Determine a sequencing for the development of knowledge and skills

The following material was taken from M3 - Planning and designing lessons Links to an external site. within the AustMS unit.

In many cases, the new content knowledge and transferable skills that you wish the students to develop will be built on prior knowledge (which the students may or may not have), and might be impaired by existing knowledge that is either incorrect or limited. If threshold concepts (see, for example, Breen & O’Shea, 2016) are involved, you will need to both build on existing knowledge and possibly transform it. You need to consider how interest will be developed and prior knowledge activated.

This problem can, in part, be addressed by the design and delivery of logically sequenced material that is contextualised – especially in service teaching (refer to the Teaching in service units: The need for relevance section later in this module).

It is fundamental and indeed imperative to the preparation of material for a mathematics lesson, be it a lecture or tutorial, that students are exposed to tasks based on a logical sequencing of material that builds the solid basis for their mathematical understanding. Note that at this stage, we are still considering the sequencing of ideas, and not specifying what the 'tasks' might be. These will include activities done by the teacher for the benefit of the students, and activities for the students to do themselves.

The careful choice of graduated examples will help students attain a range of skills and understand key mathematical concepts. These examples should also form a basis for student activities that enhance skills, such as analytical thinking, logical presentation, mathematical communication and problem solving. Make sure your planning includes an allocation of time to each task, thus ensuring that students fully grasp the topic, but at the same time not swamping them with too much detail. Looking at an example of a lesson on differentiation from first principles, in a lecture one might decide to sequence the ideas as follows:

• revise function notation and introduce any other required notation;
• revise the concept of the gradient of a straight line;
• connect the concept of the gradient of a straight line to the average rate of change of a function over a given interval;
• develop a formula for the average rate of change over a given interval;
• introduce the concept of taking a limit to approximate the instantaneous rate of change of a function;
• develop a formula for the concept of a derivative using the above derivation;
• implement the above for linear functions;
• implement the above for other polynomials;
• emphasise the importance of the continuity and smoothness of the function; and
• apply knowledge to real-world problems.