These methods for teaching math are based on a better understanding of how children learn. The intention is for the students to understand the “why” behind the shortcuts in math that we are taught to memorise. As a result, they help more students develop their confidence in math.

#### 3 Stages of Learning

Jerome Bruner identified three stages of cognitive representation:

**Enactive**, which is the representation of knowledge through actions.**Iconic**, which is the visual summarization of images.**Symbolic representation**, which is the use of words and other symbols to describe experiences.

Children need to go through each stage before they can move on to the next. In Math, this is referred to as the concrete-representational-abstract sequence of instruction:

- Each math concept is first modeled with concrete materials.
- Students are given opportunities to practice new skills using concrete materials.
- When students have mastered the concept using concrete materials, the math concept is then modeled at the representational level. Concrete materials are replaced with images that represent the concrete objects previously used.
- Students practice the math concept using the representational drawing solutions.
- When students have mastered the math concept using representational drawing solutions, the math concept is finally modeled at the abstract level.
- Students practice and master the concept at the abstract level before moving to a new math concept.

#### Support Math Learning: Concrete Stage

This stage involves the use of physical math manipulatives, such as:

#### Support Math Learning: Representational Stage

This stage involves the use of image representations, such as:

#### Support Math Learning: Abstract Stage

This final stage involves the use of abstract representations with numbers and symbols, for example:

#### Realistic Math Education

The philosophy underpinning Realistic Mathematics Education (RME) is that students should develop their mathematical understanding by working from contexts that make sense to them. Initially, they devise their own intuitive methods for working on problems but, using a carefully chosen sequence of examples and appropriate teacher interventions, they then generalise and develop a more formal understanding.

An important stage in RME is when students move from their own intuitive mathematical strategies to more sophisticated and formal ways of working. Because the students’ understanding is rooted in contexts and mental images, it is secure.

Exposing children to a range of methods to solve calculations ensures that they have a good understanding of math and that subsequent mathematical knowledge is built upon a strong foundation.

#### See also:

#### Useful Math Links:

- Khan Academy
- Oxford Owl – free reading and Math resource site
- Math Resource Books for Students
- Brilliant Minds Montessori Math Kit
- JUMP Math – Nurturing Mathematical Talent in Every Child
- Free Online Math Programs
- Free Online Math Sites for Kids
- Splash Math
- Furbles for shape classification and bar charts
- Math is Fun – learning about symmetry
- Interactive Math Games and Activities
- Mathletics
- Kenken

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