Make Sense Of Mathematics Research Articles

Assessing Learning in Mathematical Sensemaking Electromagnetism Instructions among Rwanda Polytechnic Students at Huye College

In the realm of electromagnetic (EM) courses in engineering, many studies have reported students’ learning difficulties related to mathematical frameworks representing physical phenomena. Students’ learning engagement and learning gains are not satisfying. The present study assessed first-year engineering students’ behavioural engagement and perceived learning gains in mathematical sensemaking electromagnetism instructions at RP-Huye College. Within a single-case research design, a six-weeks intervention incorporating mathematical sensemaking instructions, supported by physical experimentation and computer simulations, was implemented to 61 first-year engineering students who were enrolled in the department of electrical and electronics engineering. All enrolled students were purposively recruited to participate because this target population was less than 100. Data were collected through classroom observations, which used the behavioural engagement related to instruction (BERI) and a post-topic evaluation, which used a semi-structured questionnaire. Data analysis involved the use of graphs, descriptive statistics and inductive thematic analysis. Findings revealed that students were mostly engaged during mathematical sensemaking by hands-on and simulation-based activities, particularly in topics related to electromagnets, where engagement levels peaked at 7.5 in average. Conversely, lecture-based tasks, especially on magnetic forces and electromagnetic induction, recorded the lowest engagement at 6.2 in average. The post-topic assessment on perceived learning gains showed that students had highly positive perceptions on their learning experiences (M=4.82, SD=0.48) and recognized the significance of EM in engineering (M=4.85, SD=0.38). These numerical results were complemented by students’ narrations, which indicated that they gained particular attention about specific EM formulas and how they can apply them in engineering. However, the present study also noted that further refinement in instructional design, particularly by incorporating specific dimensions of mathematical sensemaking, could optimize learning outcomes for EM courses in engineering. Additionally, formal assessments of students’ mastery and experimental studies can benefit future work.

Research-informed translation of mental strategy teaching materials into isiXhosa

Background: This article critically examines the nature of isiXhosa translations in mathematics learning materials, specifically focusing on the doubling and halving unit within the ‘South African Grade 3 Mental Starters Assessment Project (MSAP): Teacher Guide’. Teaching in home languages is encouraged in the Foundation Phase, but unfamiliar standardised isiXhosa translations in support materials often contain barriers to understanding and/or distortion in meaning.Aim: The article addresses three questions: (1) To what extent is there fidelity and alignment of the translated isiXhosa materials to the original mathematical meaning? (2) To what extent do the selected terms align with the everyday isiXhosa that learners are accustomed to? (3) What are the implications for future translation of such materials?Setting: Analysis of the isiXhosa translation of a doubling and halving teaching unit and adapted translation for use in a Grade 3 classroom in the Eastern Cape.Methods: This qualitative research uses Toury’s Descriptive Translation Studies theory.Results: The findings highlight several ambiguities and incoherent translations. The article advocates for a comprehensive approach to translation, emphasising the importance of maintaining conceptual fidelity and clarity.Conclusion: The authors suggest the need for transliteration techniques in translations to support teacher and learner access to mathematical sense-making.Contribution: This research provides insights for translators of materials and implications for teachers, proposing an approach to translation. In particular, it argues that the involvement of Foundation Phase teachers in the translation of such materials is crucial.

Student understanding of eigenvalue equations in quantum mechanics: Symbolic blending and sensemaking analysis

As part of an effort to examine students’ mathematical sensemaking (MSM) in a spins-first quantum mechanics course during the transition from discrete (spin) to continuous (position) systems, students were asked to construct an eigenvalue equation for a one-dimensional position operator. A subset of responses took the general form of an eigenvalue equation written in Dirac notation. Symbolic blending, a combination of symbolic forms and conceptual blending, as well as a categorical framework for MSM, were used in the analysis. The data suggest two different symbolic forms for an eigenvalue equation that share a symbol template but have distinct conceptual schemata: A transformation that reproduces the original and to operate is to act. These symbolic forms, when blended with two sets of contextual knowledge, form the basis of three different interpretations of eigenvalue equations modeled here as conceptual blends. The analysis in this study serves as a novel example of, and preliminary evidence for, student engagement in sensemaking activities in the transition from discrete to continuous systems in a spins-first quantum mechanics course. Published by the American Physical Society 2024

Beyond mere persistence: a conceptual framework for bridging perseverance and mathematical sensemaking in teaching and teacher learning

Beyond mere persistence: a conceptual framework for bridging perseverance and mathematical sensemaking in teaching and teacher learning

Investigating the epistemology of physics students while reflecting on solutions

Reflecting on one’s solution is widely recognized as an important part of the problem-solving process, though the cognitive processes underlying this action are not well understood. In previous work, we identified certain strategies students used while reflecting on a solution but found that strategies most often used by experts were not often used by students and were often used incorrectly. In this paper, we present the results of a study that more carefully examines why this could be the case. We conducted think-aloud interviews with students from a variety of physics backgrounds and asked them to check the answer to a particular static equilibrium problem (there were two different problems, but each student saw only one). We found that students’ strategy use varied both by problem features and level of experience. We found students’ epistemological framing to be more stable across problem-difficulty but still correlated with experience. We further noticed more frequent shifting of epistemological frames among intermediate students. Altogether, the results point to an epistemological transition from solution reflection as an algorithmic procedure to be performed using whatever strategy is most useful, to a conceptual procedure more aligned with sense-making as students gain more experience with physics. This will be useful for instructors when thinking about the best ways to encourage novice students to engage in meaningful solution reflection or mathematical sense-making more broadly.

Guided Inquiry into a Physics Equation

We present the theoretical argument that the use of mathematics in physics can be productively conceptualized as using a language and that learning to make sense of physics equations and appropriating them into novel scientific inquiry can be understood as a process of learning to read fluently with a high level of reading comprehension and to express one’s thoughts in writing. The data are drawn from a longitudinal ethnographic account of a high school student’s year-long research apprenticeship, in which a significant part of the work required the derivation and interpretation of an advanced physics equation (the phase velocity of waves in liquids). The analysis examines in different timescales the extended learning process as a complex activity system. The findings illustrate empirically how this type of learning can be productively analyzed by researchers and instructionally implemented by teachers or mentors as engagement in practices, a perspective that complements more standard cognitivist accounts of mathematical sensemaking. We highlight productive pedagogical and discursive instructional moves that facilitated this process and explicitly connect them to the learning they induced in terms of change in the student’s participation. While constrained by the scope of this case study, the concrete, microanalytic account of the learning process concretizes some of what physicists do with equations and suggests pathways to teaching students to meaningfully participate in this practice.

Cognitive framework for blended mathematical sensemaking in science

BackgroundBlended mathematical sensemaking in science (“Math-Sci sensemaking”) involves deep conceptual understanding of quantitative relationships describing scientific phenomena and has been studied in various disciplines. However, no unified characterization of blended Math-Sci sensemaking exists.ResultsWe developed a theoretical cognitive model for blended Math-Sci sensemaking grounded in prior work. The model contains three broad levels representing increasingly sophisticated ways of engaging in blended Math-Sci sensemaking: (1) developing qualitative relationships among relevant variables in mathematical equations describing a phenomenon (“qualitative level”); (2) developing mathematical relationships among these variables (“quantitative level”); and (3) explaining how the mathematical operations used in the formula relate to the phenomenon (“conceptual level”). Each level contains three sublevels. We used PhET simulations to design dynamic assessment scenarios in various disciplines to test the model. We used these assessments to interview undergraduate students with a wide range of Math skills. Interview analysis provided validity evidence for the categories and preliminary evidence for the ordering of the categories comprising the cognitive model. It also revealed that students tend to perform at the same level across different disciplinary contexts, suggesting that blended Math-Sci sensemaking is a distinct cognitive construct, independent of specific disciplinary context.ConclusionThis paper presents a first-ever published validated cognitive model describing proficiency in blended Math-Sci sensemaking which can guide instruction, curriculum, and assessment development.

Developing mathematical patterning in ECE classrooms: participatory research with teachers of 3–5-year-olds

ABSTRACT Research shows that attention to pattern and structure is fundamental to mathematical learning and attainment yet early mathematics curricula in England underplay the importance of patterning. In a critical realist notion of powerful knowledge, pattern teaching has the potential to empower children to notice patterns, mathematise their everyday experiences and engage in mathematical sense-making. This study investigated how to harness this potential. It reports on participatory research with ten teachers of three to five-year-old children in England as they developed pattern teaching in their classrooms. Findings indicate that teacher knowledge, pedagogic interactions and pattern-rich environments (all underpinned by an appropriate developmental progression and extended to form a setting-wide shared approach) support the development of patterning praxis in early childhood classrooms. These offer potential priorities for ECE teachers in developing their patterning praxis in order to support children’s mathematical learning.

Politics and aesthetics of museum mathematics: the dissensual curriculum of early 21st century mathematics exhibitions

ABSTRACT Museum-based mathematics exhibitions are increasingly prominent but under-theorized learning environments. In this study, we analyse the curriculum of United States mathematics exhibitions developed in the early 21st century in terms of their complex suggestions about the nature of mathematics and mathematical sense-making. We apply Rancière’s notions of politics and aesthetics to explore what we describe as dissensus present in the texts, images, and multi-sensory exhibits of several major mathematics exhibitions. Our analysis characterizes this dissensus as a paradoxical mix of alternative and familiar mathematical aesthetics. On the one hand, we identify an alternative aesthetic emphasizing everyday ubiquity, sensuality, and informal sense-making. At the same time, we identify a countervailing emphasis on dominant notions of mathematics as esoteric, immaterial, and formal-symbolic. Museum mathematics efforts sometimes describe themselves as expanding how the public views and defines mathematics. A close examination of the exhibitions in this study reveals a complex picture, in which dominant and alternative forms of mathematics are co-present. The analysis suggests that museum-based mathematics researchers and practitioners view their work as containing political and aesthetic dimensions that can disrupt or reify what society counts as mathematics.

Using Math in Physics: 3. Anchor equations

An important step in learning to use math in science is learning to see symbolic equations not just as calculational tools, but as ways of expressing fundamental relationships among physical quantities, of coding conceptual information, and of organizing physics knowledge structures. In this paper, I propose “anchor equations” as a construct to support teaching and learning in introductory physics. I define anchor equation, provide examples, and suggest ways anchor equations can be used in instruction to support the development of students’ mathematical sense-making.

Applying a mathematical sense-making framework to student work and its potential for curriculum design

A mathematical sense-making framework developed in the context of extended episodes of collaborative student reasoning can be adapted to interpret multiple-choice and short-answer responses.

#Mathathome during the COVID-19 Pandemic: Exploring and Reimagining Resources and Social Supports for Parents

During the COVID-19 pandemic, schools abruptly transitioned to emergency remote instruction. Consequently, expectations for parental involvement in school mathematics rose to unprecedented levels. We sought to understand the experiences of parents to reimagine possibilities for engagement in mathematics during and beyond the pandemic. Leveraging data from tweets using #mathathome and survey responses from parents, we identified who supported continued mathematics learning at home and explored the nature of the mathematics taught there. We found that Twitter and survey data sources described two largely distinct groups of those supporting parents to continue mathematics education at home, but similar findings emerged from analyses of each data source, suggesting that themes were common among different groups. Namely, we saw a commitment to continued mathematics learning and engagement with a range of mathematics topics. These topics mostly focused on elementary-level content, especially counting, through everyday activities/objects and mathematical sense-making. Most parents used resources provided by the school alongside resources they identified and provided on their own. School responses to emergency remote instruction were mostly asynchronous, and parents expressed a need for more opportunities to interact directly with their children’s teachers. We discuss what the mathematics education community might learn from these experiences to support parental engagement during and beyond periods of remote emergency instruction.

Stepping out of rhythm: an embodied artifact for noticing rate of change

Making sense of data is fundamental to science. Yet, the form in which data are represented can make their interpretation challenging for students. We introduce an activity that transformed a data table into an embodied artifact. The activity helped 8th grade students interpret the data to find that cold water warmed “fast-then-slow,” in other words, that the rate of warming decreased over time. For the activity, students were invited to “act out” the temperature change by walking along a giant horizontal thermometer while their classmates clapped out the passage of time. To arrive at the correct temperature at the right time, students had to continually slow their pace. We offer a microgenetic analysis of the activity to illuminate how it helped participating students notice that the cold water’s temperature warmed fast-then-slow, developing an account of the students’ passage from action to understanding. We argue that embodied representations can be effectively coupled with tools and traditional symbolic systems to make abstract concepts more accessible, opening an alternative route for students to engage in rich mathematical reasoning and scientific sense-making.

The “tricky business” of genre blending: Tensions between frames of school mathematics and video game play

ABSTRACT Background Educational video games are increasingly used in classrooms because they can offer meaningful contexts for problem solving. However, educational video games bring together two historically disparate activities: school mathematics and video games. How these two activities complement, compromise, or contradict each other influences how mathematical activity takes shape during game play. Methods This paper offers a case analysis of two students: one who engages with the mathematics as intended by the game and is easily seen as on task, and a second who seems to reject the mathematics as intended by the game and is easily seen as off task. The analysis focuses on how each student’s frame of activity influences their mathematical activity during game play. Findings Findings suggest that, considered from their own frame of activity instead of the frame of the design, both students appear engaged in mathematical sensemaking, albeit in different ways: one as intended by the designer, the other as emerging from game play. Contribution By highlighting potential tensions between these official and unofficial frames, this paper contributes to continued reflections on task designs that incorporate youth culture such as video gaming to make mathematics classrooms more inviting to students.

Preservice Secondary Science Teachers’ Implementation of an NGSS Practice: Using Mathematics and Computational Thinking

ABSTRACT We investigated six preservice secondary science teachers’ implementation of reform-based science, in particular, their teaching of the Next Generation Science Standards’ (NGSS) science and engineering practice of using mathematics and computational thinking. A modified version of the Task Analysis Guide in Science served as our conceptual framework: It assesses both the integration of practices and content (i.e., the kind of thinking required), and the cognitive demand of tasks (i.e., the level of thinking required) in teachers’ lessons. We used this framework to qualitatively analyze our preservice teacher participants’ edTPA (teacher performance assessment) lessons—including their written commentaries, video-recorded lesson excerpts, and student work samples—for their implementation of the NGSS using mathematics and computational thinking practice. We examined (1) the integration of the mathematical content and practices outlined in the Common Core State Standards for Mathematics within the target NGSS practice, and (2) the cognitive demand of the mathematics in relation to science and mathematical practices. We found that four of our six preservice teachers implemented lessons that were integrated and cognitively demanding: These participants used the mathematics to move students’ understanding of the science phenomena forward. However, the other two participants implemented lessons that integrated mathematical content and practices but were low in cognitive demand. We conclude with implications for how teacher education programs can better support preservice teachers’ implementation of lessons that are both integrated and cognitively demanding so as to promote students’ mathematical reasoning and scientific sensemaking.

Assessing mathematical sensemaking in physics through calculation-concept crossover

What kind of problem-solving instruction can help students apply what they have learned to solve the new and unfamiliar problems they will encounter in the future? We propose that mathematical sensemaking, the practice of seeking coherence between formal mathematics and conceptual understanding, is a key target of successful physics problem-solving instruction. However, typical assessments tend to measure understanding in more disjoint ways. To capture coherence-seeking practices in student problem solving, we introduce an assessment framework that highlights opportunities to use these problem-solving approaches more flexibly. Three assessment items embodying this calculation-concept crossover framework illustrate how coherence can drive flexible problem-solving approaches that may be more efficient, insightful, and accurate. These three assessment items were used to evaluate the efficacy of an instructional approach focused on developing mathematical-sensemaking skills. In a quasi-experimental study, three parallel lecture sections of first-semester, introductory physics were compared: two mathematical sensemaking sections, with one having an experienced instructor (MS) and one a novice instructor (MS-nov), and a traditionally-taught section acted as a control group (CTRL). On the three crossover assessment items, mathematical sensemaking students used calculation-concept crossover approaches more and generated more correct solutions than CTRL students. Student surveyed epistemological views toward problem-solving coherence at the end of the course predicted their crossover approach use but did not fully account for the differences in crossover approach use between the MS and CTRL groups. These results illustrate new instructional and assessment frameworks for research on mathematical sensemaking and adaptive problem-solving expertise.

The lived experience of linear algebra: a counter-story about women of color in mathematics

This paper focuses on the mathematical sensemaking by women of color in the USA as part of the global effort of dismantling deficit narratives about historically marginalized groups of students. Following Adiredja’s anti-deficit framework for sensemaking, this cognitive study invited a group of women of color to share their understanding of basis from linear algebra to construct a sensemaking counter-story. Extending the framework, this study examines a task that explores the boundaries and nuances of a concept to support the effort of going beyond students’ deficits. Eight women extended the concept of basis (and vector spaces) to 22 distinct everyday contexts, drawing from their everyday lives as well as topics from their academic experiences. Their explanations revealed analytical codes describing roles and characteristics of a basis. These codes suggest ways that students can mobilize the concept of basis beyond its logical underpinnings. Contrasting interpretations using a deficit and an anti-deficit perspective construct a counter-story that showcases these women’s creativity and flexibility in understanding the concept, and potential resources for the teaching and learning of linear algebra.

A mathematics teacher’s response to a dilemma: ‘I’m supposed to teach them in English but they don’t understand’

Background: English is the dominant language in South African schools although it is the home language for less than 10% of the population. Many schools have yet to embrace the Language in Education Policy’s advocacy of additive bilingualism. This has led to a majority of the country’s children learning and being assessed through a language in which they lack proficiency. Aim: This article draws on second language teaching and learning theory to make a case for more systematic support for learners’ second language development and for legitimation of use of home language in mathematics classrooms where a different language is the official medium. The article shares empirical data from a South African Grade 4 mathematics teacher’s classroom to illuminate arguments in favour of additive bilingualism. Setting: A non-fee-paying public school in Eastern Cape province of South Africa. Methods: Data were collected through lesson observations, teacher interviews and assessment data generated by a professional development project initiative. Results: The ‘illuminatory’ lesson data suggest that allowing learners to use their home language alongside English facilitated their mathematical sense-making. This suggestion is strengthened by assessment data from a larger development project mandated with exploring ways for improving the quality of primary mathematics teaching and learning. Conclusion: Insights from this article add to many other calls made for more sustained and serious consideration of the pedagogical and epistemological value of multilingual approaches for South African classrooms.

Initiation-entry-focus-exit and participation: a framework for understanding teacher groupwork monitoring routines

In this paper, we offer a framework for teacher monitoring routines—a consequential yet understudied aspect of instruction when teachers oversee students’ working together. Using a comparative case study design, we examine eight lessons of experienced secondary mathematics teachers, identifying common interactional routines that they take up with variation. We present a framework that illuminates the common moves teachers make while monitoring, including how they initiate conversations with students, their forms of conversational entry, the focus of their interactions, when and how they exit the interaction as well as the conversation’s overall participation pattern. We illustrate the framework through our focal cases, highlighting the instructional issues the different enactments engage. By breaking down the complex work of groupwork monitoring, this study informs both researchers and teachers in understanding the teachers’ role in supporting students’ collaborative mathematical sensemaking.

Language as an including or excluding factor in mathematics teaching and learning

This article explores the power of language to either include or exclude certain groups of students from genuine opportunities for mathematical sense-making. The substantial increase worldwide in the number of students learning mathematics through a language other than their primary language makes this a particularly urgent issue. This paper focuses on the South African situation, where, because English is widely perceived as the language of opportunity, it is, by grade 4, overwhelmingly the chosen language of learning and teaching. The epistemological and pedagogical consequences of this choice are evidenced in the poor performance of the country’s students on national and international assessments of mathematical proficiency. Drawing on research literature around language immersion education models and the extent to which these align with certain key principles of second language acquisition, this position paper motivates for a stronger and more sustained commitment to providing students, particularly those from marginalized and vulnerable communities, with opportunities for becoming both bilingual and biliterate. Empirical data from two South African grade 4 mathematics classrooms are used to illuminate aspects of the mathematical sense-making challenges students and their teachers face without such commitment.