# Mastering calculus

## Concrete Pictorial Abstract

Concrete Pictorial Abstract

### Using concrete material as a first step before moving into visualising and extending ideas.

Using concrete material as a first step before moving into visualising and extending ideas.

## Bar Model method

Bar Model method

### Using double number line to teach concepts such as percentages, ration and proportion helps students to develop a better understanding.

Using double number line to teach concepts such as percentages, ration and proportion helps students to develop a better understanding.

## The two basics: basic knowledge and basic skills taking a spiral approach

The two basics: basic knowledge and basic skills taking a spiral approach

## Conceptual understanding and Procedure Skills mutually reinforce each other

Conceptual understanding and Procedure Skills mutually reinforce each other

### Keep a balanced view about the Two Basics and other aspects of maths learning (ex, high-order thinking skills).

Keep a balanced view about the Two Basics and other aspects of maths learning (ex, high-order thinking skills).

### Teaching with Variation can help expand the Two Basics.

Teaching with Variation can help expand the Two Basics.

### Students need to experience variation for the new concepts to be learnt. Varying each task within a sequence of tasks will help learning. This can be divided in two types:conceptual variation and procedural variation.

Students need to experience variation for the new concepts to be learnt. Varying each task within a sequence of tasks will help learning. This can be divided in two types:conceptual variation and procedural variation.

## 1. Contrast.

1. Contrast.

## “… In order to experience something, a person must experience something else to compare it with…”

“… In order to experience something, a person must experience something else to compare it with…”

## 2. Generalisation.

2. Generalisation.

## “… In order to fully understand what “three” is, we must also experience varying appearances of three …”

“… In order to fully understand what “three” is, we must also experience varying appearances of three …”

## 3. Separation.

3. Separation.

## “… In order to experience a certain aspect of something, and in order to separate this aspect from other aspects, it must vary while other aspects remain invariant.”

“… In order to experience a certain aspect of something, and in order to separate this aspect from other aspects, it must vary while other aspects remain invariant.”

## 4. Fusion.

4. Fusion.

## “…if there are several critical aspects that the learner has to take into consideration at the same time, they must all be experienced simultaneously.”

“…if there are several critical aspects that the learner has to take into consideration at the same time, they must all be experienced simultaneously.”

### Teachers should try to accumulate a large number of examples of teaching different topics using variation.

Teachers should try to accumulate a large number of examples of teaching different topics using variation.

### For more information visit Future Learn online course.

For more information visit Future Learn online course.