Educational Studies in Mathematics最新文献

Abstract

Previous studies on Bayesian situations, in which probabilistic information is used to update the probability of a hypothesis, have often focused on the calculation of a posterior probability. We argue that for an in-depth understanding of Bayesian situations, it is (apart from mere calculation) also necessary to be able to evaluate the effect of changes of parameters in the Bayesian situation and the consequences, e.g., for the posterior probability. Thus, by understanding Bayes’ formula as a function, the concept of covariation is introduced as an extension of conventional Bayesian reasoning, and covariational reasoning in Bayesian situations is studied. Prospective teachers (N=173) for primary (N=112) and secondary (N=61) school from two German universities participated in the study and reasoned about covariation in Bayesian situations. In a mixed-methods approach, firstly, the elaborateness of prospective teachers’ covariational reasoning is assessed by analysing the arguments qualitatively, using an adaption of the Structure of Observed Learning Outcome (SOLO) taxonomy. Secondly, the influence of possibly supportive variables on covariational reasoning is analysed quantitatively by checking whether (i) the changed parameter in the Bayesian situation (false-positive rate, true-positive rate or base rate), (ii) the visualisation depicting the Bayesian situation (double-tree vs. unit square) or (iii) the calculation (correct or incorrect) influences the SOLO level. The results show that among these three variables, only the changed parameter seems to influence the covariational reasoning. Implications are discussed.

For decades, mastery ambitions related to processes like problem-solving, modelling, and reasoning have been incorporated in mathematics curricula around the world. Meanwhile, such ambitions are hindered by syllabusism, a term I use to denote a conviction that results in mastery of a subject being equated with proficiency in a specific subject matter and making that equation the fulcrum of educational processes from teaching to curriculum development. In this article, I argue that using an open two-dimensional structure for curricular content that comprises a set of subject-specific competencies and a modest range of subject matter can help fight syllabusism. I explore and motivate the concept of syllabusism, using the development of a width-depth model of possible curricular ambitions within a given period of time to visualise the detrimental consequences for the attained depth of student learning. In the final part of the article, I illustrate the use of the width-depth model by analysing a specific mathematics curriculum. This analysis leads to two conclusions. Firstly, by highlighting mastery ambitions at the structural level, an open two-dimensional content structure is a powerful means to fight syllabusism. Secondly, using such an approach requires the explicit expression of these mastery ambitions and their conceptualisation independent of the subject matter. In the case of mathematics education, this has taken the form of a set of mathematical competencies.

Black women teachers have a legacy rooted in resisting and disrupting racism and racialization in schools. Yet, stories of Black women teachers enacting their liberatory pedagogy in mathematics go untold. This study centers Black women mathematics teachers’ liberatory stances towards teaching mathematics to Black, Latinx, and Southeast Asian students. I use a Black feminist lens to conduct a critical narrative study of five Black women elementary teachers that explores how their racialized mathematics experiences informed their liberatory stances of personal accountability, caring, and being a role model for students in their mathematics classrooms. These liberatory stances resisted normalizing whiteness and anti-Blackness in mathematics classrooms within teachers’ schools. Implications include learning about Black women mathematics teachers’ liberatory stances in different racialized social systems as a starting place to transform mathematics education for liberation.

This article examines how making mathematics responsive to perceived differences in students’ real-life needs historically produced racializing distinctions in school mathematics. Decentering social actors and their intentions, we analyze pedagogical techniques in social mathematics courses (1930s–1940s) and social justice mathematics education studies oriented to health and civic participation (1990s–). Despite shifts in ethico-political principles—from enlightening to empowering—racializing effects of these pedagogies persist by projecting relative distances between populations and cultural norms of proper living and well-ordered public life. Juxtaposing past and present, we highlight dangers in how pedagogical interventions to improve malleable differences in children and communities also racialize target groups as yet-to-develop the evidence-based reasoning or mathematical consciousness deemed necessary to attain what are imputed as healthy private and public life. We offer ordering pedagogies as an analytical tool to interrogate practices of racialization and to account for inherited regimes of truth operating in mathematics education by scrutinizing how pedagogical practices produce difference—dividing and ordering students along a hierarchy of perceived needs.

This paper adopts a biopower lens to examine emergency declarations that posit race or racism as problems to be addressed through mathematics education. We argue that attending to “slow emergencies” of racism must avoid sustaining mathematics education as a self-evident cause and cure for societal problems. We analyze how declarations of emergency reanimate racializing hierarchies by reordering spaces, temporalities, and subjectivities. To explore these concerns, we compare three race-explicit emergency declarations in US mathematics education during World War II with recent emergency declarations of pandemic-related learning loss, disengagement, and a racial reckoning. We juxtapose past and present to spotlight what we outline as distinct biopolitical working arrangements. The analysis maps how emergency declarations rearrange hopes, fears, diagnostic techniques, and intervention strategies—sometimes inadvertently reracializing students in attesting to damage or demanding redress. Our purpose is to foster deliberation over paradoxes and possibilities of addressing racialization and racism in mathematics education transnationally.

Distributing opportunities to participate in talk-in-interaction during whole-class mathematics discussions is an important equity issue, with multiple studies reporting pervasive inequitable participation patterns in mathematics classrooms. Less attention, however, has been given to the underlying interactional practices that can initiate and support minoritized students’ participation. This article examines how and what kinds of opportunities for participation are interactionally generated for minoritized students in whole-class mathematics discussions in two US high school classrooms. Through the lenses of turn-taking organizations and epistemic dimensions from conversation analysis, this study details the interactional features of turn-taking by three Black students in predominantly White classrooms. The analysis shows the importance of establishing epistemic congruence about the nature of students’ knowledge before inviting them to take up the conversational floor. The findings imply that locally achieved, mutual understanding of what minoritized students know in the moment-by-moment classroom interaction is an important interactional feature for making minoritized students’ brilliance more visible during whole-class discussions.

Abstract

The decomposition of numbers when solving subtraction tasks is regarded as more powerful than counting-based strategies. Still, many students fail to solve subtraction tasks despite using decomposition. To shed light upon this issue, we take a variation theoretical perspective (Marton, 2015) seeing learning as a function of discerning critical aspects and their relations of the object of learning. In this paper, we focus on what number relations students see in a three-digit subtraction task, and how they see them. We analyzed interview data from 55 second-grade students who used decomposition strategies to solve 204 − 193 = . The variation theory of learning was used to analyze what number relations the students experienced and how they experienced them, aiming to explain why they made errors even though they used presumably powerful strategies in their problem-solving. The findings show that students who simultaneously experienced within-number relations and between-number relations when solving the task succeeded in solving it, whereas those who did not do this failed. These findings have importance for understanding what students need to discern in order to be able to solve subtraction tasks in a proficient way.

There is growing evidence that the ability to perceive structure is essential for students’ mathematical development. Looking at students’ structure sense in basic numerical and patterning tasks seems promising for understanding how these tasks set the foundation for the development of later mathematical skills. Previous studies have shown how students use structure sense in enumeration tasks. However, little is known about students’ use of structure sense in other early mathematical tasks. The main aim of this study is to investigate the ways in which structure sense is manifested in first-grade students’ work across tasks, in quantity comparison and repeating pattern extension tasks. We investigated students’ strategies in quantity comparison and pattern extension tasks and how students employ structure sense. We conducted an eye-tracking study with 21 first-grade students, which provided novel insights into commonalities among strategies for these types of tasks. We found that for both tasks, quantity comparison and repeating pattern extension tasks, strategies can be distinguished into those employing structure sense and serial strategies.

This paper describes how the notion of the strongly didactic contract can serve to characterize the teaching adopted to implement a task in probability. It is particularly focused on the reality of mathematical work performed by students and teachers. For this research, classroom sessions were developed in an in-service teacher training course designed (and adapted) according to the Japanese Lesson Study model. Through the combined use of the Theory of Didactical Situations (TDS) and the Theory of Mathematical Working Spaces (ThMWS), a coding of the sessions observed was developed. Based on this coding, different patterns emerged which gave each session a specific rhythm and identity from which it was possible to recognize and characterize different strongly didactic contracts. The study highlights the difference between the potential contracts intended by the teachers and those observed in practice. The tools, and especially the coding, developed for the study could be used for future research on instructional situations or in-service teacher training.