Teaching Mathematics

I would like to get in the ear of every teacher who teaches Mathematics and make sure they understand this:

The subject matter of Mathematics is how to think clearly about problems.

There are no facts about reality in mathematics.

There are a lot of definitions, of terms and of notation, that make it easier to talk and reason about problems. There are particular techniques for particular types of problems, but it is highly counterproductive for students to learn techniques if they don't understand how they work. [Well it might conceivably help a student pass an exam which might help them in some way, but I am assuming here that we are learning Mathematics for some genuine reason.] Indeed if the student understands how a technique works then they will easily remember it, but they will probably not need to remember it.

There are particular facts about reality for which mathematical techniques are particularly useful. Dealing with measurement systems (like money, lengths, area) is a case. So such real world stuff often gets put in with mathematical education where it serves as a valuable source of useful examples. However the facts about coins or the metric system of measurement are not themselves part of Mathematics.

The particular case of beginning Primary School (and before) is important. Thinking clearly about integers is the core of Mathematics. Human thinking about integers is closely integrated with thinking about our fingers. Students need to go through the process of counting on their fingers. We know this because the part of the human brain that does arithmetic is heavily intertwined with the part that deals with the fingers. Before calculators students needed, for practical purposes to do sums quickly. Now it is more important that they make that crucial understanding step. Don't teach them their tables by rote. Teach them to do sums with their fingers.