Weak fundamentals, but easily understand hard concepts

[WannabeFeynman]

Hello everyone. I have a rather unusual issue in which I do not seem to some know/understand basic concepts in prealgebra, but do rather well in something like calculus. By basic concepts, I don't mean something like solving equations, but I don't understand why certain things work the way they do. I accept that it's probably partly my fault, but at school they are too focused on teaching us "tricks" rather then showing why they work. So, can anyone recommend some resources which actually show how the fundamentals of math (mainly prealgebra) actually work? I am going through AoPS and KhanAcademy at the moment.

Thanks.

 

[Dishsoap]

Is there something in particular that you don't understand?

Many people have this problem, including myself. And yes, as you learn more and more math, the early principles make more sense.

 

[WannabeFeynman]

Just a few basic things, like why does multiplying linear expressions work like that. I was able to do things like 3x^4 * 2x^6, but only recently I learned why that works.

 

[Lavabug]

You might try proof-based middle school mathematics textbooks/manuals. I'm not sure if the editorial house publishes versions in English, but MIR/URSS has a huge selection of this type of book. They're all great and dirt cheap where I live, their Spanish translations were and in some cases still are pretty much the standard in most of the Spanish speaking world. I think Dover has republished many of them.

Maybe some of the American-born posters here who were educated in the 60's-70's could also recommend similar books more easily found in English.

 

[Darth Frodo]

If micromass were here he'd most likely recommend: https://www.physicsforums.com/showthread.php?t=665174

 

[homeomorphic]

I think students sometimes get upset if you try to explain the why behind things. That may be part of the reason why things are so dumbed-down. The first time I taught, I was very idealistic and tried to explain the why behind things, but the very concept of even trying to do that was so foreign to the students that it wasn't well-received at all (I was also nervous teaching the first time and not very aware of the students' lack of understanding, so this was not the only reason why it didn't go well). I learned that I had to choose my battles, and the best I could really do was to try to get people to understand the why behind some of the easiest stuff. Even with that, I was not the best at it because so much of it is just so obvious to me, it's hard for me to swallow the fact that it isn't obvious to others or that it would need an explanation at all. By practice with tutoring, I have gradually started being able to communicate with low-level students. These students might be able to understand more complicated things, but they would need to put a lot of time and effort into it and think in ways they have never thought before, so it's not really feasible to explain everything to them if you are being forced to cover a certain amount of predetermined material in the class (besides which, it would take considerable teaching skill). This is unfortunate because a lot of the smartest students will be screwed over, cheated out of explanations, and lead astray.

One factor here might be cognitive load. Sometimes, things that are trivial to a teacher are that way precisely because they have so much of it that is just automated through practice, so that it doesn't take any additional conscious effort. The students have to expend too much conscious effort on each thing to be able to grasp the whole process. So, it's important to do a lot of drill, and not just do it blindly, but think about why it's working as you do it.

One big piece of advice is to think for yourself and to prove things for yourself. Not to always try to find a book to read that gives you the answers, although you should if it's too hard to do on your own.

One of the most irritating things to me in low-level math was the way non-integer exponents were introduced without any motivation. This is one of the great crimes of middle-school and high-school math education. However, when I attempted to explain it to my first class, they didn't seem to take very kindly to it. Since then, I have come up with better explanations, but still, I think it would take quite a bit of skill to convey them to that audience without upsetting them greatly, since they don't even have the concept of what it is like to understand more than the most trivial math.