we identify remaining challenges to coordinating different theoretical accounts of model construction.
We are interested in understanding how university students learn to use programming as a tool for “authentic” mathematical investigations (i.e., similar to how some mathematicians use programming in their research work). The theoretical perspective of the instrumental approach offers a way of interpreting this learning in terms of development of schemes by students; these development processes are called instrumental geneses. Nevertheless, how these schemes evolve has not been fully explained. In this paper, we propose to enrich the theoretical frame of the instrumental approach by a model of scheme evolution and to use this new approach to investigate learning to use programming for pure and applied mathematics investigation projects at the university level. We examine the case of one student completing four investigation projects as part of a course workload. We analyze the productive and constructive aspects of the student’s activity and the dynamic aspect of the instrumental geneses by identifying the mobilization and evolution of schemes. We argue that our approach constitutes a new theoretical and methodological contribution deepening the understanding of students’ instrumented learning processes. Identifying instrumented schemes illuminates in particular how mathematical knowledge and programming knowledge are combined. The analysis in terms of scheme evolutions reveals which characteristics of the situations lead to such evolutions and can thus inform the design of teaching.
In India, a curriculum reform inspired by critical perspectives has sought to transform primary mathematics teaching and learning. It is aimed at strengthening socio-cultural-political connections between school mathematics and students’ life experiences, thereby challenging traditional textbook culture. At the same time, this initiative has retained the textbook as a vehicle of reform while seeking to subvert many of its established conventions. Guided by Remillard’s idea of modes of engagement, this paper analyses the innovative Math-Magic textbooks associated with the Indian National Curriculum Framework. It investigates how these textbooks represent and communicate the framework ideas, focusing on key curricular elements and on the teacher as reader. Analysing the ‘voice’ and ‘structure’ of the textbooks as well as the ‘contexts’ used, it is revealed that they use a radically unique voice to introduce school mathematics while also attempting to use authentic and socially relevant contexts within their tasks. However, they have limited structural support to communicate these ideas clearly to the teacher-reader. The paper has implications for studying reformed textbooks in primary school mathematics in the Global South, where they remain the main teaching resource for teachers. Further, by focusing on ‘context’, the notion of modes of engagement within textbooks is extended through socio-cultural perspectives.
Understanding the intricate quantifier relations in the formal definitions of both convergence and continuity is highly relevant for students to use these definitions for mathematical reasoning. However, there has been limited research about how students relearn previous school mathematics for understanding multiply quantified statements. This issue was investigated in a case study in a 5-week teaching unit, located in a year-long transition course, in which students were engaged in defining and proving sequence convergence and local continuity. The paper reports on four substantial changes in the ways students relearn school mathematics for constructing quantified statements: (1) endorse predicate as formal property by replacing metaphors of epsilon strips with narratives about the objects ε, N ε , and ∣a n − a∣; (2) acknowledge that statements have truth values; (3) recognize that multiply quantified statements are deductively ordered and that the order of its quantifications is relevant; and (4) assemble multiply quantified statements from partial statements that can be investigated separately. These four changes highlight how school mathematics enables student to semantically and pragmatically parse multiply quantified statements and how syntactic considerations emerge from such semantic and pragmatic foundations. Future research should further investigate how to design learning activities that facilitate students’ syntactical engagement with quantified statements, for instance, in activities of using formal definitions of limits during proving.
This paper reports from a case study which explores kindergarten children’s mathematical abstraction in a teaching–learning activity about reflection symmetry. From a dialectical perspective, abstraction is here conceived as a process, as a genuine part of human activity, where the learner establishes “a point of view from which the concrete can be seen as meaningfully related” (van Oers & Poland Mathematics Education Research Journal, 19(2), 10–22, 2007, p. 13–14). A cultural-historical semiotic perspective to embodiment is used to explore the characteristics of kindergarten children’s mathematical abstraction. In the selected segment, two 5-year-old boys explore the concept of reflection symmetry using a doll pram. In the activity, the two boys first point to concrete features of the sensory manifold, then one of the boys’ awareness gradually moves to the imagined and finally to grasping a general and establishing a new point of view. The findings illustrate the essential role of gestures, bodily actions, and rhythm, in conjunction with spoken words, in the two boys’ gradual process of grasping a general. The study advances our knowledge about the nature of mathematical abstraction and challenges the traditional view on abstraction as a sort of decontextualised higher order thinking. This study argues that abstraction is not a matter of going from the concrete to the abstract, rather it is an emergent and context-bound process, as a genuine part of children’s concrete embodied activities.
Visuospatial representations of numbers and their relationships are widely used in mathematics education. These include drawn images, models constructed with concrete manipulatives, enactive/embodied forms, computer graphics, and more. This paper addresses the analytical limitations and ethical implications of methodologies that use broad categorizations of representations and argues the benefits of dynamic qualitative analysis of arithmetical-representational strategy across multiple semi-independent aspects of display, calculation, and interaction. It proposes an alternative methodological approach combining the structured organization of classification with the detailed nuance of description and describes a systematic but flexible framework for analysing nonstandard visuospatial representations of early arithmetic. This approach is intended for use by researchers or practitioners, for interpretation of multimodal and nonstandard visuospatial representations, and for identification of small differences in learners’ developing arithmetical-representational strategies, including changes over time. Application is illustrated using selected data from a microanalytic study of struggling students’ multiplication and division in scenario tasks.
The multi-faceted nature of mathematics knowledge for teaching, including pedagogical content knowledge (PCK), has been studied widely in elementary classrooms, but little research has focused on senior secondary mathematics teaching. This study utilised the Knowledge Quartet (Rowland et al., Research in Mathematics Education, 17(2), 74–91, 2005) to analyse mathematics teaching at the senior secondary level using excerpts from a lesson on differential calculus and another on discrete probability distributions. The findings reveal that, at this level, there is a complex interplay among aspects of the Knowledge Quartet, including the impact of foundational knowledge on contingent moments. Horizon content knowledge is shown to play an important role in teaching decisions, as do perceived constraints. This has implications for future research into how teachers’ horizon knowledge might be expanded and into teachers’ perceptions of mathematics course constraints on the enactment and development of their mathematics knowledge for teaching.
The aim of this study is to develop a descriptive phase model for problem-posing activities based on structured situations. For this purpose, 36 task-based interviews with pre-service primary and secondary mathematics teachers working in pairs who were given two structured problem-posing situations were conducted. Through an inductive-deductive category development, five types of activities (situation analysis, variation, generation, problem-solving, evaluation) were identified. These activities were coded in so-called episodes, allowing time-covering analyses of the observed processes. Recurring transitions between these episodes were observed, through which a descriptive phase model was derived. In addition, coding of the developed episode types was validated for its interrater agreement.
Although strong references to expertise in different theoretical approaches to teacher noticing have been made in the last decades, empirical knowledge about the development of teacher noticing from novice to expert level is scarce. The present study aims to close this research gap by comparing three different groups of mathematics teachers with different degrees of professional teaching experience—pre-service teachers at the master’s level, early career teachers, and experienced teachers—using data sampled in the frame of the research program from the Teacher Education and Development Study in Mathematics (TEDS-M). Furthermore, the construct of teacher noticing is assessed in a differentiated way by analyzing different noticing facets. Findings confirm that three facets of teacher noticing can be empirically distinguished—perception of important classroom events, their interpretation, and decisions regarding further developments. The results reveal a considerable increase in professional noticing between master’s students and practicing teachers. However, in contrast to other studies, among examples from East Asia, a stagnation or decrease in professional noticing between early career teachers and experienced teachers could be observed. Overall, the study highlights the cultural dependency of expertise development regarding teachers’ noticing.
The COVID-19 pandemic brought with it a new way of being in a changed and uncertain world. Aotearoa/New Zealand took a well-being approach and in turn, we share the positive outcomes which resulted for some low socio-economic schools and communities in relation to teacher learning and relationships with families. In this article, we report on how teachers and schools connected with diverse students and their families during the period of remote learning. We draw on the responses from 20 teachers and school leaders who participated in interviews. Following the wider government focus, schools took a well-being first approach which led to increased connections and positive home/school relationships. The results highlight how a disruptive event such as COVID-19 can also be a time to focus on strengths of diverse communities and gain insights. We demonstrate that while focusing on mathematics, teachers and school leaders gained insights related to their students’ funds of knowledge and saw opportunities for learning for students, parents, and the teachers themselves.
The give-n task is widely used in developmental psychology to indicate young children’s knowledge or use of the cardinality principle (CP): the last number word used in the counting process indicates the total number of items in a collection. Fuson (1988) distinguished between the CP, which she called the count-cardinal concept, and the cardinal-count concept, which she argued is a more advanced cardinality concept that underlies the counting-out process required by the give-n task with larger numbers. One aim of the present research was to evaluate Fuson’s disputed hypothesis that these two cardinality concepts are distinct and that the count-cardinal concept serves as a developmental prerequisite for constructing the cardinal-count concept. Consistent with Fuson’s hypothesis, the present study with twenty-four 3- and 4-year-olds revealed that success on a battery of tests assessing understanding of the count-cardinal concept was significantly and substantially better than that on the give-n task, which she presumed assessed the cardinal-count concept. Specifically, the results indicated that understanding the count-cardinal concept is a necessary condition for understanding the cardinal-count concept. The key methodological implication is that the widely used give-n task may significantly underestimate children’s understanding of the CP or count-cardinal concept. The results were also consistent with a second aim, which was to confirm that number constancy concepts develop after the count-cardinal concept but before the cardinal-count concept.
This paper provides an empirical exploration of mathematics teachers’ planned practices. Specifically, it explores the practice of foreshadowing, which was one of Wasserman’s (2015) four mathematical teaching practices. The study analyzed n = 16 lessons that were planned by pairs of highly qualified and experienced secondary mathematics teachers, as well as the dialogue that transpired, to identify the considerations the teachers made during this planning process. The paper provides empirical evidence that teachers engage in foreshadowing as they plan lessons, and it exemplifies four ways teachers engaged in this practice: foreshadowing concepts, foreshadowing techniques, foregrounding concepts, and foregrounding techniques. Implications for mathematics teacher education are discussed.
This paper explores how different schools of thought in mathematics education think and speak about preparing mathematics for teaching by introducing and proposing certain metaphors. Among the metaphors under consideration here are the unpacking metaphor, which finds its origin in the Anglo-American school of thought of pedagogical reduction of mathematics; the elementarization metaphor, which has its origin in the German school of thought of didactic reconstruction of mathematics; and the recontextualization metaphor, which originates in the French school of thought of didactic transposition. The metaphorical language used in these schools of thought is based on different theoretical positions, orientations, and images of preparing mathematics for teaching. Although these metaphors are powerful and allow for different ways of thinking and speaking about preparing mathematics for teaching, they suggest that preparing mathematics for teaching is largely a one-sided process in the sense of an adaptation of the knowledge in question. To promote a more holistic understanding, an alternative metaphor is offered: preparing mathematics for teaching as ecological engineering. By using the ecological engineering metaphor, the preparation of mathematics for teaching is presented as a two-sided process that involves both the adaptation of knowledge and the modification of its environment.
This study investigated how 46 pre-service teachers (PSTs) planned for differentiation of instruction in mathematics. Content analysis was utilized to explore the differentiation strategies included and student characteristics considered in PST plans and how PSTs used differentiation strategies and student characteristics to differentiate lesson content, process, product, and environment. In addition, a rubric was designed and utilized to analyze the level of detail PSTs provided in their plans. Results indicated that overall, PSTs were developing in their planning for differentiation of instruction, using general terms to describe the modifications they would make to meet student needs. Moreover, results revealed that PSTs included strategies to differentiate lesson content and environment most often in their plans and frequently considered student readiness levels when planning for differentiation. Findings also revealed that PSTs need support in learning how to plan for differentiation based on student cultural backgrounds. Implications for mathematics teacher preparation are discussed.
From the socio-critical perspective of mathematical modeling, reflexive discussions about the nature, criteria, and consequences of mathematical models are not a natural consequence of modeling in school. This report is part of a larger study focused on stimulating reflexive discussions in practice employing constructivist grounded theory as a research method. Twenty-seven college algebra students engaged in a 3-week modeling project at a community college in the USA. Audio-recorded group discussions and written reflections were collected to determine how reflexive discussions were taking place. Analysis of students’ actions and reflexive discussions during the modeling project produced four concepts: voicing mathematics, personalizing mathematics, challenging mathematics, and negotiating mathematics. These concepts are integrated into an overall process for stimulating reflexive discussions and are conceptualized as unboxing mathematics. The overarching concept of unboxing mathematics represents one interpretation of how reflexive discussions may be constituted during modeling activities and identifies classroom mathematical practices specific to the socio-critical modeling context of this study.
This study is part of the international survey studies on teachers’ beliefs concerning the integration of children’s literature in mathematics teaching and learning, and this paper reports the findings of the thematic analysis of open-ended survey responses elicited from 287 primary school teachers and teacher trainees in Taiwan. Using the seminal social psychology theory, the Theory of Planned Behaviour (Ajzen in Organizational Behavior and Human Decision Processes, 50 , 179–211, 1991) to frame the findings, this study highlights 11 perceived barriers and 11 perceived enablers that are thought to influence the teachers’ intention to integrate children’s literature in their mathematics teaching. More specifically, we identified time constraint, lack of pedagogical knowledge and confidence, and resource constraint as being the most-cited perceived barriers, while pedagogical benefits, desire to improve teaching, and enabling social norms were identified as the top perceived enablers. Ultimately, this article offers several recommendations to address some of these key perceived barriers.
The investigation at scale of the tensions that teachers need to manage when deciding to follow recommendations for practice has been hampered by the problem of occurrence: The conditions in which those decisions could be made need to occur during an observation in order for observers to document how teachers handle them. Simulations have been recommended as a way to immerse teachers in instructional contexts in which they have the opportunity to follow such recommendations and observe what teachers choose to do in response. In this article, we show an example of how a teaching simulation may be used to support such investigation, in the context of policy recommendations to open up classroom discussion and consider multiple solutions and in the instructional situation of doing proofs in geometry. A contrast between the decisions made by expert and novice teachers (n = 59) in the simulation, analyzed using multiple regression, adds empirical evidence to earlier conjectures based on qualitative analysis of classroom teaching experiments that revealed teachers to be particularly attentive to epistemological and temporal constraints. We found that expert and novice teachers differed in how likely they were to prefer practices recommended by policymakers. Specifically, expert teachers were significantly more likely than novice teachers to open up classroom discussions when they had the knowledge resources to correct a student error. Similarly, expert teachers were significantly more likely than novice teachers to explore multiple solutions when there were no time constraints.