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.

I was told by a Principal that with the new Common Core they don't care if you get the ANSWER RIGHT- they care that you know HOW you got it! They want the child to show they have the critical thinking skills.

My DS6, in 2nd but doing 3rd grade math, does everything in his head and dislikes showing work- but he is able to state how he comes up with answer, if asked.

I was told they don't want you to MEMORIZE facts, but be able to "do" them.

These things are much a matter of the interpretation and implementation of the CC by the curriculum. Third grade includes standards that say "fluently add and subtract" and fifth grade includes "fluently multiply" statements. The degree of fluency is not stated. Our district has an interpretation that they claim comes from the standards of 3 seconds per problem.

Right answers vs process is a debate that's been raging much longer than the CC, and again, it appears to arise largely through the implementation of the curriculum.

CC has long appeared to me as an issue of rearranging the deck chairs on the Titanic. Same chairs, same boat, same iceberg.

If ever a statement were worthy of gilt calligraphy and a frame...

though I like MoN's assessment of the situation, too. It makes no sense to spend all that time in elementary worrying about one-to-one correspondence and moving from physical manipulatives into symbolic math if you just IGNORE the 'why this works' part and never again pull out to look at "big picture" stuff.

Sheesh. No wonder kids loathe and fear geometry and calculus now. It's all a Busby Berkeley spectacular with dancing variables and symbols to most of them by that point. [sighhhh]

DD did intuitive math at 2 but you creating a framework for solutions does not mean diminishing talents. And early algebra was in grade 3 math with CTY.

No reason not to pursue online math and let them get challenged. Having been a kid that found school work easy, I suggest let them do the online math to a point of curiousity and challenge. Taking the easy road of knowing the answer, in my experience leads to bad habits and laziness. Though math was pretty easy, even in university engineering. But heat, math and momentum was another story. Good habits would have been welcome then.

I miswrote, it was heat, mass and momentum. Freudian slip. The thought of the course still makes me shiver.

MoN brings up a great point - why worry about 'deep understanding' of self evident facts and then abandon that when things get a bit more involved?

I continually stress to DD that numbers are just instances of abstract thoughts as opposed to something in their own right ( I hope that wasn't too incoherent ) to try to get her to see the principles and not the mechanical steps.

We use pictures and diagrams a lot at home. Things like; x^2 + 2xy + y^2 boiling down to (x + y)(x + y) become easy to understand with some squared paper, a ruler and some colored pencils. Similarly, several different size cylinders ( like jars ), some string and a ruler help drum the constant Pi in.

The world around you becomes SO much more interesting when you begin to realize how much Maths and physics work together all around you - well it did for me at least LOL and so far, DD appears to learn things a lot like me.

I agree 100% with Wren about learning from being challenged - I never was as a school boy and I didn't really learn how to learn/study as a kid at all. Not a situation I want to see repeated with my DD at all.

I did math intuitively as a child and nobody stopped me- probably because nobody expected me (as a girl in a southern town) to really need math beyond balancing a checkbook. I actually adored math and loved just knowing the answer and then proving the answer to be true by plugging it in- particularly in Algebra.

When I got to college and thought it would be fun to take Calculus as a freshman English/Theatre major, my math world came crashing down a bit. The professor there used a model that involved programming the computer to solve the problems. In other words, we had to teach the computer how to get there. I took a 5 credit B hit to my GPA and considered math done.

I thankfully still love math. But I do wonder what I could have gotten from it if I'd known better how to get to where I was getting. I've gone back and taught myself some and continue to do so, but I sure wish someone had insisted on my knowing the process earlier on.

When my DD was a five year old in first grade, she was finishing up a homework assignment, and the last question in the textbook instructed her to explain how she arrived at her answer to the previous exercise. She didn’t know what to write because she just knew the answer. I thought the question was asinine and told her it was o.k. to write that she just knew. She wrote, “because my spirit told me so.” Her teacher wrote that she loved the answer and explained to me that she didn’t expect students to do the “write about it” questions at the end of the exercises. She thought they were silly at this point too. Luckily, DD has had great math teachers who realize the shortcomings of the textbooks they have to use.

DD is now 10 and will be taking seventh grade advanced math this year. She is just starting to really be able explain her thinking. I think this is a result of her brain naturally beginning to develop new cognitive skills. I also find that these explanations don’t emanate from the problems she knows the answer to, but from the problems she has trouble solving. Whenever she faced a problem she wanted help with, I always responded with questions: “what do you need to find out?” “what do you already know?” “what do you think you should do first?” Lately when she’s been stuck, she’ll start explaining her process, explaining what approaches she’s eliminated and why they won’t work, and ultimately explaining why her answer is right without me ever asking a question. What begins with a “mom, come here I’m stuck,” ends with her giving me a mini-lesson on problem solving.

There are two dangers in pushing kids to “understand and explain” before their brains are ready to do so. One is that they can become so frustrated they wind up hating math and believing that they’ll never “get it.” The other is that in trying to explain complex concepts too early many teachers and textbooks try to simplify these complicated concepts for young students and get the math wrong.

This is a great explanation from an old Keith Devlin piece in his “Devlin’s Angle” column sponsored by the Mathematical Association of America. In it, he makes the distinction between “functional” and “conceptual” understanding of mathematics.

“I think it probably is possible to achieve understanding along with skill mastery for any mathematical topic, but it would take far too long, with a likely result that the student would simply lose heart and give up long before achieving sufficient understanding. . . . Thus, whereas conceptual understanding is a goal that educators should definitely strive for, we need to accept that it cannot be guaranteed, and accordingly we should allow for the learner to make progress without fully understand the concepts.”

Here’s the link to the article and the follow up articles that resulted from his suggestion at the end that teachers stop “saying that multiplication is repeated addition,” one of my own pet peeves.

“What Is Conceptual Understanding?”
http://www.maa.org/external_archive/devlin/devlin_09_07.html
“It Ain't No Repeated Addition”
http://www.maa.org/external_archive/devlin/devlin_06_08.html
“It's Still Not Repeated Addition”
http://www.maa.org/external_archive/devlin/devlin_0708_08.html
“Multiplication and Those Pesky British Spellings”
http://www.maa.org/external_archive/devlin/devlin_09_08.html
“What Exactly Is Multiplication?”
http://www.maa.org/external_archive/devlin/devlin_01_11.html

Perhaps we should teach kids that multiplication is an Abelian group endomorphism.

OMM, tremendous articles you posted links to - thank you very much!

Devlin seems to believe that math is all abstract and the goal of math education is to study the abstractness, whereas other people try to relate math to the real world to make math concrete and useful.

Devlin's often interesting, but he doesn't half have some strange ideas, and I say that as a mathematician.

I read the articles along with some of his other stuff and concluded that he thinks that the goal of maths education ought to be in adequately preparing people for being able to think with rigour and to abstract the essential principles at work in any given situation.

While I don't particularly agree with everything that he writes, I do like the pithy way that he expresses it.

I have just started homeschooling math with my 7 yr old, who is not super "mathy" at all. She's about 18-24monthd ahead of appropriate grade (she's skipped), so she's certainly not struggling, she's just not rocketing along naturally the way many kids here do. I'm finding it interesting that she is far better conceptually and at problem solving than she is at facts/arithmetic, it's hard at times to balance what she understands against the fact that she doesn't "just know" things that seem like they should be "self evident" (like addition and subtraction within 10). So she has struggles with fairly simple problems that are not about the concept or the idea of the method so much as the calculation, even when broken down to simplest facts.

She can be faced with "81-20" and mutter out loud "8-2...5...50..51" so no trouble getting that she should knock the zeros off, or deal with the ones later, but stumbles with the most basic computation.

I've been reading Liping Ma's book "knowing and teaching elementary mathematics" and finding it fascinating, I'm planning I read the articles above too. It seems like there are endless trends about how to best teach basic math that make little or no difference when the people teaching have no more understanding of the concepts than they did during the previous fad. There are some startling examples in the book of manipulative or concrete examples being hopelessly counter productive.

My DDs teacher has just sent home the class newsletter and has apparently been at a conference on teaching math and is now convinced that its detrimental for young students to learn math by writing down algorithms before they can do double digit addition and subtraction in their heads... So now she's all about making sure she avoids writing things out with her 2nd graders... Whih will suit some kids really well, but seriously why do schools follow each new trend with such reckless abandonment? Where is the middle ground and balanced multi pronged approach?

This seems timely-- Devlin just had a Coursera course go live. Probably not suitable for the youngest kiddos here, but could be a good fit for the 10y+ set.

Intro to Mathematical Thinking

It looked interesting, so I thought I'd give it a whirl. I'd really like for my DD to take a look at it if it is a good course, as the ideals espoused in the mission seem positive.



There is fairly extensive coverage of proofs, which makes me pretty pleased.

I prefer to stick to axioms like the distributive property:

http://www.artofproblemsolving.com/Videos/external.php?video_id=21

I was parodying Devlin's gratuitous smartypantedness.

I know, I was parodying yours.

How timely. After reading Devlin go on and on about addition and multiplication being different operations, DD brought home some vocabulary homework from her math class, and needed help looking up the various properties of operations, since they don't show up in her dictionaries.

And so we came to "identity property," and I immediately exclaimed, "Which one?"

Addition: n + 0 = n
Multiplication: n x 1 = n

They're totally different rules. Here was an immediate illustration of how thinking of them as totally different operations can help make sense of them.

Product and divisor were on the word list, addend and sum were not, so I assumed they wanted the properties of multiplication.

I don't understand your use of the word "prefer" here.

Since Abelian groups are commutative, I prefer the distributive property because it distinguishes multiplication from addition and the commutative property doesn't. Also the distributive property is a concept elementary students can understand, so we don't need to get into endomorphism to explain it. Sorry to be so confusing, I was just trying to make the point that we don't have to read Devlin as arguing that we should teach college concepts to elementary students; we can explain multiplication in ways students can understand without teaching them an incorrect definition that has to be undone later.