The Detailed Analysis of an Established Teacher's Non-Traditional Lesson

This article presents the fine-grained analysis of an experienced teacher conducting a highly interactive, non-standard lesson of his own design. The analysis, often carried out on a line-by-line level, seeks to explain how and why the teacher made the decisions he did while interacting with his students. The analysis indicates that much of the lesson, in which the teacher is truly responsive to the ideas generated by the students, can be modeled closely using a small number of contingency-based teaching strategies. Even in a case where a student makes a rather unusual comment, the model—which includes the detailed characterization of the teacher's knowledge, goals, and beliefs—is capable of predicting with some precision the nature of the teacher's response.     

Student Difficulties with Accessing and Using Mathematical Knowledge

This article examines the issue of why students fail to activate and use mathematical knowledge during problem solving when it is known that they possess the required knowledge. This issue is explored by analyzing problem-solving attempts of a high-achieving student and a low-achieving student in the domain of plane geometry. On the basis of these data and other literature, three major sources of mathematical knowledge-access difficulties are identified that might be considered by classroom teachers, including a student's (1) dispositional state, (2) management of the problem-solving process, and (3) state of organization of his or her mathematical knowledge. It is argued that teaching practices that place emphasis on careful management of problem-solving activity could help students activate and extend the use of mathematical knowledge acquired in lesson activities.     

Teaching and Learning Number Sense: One Successful Process‐Oriented Activity With Sixth Grade Students in Taiwan

The aim of this article was to (a) describe how a teacher helped his students develop fractional number sense through a process‐oriented activity, and (b) illustrate how a teacher included a worthwhile, interesting, and challenging mathematics question in his class to create a good learning environment for children. The results indicate that students' fractional number sense can be promoted through a process‐oriented activity, effective teaching, and good learning environment. It also illustrates how teachers can help children promote their understanding of number sense from a pictorial representation to a symbolic representation.     

Teaching and Learning Number Sense: One Successful Process‐Oriented Activity With Sixth Grade Students in Taiwan

The aim of this article was to (a) describe how a teacher helped his students develop fractional number sense through a process‐oriented activity, and (b) illustrate how a teacher included a worthwhile, interesting, and challenging mathematics question in his class to create a good learning environment for children. The results indicate that students' fractional number sense can be promoted through a process‐oriented activity, effective teaching, and good learning environment. It also illustrates how teachers can help children promote their understanding of number sense from a pictorial representation to a symbolic representation.     

Bridging the gap between mathematical conjecture and proof through computer-supported cognitive conflicts

In many mathematical problems, students can feel that the universalityof a conjecture or a formula is validated by their experimentand experience. In contrast, students generally do not feelthat deductive explanations strengthen their conviction thata conjecture or a formula is true. In order to cope up withstudents’ conviction based only on empirical experienceand to create a need for deductive explanations, we developeda problem-solving activity with technology support intendedto cause cognitive conflict. In this article, we describe theprocess conducted for this activity that led students to contradictionsbetween conjectures and findings. The teacher could create familiarproblem-solving situations and use students’ naïveinductive approaches to make students think mathematically andestablish the necessity for proof via computer support.     

Knowing students as mathematics learners and teaching numbers 10–100: A case study of four 1st grade teachers from Romania

Researchers have increasingly linked teacher effectiveness with teacher knowledge of subject matter, curriculum, and teaching. Moreover, teacher knowledge of students has been regarded as another very significant component of teacher knowledge, influencing the classroom practice and student performance. Knowing students as mathematics learners means being aware of the ways students learn certain topics. This study examined the knowledge of students as mathematics learners displayed by four 1st grade teachers from Romania when designing and implementing a lesson on numbers 10–100. Findings show that knowledge of students as mathematics learners influenced the ways teachers planned and implemented their lesson. Teachers learned about students as mathematics learners from one series to another, and they tailored their use of manipulatives and classroom activities to meet the needs of their current students.     

Teachers' Considerations of Students' Thinking During Mathematics Lesson Design

Teachers' abilities to design mathematics lessons are related to their capability to mobilize resources to meeting intended learning goals based on their noticing. In this process, knowing how teachers consider Students' thinking is important for understanding how they are making decisions to promote student learning. While teaching, what teachers notice influences their decision‐making process. This article explores the mathematics lesson planning practices of four 4th‐grade teachers at the same school to understand how their consideration of Students' learning influences planning decisions. Case study methodology was used to gain an in‐depth perspective of the mathematics planning practices of the teachers. Results indicate the teachers took varying approaches in how they considered students. One teacher adapted instruction based on Students' conceptual understanding, two teachers aimed at producing skill‐efficient students, and the final teacher regulated learning with a strict adherence to daily lessons in curriculum materials, with little emphasis on student understanding. These findings highlight the importance of providing professional development support to teachers focused on their noticing and considerations of Students' mathematical understandings as related to learning outcomes. These findings are distinguished from other studies because of the focus on how teachers consider Students' thinking during lesson planning. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.     

A geometry teacher's use of a metaphor in relation to a prototypical image to help students remember a set of theorems

This article asks the following: How does a teacher use a metaphor in relation to a prototypical image to help students remember a set of theorems? This question is analyzed through the case of a geometry teacher. The analysis uses Duval's work on the apprehension of diagrams to investigate how the teacher used a metaphor to remind students about the heuristics involved when applying a set of theorems during a problem-based lesson. The findings show that the teacher used the metaphor to help students recall the apprehensions of diagrams when applying several theorems. The metaphor was instrumental for mediating students’ work on a problem and the proof of a new theorem. The findings suggest that teachers’ use of metaphors in relation to prototypical images may facilitate how they organize students’ knowledge for later retrieval.     

Capturing student mathematical engagement through differently enacted classroom practices: applying a modification of Watson's analytical tool

This study examined student mathematical engagement through the intended and enacted lessons taught by two teachers in two different middle schools in Indonesia. The intended lesson was developed using the ELPSA learning design to promote mathematical engagement. Based on the premise that students will react to the mathematical tasks in the forms of words and actions, the analysis focused on identifying the types of mathematical engagement promoted through the intended lesson and performed by students during the lesson. Using modified Watson's analytical tool (2007), students’ engagement was captured from what the participants’ did or said mathematically. We found that teachers’ enacted practices had an influence on student mathematical engagement. The teacher who demonstrated content in explicit ways tended to limit the richness of the engagement; whereas the teacher who presented activities in an open-ended manner fostered engagement.     

Teacher Questioning to Promote Justification and Generalization in Mathematics: What Research Practice Has Taught Us

This article explores the teacher's role in classroom environments that center on learning through student exploration, and reinvention, of important mathematics. In such environments, teachers will often work to create situations where students are invited to express their thinking, most especially to peers. How is this done? In the work reported here, both teacher questioning and teacher listening will play important parts, as does the teacher's background understanding of the mathematics and the children. This study focuses especially on teacher questioning in third- and fourth-grade classrooms associated with a longitudinal study now in its eleventh year. Analyses of videotaped data indicate a strong relationship between (1) careful monitoring of students' constructions leading to a problem solution, and (2) the posing of a timely question which can challenge learners to advance their understanding. What a teacher needs to know in order to work well with student explorations has important implications.     

Distinguishing Features of Mechanical and Human Problem-Solving

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts, only one group were problem-solving experts. Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.     

The Growing Awareness Inventory: Building Capacity for Culturally Responsive Science and Mathematics With a Structured Observation Protocol

This study represents a first iteration in the design process of the Growing Awareness Inventory (GAIn), a structured observation protocol for building the awareness of preservice teachers (PSTs) for resources in mathematics and science classrooms that can be used for culturally responsive pedagogy (CRP). The GAIn is designed to develop awareness of: how students use language in classrooms; relationships between teacher questioning patterns and student participation; messages conveyed by the classroom environment; and ways to incorporate students’ interests into lesson plans. The methodology took the form of a multiple case study design with fourteen mathematics PSTs as one case and five science PSTs as the other case. The participants' response to the GAIn and lesson plans served as data sources. Findings reveal that the GAIn scaffolded PSTs’ awareness of their students, their own attitudes, and several elements of CRP. However, there were key areas of CRP that were neither explored with the GAIn nor identified by the participants. Consistent with design‐based research, outcomes include a design framework for revision of the GAIn and a theory of action that situates it within a teacher education course that includes a field placement.     

Encouraging problem-solving disposition in a Singapore classroom

In this article, we share our learning experience as a Lesson Study team. The Research Lesson was on Figural Patterns taught in Year 7. In addition to helping students learn the skills of the topic, we wanted them to develop a problem-solving disposition. The management of these two objectives was a challenge to us. From the lesson observation and the students’ classwork, it turned out better than we expected.     

The Road To Digitopolis: Perils of Problem Solving

How students solve problems is a topic of central concern both to educational researchers and to math/science teachers: What is the nature of good and poor problem solving? How can students improve their problem-solving capacities? Teachers are in a unique position to witness problem solving in action, and to draw connections between the classroom experiences of their students and the findings of research. This article presents an instance of problem solving (drawn from a popular children's book) annotated with references to current research in cognition and education. The annotations explore issues such as the effect of performance anxiety on problem solving, how problem solvers handle the experience of confusion, and the role of self-monitoring and metacognition in problem solving.     

A teacher’s learning process in dual design research: learning to scaffold language in a multilingual mathematics classroom

In this paper, we argue that dual design research (DDR) is a fruitful way to promote and trace the development of a mathematics teacher’s expertise. We address the question of how a teacher participating in dual design research can learn to scaffold students’ development of the language required for mathematical learning in multilingual classrooms. Empirical data were collected from two teaching experiments (each with 8 lessons, and 21 and 22 students, aged 11–12 years), for which lesson series about line graphs were co-designed by the researchers and the teacher. The teacher’s learning process was promoted (e.g. by conducting stimulated recall interviews and providing feedback) and traced (e.g. by carrying out 5 pre- and post-interviews before and after the teaching experiments). An analytic framework for teachers’ reported and derived learning outcomes was used to analyse pre- and post-interviews. The teacher’s learning process was analysed in terms of changes in knowledge and beliefs, changes in practice and intentions for practice. Further analysis showed that this learning process could be attributed to the characteristics of dual design research, for instance the cyclic and interventionist character, the continuous process of prediction and reflection that lies at its heart, and the process of co-designing complemented with stimulated recall interviews.     

The definition of the scalar product: an analysis and critique of a classroom episode

In this paper, we present, analyse and critique an episode from a secondary school lesson involving an introduction to the definition of the scalar product. Although the teacher attempted to be explicit about the difference between a definition and a theorem, emphasizing that a definition was just an arbitrary assumption, a student rejected the teacher's definition in favour of his own alternative. With reference to this particular case, we seek to explore some ways in which teachers can introduce mathematical definitions to students so as to support, rather than attempt to circumvent, their mathematical sense making. In this regard, we believe that it is important to develop learning opportunities for students which help them to gain some appreciation of important structural and historical reasons that underpin the definitional choices made.     

Capitalizing on Emerging Technologies: A Case Study of Classroom Blogging

The challenge many teachers face is how to incorporate new technology into their classrooms that strengthens classroom learning by capitalizing on students’ media literacies. Blogs, a new and innovative technological tool, can be used in math and science classrooms to support student learning by capitalizing on students’ interests and familiarity with on‐line communication. This study explores the emerging blogging practices of one high school mathematics teacher and his class to explore issues of intent, use, and perceived value. Data sources for this case included one year's worth of blog content, an interview with the facilitating teacher, and students ‘perceptions of classroom blogging practices. Findings indicate that (1) teachers’ intentions focused on creating additional forms of participation as well as increasing student exposure time with content; (2) blogs were used in a wide variety of ways that likely afforded particular benefits; and (3) both teacher and students perceived the greater investment to be worthwhile. The findings are used to critically consider claims made in the literature about the potential of blogging to effectively support classroom learning.     

From Curve Stitching to Epicycloids

The exploration of curves that seem to appear in drawings made of straight lines is fascinating and interesting to learners from the very young to the quite mature. Consistent with the Curriculum and Evaluation Standards for School Mathematics, problem solving, reasoning, communication, and connections are integrated into mathematics learning activities that permit students to know mathematics by doing mathematics. Examples in this article show how students may create drawings of sets of lines whose envelopes are parabolas, cardioids, epicycloids, or hypocycloids.