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Using Multiple Representations in Mathematics to promote Deep Learning

Introduction

One of the Mathematics syllabus’ core objectives is to develop understanding and fluency in mathematics through inquiry, exploring and connecting mathematical concepts, choosing and applying problem-solving skills and mathematical techniques, communication and reasoning (Working Mathematically). The key part of this statement that is going to be explored in this journal is the idea that students are given opportunities to explore and connect mathematical concepts.

Deep Learning is about making connections. A powerful way for students to make connections is when they notice, explore and talk about the similarities and differences between mathematical representations. Five types of mathematical representations are shown and explained below. Consider which of these you tend to use when presenting concepts - would your students benefit from a range of representations? Will this help them to connect mathematical ideas?

Image: Representations Model - Clement, 2004

Five Mathematical Representations

Manipulatives, or physical representations, refer to the use of concrete materials when solving problems. Base ten blocks, unifix cubes, counters and fraction circles are just some of the materials that students should have access to during all Mathematics lessons. By using hands on materials, students are able to compare relative sizes, explore patterns and improve their estimation skills. This representation of mathematical concepts is not age specific and should be modelled and applied in classroom K-6.

Pictures, or visual representations, are sketches that represent mathematical ideas. When children draw pictures (spontaneously or at a teacher’s request), the teacher can learn more about what they understand about a particular mathematical idea and can use the different pictures that children create to provoke a discussion about mathematical ideas. By modelling pictures of mathematical concepts and inviting students to draw their own, teachers are provided with opportunities to prompt mathematical discussion by questioning and challenging ideas.

Written symbols, including number sentences, are the abstract forms of mathematics that we, as adults, most often use. A suggestion is to introduce the abstract after students have made connections with other representations. The written symbols must be comprehensible to the students. For example, they need an abstract understanding that 25% is a part of a whole and can be seen as one quarter of a circle.

Spoken language, or verbal representations, are incredibly important for teachers to use as they ‘think aloud’ their mathematical thought process and model making connections between concepts. This is the language we use to talk about the physical model of mathematics. For example, 5-3=2 might be represented verbally as “three less than five is 2” or “five take away three makes two”.

Contextual representations, or relevant situations, is my personal favourite and links nicely with our whole school focus on problem solving and the use of the QUEST problem solving process. By situating the mathematics in a scenario, students can explore mathematical concepts in the context of a “story”. A relevant situation can be any context that involves appropriate mathematical ideas and holds interest for children; it is often, but not necessarily, connected to a real-life situation (e.g. unicorns and fairies!)

Towards Deeper Learning

By incorporating these representations into Mathematics lessons, we are more likely to promote deep learning in students. In the deep phase of learning, teachers should select mathematical tasks and tools that allow students to notice, explore and talk about the progression from physical representations to symbolic representations.

Food for thought:

• Which representation am I more confident presenting to students?

• How do my students learn best?

• How can I provide more opportunities for my students to explore, notice and talk about mathematical concepts and the ways in which they are connected?

Please comment below any thoughts or comments you might have on this topic!

Sam x