BrianMickelthwaitOnKumon < Kitchen

03 Nov 2005 - 18:07

Brian Micklethwait on KUMON & hand-made math

I'm in the middle of reading Brian Micklethwait's terrific article about his experience as a KUMON instructor, but had to stop and post this passage:

There is also in Kumon what I think of as a very Japanese emphasis on the physical process of drawing the numbers and on physically handling the world generally. (Think of the Japanese fascination with hand-done graphics.) One of the ancillary games we get the children to play is simply placing numbers on a number board. This doesn’t just help them to understand numbers. It also helps them to get better at simply handling things, while thinking at the same time. As with so much of Kumon, doing the number board so that every number is where it should be is in principle very easy, so no child is humiliated by not being able to do it. But doing it fast isn’t so easy, so the cleverer ones are kept interested. (We also give the cleverer ones more complicated things, like “leave on the board only those numbers divisible by 3”.)
This emphasis on the physical handling of the world also explains, I think, why the Kumon people are so reluctant to get involved with computers. To me, an Anglo-Saxon techno-nerd, Kumon absolutely shouts computers. Each child doing an individually selected clutch of repetitive problems. Relentless and potentially very tedious marking. Even more tedious analysis to tell you what each child should be doing next. A huge apparatus of collective, centralised analysis to see which methods work best and to tell the rest of the world. This is surely the sort of stuff that computers — and their recent combined offspring, the Internet — were invented to supervise. But I sense that the Kumon people resist such notions. There’s so far been no mention of computers in any of the Kumon back-up or sales literature that I’ve seen. Computers, I hear them saying, would only complicate things.

I've come to believe that paper-and-pencil math is math—that there's something necessary, at least when you're learning,^* about the experience of actually holding a pencil or a pen in your hand and solving problems.

Carolyn talks about the craft of math; Temple repeatedly & chronically encounters people who've learned to create scale drawings on computers and, as a direct result, cannot construct scale drawings. (Temple believes that the visusal processing and motor systems in the brain are connected. I won't be surprised to learn that she's right.)

I've been surprised at how unmoved Americans are by the Singapore bar models. I fell in love the instant I saw them, and wanted to draw them. With Sybilla Beckmann, I think the bar models are probably the reason for the Singapore curriculum's success.

I've mentioned several times that I've worked at least 300 bar model problems. I've said, too, that doing this changed my brain. I'd put money on it.

The thing is, I really don't know why this should be the case. I'd been thinking maybe they develop spatial reasoning, which is connected to mathematical ability.

It hadn't occurred to me that bar models might work simply because they involve lots more pencil-and-paper work than the traditional U.S. math curriculum.

But the explanation may be as simple as that.

When I first started drawing bar models, I badly wanted to paint one. I wanted to do a big, bold 'blow-up' of a Singapore bar model in oil, and hang it on the wall.

Maybe one day I will.

what is the opposite of a fount of wisdom?

Here's Steve Leinwand:

Shouldn't we be as eager to end our obsessive love affair with pencil-and-paper computation as we were to move on from outhouses and sundials?

The answer is no.

^*Temple says that older people who learned to draw by hand & then switched to CAD have no problems at all. The problems turn up strictly in the work of younger employees, who've never done physical scale drawing using pencil and paper.

Swoop and Swoop
the craft of math

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what is the opposite of a fount of wisdom?

The toilet of crap.

-- SteveH - 03 Nov 2005

When I first started drawing bar models, I badly wanted to paint one. I wanted to do a big, bold 'blow-up' of a Singapore bar model in oil, and hang it on the wall.

Piet Mondrian beat you to it...

-- DanK - 03 Nov 2005

One of the things I've found interesting about the bar models is how much of a bridge it is from the pictures to the abstract. I know we've said that it is abstract, yet it is a picture. Like the number line, it walks you one step closer to where you're going. I think mathematical people might have trouble realizing what a lightbulb moment they can provide.

I'll use an example that is and isn't very typical. My 8th grade LD son works at anywhere from a 6th grade level in math to a 3rd grade, depending on what mood he's in. He has an expressive language disorder and so language often confuses him much more than it would the regular kid. He will never, ever ask for an explanation because he already feels stupid, so we have a lot of trouble trying to find out what it is about something that he doesn't understand.

While I've been using the Saxon 6/5 consistently, I supplement that with Singapore 3 word problems out of the textbook, workbook, or word problem book. I think he could go to 4, but I want him to feel confident for a bit longer and I also want him to get used to drawing out the bar models for easier problems so that when we "graduate" to more difficult problems he will have a good familiarity with them.

The other day he had a word problem about finding the profit of something. It was something about some kid having something for a certain price and then selling it for another. What was his profit? Well he saw the increased price and the word profit. This was out of Saxon and so the the book said something like, "Use a subraction pattern to solve." This confused him even more because he was focused on the number increasing in size and so got lost in the words.

It would have been easy to just say, "You find the profit by subracting the original price from the new one," but I decided to just draw a comparison bar model showing the two prices and the unknown. Immediately, he saw what it was because he's done a million bar models for subraction problems. It was interesting how completely he understood his mistake by the visual. It would have taken me several minutes of explaining and doing more "profit" problems before he would have gotten it.

We're moving into division and fraction word problems and I think his understanding is going to be highly enhanced due to the constant bar model use. I wish it had been done all along.

-- SusanS - 03 Nov 2005

Dan

LOL!

You're right!

The funny thing is, thought, that I've never liked Mondrian!

btw, I'm still majorly impressed that you were able to invent a Singapore bar model for that problem just from my fractured description of it.

Kids spend a lot of time trying to learn that particular representation.

-- CatherineJohnson - 03 Nov 2005

toilet of crap

Yes, I believe that is Logically Consistent

-- CatherineJohnson - 03 Nov 2005

I know we've said that it is abstract, yet it is a picture.

I wish I had the Hierarchy of Abstraction talk some autism expert gave here.

I didn't hear it myself, but it was obviously extremely helpful, because she laid out levels of abstraction.

In the autism world, people had been assuming that an image—any image at all—was by definition concrete. (I talked about all the trouble they had with kids who couldn't 'read' these supposedly concrete images. Jimmy couldn't decode a Meyer-Johnson PECS card to save his life, while Andrew was a freaking PECS genius, I kid you not.)

Anyway, my guess is that the bar models would be at the top of the abstraction hierarchy for images.

They'd be below mathematical symbols.

-- CatherineJohnson - 03 Nov 2005

I think mathematical people might have trouble realizing what a lightbulb moment they can provide.

HD!!

(highly diplomatic!)

-- CatherineJohnson - 03 Nov 2005

I think he could go to 4, but I want him to feel confident for a bit longer and I also want him to get used to drawing out the bar models for easier problems so that when we "graduate" to more difficult problems he will have a good familiarity with them.

Stick with 3!

I backed Christopher all the way to the beginning of 3A, and it's been great.

The funny thing is, I was anticipating KUMON in this. They start you out way easy, and it works.

But boy, American kids & American parents (and apparently British parents) HATE THAT!

-- CatherineJohnson - 03 Nov 2005

wow

Interesting.

Again, the bar models have radically improved my own understanding of elementary math....uh-oh, gotta go (nagging around here!)

-- CatherineJohnson - 03 Nov 2005

Catherine,

I'm pretty sure I worked backward to get to that "bar model." As I mentioned in the other thread, that problem reminded me of the math exercise involved in hanging a picture or centering a shelf on a wall. So, I think I should launch a curriculum of Handyman's Math.

The "bar model" illustration was then just my attempt to illustrate the handyman's logic that I had used.

-- DanK - 03 Nov 2005

Dan

I have to re-read that problem.

Here's a question.

You know the kind of question about 'a repair shop has x vehicles, some motorcycles & some cars; 100 wheels altogether; how many cars & how many motorcycles....'

Carolyn was saying, awhile back, that she was pretty sure you can't model that kind of problem.

Naturally I don't think you can model it (bar model it).

Is that right?

-- CatherineJohnson - 03 Nov 2005

car & motorcycle problem

Cars and Motorcycles

A repair shop fixes both cars and motorcycles. Last month, it repaired a total of 40 vehicles. The total number of wheels on those vehicles was 100.

How many cars and how many motorcycles were repaired?

Is it possible to draw a bar model for this??

-- CatherineJohnson - 03 Nov 2005

thought experiment

Could a quadriplegic learn math?

-- CatherineJohnson - 03 Nov 2005

I'm trying to imagine learning math purely with my eyes, and it's not seeming possible.

On the other hand, in fact, you can imagine movement inside your head, and have it 'work.'

One of the best books I've ever read is Mind Sculpture by Ian Robertson.

He goes over all kinds of research in which injured athletes spent their down-time running the race in their minds.

They kept most of their fitness as a result.

So....you could 'do math' in your mind by imagining holding a pencil and moving it.

What I don't get is why the keyboard doesn't seem to do the same thing for people.....(though in the case of CAD, I do see why it's not the same).

-- CatherineJohnson - 03 Nov 2005

oh my gosh!

Robertson has written a sequel:

Opening the Mind's Eye How Images and Language Teach Us How To See

Unfortunately, he has apparently included a lecture on the Correctness of agnosticism.

sigh

Hey!

None so blind as those who will not see!

Open up your mind's eye, Ian!

I'm getting the book.

-- CatherineJohnson - 03 Nov 2005

I'm going to have to dig out my copy and re-read:

It is well known that repetitive actions or tasks become “automatic” after a while. Some skills, like walking or chewing, we take for granted; others, like touch-typing, must be learned. When a task is executed automatically, it is said to be done “subconsciously.”

Anything we do without conscious thought is done subconsciously. The average person performs many subconscious actions in the course of his or her daily routine (an amazingly large number, if one really thinks about it). Yet few people realize how the brain handles these subconscious actions.

The first few chapters of “Mind Sculpture” explain how the brain physically reconfigures itself (a process called Hebbian Learning) in response to repetitive actions or stimuli. Our brains are not “hard wired” from birth. New connections between previously unconnected brain cells are formed each time we learn a new skill or form a new association. As we learn new skills, we really are physically “sculpting” our minds.

Hebbian learning occurs because “cells that fire together, wire together.” Take the example of a billboard advertisement which shows a particular automobile alongside a beautiful woman. Two sets of brain cells will fire in response: the brain cells that control sexual arousal and the brain cells that recognize the auto. After repeatedly viewing the billboard, these two unrelated sets of brain cells become physically wired together simply because they have fired together again and again. Eventually, seeing the auto by itself will cause the arousal brain cells to fire and the prospective buyer to feel “turned on.”

The ramifications of the brain’s ability to re-wire itself are the subject of the rest of the book. How the brain can repair itself after suffering damage is explained in detail, as well as the brain’s response to loss of limb or sight, etc.. Learning and emotional trauma are also examined.

The author includes real life examples of people who have recovered from injury or trauma. He also points out that many victims could probably realize a fuller recovery if only more emphasis were placed on using the brain’s untapped potential: our mind’s ability to sculpt itself.

He also had all the fabulous data on black children in Southern schools in the 50s.

Their IQs would decrease the longer they were in school.

When the families moved north, their IQs would start going up again.

Great, great stuff.

-- CatherineJohnson - 03 Nov 2005

As one of those "mathematical people", I draw pictures all of the time. I am quite pragmatic in believing that you use whatever works. I am also concerned about forcing a tool to do what it really cannot.

-- SteveH - 03 Nov 2005

As one of those "mathematical people", I draw pictures all of the time.

How do you use pictures?????

I'm completely fascinated by this.

For awhile I was emailing with a teacher who specializes in kids with problems learning math (these are quite young kids, in the 1st grade).

She herself can't do much math beyond elementary math, and found algebra difficult, although she's very smart.

I sent her an algebra problem one time (can't remember what it was or where it was from) and she naturally drew a picture to solve it, and in fact solved it correctly.

I was blown away by that, because I wouldn't have been able to draw a picture to help, I don't think.

She said she'd always done it. She was 'bad' at algebra, and the thing that had gotten her through was drawing pictures. I say 'pictures,' but the one she showed me was highly abstract. It was kind of a cross between a Venn diagram and a bar model. She'd created an intersecting-set kind of thing, IIRC.

-- CatherineJohnson - 03 Nov 2005

I'm curious as to why the car/motorcycle problem can't be drawn, assuming it can't be.....

-- CatherineJohnson - 03 Nov 2005

I'm afraid that I neither intuitively understand nor have procedural proficiency with bar models. So I'll try solving the motorcycle problem without explicit algebra in a different way.

Had all 40 vehicles been motorcycles, there would have been only 80 wheels. For each vehicle that is a car instead, there would be 2 more wheels. To get the 20 wheels more, you'd need to have 20/2 = 10 vehicles be cars rather than motorcycles: 30 motorcycles, 10 cars.

This isn't precisely guess and check (cut and try) mathematics -- perhaps guess and correct?

-- DougSundseth - 03 Nov 2005

Steve, you have a good point, but a bar model is not forcing a tool if they haven't been taught algebra. It's just using what they know.

When I looked at some of the word problems of Singapore 5 and 6 I immediately used algebra to try to solve them. And it was much quicker and to the point. I was put off by the elaborate bar models and the units. Finally, I just tried a couple and it was interesting that it worked and was clearer somehow.

I just find it interesting that Singapore does that right before officially going into algebra. Is it just for conceptual grounding or is there something else that my limited math background won't let me see?

-- SusanS - 03 Nov 2005

"I'm curious as to why the car/motorcycle problem can't be drawn, assuming it can't be....."

It has to do with units again, and what units the two bars stand for. The two-step bar model problems all (in my experience) require setting up two different bars that have the same units. For example, one statement will be about the difference in cost between two items, so the corresponding bar stands for money. The second statement will also be about money.

Pieces of the first bar can therefore be compared to pieces of the second bar, which allows you to solve the problem.

With the car/motorcycle problem, one bar is naturally going to represent vehicles (cars and motorcycles add up to 40), and the other bar is going to represent wheels (car and motorcycle WHEELS add up to 100). It's actually not impossible to solve with a bar model, I think -- but you have to convert the second bar from 'wheels' to 'vehicles', and that's tricky and not elegant.

Basically, once it gets as tricky as this, the child should be solving the problem using algebra.

-- CarolynJohnston - 03 Nov 2005

I am quite pragmatic in believing that you use whatever works.

Spoken like a true engineer.

Engineers may have pet theories, but anything that doesn't work in practice gets quickly abandoned.

If only our teachers thought like this.

-- KDeRosa - 03 Nov 2005

I believe they use them to expose young kids to algebraic concepts well before they may be ready to learn full blown algebra.

Since we adults have so much procedural fluencey with basic algebra, we find it a waste of time to go through the modeling exercise while the algebra is so simple. We understand the concept already, that's why we can write the abstract equation with ease. The bar models help children gain the conceptual understanding necessary to do break the problem down into mathematic terms in the first place.

-- KDeRosa - 03 Nov 2005

Basically, once it gets as tricky as this, the child should be solving the problem using algebra.

Carolyn, I was working through Singapore Math's Challenging Word Problems 4 and there is a very similiar problem:

"There are three times as many ducks as sheep on a farm. All the ducks and sheep have 2400 feet altogether. How many more ducks than sheep are there?" p 11, problem 24

Here's one I can't figure out:

"17 cookies have 131 chocolate chips altogether. Most of them have 8 chocolate chips each but some of them have 1 chip less. How many cookies have 8 chocolate chips." p 125, problem 19 Huh?

Nothing makes me feel stupid/smart like Singapore Word Problems, depending on my answer of course.

Nicksmama (trying to stay one step ahead of my 9 yo)

-- KtmGuest - 03 Nov 2005

"There are three times as many ducks as sheep on a farm. All the ducks and sheep have 2400 feet altogether. How many more ducks than sheep are there?" p 11, problem 24

That's the same darn problem!

OK, I'm going to figure out how to do this with bar models. I need a challenge.

-- CarolynJohnston - 03 Nov 2005

The two-step bar model problems all (in my experience) require setting up two different bars that have the same units.

That's been my experience, and that's what you & I talked about the first time we went over this.

Last night I struggled with a problem....I'll find it.

Again, it was a problem I could do easily with algebra, but I wanted to see if it could be done with a bar model.

I finally had to reverse-engineer it the way Dan (wait! was it Dan or Doug???--I better check) reverse-engineered the Science News problem.

In fact, it could be modeled, and it was intriguing....in terms of creating a unit.

To solve the problem, or simply model the problem, using a bar model, I had to create a unit out of two units, which were rates of work.

I'll find the problem.

-- CatherineJohnson - 04 Nov 2005

I discovered topology for the first time ever reading Keith Devlin's book, and I was riveted.

Are the bar models like a topology of algebra?

-- CatherineJohnson - 04 Nov 2005

This is the way I've started to think of bar models, as Maps of Abstractions

(I haven't read this page; the chapter heading caught my eye.)

-- CatherineJohnson - 04 Nov 2005

Five turnips and 7 potatoes cost $1.31.
Six turnips and 6 potatoes cost $1.38?
How much does a potato cost?

source:
Brain Maths

This is very easy to do with algebra, but I ended up not being able to model it until after I'd done the algebra.

-- CatherineJohnson - 04 Nov 2005

After struggling with this for awhile, I was able to figure out, and to demonstrate to myself, that the difference between $1.38 and $1.31 also had to be the difference in price between one potato and one turnip.

-- CatherineJohnson - 04 Nov 2005

I'm going to print this out and mull.

-- CatherineJohnson - 04 Nov 2005

This is what's not true for me.

You may not have been reading ktm back when I was talking about this.....I have some kind of almost-purely procedural fluency in basic algebra.

I can solve two & 3 variable linear equations until the cows come home, but I don't have a clue why setting up two or three equations works, or how anyone could have invented algebra in the first place.

That's what I'd really like to know: how could a person invent algebra?

What were they thinking about when they did it?

(I hope I'll be able to read Bernie's book at some point.)

-- CatherineJohnson - 04 Nov 2005

When I was first doing bar models my friend Debbie (another bar model skeptic—they're everywhere!) dismissed them, saying, 'they're just a visual representation of the equations.

Well, I hadn't exactly noticed that, and once I did notice it, it was incredibly cool.

-- CatherineJohnson - 04 Nov 2005

I just realized that there is an assumption built into the language of the turnip and potato problem that anyone familiar with the language of school-book math will probably skip right over.

What if the potatoes and turnips are sold by weight rather than by piece? Without the statement that the vegetables are sold by the piece, the problem is underspecified.

In the language of the problem, there is a hidden assumption that they are both sold by the piece, but that is nowhere stated explicitly. I suspect this might be a problem for at least some people who have actually bought vegetables.

-- DougSundseth - 04 Nov 2005

This is what's not true for me.

Hey. No fair using yourself as a counterexample to my theory.

-- KDeRosa - 04 Nov 2005

No fair using yourself as a counterexample to my theory.

THAT'S WHY THEY CALL ME RUTHLESS!!!!!!!

-- CatherineJohnson - 04 Nov 2005

"17 cookies have 131 chocolate chips altogether. Most of them have 8 chocolate chips each but some of them have 1 chip less. How many cookies have 8 chocolate chips." p 125, problem 19 Huh?

-- CatherineJohnson - 04 Nov 2005

Good Lord

-- CatherineJohnson - 04 Nov 2005

"There are three times as many ducks as sheep on a farm. All the ducks and sheep have 2400 feet altogether. How many more ducks than sheep are there?" p 11, problem 24

That's the same darn problem!

OK, I'm going to figure out how to do this with bar models. I need a challenge.

I can't wait!

-- CatherineJohnson - 04 Nov 2005

I solved the duck and sheep problem. Here goes, "There are three times as many ducks as sheep on a farm. All the ducks and sheep have 2400 feet altogether. How many more ducks than sheep are there?"

sheep /xxxx/ ducks /xxxx/xxxx/xxxx/

There are a total of 2400 feet divided by 4 equal parts. I arrived at this because there are 3 times as many ducks as sheep. Divide the 2400 feet by the 4 units and arrive at 600 per unit or bar. I now have 600 sheep feet and 1800 duck feet. Divide sheep feet by 4 and duck feet by 2 and I have 150 sheep and 900 ducks. All done the Singapore way without algebra.

-- LoneRanger - 04 Nov 2005

OK, let's see.

What did I have to do for turnips & potatoes.

I had to create a unit that was 1 potato + 1 turnip.

Then I did a subtraction-as-comparison bar model

One bar model had 6 potato/turnip units

The other bar model had 5 potato/turnip units, and then two separate little potato units

First you solve for potato-turnip unit: divide $1.38 by 6 & get $.23. The unit one-potato-one-turnip = $.23.

Then you plug the $.23 into each of the 5 one-potato-one-turnip units in the second bar model.

That equals 5 x $.23, or 1.15

$1.31 - $1.15 = .16 So the two left-over potatoes equal $.16

which means 1 potato = $.08

-- CatherineJohnson - 04 Nov 2005

17 cookies have 131 chocolate chips altogether. Most of them have 8 chocolate chips each but some of them have 1 chip less. How many cookies have 8 chocolate chips." p 125, problem 19 Huh?

Divide the 131 chips by the 17 cookies, and you get 7 and a remainder. What is the significance of the remainder? Those are your 'extra' chocolate chips.

-- CarolynJohnston - 04 Nov 2005

Here's my question!

Shouldn't all algebra problems, or certainly all of these simple 2 or 3-variable linear-equation-type algebra problems, be 'modell-able' via bar models?'

On another thread, someone (Paul Miller??) said that bar models are just another symbol representing the equations we set up using letters and numerals.

Well, very often that is the case.

But if that is the case, shouldn't all algebra problems be bar-modellable?

And btw, this goes beyond the issue of pedagogy & when or whether to use bar models, etc.

I want to know!

-- CatherineJohnson - 04 Nov 2005

Lone Ranger is a bar modelling GGGGOOOOODDDD

(inside wrestling joke)

-- CatherineJohnson - 04 Nov 2005

OK, I'm gonna print this out, and go learn unit conversion from Saxon Math.

-- CatherineJohnson - 04 Nov 2005

I solved the duck and sheep problem. Here goes, "There are three times as many ducks as sheep on a farm. All the ducks and sheep have 2400 feet altogether. How many more ducks than sheep are there?"

sheep /xxxx/ ducks /xxxx/xxxx/xxxx/

LoneRanger
The answer is not so simple. There are THREE times as many ducks as sheep. Your answer doesn't agree with this statement. I think this problem requires two bar models. You have the first which refers to the ratio of sheep to ducks, the ratio of feet 2:4 would be the second bar model. I'm not sure how to what to do with the models at this point. I hope Carolyn will post her bars/answer soon.

nicksmama

-- KtmGuest - 04 Nov 2005

Steve, over on the other thread, has another idea about the limits of bar models: Puzzle Zone Question.

And, it was Paul who said bar models are symbols with more redundancy.

-- CatherineJohnson - 04 Nov 2005

17 cookies have 131 chocolate chips altogether. Most of them have 8 chocolate chips each but some of them have 1 chip less. How many cookies have 8 chocolate chips." p 125, problem 19 Huh?

Divide the 131 chips by the 17 cookies, and you get 7 and a remainder. What is the significance of the remainder? Those are your 'extra' chocolate chips.

-- CarolynJohnston^? - 04 Nov 2005

Carolyn, "some of them have one chip less" - wouldn't that mean that some cookies have only 7 chips? The answer is in the back of the book, and it is not 7.

nicksmama

-- KtmGuest - 04 Nov 2005

There are 3 times as many since 1 times 3 equals 3. Work the problem backwards from the solution and you will see that it proves itself.

-- LoneRanger - 04 Nov 2005

Nope, Lone Ranger, you ended up with 6 times as many ducks as sheep.

I think you have to say that it takes two ducks' to make four feet. So, if you have three times as many ducks, you will have only 1.5 times as many ducks' feet. So, you get 3 duck bars and 2 sheep bars that add to 2400 feet. 2400 / 5 = 480 feet per bar.

3 * 480 = 1440 ducks' feet.

2 * 480 = 960 sheep's feet.

1440 / 2 = 720 ducks

960 / 4 = 240 sheep.

-- DanK - 04 Nov 2005

17 cookies x 8 chips/cookie = 136 chips

136 - 131 = 5 too many chips

so 17 - 5 = 12 cookies with 8 chips

the rest, 5, have 7 chips

-- KDeRosa - 04 Nov 2005

"That's what I'd really like to know: how could a person invent algebra? What were they thinking about when they did it?"

You sound like my wife, who has a very high IQ by the way, but doesn't like math. You think too much. I gave that up when I was in High School.

-- SteveH - 04 Nov 2005

17 cookies x 8 chips/cookie = 136 chips

136 - 131 = 5 too many chips

so 17 - 5 = 12 cookies with 8 chips

Nope, you've reversed it

If you have 5 extra chips, then 5 cookies will have 8 chips because they got one extra

-- CarolynJohnston - 04 Nov 2005

Carolyn,

It's the same either way. Divide 131 by 17, and your remainder is 12. Ergo 12 eight-chip cookies, which agrees with Ken's answer.

-- DanK - 04 Nov 2005

Oh, I thought the remainder was 5

never mind... I should read more carefully

-- CarolynJohnston - 04 Nov 2005

OK, back to the motorcycle and car bar model. I made a comment in the other thread that bar models remind me of subtracting equations to get an answer.

1) M + C = 40

2) 2M + 4C = 100

If I want to subtract the first equation from the second, I need to either multiply the first equation through by 2 or 4. If I use 2, I get the following equivalent equation:

2M + 2C = 80

Subtracting this modified equation 1 from 2, I get:

2M + 4C = 100

2M + 2C = 80

- - - - - - -

2C = 20

C = 10

M = 30

Notice that if you create a bar model from these two equations (I wish I could do this graphically) you get the 2 M boxes and 2 of the 4 C boxes from the second equation to overlap with the 2 M and 2 C boxes of the first equation to leave 2 C boxes equal to 100 - 80 = 20.So, 2C = 20.

Of course, I figured this out by looking at the equations and how it reminded me of the standard process of subtracting equations. I don't think that I could have figured out this bar model without working backwards.

It seems to me that bar models will only work (naturally) if the bars overlap and eliminate one of the variables. In the above case, the M variable boxes were eliminated, leaving only the C boxes that I could relate to a constant.

In the Raincoat - Hat - Boots problem, two of the three variables (box types) overlapped (conveniently) so that it reduced to a single variable problem. If this condition doesn't happen normally, then I don't know how you can fix it graphically. I suppose you could come up with rules for multiplying boxes like you multiply equations, but that sounds like a lot of trouble.

With the equations, I can see exactly how to modify one equation to get it to subtract nicely from the other, leaving just one variable.

-- SteveH - 04 Nov 2005

My contention is that any set of linear equations can be modeled with bars - just don't ask me to do it if it has more than 2 variables.

-- SteveH - 04 Nov 2005

Steve

and

My contention is that any set of linear equations can be modeled with bars - just don't ask me to do it if it has more than 2 variables.

That's what I think, based on practically nothing (based on 'gut'??)

Actually, that's what I think, based in my cognitive unconscious.

But I have no idea whether my cognitive unconscious has competency to hold an opinion.

-- CatherineJohnson - 04 Nov 2005

A life-altering book:

One of the fascinating results of research on the cognitive unconscious is that your cognitive unconscious is quite a bit more accurate than consciousness.

They've done all kinds of interesting experiments in which people's behavior shows that they've figured out the rules governing, say, an artificial grammar.

But when you ask people what the rules are they come up with off-the-wall explanations.

-- CatherineJohnson - 04 Nov 2005

WebLogForm
Title:	Brian Micklethwait on KUMON & hand-made math
TopicType:	WebLog
SubjectArea:	AboutCurricula, ElementaryMath, KumonProgram, SingaporeMath
LogDate:	200511031306

Attachment	Action	Size	Date	Who	Comment
mon310_b.jpg	manage	2.0 K	03 Nov 2005 - 18:59	DanK

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Brian Micklethwait on KUMON & hand-made math

what is the opposite of a fount of wisdom?

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