Gifted Issues Discussion homepage Forums General Discussion Intuitive math

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!