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He just doesn't understand why he should write "Let x equal the number of pears" at the top of the problem. I didn't either when I was his age, but I did it because I had no choice (that's the way math was taught back then). Now, I understand why, and that's why I'm passing it on to him.
Catherine responded:
What is the reason?!
It seems like a good thing to do, but that's all I know.
Because math isn't the only reason to learn math. We benefit at least as much from the sequential, linear, logical thought process, because we can apply it to nearly every facet of our lives, and not just the quantitative ones.
The traditional formalism of math is the embodiment of that process. By llearning it, and being forced to reproduce it every time we do a problem, we learn the process itself, of breaking a problem into its component parts, and creating a step by step solution, where each step follows from the previous steps.
It's discipline for the mind.
This is one of my major objections to "fuzzy" math, that students never learn this logical process.
8 comments:
Hey prof,
Thanks for that post.
Since the mathies in my life have always said that math is "justification, justification, justification" rather than simply the answer, I always thought that THAT sounded remarkably like fuzzy math.
But there is a difference and maybe in some future post you can expand on that.
I guess the $4 question is what constitutes justification, i.e, the logical steps. What can one leave out and what must one include? I guess if we all knew exactly what it meant to justify each step we'd all be mathematicians.
"It's discipline for the mind."
I have found some of the kiddies (usually the abler ones) to be highly resistant to this discipline of the mind. For example, teaching the formalism of dimensional analysis to eighth graders was a circus. The abler ones could convert using their own methods and didn't see the need for this formalism that looked more like an exotic ritual. The less able ones had a supremely hard time with it.
Another area of resistance by some is isolating a variable. One kid refuses to do to the left side what he does to the right side in a simple equation like 2x = 12. He says just dividing 12 by 2 gets him the answer. No need to monkey with 2x. Could this have something to do with frontal lobe development?
COOL!
Thanks.
"Since the mathies in my life have always said that math is "justification, justification, justification" rather than simply the answer, I always thought that THAT sounded remarkably like fuzzy math."
Simply the answer is what kids think is the purpose of exercises. They don't understand that they need to understand what they are doing. If understanding what they are doing were the purpose of fuzzy math it would be great. But fuzzy math deprives pupils of foundational knowledge and skills. That's its perniciousness.
I have found some of the kiddies (usually the abler ones) to be highly resistant to this discipline of the mind.
Absolutely!
Susan had a fantastic comment about that on the old ktm.
I would give a purely "brain-based" justification for writing these things out, which is that kids at this age really, truly have no idea which steps they can safely skip (perform mentally) without making errors.
I find, studying Algebra 2, that I don't know this either.
I also thank you for this post. It is my opinion that the logical thought process involved in learning the discipline of math transfers to other fields as well, and in particular, the study of law.
Learning and understanding the law requires the ability to engage in analytical thinking and it has been my sense that math courses (particularly geometry proofs) help to discipline and train the mind for this.
There is a practical reason for writing "Let x =...". It helps keep track of what you're solving for, particularly when you are solving for two numbers, using one variable. So if you have a problem where Julie is 4 years older than Jack, writing "Let Jack's age = x, and Julie's age = 4 + x" makes it easier when you solve for x, assigning the right numbers to the correct person. If they object to the word "let", then just have them write "Jack's age = x,etc" . The important thing is they are keeping straight what it is they are solving for. Saying "Jack = x" is not as good as "Jack's age = x". What are the units, what are you trying to find?
Reminds me of the joke I've probably told a million times on this site, so bear with me:
Teacher: Suppose x equals the number of sheep.
Student: But suppose x is NOT the number of sheep!
"It's discipline for the mind."
I agree 100%. Fuzzy math is anti-discipline because discipline is drill-and-kill.
My sister-in-law, a high school English teacher, told me once about 6 different types of people she learned about in some sort of in-service teacher training class. I don't remember them all, but she concluded that I was "concrete-sequential". The point was that this was some sort of learning style. Teaching had to be built around meeting the needs of each of these learning styles. In situations like this, I usually just clam up. I don't argue. I just wondered how you teach concrete-sequential thought process to someone who is a creative (random) free thinker? "Sorry, you don't have to try?"
Well, you know those concrete-sequential thinkers just can't be creative. So we get reform math built around the idea that there is some path to math without discipline.
I hate pop psychology.
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