The Mathematics Educator strives to provide a forum for collaboration of mathematics educators at varying levels of professional experience. Its purpose is to promote the interchange of ideas among the mathematics education community, locally, nationally, and internationally and to present a variety of viewpoints on a broad spectrum of issues related to mathematics education.
Mathematics Education Student Association
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Effective school mathematics tasks: Perspectives of student teachers
This study investigated the features that primary and post-primary student teachers (STs) consider important in mathematics classroom tasks. Quantitative data revealed that STs generally valued tasks which supported higher order thinking, could be easily differentiated, and were relevant to the curriculum. Aesthetics, use of technology and ease of preparation were considered less important. Qualitative data explored STs’ reasons for these choices and for any changes which occurred over the course of their Initial Teacher Education (ITE). STs expressed growing understanding of the importance of differentiation, pupil engagement and the role of the teacher. Findings underscored the influence of school placements in shaping STs’ pedagogical beliefs. Implications point to the role of ITE courses in developing STs’ professional judgement.
Copyright (c) 2026 Julie Hamilton, Dr Barry Doherty, Gerry Trainor http://creativecommons.org/licenses/by-nc-nd/4.0
An Alternative Quantitative Reasoning Approach to Integration by Substitution
In this paper, we present a conceptual analysis for integration for substitution that centers major ideas of quantitative reasoning, including accumulation rates, relationships between measures and unit magnitudes, and the multiplicative dependency of quantities. Our centering of these idea enables integration by substitution to occur through coordinating accumulation rates and intervals to reconstruct a desired integral structure. Our approach was inspired by the conceptual analysis provided by Jones and Fonbuena (2024), and thus we compare our approach with theirs throughout in order to highlight similarities and differences between the two. We close by acknowledging that a conceptual analysis is only as good as its use in working to support learning, and thus call for future work that transitions the analysis to work with students.
Copyright (c) 2026 Kevin Moore, Sohei Yasuda, Webster Wong http://creativecommons.org/licenses/by-nc-nd/4.0
This is the list of our recent reviewers.
Jennifer Kleiman & Sohei Yasuda TME Editors
Copyright (c) 2025 Jennifer Kleiman & Sohei Yasuda TME Editors http://creativecommons.org/licenses/by-nc-nd/4.0
Our introduction to the 33rd volume of The Mathematics Educator includes brief summaries of the three articles in this issue, as well as editorial commentary and a thanks to the reviewers and others who made this issue possible.
Dru Horne & Jennifer Kleiman TME Editors
Copyright (c) 2024 Dru Horne & Jennifer Kleiman TME Editors http://creativecommons.org/licenses/by-nc-nd/4.0
A Study of Students’ Help-Seeking Behaviors in Undergraduate Mathematics Tutoring
This observational study considers the help-seeking behaviors of students who drop in to receive free tutoring at a university’s mathematics tutoring center. It reports on how these students enter the tutoring space, act, and interact with others comparing the students in two different areas of the center. One of the areas serves students taking math classes for science, technology, engineering, and mathematics (STEM) students. The other area serves students taking applied mathematics classes for business, life science, and social science majors. Findings suggest that most students enter the center alone, stay for over an hour, and are industrious, no matter the area they visit. However, students in the STEM area were more social with others in the tutoring center, more focused on gaining a conceptual understanding, and less likely to be dependent on tutors than the students in the applied area. These results add to research literature on what is known about student actions and interactions in university tutoring centers and has implications for those who organize and lead tutor training that might help them provide better support to students.
Copyright (c) 2025 Deborah Moore-Russo, Henry Christiansen, Emilee Coxsey http://creativecommons.org/licenses/by-nc-nd/4.0
Navigating Math Grading Reforms: Key Considerations for Educational Leaders
As grading reform efforts such as standards-based grading and grading for equity gain popularity, school leaders play a key role in supporting mathematics teachers' implementation efforts. This paper will describe what mathematics leaders need to know about grading reform and what mathematics leaders should do to support teachers in implementing grading reform. By developing teachers’ deeper understanding of curriculum, instruction, and assessment, assisting teachers in shifting mindsets from quantifying to qualifying learning, and improving math proficiency communication, leaders can better navigate grading reform in their schools. With proper support and guidance, teachers can assign grades that better reflect students' mathematical knowledge and skills.
Copyright (c) 2025 Matt Townsley, Chad Lang http://creativecommons.org/licenses/by-nc-nd/4.0
A Framework for Time and Covariational Reasoning
Covariational reasoning has emerged as a productive construct to characterize students’ mathematical development. Researchers have illustrated its importance for major middle, secondary, and undergraduate mathematical concepts including rate of change, accumulation, and modeling. Within this line of work, several researchers have indicated differences between experiential and conceptual time with respect to the covariational relationships students construct. I draw on this body of literature and return to Piaget’s perspective of time to provide a framework for the role of time in students’ (co)variational reasoning. The framework also clarifies the nature of the multiplicative objects underlying students’ (co)variational relationships. To illustrate the framework and capture its emergence from second-order models of students’ mathematics, I also describe the framework as it relates to students’ engagement in a task.
Copyright (c) 2025 Kevin Moore http://creativecommons.org/licenses/by-nc-nd/4.0
Semantic Structure of Word Problems: A Content Analysis
Solving word problems is a standard part of the mathematics classroom. However, many students struggle solving word problems due to the variety in types of problems. One word problem can require a different solution approach than another form with few semantic changes. The Common Core State Standards Initiative (CCSSI) released a taxonomy for classifying types of one-step addition and subtraction word problems. The present study is a content analysis of three widely distributed mathematics textbooks, one each from three separate textbook publishers. The researchers analyzed the textbooks to classify the types of word problems they contained according to the CCSSI taxonomy. The findings show that variation of semantic problem structure is not proportionate within a single textbook; however, problem category frequency is similar across textbooks.
Copyright (c) 2024 Julia Calabrese, Jamaal Young http://creativecommons.org/licenses/by-nc-nd/4.0
In this conceptual paper, I trace the role of agency in mathematics learner identity development across several empirical studies to illuminate understanding of how researchers have conceptualized agency as it relates to identity in mathematics education research. Building on this line of research, I propose an adapted conceptual framework for examining the role of agency in mathematics learner identity development through attention to both micro and macro influences on mathematics learners. I argue that such a conceptual framework is needed to support novel research involving equity in mathematics education. Finally, I offer the metaphor of “the mathematics identity bicycle” to aid in understanding how this conceptual framework might be taken up by other researchers in the pursuit of equity in mathematics education.
Copyright (c) 2024 Stephen L. Caviness http://creativecommons.org/licenses/by-nc-nd/4.0
Professional Noticing in the Context of Mathematical Modeling
The purpose of this study is to examine how professional noticing evolves in the context of modeling and what is unique about the context of modeling for developing noticing. Eight middle and secondary school teachers participated in this one-semester long study and received three training sessions on professional noticing. The results support modeling as a desirable context for developing noticing. Teachers provided more substantial detail about the mathematical aspects of student strategies after participating in the training, and they benefited from conversations with colleagues scaffolded by the three phases of professional noticing: attending to, interpreting, and responding to student thinking. Teachers also identified some unique aspects of student thinking in the context of modeling, especially local conceptual development experienced by some students during the study.
Copyright (c) 2024 Rui Kang http://creativecommons.org/licenses/by-nc-nd/4.0
Facilitating Mathematical Reasoning through Team-based Learning
Team-based learning (TBL) is a flipped classroom model, where small group discussions and peer learning play a central role. Some of its features, such as scalability to large classes and a high degree of structure, together with a well documented success rate in other fields, could make TBL an attractive option for the mathematics educator wishing to transform their teaching. This article surveys available peer-reviewed literature to provide an overview of current use of TBL in mathematics, summarizes findings and based on these, discusses TBL’s potential to support mathematics learning. We pay particular attention to if and how TBL can be leveraged to shift student focus from procedural towards conceptual learning and more creative forms of mathematical reasoning.
Copyright (c) 2024 Anders Gustafsson http://creativecommons.org/licenses/by-nc-nd/4.0
Copyright (c) 2023 Aida Alibek & Dru Horne TME Editors http://creativecommons.org/licenses/by-nc-nd/4.0
Rebuilding Readers’ Relationships with Mathematics: A Review of The Psychology of Mathematics: A Journey of Personal Mathematical Empowerment for Educators and Curious Minds
Copyright (c) 2023 Amanda Jansen http://creativecommons.org/licenses/by-nc-nd/4.0
Why I Won't Teach Mathematics for Social Justice (Even though I Want to)
This paper reports on a research study in which seven New York City high school mathematics teachers participated in a professional development opportunity around the teaching of mathematics for social justice. The teachers saw value in teaching math for social justice and were philosophically aligned with the pedagogy. Despite this and despite recognizing various benefits of its use, they all indicated that going forward they would implement the pedagogy infrequently if at all. This paper explores the reasons the teachers gave for why they would not implement the pedagogy fully as it explores barriers to teacher implementation of teaching mathematics for social justice.
Copyright (c) 2024 Lidia Gonzalez http://creativecommons.org/licenses/by-nc-nd/4.0
Determining the Multiplicative Reasoning of a High School Junior and Senior
Our study was guided by the question, how does the multiplicative reasoning of upper high school students give insight to their performance on a grade-level standards-based assessment? After giving a group of high school students a sample ACT assessment, we identified students to make comparisons between low and high scoring students on the sample assessment. Through a written assessment targeted towards assessing students’ unit coordination schemes, and through semi-structured interviews with two students, we documented a correlation between students’ level of unit coordination and their performance on the sample assessment. The evidence that students showed of limited multiplicative reasoning skills help explain some of their challenges in responding to prompts on an assessment like the ACT. This study reflects the need to give more focused attention on the multiplicative reasoning skills of secondary students and to design interventions that might develop these students’ multiplicative reasoning.
Copyright (c) 2024 Nick Shaver, Anna Dejarnette http://creativecommons.org/licenses/by-nc-nd/4.0
Preservice Secondary Mathematics Teachers’ Opportunities to Learn Reasoning and Proof in Algebra
This study examined opportunities provided for preservice secondary mathematics teachers (PSMTs) to learn reasoning and proof in algebra from the perspective of college instructors. We analyzed interview transcripts of 15 course instructors recruited from three teacher education programs in the United States. We examined the reported opportunities provided for PSMTs to engage in proving- related activities, including making conjectures, investigating conjectures, developing arguments, evaluating arguments, and disproving by using counterexamples. We also analyzed instructional strategies reported by the instructors. We found the inconsistency between instructors’ perceptions of the importance of reasoning and proof in algebra and instructor-reported opportunities to learn. Findings also indicated that developing arguments was reported the most frequently. In addition, instructors reported more pedagogy- focused general teaching strategies than proof-specific teaching strategies.
Copyright (c) 2023 Jia He, Tuyin An http://creativecommons.org/licenses/by-nc-nd/4.0
Intellectual need, covariational reasoning, and function: Freeing the horse from the cart
Authors (year) contended a quantitatively sophisticated image of a dynamic situation can provide students with the horse needed to pull the cart that is the mathematical properties important for the set-theoretic definition of function. In this paper, we extend our argument in two ways. First, we adopt Harel’s (2008, 2018) constructs of intellectual need and epistemological justification to describe how a student can develop a quantitatively sophisticated image of a dynamic situation. Second, we exemplify that constructing an epistemological justification can support a student in subsequently making determinations regarding ‘function’, yet there is an apparent lack of intellectual need for differentiating between functional and non-functional relationship. This latter factor leads us to call into question the value of focusing on the metaphorical cart that is the basis for the set-theoretic definition of function.
Copyright (c) 2024 Teo Paoletti, Kevin Moore, Madhavi Vishnubhotla http://creativecommons.org/licenses/by-nc-nd/4.0
What Makes Pedagogical Content Knowledge “Pedagogical”? Reconnecting PCK to Its Deweyan Foundations
In this theoretical paper, I review the history of research in educational psychology that inspired Shulman’s notion of pedagogical content knowledge (PCK) and critically examine interpretations of PCK reflected in prominent theoretical frameworks for mathematical knowledge for teaching (MKT). I propose a theory of PCK—grounded in radical constructivism, Piaget’s genetic epistemology, and empirical research results—that addresses limitations of these prominent frameworks. I conclude with a description of what makes PCK in the proposed theory “pedagogical” and describe a research agenda that reconnects MKT scholarship to its Deweyan philosophical foundations.
Copyright (c) 2023 Michael Tallman http://creativecommons.org/licenses/by-nc-nd/4.0
Copyright (c) 2022 TME Editors http://creativecommons.org/licenses/by-nc-nd/4.0
Contemporary, emergent mathematics for teachers
In this paper, we present a qualitative case study on an online graduate program for practicing teachers and educators that explores contemporary mathematics and its integration into K to 12 education. Data for the study comprised students’ work in the courses, student feedback, and notes from instructor debriefings for two cohorts of the course. The findings are organized in terms of (i) specialized knowledge for teaching mathematics, (ii) online mathematics teacher education, (iii) online professional learning communities, and (iv) online teaching in general. We conclude the paper with suggestions for online mathematics teacher education and the identification of venues for future research.
specialized knowledge for teaching mathematics
online professional learning communities
Copyright (c) 2023 Ayman Aljarrah, Armando Paulino Preciado Babb http://creativecommons.org/licenses/by-nc-nd/4.0
Young Students Exploring Measurement Through Problem Solving and Problem Posing
The empirical data in this study are from a series of two lessons on measurement implemented in seven classes with 119 six-year-old students in Sweden. Both problem solving and problem posing were shown to be important in early mathematics when students in this study worked on one problem-solving task and one problem-posing task on measurement. As there are few studies specifically on problem posing in early mathematics and on young children’s understanding of measurement, this study adds knowledge of value for both teachers and researchers. In the study, paper-and-pen work from the students was analysed together with interviews conducted after the students had worked on the two tasks. When solving the task on measurement, the students discerned shape, size, distance, and number as mathematical aspects of measurement. When asked to pose a similar task, only size and number reoccurred as mathematical aspects of measurement. However, other features from the problem-solving task reoccurred in the posed tasks: similar drawings were used in combination with questions on measurement as the mathematical content.
Copyright (c) 2023 Hanna Palmér, Jorryt van Bommel http://creativecommons.org/licenses/by-nc-nd/4.0
Using Bafa Bafa to help pre-service teachers experience microaggressions in the classroom
Implementing an equity agenda in the classroom is both necessary and challenging for classes containing pre-service teachers. For this intervention, we chose to begin our History of Mathematics course for pre-service middle school teachers with a cultural simulation training exercise: Bafa Bafa. After participating in the exercise, pre-service teachers were asked to write a reflection paper and were later interviewed about their experiences in the course. Although participants found Bafa Bafa an uncomfortable experience, it was not an unsafe one, and the pre-service teachers agreed that this exercise helped them better understand, articulate, and notice experiences with microaggressions.
Copyright (c) 2023 Rebecca Anne Dibbs, Kelly Lewis, Jennifer Moon, Rebecca Steward http://creativecommons.org/licenses/by-nc-nd/4.0
James Drimalla & Yufeng Ying TME Editors
Copyright (c) 2021 James Drimalla & Yufeng Ying TME Editors http://creativecommons.org/licenses/by-nc-nd/4.0
This paper presents research that exists at the crossroad of scholarly practice and scholarly inquiry. We outline the process in the design, enactment, and empirical examination of an elementary methods course activity, Exploring and Supporting Student Thinking (ESST), which engaged 18 elementary prospective teachers (PTs) in two sessions of one-on-one problem posing with 3rd grade students. Our results mirror outcomes from existing literature focused on student interviews and letter exchanges as well as reveal other potential PTs experiences from such interactions. We end by describing implications for future activity design and with a call for researchers to continue to contribute to scholarly inquiry in this area.
Copyright (c) 2022 Andrew Tyminski, McKenzie H. Brittain http://creativecommons.org/licenses/by-nc-nd/4.0
Preservice Teachers’ Conceptualizations of Mathematical Tasks
This study reports how 12 secondary mathematics preservice teachers (M-PSTs) described characteristics of Mathematical Tasks after participating in instructional activities, including reading, reflecting, and discussing task characteristics from two mathematics task frameworks and related book chapters. The findings demonstrated that after engaging in these activities, M-PSTs used formal, research-informed language in a way that highlighted student-related factors (e.g., student prior knowledge) and contextual factors (e.g., class time constraints), suggesting that the language and concepts offered through the task frameworks initiated M-PSTs’ nuanced task descriptions. The implications of the instructional activities in teacher education programs to enhance M-PSTs’ conceptions of tasks are discussed.
Copyright (c) 2022 Bima Kumari Sapkota http://creativecommons.org/licenses/by-nc-nd/4.0