I haven't read through yet myself, but skimming I found this sheet on the distributive property:





What do you think of this visual mode of teaching & representing the distributive property?

I'm thinking.....it's kind of cool.

otoh, I don't think this way of drawing it is quite 'sharp' enough, but I'm not immediately seeing how to alter it to make it work (or possibly work).

So....yoo-hoo, Doug!

If you feel like taking a crack at this, you could be doing the World of Pre-algebra a Major Service. And Lord knows, the World of Pre-algebra needs help.

homage to Russian Math, too

Another thing that will have to wait.

However, since I've begun using one today, I'll mention it now.

RUSSIAN MATH uses 'Out loud' problems to teach concepts.

'Out loud' problems are problems the student solves without pencil and paper (no calculator, either); technically they are mental math.

However, they are quite different from the mental math problems I've seen.

I've seen mental math used in two ways:

• to develop number sense
• to overlearn math facts

Instead, Out loud problems are a teaching tool in which the problem or concept to be mastered is presented in its simplest form so that the student is practicing the concept, not the calculation.

A simple example. In a lesson on multiplying fractions, the Out loud problems, which always appear at the beginning of a problem set, would be super-easy problems of the 1/2 x 1/2 variety. They are so easy you barely see the math; you 'see' the procedure or the concept instead.

Recently, I hit on the idea of using RUSSIAN MATH to supplement PRENTICE HALL PRE-ALGEBRA by using the Out loud problems. Christopher can't do any more problems than he's already doing (he could, but I'm not going to ask him to); he doesn't have much time to be studying a second math textbook, either.

Out loud problems solve part of that problem. For one thing, kids like Out loud problems. I don't know why, but they do.

For another, an Out loud problem takes all of the extra handwriting/copying/keeping columns of numbers lined up straight/remembering the numbers/etc. burden off of a student's frontal lobes. Out loud problems are incredibly 'clean' in that way.

I assume that's why Christopher, who fights me tooth and nail on extra work I ask him to do in any subject, is perfectly happy to do a set of Out loud problems.

So I'm going to see what I can do with Out loud problems for the distributive property.

what is the proper sequence of Out loud problems for the distributive property?

For example. Christopher simply can't distribute a negative factor to a negative addend. It's impossible.

I've written an Out loud sheet with four columns:

First column: All problems are of the form: 6 (x – 3)
Second column: All problems are of the form -3(x + 2)
Third column: All problems are of the form -4 (3 - x)
Fourth column: All problem are of the form -3 (-x - 2)


This will probably help, but it's not enough. I think I also need a sheet of problems demonstrating the fact that -x can also be expressed as -1x.

I'm also thinking I need to back up even further and do a sheet of strictly numerical distributive property problems:
3 (2 + 4) = 3(2) + 3(4)

and a second sheet (or column of problems) reversing this formulation:
3(2) + 3(4) = 3(2 + 4)


I'm thinking I may also need sheets like this using negative numbers.

And at this point I'm starting to get addled.

What do I actually need in order to teach the distributive property to Christopher in a way that makes sense?

If any of you have ideas, let me know.

a Saxon failure

Saxon 6/5 teaches the distributive property, but Christopher never came close to getting it, either conceptually or procedurally. This was, I think, a case of incoherence. There would be a lesson on the distributive property, 6 problems of Lesson Practice, and then you wouldn't see it again for weeks. By the time a lesson formally touching on the distributive property came up again, Christopher would have forgotten the first lesson entirely. The small bits of distributed practice didn't help.

In this case, Saxon's mode of teaching the distributive property was akin to spiralling, only within the same book. Christopher kept getting 'exposed' to the distributive property, but he never learned or remembered it. For distributed practice to work, you have to achieved some preliminary level of mastery. (I don't know what that level is to write about here, but I probably 'know it when I see it.')

I remain a Saxon fan, by the way; I'm working my way through 8/7 now.

But the Saxon Homeschool Edition can be incoherent at times.

Back to main page.

Comments

Sounds like you need a KTM intern.

-- KDeRosa - 26 Oct 2005

oh boy, that's for sure

also a book proposal intern, and a student teacher intern

-- CatherineJohnson - 26 Oct 2005

Certainly, this is exactly the sort of stuff that any good elementary mathematics teacher needs to have overlearned to automaticity to be an effective teacher.

However, the devil is still in the details. The big hurdle is getting this stuff into the heads of elementary school students so that they can master it. These major conceptual areas need to be broken down into dozens, if not hundreds, of tiny steps taught seriatim until the kids finally understand the big picture.

Reading the DI stuff, I've noticed that once a student has failed to master a crtical step in the process, a snowball effect begins in which the student gets more and more lost in subsequent steps until he stops learning altogether.

It is very dificult to achieve mastery even in a highly structured program like DI. Much trial and error is needed before the process is successful. Fequently, the DI people have to scrap entire sequences because kids weren't learning the material. Soewhere along the process the train derailed and it was necessary to find that problem spot and fix the problem and retry the sequence. Correct, revise, until mastery.

Now imagine trying to teach to mastery in a unstructured discovery learning / constructivist environment. When the train derails they have no idea where the derailing began let alone how to fix it. Their entire unstructured system precludes easy fixes. That's why their band-aids never work. Kids can't compute correctly or solve problems -- supplement with haphazard worksheets. When the worksheets don't work, now what do you do? How do you go back and find where the teaching went wrong in a poorly designed system? how do you correct the problem after you've found it? Without introducing more problems? And screwing something else up?

This is why the effects of constructivist math are random or worse, detrimental, the system was flawed from day one. Nothing wrong with that per se except that it was designed (or rather undesigned) in such a way that it can't be systematically fixed.

A perfect example of this is computer programming. One of the most difficult things about programming is going back, finding bugs and fixing them. Anyone who has ever done this knows how important it is to break down the programming into small self contained modules and steps that can be isolated so they can be debugged.

You can't easily fix errors unless you can isolate and test the variables causing the error. This is especially so in complex systems with many variables like education.

This iswhat I was thinking when I reviewed the math research study the other day. After weeding out the 98% of the unscientific studies, you were left with the remaining 2% (110) scientific studies. But, then when I read the summaries of those studies I knew that most of them weren't any good either because the hypotheses posed still contained too many confounding variables. Even if the study proved the hypothesis true the answer wouldn't tell the researcher anything useful. The problem wasn't being sufficiently isolated.

Feyman identified this problem and Hirsch has written about it in the context of education research.

-- KDeRosa - 26 Oct 2005

"A perfect example of this is computer programming. One of the most difficult things about programming is going back, finding bugs and fixing them. Anyone who has ever done this knows how important it is to break down the programming into small self contained modules and steps that can be isolated so they can be debugged."

Even unit (module) testing and integration testing has its limitations. I prefer to use prototyping, where you start with only a stub of a program and add and test only one new thing at a time. The program is then perfectly correct at each stage of development. Kind of like Saxon or Kumon Math.

-- SteveH - 26 Oct 2005

How about this image:

-- DougSundseth - 26 Oct 2005

It is very dificult to achieve mastery even in a highly structured program like DI. Much trial and error is needed before the process is successful.

This is why I'm interested in Kumon.

Those worksheets have been around for 50 years (I think it is), used by I don't know how many millions of children.

They've had a long test-run.

-- CatherineJohnson - 26 Oct 2005

I prefer to use prototyping, where you start with only a stub of a program and add and test only one new thing at a time. The program is then perfectly correct at each stage of development. Kind of like Saxon or Kumon Math.

That's Kumon to the max, I'd say.

(Saxon, not so much, only because the Homeschool edition is so darn splintered. Apparently the school version is much better.)

-- CatherineJohnson - 26 Oct 2005

OK, looking at Doug's chart....I get confused about what the problem being solved is.

Is it 2 times 8, or 2 times 16?

-- CatherineJohnson - 26 Oct 2005

Let's see....oh, OK.

It's TWO TIMES, so we have TWO of the THREEPLUSFIVE....

-- CatherineJohnson - 26 Oct 2005

I'm thinking.....the chart (which is incredibly beautiful! Doug's charts make me WANT to learn math!)....I'm thinking the numerical version of the problem should probably be written to the left...

I have to draw this myself, and see why I'm getting confused.

-- CatherineJohnson - 26 Oct 2005

2 x 8

I was reading recently that Microsoft recently scrapped all its new Longhorn Windows code because it had gotten so complex that it was impossible to debug properly. They started over from scrath and rewrote all the code in small steps so that it could be tested and error free before it was incorporated into the new OS. Now windows is a complicated beast, but education is far worse. Can't find the link from quick googling.

-- KDeRosa - 26 Oct 2005

How about this one:

I think the curly braces were more confusing than helpful. I also added an explicit statement of the distributive property in two places.

-- DougSundseth - 26 Oct 2005

oops--haven't seen your new effort yet--here's what I drew, just sitting here:

(Editorial comment: We seem to be using a different way of including graphics in the comments. My way shows up in both Firefox and IE, whereas your way is only showing up in IE; in Firefox I just see the name of the graphic file.)

-- DougSundseth - 26 Oct 2005

Good grief.

Why don't I give up and go look at your second draft?

-- CatherineJohnson - 26 Oct 2005

OK, NOW I am going to look at your second draft.

-- CatherineJohnson - 26 Oct 2005

Hey!

That's pretty good!

-- CatherineJohnson - 26 Oct 2005

Should I have included an explicit note that "3 + 3 = 2 x 3" in the bottom bubble? I didn't think about it until now.

-- DougSundseth - 26 Oct 2005

wow

I'm going to try yours out on Christopher when he gets home.

I still think you need to write in the problem on the left OR put in blank lines for the child to write in the problem!

-- CatherineJohnson - 26 Oct 2005

I drew mine the way I did to show that we are either adding two 7s or we are adding two 3s PLUS 2 4s....

-- CatherineJohnson - 26 Oct 2005

Should I have included an explicit note that "3 + 3 = 2 x 3" in the bottom bubble? I didn't think about it until now.

Probably, ALTHOUGH having a place where the child can write the equation himself might be a very good idea.

-- CatherineJohnson - 26 Oct 2005

I'll tell you.....I'm wondering about manipulatives, like your fractions.

I can easily imagine that having a child play around with simple squares-representing-ones, and forming them into different multiplication arrays, could be very helpful in terms of finally getting the idea that we can multiply the addends first, and THEN add the sums.

I'm also thinking a simple piece of paper with one-inch squares as the array would be great.

You could say to the child, show me 2 ( 3 + 5 ) and have him arrange the chips on the board.

btw, I have colored 'counting chips'; Saxon sends them along with their package for Kindergarten.

So this might work just using a large grid, the same way the fraction manipulatives work.

DISTRIBUTIVE PROPERTY MANIPULATIVES!

OUR FIRST KITCHEN TABLE MATH INVENTION!

-- CatherineJohnson - 26 Oct 2005

Let's see if I can get this right this time:

-- JdFisher - 26 Oct 2005

WHOA!!!!!

-- CatherineJohnson - 26 Oct 2005

Let's try one more:

Since we're not trying to answer a question but to demonstrate a proof, I decided to change the drawing a bit differently than you suggested.

"2 x (3 + 5)" is just an example, so I noted that explicitly at the top, added explicit statements next to the bubbles in the second steps, and added a box to link the general statement at the top to the general statement at the end. I also used a dot for multiplication, because the x was too easy to confuse for a variable.

-- DougSundseth - 26 Oct 2005

Hi--Just got back, but now we must plunge into TEST PREPARATION.

Back soon--

-- CatherineJohnson - 26 Oct 2005

I like the explicit pointing out that this is an example of a general principle.

-- CatherineJohnson - 27 Oct 2005

Yes, that works great, up there at the top.

You get oriented going in: we are doing 2(3 +5)

-- CatherineJohnson - 27 Oct 2005

oh golly, I think this is it!

yes, I'm seeing this one....easily, I guess is the word.

My eye is going where it should go, and the comparison-and-contrast between 2 eights versus 2 threes added to 2 fives is 'popping.'

This is terrific!

-- CatherineJohnson - 27 Oct 2005

I'll get it pulled up front first thing tomorrow!

-- CatherineJohnson - 27 Oct 2005

Beautiful!

Thank you!

-- CatherineJohnson - 27 Oct 2005

By the way, I bet both you & J.D. (and probably Dan) would love Sawyers VISION IN ELEMENTARY MATHEMATICS.

I think it's incredible, though it's very slow-going for me.

You guys would whip through it.

-- CatherineJohnson - 27 Oct 2005

-- CatherineJohnson - 27 Oct 2005

Here's a terrific example of why rectangular fraction tiles are superior to circles:

-- CatherineJohnson - 27 Oct 2005

fraction division & forming a unit

-- CatherineJohnson - 27 Oct 2005

Attachment	Action	Size	Date	Who	Comment
DistributiveProperty.gif	manage	9.5 K	26 Oct 2005 - 18:40	DougSundseth	Distributive property image
DistributiveProperty2.gif	manage	11.1 K	26 Oct 2005 - 19:04	DougSundseth
distributive_property.jpg	manage	33.5 K	26 Oct 2005 - 19:29	JdFisher
DistributiveProperty3.gif	manage	13.4 K	26 Oct 2005 - 19:37	DougSundseth