There is a story often told about maths teaching. It is a story of how, in olden times, children were taught rote mathematical procedures. They were never taught conceptual understanding of the principles involved. These days, we have computers to perform mere procedures for us and so, instead, we should focus on conceptual understanding.
This is flawed logic.
Take the principle of equivalence. This is an idea that is often investigated in educational psychology experiments as an example of conceptual understanding. When children first meet mathematical equations, they are of the form 2 + 3 = ?. This means that they reasonably, but incorrectly, infer that an equals sign (=) is a command to write a correct answer. In fact, an equals sign means ‘the same as’ and a failure to grasp this may cause problems later when students have to solve problems of the form 2 + ? = 5.
The principle of equivalence is also important in solving simultaneous equations by elimination. The basic logic is as follows:
If A = B and C = D then A – C = B – D
I remember being quite mystified by this seemingly magical move at school. This may be because nobody ever pointed out the logic and how it utilises the principle of equivalence, or it maybe that they did do this but I have forgotten.
That fact is that a lot of these revered items of conceptual understanding are straightforward items of declarative knowledge. It is the application of these principles that is important and difficult to learn and that’s why maths lessons traditionally devote the majority of time to this.
In contrast, I could train a parrot to give me the correct dictionary definition of what the equals sign means. Does the parrot understand? Maths teachers who focus on conceptual understanding don’t tend to ask students to recite dictionary definitions, but they may as well do so. Making posters or having discussions about a relatively simple item of declarative knowledge is not much different.
The power of mathematics is the ability to move from simple, straightforward axioms, step by step towards something that is not at all obvious but incredibly useful to know.
Yes, it is important to make these axioms clear, and I am quite prepared to accept that maths teachers do not always do this or do not return to them often enough to reinforce them. But to focus on these axioms at the expense of procedures is to completely misunderstand what mathematics is and why it is so powerful.