10 Nov 2005 - 15:57

discovery learning with KUMON


picking up speed faster than you expect
Interesting.

I've mentioned before that, in my experience, kids pick up speed on timed tests unbelievably quickly.

I can't find the post I wrote about this, so I'll tell this story again.

A friend of mine has a son who has math talent that isn't being recognized because he's a high-level special needs kid (so high that the school is trying to declassify him). He came to my after-school class last year, where I gave him a Saxon Math Fast Facts sheet to do. The time limit is 5 minutes.

The first time he did one it took him 10 minutes. Something like that. He was so slow, I decided I would do his handwriting for him. I told him just to say the answer & I'd write it down. (He'd been in the multi-sensory class, which emphasizes handwriting, and now had beautiful but very slow & deliberate penmanship.)

So the next time he came to class, probably a week later, I did the writing, and it took him 8 minutes to do the sheet.

It didn't seem to me that my being the answer-writer was helping matters any, so the next week I had him do his own writing.

He did the sheet in 5 minutes.

He starts out with a time of 10 minutes; 2 sessions later he's hitting 5 minutes.

With no practice in between.

♦ ♦ ♦ ♦ ♦

That is now happening to me.

Last night it took me 7 minutes to do one of the KUMON fraction reduction sheets, which are supposed to be completed in 3 to 4 minutes.

Today I'm down to 3 to 4 minutes.

I got faster overnight.


efficient drill
The notion of efficient drill is something I never hear people talking about.

Constructivists oppose timed tests & an emphasis on speed on principle:

De-emphasis of rote work. We believe that children must indeed learn their math facts, but we de-emphasize rote memorization and the frequent administration of timed tests. Both of these can produce undesirable results. Instead, our goal here is that students learn that they can find answers easily using strategies they understand.
source: TRAILBLAZERS Teacher Implementation Guide

Math-content types like us, along with cognitive scientists, believe in practice, practice, practice:

Q: How long does it take for Mozart to become Mozart?

A: 10 years of 12-hour a day practice.

Remember this?


cogsci.jpg


But this is looking at things the wrong way, I think.

We're talking about math facts and procedures, not symphonies. The goal of practice in math, at least in the early years, isn't to become brilliant fraction-reducers.

The goal is to become efficient and accurate fraction reducers.

There's no reason to be talking about 10-year timelines of 12-hour a day practice in the context of students learning math procedures.

The question we should be asking is: how quickly can a student learn math procedures to mastery?

And: do some worksheets produce more efficient learning than others?

I don't know the answer to either one, but I think I do know, at this point, that the focus on 'hard work' and 'practice, practice, practice' is distracting us from the question of efficiency and acceleration.

I'm thinking the KUMON approach to creating worksheets, which is the opposite of the randomly-generated, mixed-practice approach American texts & web sites tend to take, may promote faster learning.


discovery worksheets
I think I've run into a snag with my KUMON worksheets, which is that I'm having to do a fair amount of discovery learning.

This week's worksheets require the student to reduce fractions 'in one step.'

I was never taught how to do that systematically.

(Apparently, I was also not taught to READ THE DIRECTIONS FIRST. On sheet D195b, I didn't notice the line saying "Some fractions can be reduced by 13" until I posted it here on ktm.)

The worksheets are designed to be supplemental, so I'm assuming that, in Japan, students must be taught to reduce fractions rapidly and systematically.

I can reduce fractions rapidly, just because I'm fast. But I do it in two steps. If I don't instantly see the GCF, I start with the factor I do see, then work on from there.

These worksheets expect a student to instantly see the greatest common factor.

Last night, after my dismal 7-minute performance, I started thinking how a person would go about instantly seeing the greatest common factor. I came up with various things, none of which mean that I instantly see the greatest common factor, but all of which help. (I didn't get a chance to print out people's Comments on prime factorization last night, because the computer had locked up. So I was on my own.)

For instance, First check to see if both numbers are even. Second, check to see if one number, when divided by 2, becomes a prime incapable of further reduction. Another one: if you've still got two even numbers after the first division, then the GCF definitely has a 4 in it. Look to see if the numerator itself is the GCF. And so on.

This is all to the good; my conceptual understanding of prime factorization—which was already pretty good, thanks to Russian Math, is jumping.

But it's haphazard, and I don't think it's probably what the KUMON folks had in mind.


195afractionreduction.jpg


195bfractionreductionsm.jpg


teaching efficiency
I've mentioned any number of times now that it's extremely difficult to get a child—any child—to budge from a procedure he's learned to mastery.

The problem is, all of the procedures kids learn when they're just starting out need to be pared down at some point. Christopher still, when he does the four operations, crosses out the number he's borrowing from, writes in a '9' or a '10,' etc. He does nothing inside his head.

He doesn't need to keep writing in all those digits, but he does.

I'm wondering whether Japanese curricula explicitly teach efficiency as a goal. If you tell a child, from the beginning, that the goal is to solve problems as efficiently as possible, might that be the grade school equivalent of mathematical elegance later on?

And is that part of the point of the KUMON worksheets?


Susan on Mad Minutes; Carolyn on college kids who don't know their math facts



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Catherine,

Regarding GCF, without lookng back, I believe previous Kumon worksheets concentrated on GCF so the idea is (I think) to carry that process forward. That said, I don't think it ever worked with my eldest. She could do the GCF, but I don't think she used that process in reducing fractions.

Chris

-- ChrisAdams - 10 Nov 2005


The question we should be asking is: how quickly can a student learn math procedures to mastery?

Engelmann says to teach it to reasonable firmness and move on, don't get bogged down. Future lessons will reinforce what was learn and provide sufficient opportunities for more practice and mastery. You know, spaced repetition and all that.

do some worksheets produce more efficient learning than others?

This is where good field testing comes into play. Continually test and revise the worksheets/lessons until the students are capable of mastering the material.

-- KDeRosa - 10 Nov 2005


These worksheets expect a student to instantly see the greatest common factor.

...

For instance, First check to see if both numbers are even. Second, check to see if one number, when divided by 2, becomes a prime incapable of further reduction. Another one: if you've still got two even numbers after the first division, then the GCF definitely has a 4 in it. Look to see if the numerator itself is the GCF. And so on.

I know you were just listing strategies here, but I think the way to try to get to the GCF instantly, is to start with the last thing you mentioned: Look to see if the numberator itself is the GCF. If not, then try half of the numerator, then one third of the numerator. Work through from there. It means you have to hold two division problems in your head at once: the numerator divided by N, and the denominator divided by the quotient of that first division.

-- DanK - 10 Nov 2005


Engelmann says to teach it to reasonable firmness and move on, don't get bogged down. Future lessons will reinforce what was learn and provide sufficient opportunities for more practice and mastery. You know, spaced repetition and all that.

Absolutely!

I can watch that process happening with me.

This probably accounts for some of the 'delayed getting it' phenomenon various people have mentioned here at ktm.

-- CatherineJohnson - 10 Nov 2005


I know you were just listing strategies here, but I think the way to try to get to the GCF instantly, is to start with the last thing you mentioned: Look to see if the numberator itself is the GCF. If not, then try half of the numerator, then one third of the numerator. Work through from there. It means you have to hold two division problems in your head at once: the numerator divided by N, and the denominator divided by the quotient of that first division.

Thanks!

I've actually pulled together, into one file, everything everyone has said....I think I'll post it in one place, too, with a link on the User's Page.

I'm going to ask the KUMON guy on Saturday if what exactly 'in one step' means!

(I'm wondering if kids in Japan in the 1950s learned GCFs the same way we learn math facts.....)

-- CatherineJohnson - 10 Nov 2005


This is where good field testing comes into play. Continually test and revise the worksheets/lessons until the students are capable of mastering the material.

I think that's what has happened with the KUMON sheets....

-- CatherineJohnson - 10 Nov 2005


I believe previous Kumon worksheets concentrated on GCF so the idea is (I think) to carry that process forward. That said, I don't think it ever worked with my eldest. She could do the GCF, but I don't think she used that process in reducing fractions.

How did she do it?

-- CatherineJohnson - 10 Nov 2005


And did you guys like KUMON?

Did you feel it had value?

How long did you do it?

-- CatherineJohnson - 10 Nov 2005


I'll ask her and report back...

Yes we like Kumon. The younger child, Kate, (4th grade) is grudgingly thriving on it. She is bored to tears in 4th grade math and easily crushes her classmates on timed math tests. Starting in Dec or Jan the elementary school is supplementing the standard "investigations" math with a Kumon type math facts every day. Kate has been doing Kumon for probably 1 1/2 years.

Brittany (9th) didn't start until 7th grade (2 1/2 years ago) and her confidence in math was shattered thanks to "Investigations". Her math skills have dramatically improved but she is presently rebelling against Kumon. The fractions part she coniders tedious and boring so we've been stuck for awhile as she she suffers from a bad case of MDD (Motivations Deficit Disorder) and other 14 year old "daughters knows best" type issues! Also, I think she is frustrated that Kumon is doing fractions and she is doing Algebra in school. She wanted Kumon to help her with Algebra (which it will, when she gets there). Note: Kumon is BIG on fractions.

So we as parents are busy restructuring her life and putting the heat on her to prioritize Kumon and restricting time on the computer, etc, etc.

Yes we feel Kumon has value and it's our intent to continue doing it for some time to come, probably at least through the Jr year of HS for both kids.

I simply don't trust the local school system to teach math in an intelligent fashion. They are so stuck on this "Investigations" math that it's sickening.

That said, I DO think Kumon can be or is tedious sometimes. In general it isn't a program to fix a specific set of math problems but a to learn math in a VERY systematic fashion.

-- ChrisAdams - 11 Nov 2005


Note: Kumon is BIG on fractions.

The fractions are HARD.

I was supposed to graduate to 5th grade tomorrow, but I'm going to have to ask them to keep me back in 4th so I can re-do this week's worksheets.

-- CatherineJohnson - 11 Nov 2005


Tell your daughter you know a grown-up in KUMON 4th grade.

-- CatherineJohnson - 11 Nov 2005


In general it isn't a program to fix a specific set of math problems but to learn math in a VERY systematic fashion.

Yes, absolutely.

I have the same disappoinment your daughter does; I need help with Christopher's pre-algebra. Instead, he's doing 3rd grade worksheets.

However, now that I know what KUMON is, I'm fine with that.

I just hope he can catch up to himself quickly.....

-- CatherineJohnson - 11 Nov 2005


Supposedly ages 12 to 16 are the worst in terms of parents having any influence whatsoever. After that it gets better again.

We saw that with our niece.

-- CatherineJohnson - 11 Nov 2005

WebLogForm
Title: discovery learning with KUMON
TopicType: WebLog
SubjectArea: AboutCurricula, KumonProgram
LogDate: 200511101052