Re-learning how to think is always a difficult process. My oldest son is starting the the 7th grade and he’s having to go through it. We homeschool, so we pick the curriculum. Since he’s consistently been above the 95% range with respect to math going back to ever since he started school (my wife has a M.Ed. in School Psychology and does the standardized testing for a lot of the homeschooler families we know, so this is pretty easy for us to do), and because he was obviously bored out of his mind when it came to math, I made the decision to start him in Algebra I and skip pre-Algebra altogether. This was something that my wife and I discussed before I made that decision. I have intentionally said that I made the decision. That’s because I knew that if was going to do Algebra I early, I was going to have to step up and be his primary teacher in the subject.I had to make the decision to take it on.
Up until this point, most of his learning has been concrete – seeing numbers and doing a computation or reading a word problem and putting the calculation together and solving. The latter requires some abstract thinking, but he didn’t use variables and he definitely didn’t use terms like 2xy. Of course, in Algebra you encounter that right away. An example of where he’s struggling is solving something like:
2z * (2xy + 4x) =
Because of the distributive property we know the answer is 4xyz + 8xz. Watching him as we were trying some problems last night, he was coming up with answers like 4z + 4xz + 2yz for the first part of the calculation. In his mind, he’s struggling with the grouping of 2, x, and y together as one term for the purposes of the multiplication. Now, if he saw the following, it wouldn’t be a problem with him:
2(4) * (2(1)(3) + 4(1)) =
It wouldn’t be a problem because he would immediately simplify this to 8 * (6 + 4) = 8 * 10 = 80. Those are numbers, numbers he can “touch” or “count.” Dealing with variables is a whole different matter. While the process is the same (once you grasp it), until your mind clicks over, it seems like a completely foreign concept. Now, the material I’m using with him also introduces lightly the idea of doing proofs. For instance, show that when you multiply an even number and an even number, you always get an even number. Or when you multiply an even number and an odd number, you also always get an even number. Intuitively we understand these statements are true. But proving them is a different matter. In this case, though, both are actually pretty easy to show with a single proof, especially since only natural numbers have been introduced thus far.
Given the definition of an even number as 2n where n is a natural number (1, 2, 3, 4,…), assume an even number 2x (meaning x is a natural number) and a natural number y. The product of those two numbers 2x * y = 2 * xy due to the associative property of multiplication. Let z = xy. By substitution, 2 * xy = 2 * z = 2z. Due to the closure property for multiplication, since x and y are both natural numbers, their product is a natural number as well. Therefore z is a natural number. By definition, the product of 2 and a natural number is an even number. Therefore, 2z is an even natural number and we can conclude that the product of an even natural number with any other natural number is always an even natural number.
I say pretty easy to show, but I should say easy for me, and probably easy for a lot of folks who are technically minded. But for my oldest son, this is a new challenge. His initial attempts at proving either case used concrete numbers. That’s what he knows. However, that allowed me the opportunity to present him with the situation that if we wanted to prove it by plugging numbers in, a single case would prove what we were stating as wrong. Therefore, that means we would have to test every case. I asked him if that was possible. And that’s where I saw the first steps being taken into a broader way of thinking, when he said, “No, because there’s unlimited numbers.”
So what’s the point of all this? Just that re-learning how to think takes time (another phrase is undergoing a paradigm shift). And we all do this from time to time if we’re growing in knowledge and proficiency. For instance, for someone who is used to iterative, procedural programming languages (Pascal, C, C++, Java, VB.Net, C#, etc.), facing a set-based language like SQL requires re-learning. Or moving from either an iterative or set-based programming language to a functional language like Python requires re-learning. Learning a foreign language when you’re primary language is English often requires re-learning, too. Case in point: I’m learning Koine or Common Greek, the primary language the New Testament was written in. In English we’re used to certain word orders for the structure of a sentence. For instance, the subject before the verb. But in Koine Greek, those structures don’t exist. The language is such that you put first what you want to emphasize. So that means you learn to identify subject, verb, direct object, etc., by endings on the words. These modify the roots. And that’s definitely different than what we’re used to in English.
With my son, I expect the relearning is going to take time. And I expect some initial struggling with him. I’ve built that into the lesson plan. This morning, before I headed into work, I pulled him aside to explain to him that if he started to get frustrated, to take a step back, take a deep breath, and relax. He’s having to learn to think in a way he hasn’t done much of in the past. And that means it’s going to be a challenge, he’s going to find difficulty in doing so, and he’s going to be slower at it than he’s used to with respect to completing his math work. If you’re transitioning to something that’s new, make the same sort of plans. Don’t put together an unrealistic schedule and that beat yourself up over it if you can’t keep it. Expect to struggle. Plan accordingly. If you happen to catch on quick, that’s great! You’ll get done faster. But expect that to be an exception, not the rule. And that reminds me of a conversation I had with a former boss about why you couldn’t just send a system administrator to a development course and expect to have a developer. But that’s a post for another time.