Gifted Issues Discussion homepage Forums General Discussion Intuitive math

I sometimes feel like my son learns math by osmosis, and neither he nor my husband and I can tell how he knows something. He just started 4th grade and is able to do very basic pre-algebra problems (such as 15x-10=57.5).

He hears a problem or sees it and seems to just know the answer, even though he either can't tell me how he got the answer at all or comes up with some odd, more complex than necessary methodology.

For those with children like him, how have you approached that subject? Do you teach them the "proper" way to solve the problems or just let them work on their own until they are taught this in school (in our area, my son won't formally learn pre-algebra for another 2-3 years). He wants to learn more advanced math, and creates problems to ask us on his own. I don't want to do anything to kill his love of math or even his creativity in solving the problems, but I also worry that his approach could hurt him down the road. Could it? Or is the creativity simply a sign of problem-solving/critical thinking skills?

Our oldest has crazy math instincts and I've often remarked to DH that he just knows it intuitively. He is younger than your DS, and most of the complex math we do is just for fun while driving, but i do sometimes encourage him to walk through his process and explain how he got there. I know it's a skill he'll need later on, and since it's very light-hearted, I think it also helps to draw him in a bit deeper into the magic (and method) of it all.

I think you need to separate two things that are going on here. Intuitively knowing how to tackle a problem, inventing methods for oneself, is a strength, not a problem - it may be worth demonstrating more efficient methods sometimes, and discussing different ways more generally, but that's all. So "odd, more complex than necessary" methods don't bother me, though when he gives such a method I'd try to discuss it. (Watch out for methods that are not over-complex but just different from what you were expecting!)

Not being able to explain, though, is another matter, and it mystifies me why adults sometimes seem to admire this (not you necessarily). It's actually the explanations that are the maths, tbh. I wouldn't count a question as answered unless the child can explain why the answer is correct. That doesn't have to mean "show your work" in a tedious way. It can be more useful to play an "I don't believe you" game - push for a gradually more elaborated argument by questioning selected steps. If there's really no explanation available I'd reframe the answer given as a guess: "OK, so you guess the answer's going to be ... Let's see if we can work out whether you're right."

ETA one thing that can cause impressive leaps is being really good at visualising, and then it can be hard to know where to start in explaining method. Might be worth explicitly suggesting beginning by drawing a picture, if you think that might be happening.

I agree with ColinsMum-- the explaining IS the part that must happen in order to do higher math.

There's not a right "method."

But there is a right way to communicate what you are using as a method.

It has to be valid as a set of instructions for another person in order to be correct, basically.

I was in the intuitives zone and trying to figure out the bridging between how I understood things and to explain/codify it as a pre-teen eventually became a career defining process.

The path I'd suggest is to separate the two elements even further apart. Rather than him explain or challenge him on any particular problem, I'd look for him to instruct you in how to solve a problem. That external need can sometimes sharpen the introspective skills. I also think strategy games may help because they often require projecting specific thought processes onto another person.

DS7 is in a different category as he gets very positive remarks about how well he'll explain math.

Tutoring others has improved my DD's ability here dramatically.

But it's the intuitive jumps that lead to the higher math being discovered in the first place. The explanations are how you prove to others that the jump was correct, and the road map that lets others follow in your footsteps.

DS is math-intuitive, and it does get very frustrating, trying to get him to "show his work". I just asked his new teacher, in her "syllabus questions" homework, what happens if he doesn't have any work to show? Many things happen in that boy's head, and some of them are scary. If she wants the answers out of there, she's going to have to get them herself, because when I try, it makes my brain melt.

The intuitive jumps are important, sure. But they only give you conjectures, which are worth very little without proof. The explanations are how you satisfy yourself that what you believe is true. Otherwise there wouldn't be a million dollars riding on each of the Millenial Problems, and Andrew Wiles needn't've bothered.

Don't let your kids think they've finished when they've got the answer in the back of the book - really honestly, if they can't explain, they've hardly started.

My son needed more than we would have been able to accomplish with an "I don't believe you" game. He needed a systematic teaching of the language you use to explain his thinking. He has clear language quirks in his development, so it did seem like a necessary intervention.

Recently he told me that he takes a problem and knows the answer, and with the language stuff his teacher taught him, he goes back and translates it into steps. He does this for things no one ought to be able to do in their head, and it doesn't see like he's working it out in his head, but he knows the answer somehow. He could do this for a situational problem before he even knew what the words "multiply" or "subtract" even meant. This might be an elementary/odd form of intuition -> conjecture where he needed to be taught the language of a proof.

I guess my point is that we've seen a lot of positive out of directly teaching "this is how you justify your intuitive result."

The Eides wrote about this in their Dyslexic Advantage book. Some kids really learn math backwards - they know the answers and sometimes it takes them until their teen years to understand and work out how they got the answers. They believe that kids like this should not be penalized for not showing their work. Also, many kids have been ruined by teachers and others with the attitude that they must show their work. They said that they see kids with near perfect scores on the arithmetic section of the WISC IV that are failing math and hate it because they truly cannot show their work yet.

Brains are wired differently and we should applaud the strengths of differences.