Greg Ashman also writes about conceptual understanding in math and does a good job of it.
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.
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traditional math 