Exploring the Process of Preservice Teachers’ Diagnostic Activities in a Video-Based Simulation

Formative assessment of student learning is a challenging task in the teaching profession. Both teachers’ professional vision and their pedagogical content knowledge of specific subjects such as mathematics play an important role in assessment processes. This study investigated mathematics preservice teachers’ diagnostic activities during a formative assessment task in a video-based simulation. It examined which mathematical content was important for the successful assessment of the simulated students’ mathematical argumentation skills. Beyond that, the preservice teachers’ use of different diagnostic activities was assessed and used as an indicator of their knowledge-based reasoning during the assessment situation. The results showed that during the assessment, the mathematical content focused on varied according to the level of the simulated students’ mathematical argumentation skills. In addition, explaining what had been noticed was found to be the most difficult activity for the participants. The results suggest that the examined diagnostic activities are helpful in detecting potential challenges in the assessment process of preservice teachers that need to be further addressed in teacher education. In addition, the findings illustrate that a video-based simulation may have the potential to train specific diagnostic activities by means of additional instructional support.

Mathematical Argumentation Performance of Sixth-Graders in a Chinese Rural Class

Researchers have established that solid argumentation is essential for developing, establishing and communicating mathematical knowledge, which attracted substantial attention from researchers, but few have simultaneously investigated the argumentation performance of sixth-graders and their teacher’s potential influence in Chinese rural classrooms. In this pilot study, 33 sixth graders in a Chinese rural class were examined, and the math teacher who had been teaching them for three years was interviewed. Findings related to the students’ performance revealed the need to improve their argumentation competency, including using more diverse modes of arguments and argument representation as well as developing more advanced types of arguments (e.g., deductive argumentation). The interview finding with the math teacher indicated that the teacher’s perception and knowledge might impact students’ learning opportunities to conduct argumentation and, therefore, may influence students’ argumentative performance. Implications and limitations of this study is discussed at the end.

How Ms. Mayen and Her Students Co-Constructed Good-at-Math

What does it mean to be “good-at-math,” and how is it determined? Cobb et al. (2009) defined the normative identity of mathematics classrooms as the obligations that students must meet to be considered good-at-math. Obligations are negotiated between teachers and students through series of bids. Normative identities reveal distributions of agency and authority within classrooms, which affect learning opportunities for students. Traditionally, mathematics teachers held the predominance of agency and authority in classrooms. Research supports shifting toward more equitable teaching and learning (e.g., National Council of Teachers of Mathematics, 2018). Clear examples of enacting and supporting changes are helpful. This article shares how sixth-grade students and their teacher co-constructed good-at-math to invite and obligate students to become active agents in mathematical argumentation.

Supporting Mathematical Argumentation and Proof Skills: Comparing the Effectiveness of a Sequential and a Concurrent Instructional Approach to Support Resource-Based Cognitive Skills

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as ‘resource-based,’ as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students’ work on proof construction tasks: (i) a sequential approach focusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) a concurrent approach focusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students.

PROFILE OF STUDENTS' JUSTIFICATIONS OF MATHEMATICAL ARGUMENTATION

This study investigates the aspects that influence students' justification of the four types of arguments constructed by students, namely: inductive, algebraic, visual, and perceptual. A grounded theory type qualitative approach was chosen to investigate the emergence of the four types of arguments and how the characteristics of students from each type justify the arguments constructed. Four people from 75 students were involved in the interview after previously getting a test of mathematical argumentation. The results of the study found that three factors influenced students' justification for mathematical arguments, namely: students' understanding of claims, treatment given, and facts found in arguments. Claims influence the way students construct arguments, but facts in arguments are the primary consideration for students in choosing convincing arguments compared to representations. Also, factor treatment turns out to change students' decisions in choosing arguments, and these changes tend to lead to more formal arguments.

What counts as a “good” argument in school?—how teachers grade students’ mathematical arguments

As argumentation is an activity at the heart of mathematics, (not only) German school curricula request students to construct mathematical arguments, which get evaluated by teachers. However, it remains unclear which criteria teachers use to decide on a specific grade in a summative assessment setting. In this paper, we draw on two sources for these criteria: First, we present theoretically derived dimensions along which arguments can be assessed. Second, a qualitative interview study with 16 teachers from German secondary schools provides insights in their criteria developed in practice. Based on the detailed presentation of the case of one teacher, the paper then illustrates how criteria developed in practice take a variety of different aspects into account and also correspond with the theoretically identified dimensions. The findings are discussed in terms of implications for the teaching and learning about mathematical argumentation in school and university: An emphasis on more pedagogical criteria in high school offers one explanation to the perceived gap between school and university level mathematics.

Argument Mapping to Improve Student's Mathematical Argumentation Skills

This study aims at investigating the difference in students' mathematical argumentation skills before and after the implementation of argument mapping in learning mathematics. It is a quasiexperiment with a quantitative approach. The population was the students of class X Natural Sciences Program in a state senior high school in Pasuruan, East Java, Indonesia. 36 students were involved. The instrument was a mathematical argumentation skills test. Several components were established, adopted from the Revised Bloom's Taxonomy, namely identifying (C1), explaining (C2), drawing conclusions (C3), reducing/adding premises (C4), deducing (C5), and developing/constructing (C6). Students' mathematical argumentation skills were analyzed using the Wilcoxon signed-rank test at 5% level of significance (∝ = 0.05). The findings of this study indicate that students' argumentation skills after the implementation of argument mapping is better than prior treatment (p = 0.002). It can be claimed that argument mapping is effective for improving such skills.