Instructional Tasks in Mathematics Classes in Singapore

Let reasoning and communication be an integral part of your mathematics curriculum – and use students’ insights to modify instruction

Tasks in which students are expected to reason about mathematics and communicate their understanding should be an integral part of mathematics instruction, even at the elementary level. Students’ responses provide insights into their thinking that highlight the robustness of their understanding, which in turn can help teachers modify their instruction to enhance students’ learning. With relatively minor adjustments to the current curriculum, teachers can embed reasoning and communication into their instruction. In this session, we will explore samples of reasoning tasks across content areas (number and operations, algebraic thinking, geometry, measurement, and data analysis and probability) and consider how to adapt curriculum items to focus them on reasoning and communication.

Some “what” strategies to engage students in reasoning and communication in the primary mathematics classroom

This lecture focuses on specifically four ‘what’ strategies that primary school teachers may use to develop their students’ reasoning and communication. Though it is often assumed that learning of mathematics is virtually impossible without reasoning, of particular interest in this lecture are the types of mathematical tasks that may be used by teachers to explicitly provide contexts for students to work collaboratively with their peers and engage in reasoning. The four ‘what’ strategies explored in this session are: what number makes sense?, what’s wrong?, what if? and what’s the question if you know the answer? The lecture will also showcase a primary one mathematics lesson during which the teacher provided for her students to reason and communicate using the strategy: “What’s the question if you know the answer?”

Reasoning, Proof, and Justification – It’s Not Just for Geometry Anymore!

Reasoning and proof have long been a focus of the geometry curriculum in many parts of the world. But given the central role that reasoning and proof play in mathematics, it is essential that students have numerous opportunities to engage with reasoning and proof throughout the secondary curriculum. Students’ responses to such tasks provide insights into their thinking that highlight the robustness of their understanding, which in turn can help teachers modify their instruction to enhance students’ learning. By being aware of limitations in opportunities for reasoning in the curriculum, teachers can make relatively minor adjustments to embed reasoning and communication into their instruction and assignments on a regular basis. In this session, we will explore samples of reasoning tasks in areas other than geometry and consider how to adapt curriculum items to focus them more on reasoning and proof.

Making connections through promoting mathematical thinking skills

The Singapore Mathematics Curriculum emphasizes problem-solving in the sense of going beyond applying routines and algorithms to compute answers, whether in arithmetic, algebra, statistics or geometry. Yet, our students tend to be taught topics in isolation and normally solve problems by recognizing the topic and identifying the type of problem thereby attacking the problem with known and familiar strategies. Thus, students see mathematics topics in silos and are unable to understand the bigger picture of what they are learning. This talk suggests through examples that we can make connections within topics and across mathematical topics through explicitly identifying, teaching and promoting mathematical thinking skills. Such practices, taught in small doses but on a regular basis, will deepen our students’ understanding of the essence of mathematics.

Mathematics and the world around us - helping students see connections

If you ask mathematics teachers to describe mathematics, they will often use words like beautiful and relevant. However, if you ask mathematics students, you will often hear words like boring and irrelevant. I believe that this paradox is one of the central challenges in mathematics education. In my presentation I will demonstrate how I achieve the following goals in my university courses: i) Help the students appreciate the world around them and start looking at their surroundings with different eyes. I want them to notice and question things they used to take for granted. ii) Show them connections between mathematics and culture. I want them to stop thinking about knowledge in terms of school subjects, but as part of mankind’s struggle to understand the world. iii) Demonstrate the nature and importance of mathematics by showing how it solves problem of general interest. I want them to see the beauty and centrality of mathematics. Also, based on my experience as judge at the Singapore Mathematics Project Festival, I believe that project work in mathematics can be another way to achieve the goals above, so I will make some comments about how to propose suitable projects.

Mathematical reasoning and connections across mathematical concepts at A-levels

In A-Level mathematics, different sections of the curriculum are taught topically and the various mathematical concepts are taught in isolation. Studies have shown that greater connection across different concepts helps students to learn the subject better than in isolation. In this talk, the author presents different ways to link the various mathematical concepts, within the same topic and across different topics and across different levels. In particular, how A-Level concepts are built on O-Level mathematical concepts, which in turn form the basis for undergraduate mathematical concepts. The author will illustrate with examples from different topics in the A-Level curriculum, showing how these provide the bridge between the secondary and tertiary mathematical reasoning.

Visual and spatial reasoning: The changing form of mathematics representation and communication

Until recently, most mathematics tasks presented to primary-aged students were exclusively word-based problems. Current practices, from both curriculum and assessment perspectives, have moved toward more visual and graphic forms of representation (Lowrie & Diezmann, 2009). This is unsurprising given the increased use of graphics in society and the increasing challenge of representing burgeoning amounts of information in visual and graphic forms. From a young age students are exposed to visual forms of communication with more intensity and engagement, whether playing computer games or navigating web pages. This presentation highlights the increasingly important role visual and spatial reasoning plays in how mathematics is communicated. In particular, it considers the changing nature of mathematics representation in classroom practices, and an evolution in student engagement—where students are increasingly required to decode visual and spatial information. The presentation also considers the need for young students to employ encoding strategies which effectively encourage visual and spatial reasoning.

Project-based mathematics learning through LOGO programming activities

This lecture will introduce some results of the project-based mathematics learning through LOGO programming activities, which was done for Korean 6th grade students to improve mathematical reasoning and to activate communication with peer students and teachers in the various projects to connect mathematics and visual art. In this learning, students were found to promote reasoning skills such as inference, generalization, integration and critical thinking etc. And, students could activate communication with other students and teacher in analyzing, debugging, comparing and contrasting their programming. In this learning, teacher’s role was important in the sense that without collaboration of teacher as a facilitator it was not easy for students to reach the final goals. This project-based learning through LOGO programming activities was found to be a effective way to promote higher order thinking and to facilitate interaction between students and teacher on equal level.

Numeracy: Connecting mathematics

While numeracy is sometimes regarded as elementary arithmetic, it is more appropriate to think of it in terms of the “capacity, confidence and disposition to use mathematics to meet the demands of home, work, learning and community and civic life” (Willis, 1992). Viewed in these terms, the task of the mathematics curriculum is to help students learn appropriate mathematics and make suitable connections between their mathematics and other authentic activities, including especially learning in areas other than mathematics. In this lecture, we will explore the ways in which numeracy has been constructed recently in the Australian curriculum, and the roles of teachers of mathematics to support its implementation. While it may be an overstatement to suggest that every teacher is a teacher of numeracy, it is nonetheless important to recognize that developing numeracy involves more than developing mathematics and requires effort from both mathematics teachers and others.

Building communities of inquiry in mathematics classrooms: The key to facilitating reasoning, communication, and connections in mathematics

The key to students’ rich experiences in learning mathematics is in the way teachers try to have students’ reasoning and thinking mathematically meaningful and personally making sense. This lecture focuses on building communities of inquiry in mathematics classrooms for facilitating reasoning, communication and connections in mathematics. Rationale for trying to allow mathematics to be problematic for students are discussed, as well as the importance of focusing on the methods and processes used to solve problems and of telling the right things as the right times. Examples of video clips from a 8th grade mathematics classroom in Tokyo are presented to show how an experienced teacher facilitates reasoning and communication in the classroom for achieving their goal of teaching mathematics. Finally, some practical ideas in the classroom shared by Japanese teachers are presented.

What can we do without using similarity and congruency?

Triangles are the simplest type of polygons in Euclidean plane geometry. Any polygon can be decomposed into numbers of triangles. Even circles can be approximated by triangles. Thus, it is not surprising why the theory on triangles form the foundation of the whole plane geometry. The two most important and useful relations between two triangles are the “similarity” and “congruency”. The results, exercise problems and proof techniques involving similarity or congruency form the most exciting and rich part of plane geometry. Mathematics would be very different if these two relations were removed --- many basic facts and concept , such as areas and trigonometry functions, would not be justified. The part of geometry on similarity and congruency also provides a convenient platform for training reasoning among school students. In this lecture, we examine how some parts of mathematics reply on the two relations and the role of similarity and congruency in school reasoning training.

Connecting JC Math and University Math - A perspective from NUS Math Lecturer

I have been teaching first year university math modules for many years, and have made some observations of the performance of the freshmen in these courses. In this talk, I will share with the audience how the students, especially local students who have done the JC math are doing for their math courses in the university. In particular, I will highlight some of the common problems faced by these students and the skills they are lacking in problem solving.