When I was first being taught how to solve quadratics, after factorisations, the teacher went on to show us 'completing the square'. It was a logical sequence of steps for me so I used it a lot. Then we were given the quadratic formula to memorise and told we could use that as well. My fellow students did that but they had to look up the formula every time or try to memorise it. I didn't want to do that, so I stuck with completing the square. It takes more lines but it's the same calculation. Then I tried it with ax^2 + bx + c = 0 and realised I was proving the formula.

So I switched to using the formula (which I still hadn't learnt) but I didn't look it up either. I re-proved it each time. Gradually I got better and better at this until I could imagine the steps in my head and write the formula from that.

So I'd managed to learn it by re-proving it every time. 58 years later and I can still do that. I was delighted when, in an exam, we were asked to prove the formula. Easy for me; tough for my 'memorising' friends.

So how does this address your post? Working from first principles reinforces understanding and makes it easier to remember. Things I have just memorised go straight out of my head again. If I develop a full understanding I can always do things but I sometimes I have to re-prove the formulas.

Bob

Bob,

1. Learning math formulas without knowing how to derive them is not learning mathematics correctly.

2. I never have trouble when using formulas. I just don't know how to derive them at any level. This keeps me at the amateur level of mathematics.

3. Not to step away from the topic at hand but I want to ask you 2 questions. WHAT DOES INDOCTRINATION OF STUDENTS MEAN? Does INDOCTRINATION have a lot to do with students not learning mathematics correctly?

]]>So I switched to using the formula (which I still hadn't learnt) but I didn't look it up either. I re-proved it each time. Gradually I got better and better at this until I could imagine the steps in my head and write the formula from that.

So I'd managed to learn it by re-proving it every time. 58 years later and I can still do that. I was delighted when, in an exam, we were asked to prove the formula. Easy for me; tough for my 'memorising' friends.

So how does this address your post? Working from first principles reinforces understanding and makes it easier to remember. Things I have just memorised go straight out of my head again. If I develop a full understanding I can always do things but I sometimes I have to re-prove the formulas.

Bob

]]>(y -b)/x = m is right.

m is the slope.

Thanks. I often find myself going through a textbook like walking in the park on a sunny day. I then reach a certain chapter or section that requires knowledge of all the previous math material I have learned or thought I learned. Confusion sets in to create a stumbling block that keeps me wondering if I truly learned anything at all in my previous math courses or chapters or sections that I just completed. Why does this happen? How do I keep myself from forgetting the earlier chapters or sections in math textbooks?

]]>m is the slope.

]]>Subtract b from both sides.

y - b = mx + b - b

y - b = mx

Divide both sides by x.

(y - b)/x = m

You say?

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