Choy, Ban Heng

Orchestrating productive mathematics discussions by building on students’ ideas is challenging. Although certain talk moves involving eliciting student responses are associated with this high-leverage practice, they may not be sufficient for enhancing student reasoning. Telling, on the other hand, may play an important role despite the perception they are contradictory to a more interactive stance in teaching. In this paper, we examined how an elementary school teacher orchestrated a productive whole-class discussion through the skillful interweaving of talk moves and telling.

The urgency of teaching diverse learners is aptly demonstrated in many parts of the world as the ethnic, racial, class, and linguistic diversity grows rapidly. Such diversity not only brings about opportunities for creative teaching, but also challenges for ensuring educational equity and providing high-quality teaching for all students from diverse backgrounds, especially those presently underserved by the educational system (Buehl, & Beck, 2014; Civitillo, Juang, & Schachner, 2018). Researchers have found that teachers prepared for working with students from diverse cultural backgrounds need to embrace beliefs that recognize the strengths of cultural diversity (Anagnostopoulos, 2006; Banks et al., 2005; Fives & Buehl, 2014; Gay, 2010). Thus, exploring and challenging teachers’ beliefs about cultural diversity should constitute a major objective in teacher professional learning. However, only a few studies have examined how in-service teachers’ beliefs are enacted and shaped in professional learning community practices (Little, 2003; Tam, 2015; Turner, 2011), and focused even less on teachers’ beliefs about cultural diversity (Pang, 2005; Sleeter, 1992). There are a few studies examining teachers’ cultural beliefs about diversity in Singapore, and found that Singaporean teachers are influenced by prevailing political ideologies, and have ambiguous perceptions towards students from less advantaged backgrounds (Anderson, 2015; Alviar-Martin & Ho, 2011; Dixon & Liang, 2009; Ho & Alviar-Martin, 2010; Ho et al., 2014; Lim & Tan, 2018). However, these studies discussed teachers’ individual perceptions of disadvantaged learners without further exploring how these perceptions are mediated by influences from professional development practices, where teachers’ cultural beliefs about diversity issues are in (inter)action as ideas emerge, clash, change, and (dis)agree with each other when teachers work together.

Mathematics education research has focused on developing teachers’ knowledge or other visible aspects of the teaching practice. This paper contributes to conversations around making a teacher’s thinking visible and enhancing a teacher’s pedagogical reasoning by exploring the use of pedagogical documentation. In this paper, we describe how a teacher’s pedagogical reasoning was made visible and highlight aspects of his thinking in relation to his instructional decisions during a series of lessons on division. Implications for professional learning are discussed.

Learning must change from an emphasis on knowledge acquisition to knowledge construction in order to address the possible challenges of the fast-changing world. (Perkins, 1992)

“. . . our past and current achievements are no assurance of future successes in a rapidly changing world.” (Review of the Junior College/Upper Secondary Curriculum, MOE, 2002)

It was against this backdrop of call to change, that the Singapore Ministry of Education put in place a series of educational initiatives to provide for a more diversified educational landscape that would better prepare our students for the future. Among them, there was also a call for schools to move towards a curriculum that would place more emphasis on knowledge construction and other skills necessary for the future.

Driven by the ‘Teach Less Learn More’ movement in Singapore, many schools had begun their journey of curriculum redesign. Several curriculum frameworks were considered and one of them was the Teaching for Understanding (TfU) framework. In the case of Mathematics, teaching for understanding has generated much interest among Mathematics teachers (Skemp, 1978; Greeno, 1984; Sierpinska, 1994; Meel, 2003; Watson, 2005).

In a study by Hammerness et al. (1998) on TfU implementation across four subjects, the students in the Mathematics class performed the worst despite the teacher’s longer involvement with the framework. In 2006, the researcher’s school redesigned the entire school’s curriculum using the TfU framework and this motivated the researcher to investigate the impact of TfU implementation on students’ understanding of the Cartesian Connection—the connection that the coordinates of any point on a line will satisfy the equation of the line (Knuth, 2000a).

The researcher’s key research problem statement would be

What can we say about high-ability students’ understanding of Cartesian Connection when they are taught using the TfU framework?

In particular, the study will aim to answer the following questions using the elements from the TfU framework:

1. What do high-ability students understand about the Cartesian Connection when they are taught using the TfU framework?

2. What is the level of understanding of the Cartesian Connection attained by these high-ability students?

3. Were all four dimensions of understanding as proposed by the TfU framework equally demonstrated by these students?

4. What are the factors affecting TfU implementation in Mathematics?

It is hoped that this study will offer a better idea of how the TfU framework can be used or adapted to teach for understanding in Mathematics.

This study took place in a Integrated Programme school in Singapore, which takes in the top 10 % of the national cohort from the Primary School Leaving Examination (PSLE) each year and has the flexibility to design the curriculum using TfU. The school’s student population is therefore relatively homogeneous and are of the higher ability group. They are self-motivated and relatively strong in their academic studies.

The 25 students involved in this study took a modified version of Knuth’s (2000a, 2000b) test on the Cartesian Connection. Besides scoring the test solutions, the researcher used the TfU framework which comprised of four dimensions of understanding: Knowledge, Methods, Purposes and Forms with their corresponding four levels of undertanding: Naive, Novice, Apprentice and Master, to examine their answers to the test items. The use of the TfU framework allowed the researcher to describe qualitatively and quantitatively students’ understanding beyond what was given by Knuth (2000a, 2000b).

In addition, the TfU framework was also used to examine the unit plan, the lesson plans and worksheets used in the course of this study. Interviews were conducted with 13 out of the 25 students who volunteered and their responses were also analysed using the framework. In addition, eight representative teachers who taught the students in this study were interviewed and their responses were scanned to surface any possible factors that could impact on TfU implementation.

The 25 students in this study did better than students in the study by Knuth’s (2000a, 2000b) and seemed to show a better understanding of the Cartesian Connection. Moreover, the TfU framework allowed us to determine the levels of understanding in each of the dimensions. It was found that the Knowledge dimension was the best developed and the Methods dimension needed more attention. The framework provided a language to describe clearly what the students understand and provided a means for us to help students improve their understanding.

A chi-square analysis of the students’ understanding in this study using the same methods by Hammerness et al. (1998) revealed an interesting finding. In the study by Hammerness et al. (1998), all the dimensions were found to be significantly related when results from all four subjects were pooled together. In this study that involved only Mathematics, it was found that only Knowledge and Forms and Methods and Forms were associated significantly. This would have implications for the Mathematics teachers if the associations were indeed true.

These findings were analysed and explained by looking at data collected from the students’ interviews, teachers’ interview, analysis of unit plans and worksheets. Through these data, the researcher was able to provide a richer description and deeper insight into students’ understanding of the Cartesian Connection and highlighted factors that could impact on future TfU implementation in Mathematics.

Despite recent calls to adopt practice-embedded approaches to teacher professional learning, how teachers learn from their practice is not clear. What really matters is not the type of professional learning activities, but how teachers engage with them. In this paper, we position learning from teaching as a dialogic process involving teachers’ pedagogical reasoning and actions. In particular, we present a case of an experienced teacher, Mr. Robert, who was part of a primary school’s mathematics professional learning team (PLT) to describe how he learned to teach differently, and how he taught differently to learn for a series of lessons on division. The findings reiterate the complexity of teacher learning and suggest possible implications for mathematics teacher professional development.

Teacher-designed notes and worksheets are common instructional materials used in Singapore mathematics classrooms that are critical to guiding the flow of a lesson. However, making sense of how teachers design these materials is complex and research that reports on their creation is only just emerging. In this paper, we propose a tri-lens approach for capturing teachers’ design processes by using notions of pedagogical reasoning and action, curricular noticing, and resources, orientations and goals. We demonstrate how these frameworks combine to form a tri-lens for unpacking important aspects of teachers’ design work using the example of Mrs Fung, an experienced secondary mathematics teacher. We further illustrate how a tri-lens approach can provide a more comprehensive portrait of the teacher and argue that this approach can potentially address the complexity of teachers’ design processes when crafting instructional materials.

Much of a teacher’s practice and professional learning remains unseen despite recent calls to incorporate practice-based and inquiry-based approaches to improve mathematics instruction. Although the idea of pedagogical reasoning and action can provide a way to unpack these unseen aspects of practice, it remains to be seen how a teacher’s actions and thinking can be made visible. In this paper, we present a case of how a teacher’s pedagogical reasoning is made visible through pedagogical documentation, which suggests the possibility of using documentation to unpack these unseen aspects of a teacher’s practices.

It is challenging to design and structure lessons to maximize high-quality opportunities to learn mathematics in the classrooms. This paper presents a case study of Mary, a beginning mathematics teacher in Singapore, to illustrate how she noticed opportunities to learn during the planning and enacting of a lesson on decimal fractions for Primary 4 students. The case highlights the importance of noticing affordances of typical problems and opportunities to orchestrate productive discussions to provide quality opportunities to learn.

When mathematics teachers plan lessons, they interact with curriculum materials in various ways. In this paper, we draw on Brown’s (2009) Design Capacity for Enactment framework to explore the practice of adapting curriculum materials in the case of a Singapore secondary mathematics teacher. Problems from the textbook used and the worksheets she crafted were compared to determine how she adapted the content. Video-recordings of the lessons and post-lesson interviews were used to clarify how her personal teacher resources contributed to her design decisions. The findings suggest that her seemingly casual use of problems from the textbook are in fact unique variations of adapting curriculum materials.