This study replicated an earlier study that explored the beliefs of a sample of K-8 pre-service teachers (PSTs). Results of this study position PSTs' beliefs along a traditional-reform continuum. The positioning from the current study is compared to that of the former study, to determine the level of progress made toward the National Council of Teachers of Mathematics' commitment to high-quality instruction. To position beliefs, PSTs completed a Likert survey, including open-ended questions, three times throughout their teacher preparation program (TPP): before taking any of their mathematics education courses, after taking both required mathematics education courses, and after student teaching. On the last survey iteration, PSTs completed additional Likert items and open-ended questions regarding their teaching practices. Analysis centered on the relationship between each iteration of the new survey, comparisons to results from the initial study, and the alignment between beliefs and teaching practices. Overarching results convey that even with the passage of eleven years, the positioning of beliefs and teaching practices remained relatively static. Beliefs related to the power of student ideas, mathematical processes, productive disposition, and productive struggle were categorized as being reform oriented. The perception that mathematics is mostly computation and that the expository teaching is the most effective way to teach mathematics were categorized as being strongly traditional. Beliefs that were categorized more centrally related to collaboration and the emphasis on answers. Suggestions are provided for teacher educators to continue their work toward reforming beliefs about mathematics and how to teach mathematics.
This qualitative study explored how Grade 4 students and the researcher co-created language to express statistical concepts in a multilingual Indian classroom during a weeklong instructional sequence. The analysis draws on a situated socio-cultural perspective on language in mathematics. Students' language progressed from simple claims to more refined and sophisticated expressions, enabling them to discuss central tendencies and distributions while acknowledging uncertainty. Co-created phrases such as & amacr;sap & amacr;s resh & amacr; (the nearby line) not only supported communication but also deepened reasoning. The article also discusses what role scaffolding played in refining these informal resources.
Challenges related to the teaching and learning of formal notation in school mathematics are widely documented, and specifically in relation to the underlying mathematical structures that the notation is intended to convey. In this article, we draw on embodied cognition to examine the interactions among three students working with the software Grid Algebra. Embodied cognition emphasises the role of gesture and movement in learning and understanding mathematics. Grid Algebra uses movement to direct students' attention to mathematical operations on numbers and numerical expressions within the grid and the structure of these operations, while the software takes care of the formal notation of the numerical expressions that describe these sequences of operations. We analyse how different modes of communication work together to scaffold students' fluency with operations and the formal notation representing these operations and the order in which they are performed. The dynamic between notation, speech, movement and position allows students to educate their interpretation of mathematical notation through the movements and positions that they are very familiar with.
This study explores the critical role of language and communication in fostering collaborative noticing practices that display mathematical intuitions within probability learning. Through an analysis of student interactions, I show how the joint articulation of ideas and the exploration of shared doubt enable intuition, often perceived as an object that is held and a solitary experience, to become a socially shared and co-constructed process. The observed collaborative dialogue concerns shared content and form, constructing a joint narrative architecture that enables common understanding and allows the group to notice, examine, and reshape their intuitions. Collaborative noticing when working with probability problems facilitates the surfacing, interrogation, and transformation of student mathematical intuitions, often displaying previously unconscious cognitive contents and processes. I argue that collective narratives about mathematical intuitions offer unique opportunities to communicate what would often be imperceptible individually. This research highlights that intuition is continuously reconstructed through the embodied and situated dialogue of shared mathematical experience. Findings show that language and interaction are fundamental for making explicit, noticing, and reflecting upon intuitive responses, particularly when these are challenged. Playfulness, affective safety, humour, and shared doubt emerge as catalysts, facilitating a progressive shift from individual to collective reasoning, demonstrating that intuition is not a static certainty but a dynamic, socially mediated process.
In concluding the second series of eight articles in the double special issue on 'Engaging with communication and language in relation to mathematics and its education', we take the opportunity to reflect across the articles to prompt us all to consider potential and emerging directions for future research. This final article is overall a commentary inspired by the communication-centred and language-centred contributions to mathematics education across the entire collection of articles, that is, all those comments that situate mathematics and its education as language and communication praxis. We explore this relationship between mathematics and its education with communication and language through three themes: time, theories and methodologies, and the nuances of the term 'relation' in the title of this double special issue.
In this paper, we present the rationale and results of a study conducted with 79 pre-service primary-school teachers on their sensitivities to notice the pedagogic function of naming practices in the language of teaching fractions. Our results are a contribution to noticing research on mathematics teacher education and language. Although participants were not familiar with discussions of the language of teaching mathematics in general and the language of teaching fractions in particular, they engaged in an intervention with a professional task that provided opportunities to notice potential relationships between mathematical naming and the conceptual understanding of fractions. The written answers to questions from the task show attention to mathematical naming practices that may be useful in interactions with students who experience specific learning challenges. Taking decisions about how to name in mathematically relevant ways in the teaching of fractions, nonetheless, seems to be quite complex, or at least not obvious in the participant answers. The teacher education intervention lasted two sessions of two hours each. We therefore recommend a more sustained curricular inclusion of the language of teaching mathematical content in the initial preparation of mathematics teachers.
Research has progressively considered a resource-oriented rather than a deficit-based approach towards understanding the role of language in helping students learn mathematical concepts. Notably, there has been a call for greater attention to the notion of mathematics teaching talk, which focuses on teachers' use of language as a resource to teach mathematics (and the language of mathematics). It is argued that mathematics teaching talk plays an important role in enabling teachers to communicate mathematical ideas or meanings of mathematical concepts and in turn, preparing students to participate better in mathematical discourse practices. Specifically, this paper seeks to exemplify what teachers know and do in relation to mathematics teaching talk through the lens of the Mathematics Register Knowledge Quartet (MRKQ). Consequently, we propose the utilisation of the MRKQ as a resource in mathematics teacher education to develop teachers' knowledge and use of the mathematics register for teaching and communication.
This qualitative study examines communication in family mathematics settings, focusing on how embodied communication is constituted during parent-child interaction with a multitouch technology application, TouchCounts. Moving beyond information-transmission perspectives, the study adopts an embodied and relational approach centered on affectivity-conceptualized as the circulation of attunement, resonance, and intensity of movement and feeling across human and non-human components of the parent-child-TouchCounts assemblage. Drawing on close qualitative analysis of two selected video-recorded excerpts, the study traces how mathematical events unfold moment by moment through bodily action and material engagement, and how affective flow shapes how participants respond to one another and to the digital interface. The findings show that affectivity operates as a constitutive dimension of mathematical communication, shaping how mathematical concepts (e.g., addition toward bigness, making two by V-gesture) are enacted, oriented, sustained, and transformed as lived events. By foregrounding affect and embodiment, this study offers a novel perspective on communication in family mathematics and contributes to broader discussions on mathematical meaning-making in technology-mediated contexts.
The shape of a graph arising from the covariation of variables is a crucial idea in reasoning about functions and graphs. Students' perspectives on graph shape may foreground visual features and consider the shape to be a fixed property of the function (static graphical shape thinking). In contrast, students can perceive the graph as a trace of covarying quantities, through which its shape arises (emergent graphical shape thinking). It is of interest to understand how these perspectives are supported and promoted by teachers in classroom discourse. We present findings from a study of classroom discourse in which we used systemic functional linguistics (SFL) to identify features of how teachers and students described graphs. We identified informal but vivid descriptions of graphs that may have affordances for, and even foreground, emergent graphical shape thinking.
This paper presents the findings of a study, conducted as part of a PhD thesis, exploring ways in which translanguaging can be used to support the development of conceptual understanding of division. The study focused on 18 8-9-year-old pupils learning mathematics through the instructional languages of English and French in an 'immersion-style' bilingual education context in England. The data was collected from four consecutive mathematics lessons focusing on division, the first two taught through English by a monolingual English teacher and the second two taught in French by a bilingual French teacher. Both teachers were interviewed directly following their second lesson and the pupils participated in two focus groups, following their lessons in English and then French. This paper focuses on three of the seven translanguaging cycles that were identified as part of the analysis, each with a different combination of interlocutors. The first extract presents a peer-peer discussion of division involving exchange, the second involves a teaching assistant scaffolding learning for a new-to-French pupil and the third illustrates a pupil seeking clarification from the teacher. The findings reveal that crosslinguistic translanguaging affords a flexible space in which pupils are able to draw upon their linguistic repertoires to make linguistic and conceptual connections. The use of multiple representations and reprocessing opportunities further supported crosslinguistic transfer within these translanguaging cycles.
We analyze California's gubernatorial recall system using basic ideas from the mathematical theory of social choice. We expose some pathologies of the system, use computer simulations to explore how often these pathologies might occur, investigate the effects of strategic voting on the analysis, and suggest changes to the system that cure the pathologies.
The U.S. Presidential Primary is a series of statewide primaries and caucuses used by the Democratic and Republican parties to winnow a pool of candidates to arrive at their nominees. As unsuccessful candidates leave the race, there is a question as to how voters who planned to vote for the candidate reassign their votes in the upcoming primaries/caucuses. When a profile of voters' preferences is independent of elimination under plurality, then voters whose first-place candidate has dropped out of the race have their votes distributed to the remaining candidates in the same proportion of the remaining candidates' first-place votes. For any number of candidates, we show that this notion of independence can be reframed in terms of a probabilistic view of the election. We relate the repeated dropping out of candidates to Harville's equations from probability theory. To motivate our analysis, we look at data from the 2024 Republican Presidential Primary.
This article investigates the interplay between apportionment functions and order-theoretic structures on population and allocation vectors. We define a class of functions, called apportionment maps, that respect population monotonicity, house monotonicity, and size monotonicity. A central focus is placed on the preservation or violation of majorization under these functions, with particular attention to the Jefferson method and related divisor schemes. By applying tools from convex analysis and the theory of Schur-convex functions, we characterize the extent to which fairness criteria are preserved under apportionment.
This article describes an interesting error I committed in the course of my research in voting theory. While embarrassing, the error has led to productive discussions about mathematical failure with my students in my probability and statistics course. I describe my mistake and discuss how the incorporation of such mistakes into the classroom can enhance student trust and engagement.
The venerable Secretary Problem asks how a decision maker (DM) should select one of n candidates, who come up randomly, when the only information available is each candidate's strict rank within the set of previous applicants. Moreover, DM may select only the current candidate. Our illustrative case has n = 9 candidates, for which the Standard Method is to reject outright the first 3 candidates and then choose the first of the 4th through 8th candidates who is better than all of the first three, or the 9th candidate if none of them is. We compare the Standard Method with two other selection methods that change the conditions under which DM decides: Reserve Method. Same as the Standard Method, except that the best of the first 3 candidates is held in reserve and chosen if none of the 4th through 8th candidates (Version A) or none of the 4th through 9th candidates (Version B) is better. Score Method. Each candidate receives a score between 0 and 1; scores are known to be uniformly distributed. In each round, DM decides on a numerical threshold and selects the candidate if she exceeds the threshold. If none does, candidate 9 is selected. We assume that DM chooses thresholds so as to maximize the expected score of the selected candidate. The Standard Method gives DM a probability of 41% of selecting the best candidate, whereas the Reserve Methods substantially raise this probability to 70% or 74%-depending on whether Version A or B is used-while the Score Method raises it to 55%. Thus, the other methods, especially the Reserve Methods, outperform the Standard Method in selecting the best candidate. But the Score Method requires seeing significantly fewer candidates-an average of 4.3, compared with an average of 6.3 for the Standard Method and 6.3 and 6.7 for the two versions of the Reserve Method. We also discuss a third criterion, the average rank of the candidate selected, which is about 1.5 for the Reserve Methods and the Score Method but 2.9 for the Standard Method. In sum, the superiority of the alternative methods over the Standard Method reflects the severity of the restrictions placed on the original Secretary Problem, suggesting it is time to revisit the assumptions of this method and consider realistic alternatives.
In Malta, mathematics education is accessed bilingually through Maltese and English. While Maltese is often used for verbal interaction, English is the language of written texts, including assessments. Consequently, switches between the languages is a common feature of Maltese mathematics classrooms. Anecdotal and research evidence indicate that Maltese students benefit from the use of their home language. However, to date, there is no recognised and standardised Maltese bank of mathematics Our project aims to address this lacuna. In this article, we outline the process of compiling a bilingual glossary of mathematics terms intended for kindergarten and elementary school levels in Malta (ages 3-11 years). The purpose of the glossary is to support teachers and students in their use of the two languages for the teaching/learning process. This novel resource can serve to validate the Maltese language for mathematics and to encourage consistency of terminology across school and classroom contexts. The glossary can also be helpful in the future should we choose to introduce written Maltese for mathematics. In this article, we explain how the project was developed and how word choices were made with respect to the Maltese terms.
The present study reports on analysis of how two teachers in Lebanon, and their students navigated the learning and teaching of geometric proof in English, in a context in which Lebanese Arabic is the language of home and wider society. One of the teachers taught in a public school, the other in a private school. The study focuses on language socialization practices in the two grade-seven classrooms, in relation to the specific genre of geometric proof. The focus is not directly on proving as a cognitive process; rather, the study explores the interactional process of interpreting and constructing geometric proofs as a form of textual organization. This focus is consistent with the theoretical framing in which learning mathematics is understood as a process of socialization into mathematical discourses, of which the proof genre is a part. The findings reveal four main sets of socialization practices in relation to the genre of geometric proof. These sets of practices were guiding, obtaining information, deducing and attending to accuracy and precision.
