ABSTRACT The basis for meaningful one-to-one counting is the cardinality principle (CP): The last number word used in counting a collection indicates its total. A popular view is that CP development builds on subitizing: the immediate association and labeling of a small collection with a number word. Simon etal. (2021) challenged the theoretical rationale and supporting evidence of this view, arguing in part that subitizing does not provide an adequate conceptual basis for constructing CP knowledge. The present review entails an examination of the theoretical role of subitizing in the construction of the CP and an evaluation of the evidence supporting it.

ABSTRACT This study explores the application of variation theory (VT) to provide instructional guidelines for teaching sampling distributions in higher education. Sampling distributions are often misunderstood by students, particularly in distinguishing between sample distributions and sampling distributions, and understanding the role of sample size on sampling variability. Through an educational design research approach, we developed two design principles grounded in VT. The first emphasizes systematic variation in sample size to help students discern the relationship between sample size and sampling variability. The second focuses on the distinction between sample distributions and sampling distributions. Data were collected from two iterations of a Master level course about quantitative research methods, including performance tests, interviews, and classroom observations. The study contributes to “humble theory” by offering practical, topic-specific design principles for sampling distributions while also underscoring the importance of systematic variation in instructional design to foster statistical reasoning.

ABSTRACT Quantitative reasoning, as a research program within mathematics education, is well developed with numerous studies on students’ reasoning within real-world-based tasks. However, these studies typically extend only up to students at the calculus level. To move the field forward and to assess the utility of these theories to analyze student thinking beyond calculus, further efforts are necessary. The purpose of this study is to contribute to and extend existing research on students’ quantitative reasoning by investigating students’ mathematics in a novel research setting. We accomplish this by documenting how two differential equation students quantified attributes during a mathematical modeling task sequence focused on supporting students’ construction of a model for a multi-species interaction system. In particular, we describe how our participants quantified change, total amount, and rate of change by operating on quantities. Our findings affirm previous research on students’ quantification of attributes and highlight important implications for teaching modeling of dynamical scenarios.

ABSTRACT Research shows that preschoolers can understand multiplication and division situations, yet most studies have reported children’s use of manipulatives as tools for representation. This study addresses a gap in the literature concerning young children’s interpretation of division and explores how drawing and gestures might offer insights into their problem-solving process. We hypothesize that drawing and gestures, alongside manipulatives, could serve as additional tools for children to represent division situations. We investigated how 5- to 6-year-olds from Australia and Italy represented measurement and partitive division word problems through drawings, words, and gestures, using a task-based interview. A multimodal semiotic approach was used to analyze their strategies. The findings suggest that the children were able to interpret and represent both forms of division, and their drawings and gestures helped to identify the strategies children used. We argue that children’s drawings can capture their understandingof division word problems as effectively as manipulatives.

ABSTRACT We articulate two perspectives, termed co-occurrence and same extent, to capture our understanding of what six future U.S. mathematics teachers intended to express when generating drawings and equations for proportional relationships. We illustrate these perspectives through two case studies and make three contributions. First, the future teachers used co-occurrence and same extent across different forms of inscription. In other words, general orientations to expressing proportional relationships with inscriptions influenced how they generated equations. Second, the future teachers used the two perspectives to generate both normatively correct and normatively incorrect equations. Thus, whether or not they generated normatively correct equations was a matter of where and how they combined the two perspectives. Third, we conjecture that the two perspectives help explain why, at least for some, the x/a = y/b and 1/a • x = 1/b • y forms may provide more accessible entry points into linear equations than the traditional y = mx form..

ABSTRACT Flexible and adaptive strategy use is an important objective of contemporary mathematics education. However, empirical findings show that many students hardly use different strategies in arithmetic. This raises the question what students’ learning opportunities are to acquire strategy flexibility, the use of various strategies, and adaptivity, the use of the optimal strategy for either the task or the individual. Since mathematics textbooks play a pivotal role in shaping students’ learning opportunities, we analyzed four commonly used Dutch primary school mathematics textbooks, as well as their predecessors, focusing on multi-digit number-based subtraction in Grades 2 and 3. First, all lessons in which the teacher manuals provide directions for whole-class instruction or discussion addressing this topic were identified. Next, in these whole-class moments, we coded which strategies are offered (learning opportunities for flexibility) and whether there are multiple strategies addressed (learning opportunities for adaptivity). Findings show that the learning opportunities for flexible and adaptive strategy use are limited in all textbooks, and even more limited in the current textbooks than in their predecessors. There is ample attention to one universal strategy, but other strategies are offered infrequently. Opportunities to compare and choose between different subtraction strategies are very rare.

ABSTRACT In this study, we investigated how Japanese curriculum materials represent functional relationships through the lenses of quantitative and covariational reasoning. Utilizing macro and micro analyses, we analyzed Japanese elementary and lower secondary-level curriculum materials, including course of study, teachers’ guide, and Mathematics International textbook series. We analyzed a set of statements and explanations in the course of study and the teachers’ guide. We examined tasks, questions, and representations in the textbooks. Findings showed that Japanese curriculum materials attentively focus on learners’ quantitative and covariational reasoning starting from 4th grade by the end of 8th grade. Particularly, Japanese curriculum materials have a spiral nature that supports learners’ covariational reasoning gradually building up to continuous covariation in learning functional relationships involved in proportional and linear situations. We discuss the possible implications of this study for teachers, teacher educators, curriculum developers, and textbook writers.

ABSTRACT Although the importance of developing students’ quantitatively-grounded meanings for integrals is widely acknowledged, there is limited research into how this can be accomplished in classroom settings. To address this limitation, I conducted a classroom design study in which I analyzed the collective development of classroom mathematical practices for establishing quantitatively-grounded meanings for integrals by a Year 12 calculus class. The findings suggest that the class established ways of approximating the value of definite integrals and integrals with a variable upper bound by progressing through the six layers of the Adding Up Pieces framework in ways that were largely similar to previous findings from studies conducted in clinical settings. The findings from this study also extend previous research by documenting the reasoning that students used to partition the domain and calculate target quantities using the trapezoidal rule, as well as the difficulty they encountered in developing the accumulation function.

ABSTRACT Thoughtful and purposeful displays can support students’ participation in argumentation. This report addresses how displays are used within collective argumentation. We examined a secondary mathematics teacher’s and her students’ use of displays during selected episodes of collective argumentation. From video classroom observations and interviews with the teacher, we extracted several kinds of displays according to their functions and then investigated how these displays supported collective argumentation as direct contributions to the argument or actions supporting the argumentation. Furthermore, we also considered actions by the teacher that supported her students to display. We found displays can be both part of the mathematical argument and supportive of the mathematical argumentation. This finding has direct implications for studying mathematical discussions because the literature has primarily focused on verbal contributions to the argument over nonverbal contributions such as displays.