# 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 and encourage learning and thinking across the mathematics curriculum.

The instructional routines (IR) are purposefully structured activities that help students develop procedural fluency, as well as reasoning and problem-solving skills, through meaningful practice. Often warm-ups or lesson openers/finishers, not necessarily related to the main topic of the lesson.

Teachers can also improve their pedagogical knowledge by selecting a relevant free, self-paced online course, face-to-face or virtual workshops and other professional learning resources provided by the University of Adelaide. Register <here> for this free Maths in Schools professional learning program.

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

The CRA model is a three-phased approach where students move from concrete 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.

The CRA instructional sequence

An explanation of the Concrete, Representational, Abstract learning sequence and how to apply it in the primary classroom.

Algebra tiles learning sequence

A sequence of lesson ideas on using algebra tiles, from integer operations to completing the square.

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.

### Virtual manipulatives

Virtual manipulatives are an alternative to concrete, physical manipulatives. Teachers can use them to illustrate concepts on a screen to a whole class, and students can use them on a device to develop a deeper understanding. These links are to libraries of various virtual manipulatives.

Interactive teaching resources. These resources can be adapted to meet your requirements.

### Number talks (IR)

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

A description of what a number talk is, how to plan one and how to run it in the classroom.

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

Number talks: secondary example

Jo Boaler describes a number talk for the secondary level.

### Number strings (IR)

A number string is an extended number talk that focuses students on a specific strategy.

Description of the number string routine and many classroom-ready resources.

### Other ‘talks’ (IR)

There are many variations on the number talk routine. Here is a selection.

Students are shown an image and asked what fraction is shaded. The discussion promotes the use of the language of fractions and proportion. On this site there are many prompts for all levels.

Which One Doesn’t Belong (WODB)

In this routine a series of four pictures/objects/equations etc. are shown. Students have to explain why each one could be the odd-one-out and the other three a group. This routine encourages reasoning and perseverance. This site has a wealth of WODB prompts for all year levels and topics that is constantly being updated. (You can even submit your own!)

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 and are suitable for all year levels and cover most maths topics. These promote problem-solving and reasoning skills. On this site you will find a comprehensive range of problems.

This strategy is similar in structure to a number talk, but students are shown a data visual and asked what interests them, what do they see. The discussion will develop students’ data literacy and vocabulary. This site has many prompts for all year levels.

### Math Language Routines (MLRs)

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.

### Thinking routines

Developed by Harvard's Project Zero, a thinking routine is a set of questions or a brief sequence of steps used to scaffold and support student thinking.

Here you will find a searchable list of the thinking routines. Each routine is described and has associated resources to help you implement them successfully into the classroom.

### 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.

View this resource for a detailed explanation about 3-Act Tasks, including ways to use this strategy in the classroom.

### 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.

Five principles of mathematically rich games

How to choose, use and design games to promote deep mathematical thinking.

Learning mathematics through games

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