# Teaching approaches in action

There are many approaches to teaching mathematics. Below you will find a range of instructional routines and pedagogical strategies to engage students and encourage learning and thinking across the mathematics curriculum.

Instructional routines are purposefully structured activities that help students develop procedural fluency, as well as reasoning and problem-solving skills, through meaningful practice. These activities are often used as warm-ups or lesson openers/finishers, so are not necessarily related to the main topic of the lesson.

## Teaching approaches

##### Concrete, Representational, Abstract model (CRA)

### Concrete, Representational, Abstract model (CRA)

The CRA model is a three-phased approach where students move from concrete or virtual manipulatives, to making visual representations and on to using symbolic notation. This strategy has been shown to be highly effective in mathematics in all year levels.

Manipulatives in the primary classroom

This article explores research into the use of manipulatives and offers suggestions about how using practical apparatus can support children's mathematical thinking, reasoning and problem solving.

Using mathematical representations

Guidance on how to use manipulatives in secondary year levels.

##### Explicit teaching

Sometimes called the ‘I do, We do, You do’ model, explicit teaching is a gradual-release model where the teacher explicitly teaches a concept, works through examples with the class and then the students work on their own, or in groups. The class then come together to review their questions and understanding.

Explicit teaching – numeracy example

An AITSL illustration of practice that follows a teacher in designing and implementing explicit mathematical learning experiences for her year 3-4 class.

Tried and tested – Explicit Instruction

This research article describes a set of teaching practices that have been found to improve student achievement by making instruction explicit.

##### High Impact Teaching Strategies

The high impact teaching strategies (HITS) are 10 instructional practices that increase student learning. The HITS are evidence-based, drawing on research both in Australia and internationally.

The practices include: setting goals, structuring lessons, explicit teaching, worked examples, questioning, metacognitive strategies and differentiated teaching.

A guide to the high impact teaching strategies (HITS)

The high impact teaching strategies (HITS) are 10 instructional practices that increase student learning. The HITS are evidence-based, drawing on research both in Australia and internationally.

The practices include: setting goals, structuring lessons, explicit teaching, worked examples, questioning, metacognitive strategies and differentiated teaching.

High impact teaching strategies (HITS)

This site has descriptions explaining all the HITS and provides links to research and videos.

##### Instructional routines

### Number talks

Number talks are designed to encourage students to think deeply about a problem, share their thinking and use mathematical language. They are structured conversations.

An illustration of a teacher running a number talk with a lower primary class.

In an Open Middle problem an expression is presented and students are challenged to find the numbers that will make the expression true. They are challenging, suitable for all year levels, cover most maths topics, and promote problem-solving and reasoning skills. On this site you will find a comprehensive range of problems.

### Math Language Routines

Developed at Stanford University, the eight MLRs promote the development of mathematical vocabulary and reasoning skills in secondary classrooms.

This paper describes the theory behind each of the routines and the routines themselves in detail.

### 3-Act Tasks

A 3-Act Task is a task consisting of three distinct sections:

- an engaging and perplexing Act One, often a video. Students discuss what they saw, pose questions and decide what information is needed for them to be able to answer the questions they have posed.
- an information and solution-seeking Act Two. Students are given the information they need and work, either in small groups or individually, to find a solution.
- a solution-discussion and revealing Act Three. Students discuss their findings and the solution is revealed. Students may then discuss and pose further questions.

A Three-Act task on proportion and ratio.

##### Mathematical games

Using games to explore mathematical concepts can engage even the most reluctant students. It is important that the mathematical purpose for the game is made explicit and the mathematics discussed. Designing the rules of a game and exploring how changing the rules will change the game can uncover deep mathematical concepts.

Learning mathematics through games

A series of articles about the nature of games and their role in the teaching and learning of mathematics.

##### Maths Investigations

Using a prompt to ignite students’ curiosity, a maths inquiry allows students to explore ideas and find meaning in the mathematics concepts they are learning in the classroom. They engage in questioning, conjecturing, listening and communicating their ideas.

The reSolve: Mathematics by Inquiry Protocol promotes structured and purposeful investigations of mathematical and realistic contexts. The Protocol provides a set of principles that underpins the reSolve Professional and Teaching resources, including descriptions of excellence in mathematics teaching and learning.

Here you will find a description of the ‘4D Guided Inquiry model’ with four phases – Discover, Devise, Develop and Defend. There are links to ten developed units of inquiry for Foundation to Year 6.

##### Understanding by Design (UbD)

Understanding by Design is a framework for planning that consists of three stages:

- Identify the desired outcomes – what part of the curriculum are you covering?
- Identify the evidence that will demonstrate these outcomes have been met – how will students show they have mastered the content?
- Plan learning activities that will lead to the desired outcomes – what will the lesson contain?

This page clearly explains the structure and rationale behind Understanding by Design (UbD). There is a video of Grant Wiggins, the co-creator of UbD.