1. Memorised procedures

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Some (most) novices have no mental models for the maths they are learning, and rely on memorized procedures.

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For example:

Differentiation quiz

Student’s answer: “Use the quotient rule …”

You can see that this student has memorized a procedure that says, “if I am doing differentiation and I see a fraction, I need to apply the quotient rule.”

Having no mental models and relying on memorized procedures is the worst case.

Many such students do not even realise that there is more to maths than memorizing formulae and algorithms.

They may not be aware that maths is actually a cohesive whole and they should have a mental model that joins the different formulas and concepts together and explains how they work.

When you teach something to these students, they look for a procedure or formula to memorize, because this is how their mental framework is set up. They don’t store the understanding or the big picture or the underlying process, because there is no place for these in their mental framework.

These students get through exams by learning to use cues in questions to recall the correct formula or procedure. Part of their procedure is to find trigger words or variables in questions that indicate the formula to use. They get through exams essentially without any understanding.

You could teach the student from this example that actually this is a limit problem, so you need to do the following … but this is just replacing one procedure with another. It’s not adding any understanding. We need to look at ways to change this student’s mental framework.

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