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| Thinking (Image credit: Wikimedia Commons) |
Learning something can sometimes be either difficult or easy, usually in between the two. Just what makes learning differential equations and learning simple arithmetic the difference between learning "complex" and "simple"?
Well, the direct answer would be that to learn differential equations, you must know arithmetic, you and you must know algebra. You also need to know how to use numbers, logic, and algebra to even attempt to learn calculus. Compared to simple arithmetic, all you need to know is how to count. Learning isn't some miracle happening inside your brain, it's a putting together and adding little things we know or have memorized as true. To learn differential equations, you need to put together what you learn from arithmetic, geometry, and algebra.
For example, to learn to add, a child must first learn to count. A child memorizes the numbers 1 through 10. 2 comes after 1, 3 comes after 2, 4 comes after 3, etc. Now, after memorizing the numbers, he is taught the concept of putting two numbers together called adding. 4 plus 5 is counting 5 from 4. 5 (1), 6 (2), 7 (3), 8 (4), 9 (5). There we go, the answer is 9. A child must count twice and know where to count in order to add two numbers together.
As you might guess, learning multiplication is the same thing. A child just has to add the same number some number of times. 4 times 5 is just 4 added with 4 added with 4 added with 4 added with 4.
Now imagine if a child had to learn multiplication without learning how to add or count. The seemingly simple operation would be exceedingly difficult and complex!
I know not all kids are taught the concepts of addition or multiplication. They just memorize big tables. Then again, when we ourselves multiply 4 and 5, we don't actually add 4 to itself 5 times, we just know it's 20. However, it serves as a good example. Why do so many students think math is so difficult? If you don't understand one concept, you will miss the next concept, and the workload piles up.
It isn't just math, it applies to everything. Science, sports, writing blogs, you name it. Anything to add? Post it in the comment section below.
Here is an interesting chart on another take on learning simple to complex. It divides learning a concept into multiple steps for a fuller understanding.
Hey Anthony,
ReplyDeleteI agree with your post a lot. I also wrote a post about learning math, since math is a subject that a ton of people struggle with. Is math really that hard, or are the techniques of learning making it hard? I understand the memorizing big tables part. When I was younger, my father taught me the tables up until thirteen. I would write them over and over again, and not really understand what they meant. Sure, I could do the multiplication sheets in first grade easily, but when it came to applying those to story problems, I was confused. Math is a hierarchy of concepts. If you can't do Pre Algebra, you can't do Algebra One, Algebra Two, or Pre Calculus. Then you enter Calculus, and you are crying. If one learns the concepts clearly, then math is not a struggle. It is fun actually. Do you agree with the methods of teaching math in classes these days? Should teachers focus more on concepts than the curriculum given to them? Is memorization the right way to go?
Hi Anthony,
ReplyDeleteI find it very interesting how you take something as simple as multiplication and show how learning it without the fundamentals of addition can actually make it a complex endeavor. I certainly agree that the nature of learning math or anything else involves structure and hierarchy. However, one important concept you do not mention is the idea that this hierarchy requires practice and reinforcement. We don't just learn by moving from one step to the next, we also review continuously. One frustrating thing about the human brain is how we don't just preserve information. Every time we memorize or learn a concept, the action of 'retrieval' we perform to recall that information is actually a recreation of it. To learn and remember we must relearn over and over again. I am a big fan of Radiolab at WNYC. Here is an interesting podcast about memory you might find relevant and interesting: http://www.radiolab.org/story/91569-memory-and-forgetting/.
Perhaps math is difficult for many students not because they failed to understand a basic concept but because they did not reinforce their understanding over time. How do you think reinforcement is emphasized in contrast to understanding in the classroom? Does it vary between math and other subjects?