# This is how didactic Peter Kop encourages high school students to make friends with algebraic formulas

For many high school students, mathematical formulas are nothing more than abracadabra. Didactic and mathematics teacher Peter Kop wants to change this. For his PhD research, he devised a new series of lessons full of drawing assignments that learn students to assign more meaning to the formulas. Kop: ‘Students have a full box of tricks, but don't know when to apply which trick.’ Promotion on 18 October.

## Stand-alone tricks

‘During the research, one of my students told me that he never really had a clear picture of what he had to do,’ says Kop. ‘During a maths assignment, he would look in his head for a trick he had learned and then apply it.’ In the 35 years that he teaches in high school, Kop regularly saw that students learn tricks by heart, but that they lack a thorough understanding of formulas. ‘If during a test, the assignments deviate even slightly from what such a student has learned, it immediately goes wrong.’

## Symbol sense

In technical terms, this is caused by a lack of symbol sense: a broad concept that has to do with, among other things, identifying the structure of algebraic formulas, giving meaning to those formulas, and reasoning with and about them. Kop: In class, the main focus lies on practising and manipulating formulas, but too little attention is paid to understanding them. That's why students are often just fumbling around without making any sense. It’s like you have a full box of tricks, but you don't see the connections. In that case, you also don't know when and where to apply a certain trick.’

New technological developments certainly play a role in this. For example, about twenty years ago the graphical calculator was introduced, that allows you to draw and edit graphs. Kop: ‘At the time, I think we thought that if someone keeps using the graphing calculator and keeps making more or less the same graphs, then at some point some knowledge get stuck and some recognition will occur. But compare it to driving with navigation. If you always use your navigation, you will never get a good overview of how the Netherlands is structured. That's why I think that in addition to ICT, we also need to keep practising and reasoning with pen and paper.’

## Drawing graphs

Kop devised a series of lessons to help high school students gain more insight into algebraic formulas. He based this method on expert-strategies. He interviewed three university lecturers, a lecturer involved in the development of final exams and a lecturer who co-authored a mathematics teaching method.

The series consists of five lessons in which students have to draw graphs based on formulas. ‘Without too much arithmetic,’ explains Kop. ‘They do this by looking at the structure of the formula and reasoning about it. For this, we use a repertoire of basic functions that the students can depict with arm gestures and that can serve as building blocks for more complex formulas.’

Pot tested his students before and after the special lessons, and again four months later. He also compared his own students with students from five other schools who hadn't followed the lessons. During the test he made six students think out loud. ‘We did not only saw that our students scored better, but also that they indeed applied the ingredients from our lessons, also in a broader context.’

## Educational revolution

Kops colleagues are enthusiastic: two of them are already using the new series of lessons. ‘We haven't exactly measured it, but they are achieving the same positive results,’ says Kop. At the moment, he is writing a new mathematics method at his school. ‘Of course, I'm going to incorporate this new methodology. I am also trying to enthuse colleagues elsewhere in the country, through presentations at study days and conferences. This new method can promote the symbol sense of pupils and therefore deserves a prominent place in mathematics education!’