Mastering calculus

Concrete Pictorial Abstract 

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

Bar Model method

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

Conceptual understanding and Procedure Skills mutually reinforce each other

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

Examples, non-standard examples and counterexamples play a crucial role in Variation. As with CPA, there is a role for multiple representations (see CPA). Marton, Runesson and Tsui (2004) formulate the functions of variation like this:

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

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

  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.”

  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.” 

Teachers should try to accumulate a large number of examples of teaching different topics using variation.
For more information visit Future Learn online course.