This seems to be a difficult point for people to grasp, but we notice that it usually comes up for people who, by their own admission were 'never good at math.' So we'd like to take the time to break down what mathematics really is, why so few are 'good at it' and how to support yourself to become better with math.
In the course of your math education, you were exposed, slowly but surely, to new words. Words you probably didn't hear in your day to day life. You were exposed to numerators and denominators, logarithms and exponents, variables and coefficients, graphs, quadratic equations, slopes, functions, integrals, derivatives, trigonometry, sines, cosines, Pythagorean theorems, letters that you were told were numbers, and there was that one kid who sat near you in class for whom it all seemed to just make sense. (I was NOT that kid in high school by the way, far from it. I was the kid who, perhaps like you, just memorized what I could, never did my homework, barely scraped by, sometimes got A's through what seemed like sheer luck, but really had no idea what I was doing.)
At a certain point, after I had been exposed to TechnoTutor by Bernard Poolman, I went back to university and decided to test out my new understanding of the role of vocabulary in education along with the technology to put this understanding into action. I decided, to really test myself, to take (after a 6 year hiatus from school) the most difficult courses I could qualify to enroll in: Analysis, multi-variable calculus, object-oriented programming, probability, and others.) Lo and behold, I was the TOP STUDENT in EVERY SINGLE CLASS that I took. And what I did was really quite simple. When I reveal to you the secret to doing well in school, you will laugh.
When we were presented with a new lesson, I would take the text book, skim it for new vocabulary that either jumped out at me or seemed relevant to the material (usually these are given to you at the beginning of the lesson in the book) and I would integrate them using TechnoTutor. Then... wait for it... keep waiting... here it comes... you'll never guess it in a million years...
I would read the book and work out every exercise, both embedded in the text and at the end of the section (the 'homework exercises' that usually are assigned to you, but I just did them all immediately after reading the new section).
Guess what happened?
I understood what I was doing, because I focused first on the vocabulary and then I read to understand what each word meant and how it related to the other words. Then I applied it by trying to solve problems based on the new information or techniques I had just read about it. This gave me the opportunity to see if I really understood what I had read while also allowing me to develop a more intimate understanding that wasn't just abstract but practical.
Genius, huh?
What's funny is that is the part no one wants to do. And the reason is because, by the time you are learning math, you are already outside of your natural learning ability's open window and so it becomes nearly impossible to learn new vocabulary at the level of unconscious competence, where you just absorb it and can immediately begin using it. Like, have you ever told a 4 year old something and they just from that moment on starting using what you told them and never forgot it? That's NORMAL! The problem is that we are systematically SLOWED DOWN by untrained parents and educators who (to be honest usually were the WORST students).
Ok, so maybe my example is not enough for you. Maybe you need to UNDERSTAND how math is really just vocabulary.
Part of the problem here, is that you erroneously think there is something more to it. It's like you are trying to ask me how I know the sky is blue and you are blind. Well, what makes you think otherwise?
Math is no different to anything else. It just seems different because the vocabulary is so far outside of your normal day to day life, at least the math they teach in school, and so it seems like you have to be some kind of special genius to understand it. But in fact you just have to understand what all the words mean and how they relate to each other (which is just another way of saying 'what the all mean in the context of math').
This probably won't be satisfying to many people because you are likely experiencing the cognitive dissonance of being brainwashed to believe math is amazing, extraordinary, special, etc. But it is actually quite mundane. I mean, look at an accountant. They get paid to do math all day. But beyond just addition, subtraction, multiplication and division, they understand the words of balance sheet, profit, loss, gross, net, taxes, deductions, assets, liabilities, appreciation, depreciation, etc, etc, etc.
The same goes for a physicist. The same goes for an actuary. The same goes for a mathematician. They just have vocabulary to work within specific contexts that you are not familiar with. This is also why its difficult for specialists to think outside of their profession and put things together from various fields. Their vocabulary in many cases is extremely narrow. That's why most engineers are not Elon Musk level. He knows a lot about a lot. Not just a lot about a little. And it's interesting because people who know a lot about a lot, tend to have more power in this world. They have a larger, broader vocabulary and ability to translate a vision into words that others can understand, even if they don't know all the details of the whole vision, they can at least understand what they need to do to make the powerful person's vision a reality.
And this is fundamentally the root of all inequality in this world. In a world that is MADE of words, the one with the most words is the ruler. So the solution? Support everyone to know EVERY WORD and what it means. Doesn't mean people can't still focus and develop a specific point of interest and create cool stuff, but seriously, by the time a child is 18, they can have the vocabulary of Einstein, Elon Musk, Bill Gates, and MLK Jr (assuming they use the right tools of course).
Speaking of the right tools, Max (our 4 year old) and I are reading Richard Feynman's autobiography and in it he talks about learning algebra and some other points and Max asks me to tell him more about algebra. This is at 9:30pm, mind you.
So I get out the little magnetic drawing board and I write out the following equation: 4 + 3 = 7
I ask him if this is a true statement? Does 4 + 3 in fact equal 7? He says yes. I said how do you know. He says "because, 5, 6, 7... 7 is 3 more than 4. It just makes sense."
Ok, cool. So then I write X + 3 = 7 and I explain to him that this is like a problem where we are trying to figure out what number we could replace X with so that the equation is a true statement. Clearly the answer is 4, we already know that, but I showed him how I would use algebraic techniques to figure it out. I showed him that I could subtract 3 from both sides of the equation, that I do it to both sides because it keeps the equation balanced, just like adding or removing something from both sides of a scale in equal amounts (something I know he is already familiar with). I then say, so here I have 3 minus 3 which equals? He says "zero". Ok, and what is X plus 0? He says "X".
Now how did he know that? I didn't explain that to him? But he is not thinking oh there are numbers and letters and these are different things, how can you add a number and a letter? He just doesn't have that issue. We integrated the letters, numbers, math symbols, punctuation, all symbols in the English language by the time he was 2. You can go back and watch the videos we put out and see. So at this point, these symbols are just a part of him.
Another point to consider. If I ask you what are the symbols of the alphabet, you probably list them from A to Z. But that is not correct. Take all of the letters, then follow that with the numerals 0 through 9, then all of the other symbols for math and punctuation and THAT is the real alphabet. If you only really learned part of it, well you can see why you are limited. And really, what is the difference between a letter and a number? I mean, computers really only know 0 and 1 and yet here you are reading words being displayed by a computer who only knows 0 and 1. The understanding of which I could only explain to you using words that are likely not within your current vocabulary. In any case, once they ran out of numerals in math, they started using Greek letters. Remember Pi ? That is just a Greek letter, similar to our P.
So back to Max. Then on the other side of the equation I do 7 minus 3. I say 'this part we call simplifying' and what does it equal, he says '4'. So now we have X = 4. Which we already knew but this is how we would have figured it out if we didn't'.'
So then he says, enthusiastically, let's do another one! This time with W! So I write W + 2 = 6 and then we go through the steps again, subtracting 2 from each side, simplifying, etc. He wants another one, with a different letter this time! Then he wants an equation with TWO letters! Well, I explain that now we are moving beyond the concepts that I've already explained and we need to build a little more context to do something like that, but hey, we can still do it and play around, why not! So we play around and HE starts writing out his own equations! Oh, and in case you haven't seen the videos or posts on social media, Max already knows the Greek letters and can draw them from memory. So, who do you think would have the advantage when they get to high school and start learning trigonometry? Well, of course, Max will have no reason to go to high school, but I think you get my point.
Now, why do 12 year olds have trouble understanding and doing what Max just learned in less than 10 minutes? Some of it has to do with the fact that the teacher cannot properly teach 20 different children at the same time effectively. Geez, I took the kids to the park yesterday while Katie was on the phone with some other women, and directing 2 kids requires a lot of focus. Imagine 20! or 30! And you're trying to teach them algebra!
But the bigger issue is that each child does not have the symbols and the words (which are just larger symbols made of clusters of the smaller ones) integrated physically with clear definitions and without emotional bias. And instead, at a certain age, Life becomes less about learning and rather about fitting in based on what you were brainwashed to believe about yourself based on a limited ability to learn new things. Sad, really.
I know it's difficult, emotionally, for you to accept, but Math is just Vocabulary.
Everything is just vocabulary.
Education is every word you know, can process, and comprehend. That's it.
BUT WHAT ABOUT APPLICATION?????
Ironic question given that most professors spend their lives teaching and not doing, but hey its actually a valid question.
When I ask people what is it that you think math is beyond vocabulary, usually the response is, but don't you have to know how to apply it?
Well yes, but, when you understand all the words, the application becomes a result of applying common sense, testing assumptions, and trial and error over time. After all, who taught Pythagoras how to apply the Pythagorean theorem. And before you think about the chicken and the egg, let me enlighten you. He FIGURED OUT THE APPLICATION. Einstein came up with his own way to prove it when he was 12 just because he understood all the prerequisite vocabulary and thought about it long enough. If he didn't know what a triangle was, my guess it that it would have taken him quite a bit longer, as in, he would still be trying to figure it out.
But the reason your mind jumps to 'but what about the application' is because that is ALL THAT IS TAUGHT in school. They don't really make sure you understand what things mean. They just demand that you memorize the steps of how to solve the problem, but if you're given a problem you have never seen before, you have a melt down. Well, why would THEY not want you to know how to solve problems you've never seen before. Probably because workers just need to do what they are told. And when you solve problems that have never been solved before, you gain power over others. So in a world where they want you wearing a mask and standing 6 feet apart, it's just better if you don't actually know how to think for yourself or figure things out but instead repeat what you are told, even if it changes on a week to week basis.
The reality is that, more than anything, Math is a way of thinking. It is really similar to Scientific thinking. You start with a set of assumptions and then look at how you can put together those assumptions in ways to draw conclusions beyond what was initially obvious. And then you look to see if those conclusions are useful in any way. And if you have a clear vocabulary and understand how reality works (which means the same thing as having a clear vocabulary) and then you apply mathematical thinking to the problems in this world, well, hell. You just might be able to solve things like "why do so many children go to bed without enough food?" Or "in a world of abundance, why do we have to compete to survive?"
And if enough people had the vocabulary and ability to think mathematically, in other words to put two and two together, well, we just might become our own authority and no longer place responsibility for creating a world that is best for all outside of ourselves. And if you weren't supported to develop an effective vocabulary and ability to think mathematically from a young age, you just might never realize that a world that is best for all means a world that is best for YOU.
Resources:
Very Key Thanks!
ReplyDelete