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# Entries from MathManipulatives

CalStateStudyOnManipulatives 24 May 2005 - 20:01 CarolynJohnston

Part 3 in a mini-series on a review of quality math ed research articles.

Part 1: CalStateStudyIntro

Part 2: CalStateStudyOfGroupLearning

Another surprising fact -- about math manipulatives -- comes out of the Cal State Study.

There were only four studies of manipulative use that were of high enough quality to make the Cal State cut. That really isn't enough to draw a conclusion from, especially given the studies' haphazard coverage of the range of instructional possibilities. Still, there are enough results that they suggest a pattern. See if you can detect it ('benefit' implies that kids did significantly better on normalized tests of math achievement than control groups did).

Kindergarten kids learning counting: no benefit conferred by including manipulatives.

Third graders learning multiplication: two different studies show no benefit to the use of manipulatives before teaching formal computation.

Fifth and seventh graders learning fractions: kids benefit from a fractions game played with or without other manipulatives and pictorial representations.

Elementary schoolers using fraction/ratio manipulatives with fraction/ratio instruction: no benefit.

Seventh graders using fraction/ratio manipulatives with fraction/ratio instruction: benefit.

I love what these results suggest because it is so unexpected and counterintuitive. Most of us think of manipulatives as a stepping-stone from the concrete to the abstract, as something to be used only by the very young when they are first introduced to a topic.

But these results suggest that older kids get more benefit out of manipulatives. In a way, now that I think about it, it makes sense; their relative maturity means kids have a conceptual 'hook' on which to hang the insights that the manipulatives give them. They already have half a clue, and that helps them get the point of the manipulatives.

Perhaps to a younger kid, less able to generalize from the concrete to the abstract, the manipulatives are simply toys.

This is all the evidence I need to get the fraction manipulatives out for my soon-to-be sixth grader.

For more information on math manipulatives, see our favorite math supplements for kids and FractionManipulatives

Also see:
EdResearch
CalBoardOfEdStudyPart2
FractionManipulatives
FractionManipulativesPart2
NewStudyOnManipulatives
New Study on Manipulatives Part 2

FractionManipulatives 13 Nov 2005 - 18:20 CatherineJohnson

re: CalStateStudyOnManipulatives

Over the past year I've used two kinds of manipulatives with Christopher, who is 10:

fraction manipulatives
play money

I didn't need play money and neither does anyone else. I got it only because I wanted to teach Christopher how to make change without a cash register, a lost art, and because . . . if I stacked up a pile of Real Money big enough to make change with, it was going to get raided for lunch money, bake sale money, field trip money (and that's just for starters).

We are chronically short on ONES around here, let me put it that way. So I decided to make things easy on myself and buy some fake money.

+ + +

I'm a huge fan of fraction manipulatives. Christopher and I have spent quite a lot of time using a set of fraction tiles to illustrate:

equivalent fractions
the addition and subtraction of fractions
the addition and subtraction of equivalent fractions

Nothing makes the idea that 2/12 is equivalent to 1/6 more obvious, IMO, than actually lining up two 2/12 tiles below one 1/6 tile and seeing that, yes indeed, 2/12 = 1/6.

These are the fraction tiles I use. They cost \$8.75 plus shipping:

The same company, (Rainbow Resource, a homeschooling catalogue), also carries a set of extra fraction tiles without the tray that I wish I'd had when we first started trying to learn fractions. (I have them now, but we may be past the point of needing them. We'll see.)

You need the extras because you really want the ability to demonstrate addition and subtraction of fractions with different denominators.

+ + +

There are lots of other fraction manipulatives out there, but I chose these after reading a comment from a mom on a homeschool forum somewhere. (I wish I'd kept the link.)

She said that her daughter didn't get anywhere using the more-common circular, 'pie chart' fraction manipulatives; she needed to see rectangular fractions. I have no idea why this would be, but it 'felt' right to me, so I searched for rectangular manipulatives and found these.

At the same time, SAXON MATH uses circular manipulatives, so Christopher has been exposed to both, which I think is almost certainly ideal. A core principle in teaching math, from what I gather, is to teach the same material from different angles.

+ + +

Another terrific activity to do with fraction tiles:

Show how different combinations of fractions add up to 'one whole.'

To do this you just have your child keep lining up fraction tiles on top of the bright red 'one whole' tile until he's covered the whole thing without anything hanging over the end.

So, for example, he might put 2 1/12th tiles, 1 1/6 tile, & 2 1/3 tiles on top of the 1-whole tile, illustrating the fact that:

2/12 + 1/6 + 2/3 = 1

After awhile it starts to become obvious that you can put 6ths and 3rds & 12ths together evenly to make one whole, or 8ths & 4ths & halves, or 5ths & 10ths, . . . but you can't put 3rds and halves together, or 4ths and 5ths (not unless you have a bunch of 20ths, which you don't), and so on.

You can see your child start to get a feel for multiples* and divisibility, whether he has explicitly studied multiples and divisibility yet or not.

+ + +

That's a whole other issue: is it useful to 'preview' concepts in this way? I have no idea, so offhand my answer is 'It depends.' That's one of the big gripes with constructivist math; the kids are constantly being exposed to advanced topics -- sometimes very advanced -- and then not taught the topics to mastery, because the book will be 'spiralling back' to the same topic the next year and the next year after that. Parents tend to hate this, but parents could be wrong. It happens.

Let's just say that my perception, working with Christopher and the fraction tiles, was that he was developing an intuitive grasp of numbers that are multiples of each other versus numbers that aren't. This seemed like a good thing at the time, but who knows? I'm new at this.

Come to think of it, I'm going to get the fraction tiles out again when I get back to teaching the Singapore Math lesson on Changing Ratios. (My neighbor and I team-taught this lesson to our kids two weekends ago, but it was over Christopher's head. Her son is a year older.)

Singapore teaches changing ratios in the first half of 6th grade:

+ + +

Since I never remember definitions of even the simplest terms, I am including the definition of a multiple here:

* multiple - The multiple of a number is the product of the number and any other whole number. (2,4,6,8 are multiples of 8)

QuickThoughtAboutFractionManipulatives 27 Nov 2005 - 19:23 CarolynJohnston

Catherine mentioned that she is a fan of tile fraction manipulatives over the more usual 'pie' manipulatives:

She said that her daughter didn't get anywhere using the more-common circular, 'pie chart' fraction manipulatives; she needed to see rectangular fractions. I have no idea why this would be, but it 'felt' right to me, so I searched for rectangular manipulatives and found these.

I prefer tile manipulatives too, for what I think are solid pedagogical reasons, and here is why: if you want to talk about improper fractions -- fractions greater than one -- with your kid, then the pie-shaped manipulatives add potential for confusion because you can't make a single connected object that represents a quantity greater than one. If you want to represent, for example, 3/2 with pie manipulatives, then you'll have one whole circle and a half circle. You can tell a kid that that represents a single object, the quantity 3/2, all you like; but to him it will look like two objects. Fractions are confusing enough without that.

Conversely, you can make a single line of tiles that is as long as you like.

So unless your child is really off and running with the pie manipulatives, I'd recommend the tile manipulatives.

FractionManipulativesPart2 20 May 2005 - 16:06 CatherineJohnson

Wow!

Thank you!

This is why Life Changed when I met Carolyn. She's not just a mathematician herself; she spent years teaching math, and she is actively engaged in acquiring pedagogical content knowledge.

Pedagogical content knowledge is a fancy way of saying that the things really good math teachers know are somewhat different from the things really good mathematicians know, and that the difference is important.

(This is why neither Carolyn nor I feel that simply requiring math teachers to major in math is going to do the trick when it comes to raising math achievement. But that is a subject for another post.)

While I was writing about rectangles being better than circles, I was visualizing circle manipulatives, and I was thinking:

``` Well, you can put together two 2/12 pie slices with one 1/6 pie slice with two 1/3 pie slices and still get one whole pie, the same way you can with the rectangular manipulatives. So what's the problem? `````` ```

But then I was thinking,

``` I know there's something else . . . ```

Now, here is Carolyn pointing out that it's going to be 'visually' impossible to tell a child that 3/2 represented as 1 and 1/2 circle is ONE THING, whereas it's going to be (reasonably) easy to tell a child that 3/2 represented as 1 and 1/2 of a bar is ONE THING.

This observation has opened a window for me:

I see that I hadn't progressed to the point of realizing that 3/2 should or even could be considered ONE THING.

I have a ways to go.

Still, this makes me hopeful that I'm beginning to develop some intuitive knowledge of math content and math pedagogy or teaching . . . because I could tell there was a reason why I'd grown more attached to rectangular fraction manipulatives over the year, not less. I just couldn't put my finger on it.

Veering off on a tangent here, one of my very favorite books on the cognitive unconscious (tacit knowledge, or, sometimes, intuition) is Arthur Reber's Implicit Learning and Tacit Knowledge: An Essay on the Cognitive Unconscious.

```
```
I remember Reber writing that one of the reasons the field of implicit learning got going in the first place was the question of how to make sure experts in one generation passed their knowledge on to the next generation.

As I recall, the first thought everyone had was simply to ask experts, such as surgeons, how they did what they did. They figured the experts could tell them.

It turned out the experts couldn't tell them. They were experts, not teachers.

That raised the question of what we know that we don't know we know.

I hope I'm developing some intuition about teaching math, and about the content of mathematics itself.

But while intuition about how to teach math may be good enough, intuition about math itself probably is not.

To be a good math teacher, it seems, you have to be able to put what you know about math into words and images.

Also see:
EdResearch
CalBoardOfEdStudyPart2
CalStateStudyOnManipulatives
FractionManipulatives
NewStudyOnManipulatives
New Study on Manipulatives Part 2

FractionManipulativesPart3 02 Jun 2005 - 00:20 CatherineJohnson

On the subject of buying fraction manipulatives, if it doesn't break the bank I would also get an inexpensive labeler.

Use it to label each fraction tile with the equivalent percent.

Add a "10 percent" label to each 1/10 tile, "33 1/3 percent" to each 1/3 tile, "100 percent" label to each 1-whole tile, and so on.

If you can fit the decimal representation of the number (.1, .3333, 1.0) on the tiles, put that on, too.

I got this idea from Saxon Math 6/5. 6/5 includes lots of worksheets with fraction circles printed on them, and always, on every sheet, the fractional parts are labeled with all three representations of the number: fraction, decimal, percent.

Brother PT-65 Home & Hobby III P-Touch Labeling System, \$29.95

Saxon also has the kids answer mental math questions about fractions and percents ("How much is half of 5?" "What is 50% of 50?") in virtually every lesson in the book.

At first I didn't get this.

The concepts hadn't really been taught, and it seemed like pure memorization to me.

But I found that this constant practice of simple 'recognition knowledge' -- visually and verbally recognizing 1/2 as 50% and 50% as 1/2 -- meant that whenever we studied a conceptual lesson on fractions, Christopher was ahead of the game.

At least, that's the way it seemed to me.

He could look at a pie chart divided into 10 pieces and see instantly that 50% = 1/2 = 5/10. He already had, inside his head, "50% of 10 is one-half of 5;" it just came naturally.

[*update*: OK, 50% of 10 is not one-half of 5. This is the kind of thing that drives me nuts; I am constantly popping off with statements like 50% of 10 is one-half of 5; I am starting to think I am dyslexic for numerical expressions, da***it. Thank you, Carol Morgan.]

I also began to find that Christopher was getting faster at fraction problems than I was. Faster, and more accurate.

I would ask him a Saxon fraction problem I myself was slightly confused on, he would come back fast with an answer, I'd say it was wrong, he'd say it was right -- and lo and behold, it was right.

I assume he was passing me by because I hadn't been doing all the 'memorization' he'd been doing. I hadn't been doing it because I didn't think I needed to. I already knew the concept of equivalent fractions, and I could do the calculations (which he couldn't) .

And yet by the end of the book Christopher seemed to be overtaking me on conceptual understanding (that's assuming I know what conceptual understanding of mathematics actually is, which I don't).

Christopher seemed to be developing a quicker and more reliable feeling for numbers, for the fact that a particular answer had to be right, or had to be wrong, or was or was not 'in the ballpark.'

So, for the time being, I'm convinced that we want to do solid memory work with our kids. Memorized material seems to give us the base we need to build up something . . . more.

re:
FractionManipulatives
FractionManipulativesPart2
CalStateStudyOnManipulatives

+ + +

One last thing: Saxon seems to have extended the concept of math facts to include fraction facts.

Fraction facts, decimal facts, and percent facts.

In books Saxon 7/6 & 8/7 he has kids do timed worksheets converting fractions into decimals, decimals into fractions, and so on.

Given how incredibly difficult fractions are for most all students, I think that's probably a good idea.

MentalMultiplication 20 Jan 2006 - 03:45 CarolynJohnston

I just got off the phone with an old friend. Gerry, Bernie and I all used to be colleagues in the Florida Atlantic University math department, and we more or less independently left and moved to take up new lives in the greater Denver area. Bernie and I went into industry, and Gerry went into teaching; he now teaches mathematics at a private Catholic girl's school in Denver. We see them occasionally (not often enough!).

Gerry is a great innovator when it comes to math education, and a prolific inventor of new and creative math manipulatives, including one of the largest math manipulatives ever: the Sugar Sand Park Moebius Climber, designed with the aid of Mathematica.

Gerry is an extremely thoughtful individual. We are both fascinated by developmental issues and how they affect math education, and we began a conversation tonight that I hope will continue over a long period of time on this website.

But just for tonight: here is a tip he dropped on me for teaching the essence of multidigit multiplication.

At the core of multidigit multiplication is the distributive property of real numbers: (a+b)c = ac+bc. The standard algorithm utilizes it more or less explicitly. But often, these days, the standard multidigit algorithm is not taught: either it's eschewed completely, or some variant like the lattice algorithm is taught instead. If kids are not explicitly taught the distributive property, it will come back to bite them in algebra, where it is used all the time in algebraic simplification and in factoring polynomials.

Here is Gerry's tip; if you want to be sure your kids understand the distributive property, get them to do problems where they multiply one-digit numbers by two-digit numbers entirely in their heads.

Working memory can't hold too much in storage, but it can do that much. If a kid knows his single-digit multiplication tables cold, then he can multiply a multiple of ten by a single-digit number, and add it to a multiple of two single-digit numbers, all in his head. And in doing so, he'll internalize the distributive property, because he has to use it in order to do this sort of problem.

Because unless you have an incredible visual memory, the lattice method isn't of much use for doing mental math.

Brilliant and simple. Like all of Gerry's other math ed innovations.

PanBalanceProblems 11 Jun 2005 - 18:50 CarolynJohnston

In BarModelingVsGraphing, a guest mentioned that variables and equations could be introduced using pan balance problems, in simple cases.

Catherine and I were talking about pan balances this past spring, in exactly this context. She asked how I would introduce equations to an absolute newbie; I said that with Ben, I had had luck using the analogy of a pan balance. It's a rather neat analogy, in the initial stages of learning about equations. Emphasis on initial.

Not a week later, by coincidence, Everyday Math introduced a whole unit on pan balance problems. These were problems of the following sort:

Given the diagram below, tell how many squares are equivalent to a circle.

Very neat idea, I thought; it addresses, in an intuitive way, the preservation of equality under both the addition-subtraction and multiplication-division operations. I liked it.

The pan-balance problems kept coming home. Pretty soon we had moved on to double pan-balance problems:

Given the diagrams below, tell how many squares are equivalent to a triangle.

That, I thought, was getting to be a bit over the top -- I was starting to have to coach Ben on how to approach the pan-balance problems that were supposed to be helping him to approach the problem of equation-solving.

Then we started seeing pan-balance problems that looked like this:

We were getting so close to actually doing real equations, I could feel it.. and the kids were developing such great intuition; they were so ready for the next step, the step to real equations --

and then the unit ended. Fifth grade Everyday Math ended without the kids ever having really been introduced to manipulating equations.

But they are good with pan-balances, at least virtual ones.

I guess this is a sort of a cautionary tale about the dangers of falling in love with your cool teaching tools.

NewStudyOnManipulatives 29 Jul 2005 - 17:27 CatherineJohnson

I want to follow up on Carolyn's post on Congressional math incentives, but before I do that here's a reminder:

Anne Dwyer has posted new notes on her summer math class.

And...quickly checking her page just now, noticed this comment:

So, what have I learned so far?

• they like games where they compete with one another
• they prefer pencil and paper exercises
• they like to figure out puzzles

This is fascinating, because Kevin Killion, of Illinois Loop, just pointed me to a new article in Scientific American showing that manipulatives are less effective than pencil and paper with young children. I'll write a bit more on this later (bike ride time!), but here's the critical passage:

Teachers in preschool and elementary school classrooms around the world use "manipulatives"--blocks, rods and other objects designed to represent numerical quantity. The idea is that these concrete objects help children appreciate abstract mathematical principles. But if children do not understand the relation between the objects and what they represent, the use of manipulatives could be counterproductive. And some research does suggest that children often have problems understanding and using manipulatives.

Meredith Amaya of Northwestern University, Uttal and I are now testing the effect of experience with symbolic objects on young children's learning about letters and numbers. Using blocks designed to help teach math to young children, we taught six- and seven-year-olds to do subtraction problems that require borrowing (a form of problem that often gives young children difficulty). We taught a comparison group to do the same but using pencil and paper. Both groups learned to solve the problems equally well--but the group using the blocks took three times as long to do so. A girl who used the blocks offered us some advice after the study: "Have you ever thought of teaching kids to do these with paper and pencil? It's a lot easier."

Dual representation also comes into play in many books for young children. A very popular style of book contains a variety of manipulative features designed to encourage children to interact directly with the book itself--flaps that can be lifted to reveal pictures, levers that can be pulled to animate images, and so forth.

Graduate student Cynthia Chiong and I reasoned that these manipulative features might distract children from information presented in the book. Accordingly, we recently used different types of books to teach letters to 30-month-old children. One was a simple, old-fashioned alphabet book, with each letter clearly printed in simple black type accompanied by an appropriate picture--the traditional "A is for apple, B is for boy" type of book. Another book had a variety of manipulative features. The children who had been taught with the plain book subsequently recognized more letters than did those taught with the more complicated book. Presumably, the children could more readily focus their attention with the plain 2-D book, whereas with the other one their attention was drawn to the 3-D activities. Less may be more when it comes to educational books for young children.

This perfectly supports the study Carolyn mentioned way back when, showing that fraction manipulatives are good for middle schoolers.

CA state study on manipulatives
Fraction Manipulatives
Fraction Manipulatives Part 2
New Study on Manipulatives Part 2

NewStudyOnManipulativesPart2 28 Jul 2005 - 20:15 CatherineJohnson

I'm reading the Scientific American article about manipulatives & symbolic representation now:

About 20 years ago I had one of those wonderful moments when research takes an unexpected but fruitful turn. I had been studying toddler memory and was beginning a new experiment with two-and-a-half- and three-year-olds. For the project, I had built a model of a room that was part of my lab. The real space was furnished like a standard living room, albeit a rather shabby one, with an upholstered couch, an armchair, a cabinet and so on. The miniature items were as similar as possible to their larger counterparts: they were the same shape and material, covered with the same fabric and arranged in the same positions. For the study, a child watched as we hid a miniature toy--a plastic dog we dubbed "Little Snoopy"--in the model, which we referred to as "Little Snoopy's room." We then encouraged the child to find "Big Snoopy," a large version of the toy "hiding in the same place in his big room." We wondered whether children could use their memory of the small room to figure out where to find the toy in the large one.

The three-year-olds were, as we had expected, very successful. After they observed the small toy being placed behind the miniature couch, they ran into the room and found the large toy behind the real couch. But the two-and-a-half-year-olds, much to my and their parents' surprise, failed abysmally. They cheerfully ran into the room to retrieve the large toy, but most of them had no idea where to look, even though they remembered where the tiny toy was hidden in the miniature room and could readily find it there.

Their failure to use what they knew about the model to draw an inference about the room indicated that they did not appreciate the relation between the model and room. I soon realized that my memory study was instead a study of symbolic understanding and that the younger children's failure might be telling us something interesting about how and when youngsters acquire the ability to understand that one object can stand for another.

### here's the anti-constructivist moment:

[The] capacity [to] create and manipulate a wide variety of symbolic representations .... enables us to transmit information from one generation to another, making culture possible, and to learn vast amounts without having direct experience--we all know about dinosaurs despite never having met one. Because of the fundamental role of symbolization in almost everything we do, perhaps no aspect of human development is more important than becoming symbol-minded.

### symbols aren't 'natural'

The first type of symbolic object infants and young children master is pictures. No symbols seem simpler to adults, but my colleagues and I have discovered that infants initially find pictures perplexing. The problem stems from the duality inherent in all symbolic objects: they are real in and of themselves and, at the same time, representations of something else. To understand them, the viewer must achieve dual representation: he or she must mentally represent the object as well as the relation between it and what it stands for.

A few years ago I became intrigued by anecdotes suggesting that infants do not appreciate the dual nature of pictures.

[snip]

.... the Beng babies, who had almost certainly never seen a picture before, manually explored the depicted objects just as the American babies had.

The confusion seems to be conceptual, not perceptual. Infants can perfectly well perceive the difference between objects and pictures. Given a choice between the two, infants choose the real thing. But they do not yet fully understand what pictures are and how they differ from the things depicted (the "referents") and so they explore: some actually lean over and put their lips on the nipple in a photograph of a bottle, for instance. They only do so, however, when the depicted object is highly similar to the object it represents, as in color photographs....

[snip]

it takes several years for the nature of pictures to be completely understood. John H. Flavell of Stanford University and his colleagues have found, for example, that until the age of four, many children think that turning a picture of a bowl of popcorn upside down will result in the depicted popcorn falling out of the bowl.

### Andrew makes Barney fly

A couple of weeks ago Andrew (10, autistic, nonverbal) brought me Christopher's yellow plastic airplane, on top of which he'd mounted one of his Barney's, and handed the whole big package to me with an urgent look on his face. He was on a mission.

Martine came in and said, 'He wants you to make Barney fly.' She'd been sitting in the family room when Andrew had put his Barney on top of the plane, and then flung plane & Barney up into the air, apparently thinking Barney would fly around the room.

Andrew had been very unhappy with the outcome, and was now appealing to me. Clearly he believed that making Barney fly was one of those things, like operating the TIVO, only adults know how to do.

I was flattered, but also dumbfounded. What goes on inside this child's head? was my exact thought.

I was thinking....does he not understand gravity?

Does he not understand toys?

What's with this kid???!!

The Scientific American article makes me think that Andrew, although he can read, hasn't completely figured out the dual nature of symbolic representation.

He probably couldn't understand the plastic airplane as being TWO THINGS:

• an airplane

AND

• a symbolic representation of an airplane

What I'd like to know is: what does he think about Barney?

### is this a shoe?

Here's a little guy trying to put his foot inside a picture of a shoe.

### lost in translation

I constantly have the experience of reading constructivist texts, noticing that the ideas they're advocating are good ones or at least not obviously bad ones.....and then, five seconds later, finding that they've taken a sound idea and just completely gummed it up in the application.

Assuming this work on manipulatives & symbolic representation is correct, the constructivist obsession with manipulatives looks to be another instance of a good idea lost in translation. Constructivism is majorly obsessed with manipulatives, that's for sure. I understand that the TERC curriculum is basically just a huge box of manipulatives, with no textbook or 'consumables'--workbooks--at all.

Following in Piaget's footsteps, constructivists believe children don't reach the stage of 'formal operations' until age 11; from 7 to 11 they're in the Period of Concrete Operations. (Often you'll see the word 'developmental' used to designate constructivist curricula. Apparently that's a reference to Piaget.)

Wayne Wickelgren says this is nonsense; children can handle abstract concepts long before age 11. But constructivists are the people time forgot, and they're still basing their pedagogy on work done in the 1950s.

That's bad enough in itself, seeing as how the field of cognitive science was just getting started around that time, and Piaget's work hasn't fared so well over the past 60 years.

But the more glaring misstep, it appears, is that they failed to grasp the nature of the concrete.

The reason constructivists think children should spend their grade school years working with manipulatives is that manipulatives are concrete. But they're not. Manipulatives are symbolic objects that require the child to have mastered the concept of dual representation.

Skinnies and bits are not concrete. They are symbolic representations of the Hindu-Arabic numeral system. Worse yet, they are more intellectually demanding, and hence more confusing, symbolic representations than pencil marks on paper.

They're harder to understand, not easier.

Lost in translation.

### question

I hope I'll get a chance to talk to these researchers at some point.

My question is: why should pencil and paper be less challenging than manipulatives?

I can see why pencil and paper wouldn't be any more challenging than manipulatives, but why should pencil and paper be easier? Do pencil marks somehow not involve dual representation? That's what the authors seem to imply, but they don't say so directly.

CA state study on manipulatives
Fraction Manipulatives
Fraction Manipulatives Part 2
NewStudyOnManipulatives
New Study on Manipulatives Part 2

NewDimensionalDominoes 04 Aug 2005 - 16:22 CarolynJohnston

DanK has added some more dimensional dominoes worksheets at his dimensional dominoes page. If you're interested in learning a little bit about dimensional analysis and perhaps teaching it to your kids -- it's great for checking work -- have a look (and please don't be shy about giving feedback and asking questions!).

DougSundsethNumberLines 30 Sep 2005 - 21:37 CatherineJohnson

blank number lines (pdf file)

symmetric number lines (positive numbers, negatives numbers, 0 (pdf file)

number lines: all positive numbers (pdf file)

number lines: all negative numbers (pdf file)

### update

If anyone is interested in, or has time to, critique these study sheets, that would great. (There's no pressing need for this; I'm reasonably certain these are accurate, especially since the second document came straight from the pages of Mathematics 6.

addition & subtractions of integers review sheet

RonAharoniOnTheFifthOperationOfArithmetic 14 Sep 2006 - 14:53 CatherineJohnson

Carolyn has kindly left my two favorite passages in Ron Aharoni's What I Learned in Elementary School for me to blooki.

Here's the first:

##### What Arithmetic Should Be Covered in Elementary School?

The embarrassingly simple answer is: the four basic operations—addition, subtraction, multiplication, and division.

Yet, this seemingly simple answer is deceptive in two ways. One is that there are actually five operations. In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.” This operation is at the base of much of the mathematics of elementary school. First of all, in counting, when you have another such unit you say you have “two,” and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. The decimal system is based on gathering tens of objects into one unit called a “10,” then recursively repeating it.

The forming of a unit, and the assigning of a name to it, is something that has to be learned and stressed explicitly. I met children who, in fifth grade, knew how to find a quarter of a class of 20, but had difficulty understanding how to find “three-quarters” of the class, having missed the stage of the corresponding process of repeating a unit in multiplication.

I've thought about this observation every day since reading Aharoni's article. I probably can't explain why. At least, I can't at the moment. (Good thing I'm not taking the Regents, I guess.)

But it reminded me of a post Carolyn wrote early on:

Catherine mentioned that she is a fan of tile fraction manipulatives over the more usual 'pie' manipulatives:

She said that her daughter didn't get anywhere using the more-common circular, 'pie chart' fraction manipulatives; she needed to see rectangular fractions. I have no idea why this would be, but it 'felt' right to me, so I searched for rectangular manipulatives and found these.

I prefer tile manipulatives too, for what I think are solid pedagogical reasons, and here is why: if you want to talk about improper fractions -- fractions greater than one -- with your kid, then the pie-shaped manipulatives add potential for confusion because you can't make a single connected object that represents a quantity greater than one. If you want to represent, for example, 3/2 with pie manipulatives, then you'll have one whole circle and a half circle. You can tell a kid that that represents a single object, the quantity 3/2, all you like; but to him it will look like two objects. Fractions are confusing enough without that.

Conversely, you can make a single line of tiles that is as long as you like.

So unless your child is really off and running with the pie manipulatives, I'd recommend the tile manipulatives.

These are the fraction tiles I like:

You can order extra tiles, too, which I have done. I've used these over and over again, with Christopher, and with at least two of his friends.

Worth their weight in gold.

Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)

FractionManipulativeLessonOnReciprocals 14 Sep 2006 - 14:26 CarolynJohnston

The other night, Ben was working on his math, and I was doing something else. He paused in his work and asked me: "Mom, what's a reciprocal?"

I guess it's no surprise that he doesn't know what a reciprocal is, since it wasn't taught in his Everyday Math classes in elementary school, and since (apparently) it's not introduced in the early part of Saxon 6/5, the curriculum I was supplementing from last year. But in Saxon 8/7, which he's using this year (he tested into it, I swear), knowledge about reciprocals, and the role they play in division of fractions, is assumed.

So I'm doing reactive teaching again, but at least this time I'm reacting to the curriculum of my own choosing.

Saxon has had problems in the mixed practice the last few nights that go straight to the heart of why the reciprocal gets involved in fraction division. The questions are like this: how many 3/8s are there in 1? How many 4/5ths are there in 1?

Here's a demonstration I devised for him on the 3/8ths problem, using the tile fraction manipulatives that Catherine and I have recommended here (warning: pies won't work for this very well).

This sort of question seems to throw him off, so I start by asking other questions that sound more familiar, like: How many 2s in 8? and How many 3s in 9? Then I point out that he is getting the answer by dividing, so by analogy, we'd want to divide 1 by 3/8.

Everyone knows the rule for fraction division: Ours is not to reason why, just invert and multiply. But of course, we are modern traditionalists here, and procedural knowledge is only the beginning of our demands. We want our kids to have an understanding, too, of why they are inverting and multiplying. I taught Ben the invert-and-multiply rule, but then I wanted to convince him that the answer that the invert-and-multiply rule gives you, 2 and 2/3rds, is the right answer.

We attacked the question of how many 3/8ths go into 1 directly, using the manipulatives. The manipulatives were all placed on a sheet of paper, so I could write curly braces and labels next to the tiles. I drew a diagram below of what we do with the tiles (note that the 3/8th tiles are not really single blocks, they are 3 1/8 blocks in a row; I have to tell him to think of them as a single unit.. The labels and curly braces help with this).

It's easy for him to see that two 3/8ths will fit into the 1; I stick them below the 1 tile, and label them as "2 3/8ths". A third 3/8th will overhang the end, though. So I take the extra 3/8ths unit and break it apart into thirds (pointing out that that's what I'm doing). Two of those thirds will fit into the rest of the space in the 1. So this gives us a total of 2 and 2/3rds 3/8-units that will fit into the 1.

I don't expect that this is the end of this; we'll do this a bunch more times and hopefully it will sink in. The trick is to get the kid thinking of the divisor (in this case 3/8ths), however weird a fraction it is, as being a unit. I hope Saxon keeps this sort of problem coming for a while.

FractionTilesByDoug 14 Dec 2005 - 21:39 CatherineJohnson

### ideas for using these

from Carolyn:
Print em out, paste em on foamcore, label em, cut em out with an exacto knife, and you've got some nice cheapo fraction tiles.

from Catherine:
9" x 12" self-adhesive foam boards at Office Depot
I also happen to own a laminator, which I guess places me in a Special Autism Category. But if you have a laminator, it's incredibly quick. They have those 'fake laminator' pages too....the 'cold' ones, at Staples. You don't need a laminating machine to use them
self-adhesive lamination, letter size sheets at Staples

from KDeRosa:
Rubber cement will do the trick without wetting the paper and ruining it [if you're using non-adhesive foam board].

from Doug:
Another option for sticking these down would be to print them on label paper. You can (or at least could) get 8-1/2 x 11" labels at office supply stores, then stick them down on foamcore, cardstock, or whatever.

from Susan
...for the lazy parent you can just print several times on heavy card stock in different colors. They'd be stiff enough to last for a bit. I did that with the Saxon pie manipulatives from Saxon 6/5. I just scanned the pages and bought heavy duty colored card stock and printed. Laminating sounds good, too. I totally forgot about that cold stuff at hobby stores. Just printing out works best for me, though. If they wreck it I just print again.

This is cool: brightly colored card stock at Staples

(hmm. I wonder if Doug feels like making one more fraction sheet with no color or graytone at all??)

### fraction tile pdf files attached to Comments page

I've posted a link on Math lessons, on Favorite Math Supplements for Kids, and on the book-style index page, too. (Just noticed: I need links to Doug's number lines & Dan's dimensional dominoes in the index, too.]]

### extended response

from Doug:

I chose the tile colors using a particular scheme.

What is that scheme?

(Catherine isn't allowed to answer, since I told her what it was in the email.)

I'm going to try this on Christopher!

(This strikes me as a Math Challenge that's actually FUN. Or could be, anyway.)

I say 'or could be' because my philosophy on What Is Fun For A Child is to find out from an actual child whether a thing is fun or not.

Do you ever wonder whether the people at Everyday Math have ever actually met a child?

### fraction tiles in black & white

keywords: Doug's fraction manipulatives

DougsDistributivePropertyGraphic 26 Oct 2005 - 23:04 CatherineJohnson

this is beautiful!

In my next life, I want to come back as a graphic designer.

### distributive property manipulatives?

Any thoughts on whether distributive property manipulatives might be a good idea?

I'm thinking something extremely simple, maybe a large grid and those round plastic number 'counters' (they look a little like thin Poker chips). Or you could just use paper, or pennies or buttons. Anything.

You could ask the child to show you:

3 ( 2 + 4)

as an array like Doug's, or possibly as the two different arrays, which might be best. The child would see the equivalence because he or she had just made the equivalence with his own hands.

### Temple on using your hands

I talked to Temple (Grandin) again last night. She had lots of horror stories of professional architects who can't make scale drawings. These are all young people who learned to do scale drawings on the computer.

She is adamant that there is a motor component to seeing. She thinks you can't get your visual processing right if you haven't....done things with your hands (can't be more specific than that).

The mistakes these people made were all perceptual mistakes, not cattle handling mistakes. (We're talking about the meatpacking industry.)

I really think there's something to this. (It's making me curious about that strange-looking Borenson fellow with his hands-on algebra, I must say....)

JDFishersDistributivePropertyManips 27 Oct 2005 - 13:35 CatherineJohnson

Incredible!

I can't wait to try these with Christopher!

hmm

I wonder if this image is slightly to big for the front page....

I have to go get Christopher & his friend Joe; back later. (If I need to cut this image down slightly, I'll do it when I get back.)

Extremely cool!

J.D.'s full-size graphic is here.

### J.D.'s model for distributive property with subtraction

I just noticed J.D. also included a model for the distributive property used with subtraction, also in the Comments section. (Will jpeg it & post up front ASAP.)

### NAEP analysis

And...I just discovered J.D. has posted an analysis of the new NAEP scores.

MathemagicalManipulatives 27 Oct 2005 - 13:33 CatherineJohnson

J.D. just coined a term for really bad manipulatives.

DougsWorksheetGenerator 28 Oct 2005 - 19:16 CatherineJohnson

Random Worksheet Generator

DanVisualFractionMultiplication 30 Oct 2005 - 22:11 CatherineJohnson

from Dan K

[This is] my attempt to show why you need to multiply the denominators in fraction multiplication.

### 3/4 x 5/6

In the attached powerpoint file, we have one big figure that represents 3/4. That is shaded in light blue. Then, we have a smaller figure that represents 5/6. That is shaded in orange.

In the second slide, we combine these figures by multiplying them together. Every place that was shaded light blue to begin with is "covered" with a 5/6 section. The amount of the resulting figure that is shaded orange is the answer to 3/4 * 5/6. You can see that the original large figure has now been subdivided into 24 sub- sections, 15 of which are shaded orange.

hmm

I'm having trouble 'reading' this. (I say visuals are abstract, and they are!)

The first thing I stumble over is the 5/6 being smaller....then I'm wondering what would happen if we made the 3/4 small and the 5/6 large, and we covered up the 5/6 orange with 3 of the THREEFOURTHS chunks....and I'm thinking, OK, it would be the same thing.....

But I'm still not getting why we're multiplying the denominator, apart from needing the same unit in which to express the answer.

Here's the thing that constantly stumps me, and I don't know why it should.

With addition & subtraction of fractions, it makes sense to me that we need the same unit in which to express the answer, and that the unit should be the same denominator, AND that we find the same denominator by using multiplication (or division, as the case may be) to find an equivalent fraction.

For some reason, I'm just not getting why we use multiplication to create the same denominator in the multiplication of fractions.

I don't know what my problem is, or why I'm not 'seeing' it here....

(I need to look at the other two answers.)

WhatToDoWithLinnkingCubes 17 Nov 2005 - 01:19 CatherineJohnson

I met a retired teacher at a bowling party the other day, who told me a story about math manipulatives.

He'd been a middle school teacher, and for some reason he was put in a 3rd grade class and given a crate full of math manipulatives. He had no idea what to do with them. He was especially stumped by the 'linking cubes,' which looked like Legos, but clearly were not Legos.

I had the same experience two summers ago when I orderd the Saxon Math manipulatives that go with the Saxon Kindergarten book. The Geoboard especially stumped me, but the linking cubes were a mystery, too.

Now I get it.

source:
Harcourt School Publishers Multimedia Math Glossary

I like Harcourt's glossary. The images are clean and sharp, and you can ask the website to speak the name of each concept or rule, which is fantastic for a child like Andrew who (probably) never hears the same word the same way twice.

A recorded word is much more likely to be a 'stable stimulus.'

On the other hand, their illustration of multiplication (in Grade 2) is visually wrong, I think. (Anyone else?)

DougsExcelProblemWorksheet 18 Nov 2005 - 18:20 CatherineJohnson

Doug to the rescue again.....

We were talking about the problems a lot of our kids have re-copying problems from worksheets, and....you guessed it.

They're here (scroll to the bottom of the Comments section):

Doug also left this comment:

I've promised him cash for performance (\$1 if he can get under 4 minutes with 100% correct — so largely a token), but he was pretty happy with his feeling of accomplishment and our praise last night. Now that I think about it, maybe I should cobble up a certificate too.

Needless to say, I vote yes on the certificate.

The very thought of a beautiful, professionally designed Homework Award Certificate, instead of the garish, Page Splattered reward stickers you buy at Staples—I'm feeling faint.

GeometryFractionKit 19 Dec 2005 - 20:38 CatherineJohnson

I wonder if I can still get one of these in time for Christmas?

Or this:

LegosForVolumeAndArea 03 Jul 2006 - 19:32 CatherineJohnson

After Ed did his final teaching-to-crammery unit on volume formulas, "teach conceptual understanding of volume formula" popped onto my to-do list.

I don't know whether I acquired much comprehension of volume formulas when I was in school. I don't think I did. Since I've been reteaching myself math I've spent quite a bit of time staring at array models trying to grok the fact that 3 x 4 really is 3 "of" 4 and vice versa.

I love array models.

source:

Saxon 8/7 has a number of fantastically effective lessons in which students have to figure out how many 1-inch square sugar cubes will fit into a square or rectangular box. I haven't checked Prentice - Hall (I may never be able to look at that book again) but I'm guessing Christopher didn't spend a lot of time thinking about how many small cubes can fit into a larger cube or prism.

So I was gearing up to search through Saxon 8/7, find all the lessons with sugar cubes, figure out a schedule, harrass Christopher to sit down with me and go over them, etc. when it struck me that a cottage-cheese size six-dollar plastic container of tiny Legos would do the job.

I bought two little white bases plates, each measuring 8 x 2cm and a bunch of white & see-thru red squares. Christopher likes to build Legos &mdsah; at least he's back to liking them at the moment — so he can spend some time building Lego rectangular prisms & cubes and I won't have to spend hours tracking Lessons in Saxon 8-7 and then tracking Christopher to get him to come do them.

I also found the coolest little Lego thingie for fooling around with area & circumference of circles! It's a Lego square with a Lego circle on top, that turns. don't know what it's supposed to be. I'll see if I can find a picture.

[pause]

I have no idea what I bought. It's minute — diameter is 3 cm — and we won't spend much time with it.

But it's all I need for some hands-on distributed practice.

I'll also have him take a look at this webpage for Houghton Mifflin Math:

The volume of a rectangular prism can be found by counting the number of cubic units or by using a formula. The formula for finding the volume of a rectangular prism is V = l w h.

The volume of this rectangular prism is 30 cubic meters.

Another way to measure a solid figure is to find the surface area. Surface area is the sum of the areas of all the faces of the solid figure. Since opposite faces of rectangles or cubes have the same area, you can also multiply each area by 2 and then find the sum of the areas of the faces.

The surface area of this rectangular prism is 52 yd2.

Perimeter, area, volume, and surface area are measurements of geometric figures. Perimeter and area are measurements of plane figures and surface area and volume are measurements of solid figures.

This one's nice, too:

To find the area of a complex figure, separate it into simpler figures, find the areas, and then add the areas together.

And these two:

Figures with the same perimeters can have different areas.

Figures with the same area can have different perimeters.

teachtocrammery

-- CatherineJohnson - 19 Jun 2006

DougFractionTilesRevisited 05 Jul 2006 - 23:07 CatherineJohnson

Here's a cool lesson to do with Doug's fraction manipulatives (pdf file):

source:
Primary Mathematics 2B US Edition

Doug's fraction tiles & number lines (pdf files)

PicturesOfNumbers 20 Jul 2006 - 17:23 CatherineJohnson

This blog looks fantastic.

good grief

More synchronicity. This is becoming a daily occurrence.

I just pulled up my two-year old Excel chart for Jimmy's weight, and today I find a blog with a post called Fixing Excel's Charts.

I'm going to pretend this didn't happen.

-- CatherineJohnson - 18 Jul 2006

FractionManipulativesAndNumberLinesByDougSundseth 14 Sep 2006 - 14:44 CatherineJohnson

Doug Sundseth created these fraction manipulatives a year ago. They're incredibly useful, so download when you need them and let your friends know, too. (pdf files)

Doug's manipulatives and number lines are easy to find on Our Favorite Math Supplements for Kids (link in sidebar)

Alternatively, if you search Blog Posts this post should be the first to come up.

Last but not least, I've finally added links to Doug's fraction manipulatives and number lines on the "Book-Style Index" page.

So these things are about as findable as I can make them at the moment. (Apparently I've been inspired by my purchase of a used copy of Steve Krug's Don't Make Me Think at the Mercy College bookstore.)

KTM Guest: fraction manipulative website

I'm guessing this is from Suzanne:

Math Playground has an online version of the fractions manipulative. You can change the number of parts, shade in what is needed, and toggle the fraction labels. The link is: Math Playground fraction bars.

Math Playground looks wonderful. Thank you!

Doug's number lines

These are good, too:

Links to Doug's number lines are also on these pages:

Carolyn's lesson on reciprocals

Here is Carolyn's lesson on fraction manipulatives and reciprocals.

Now I need to pull together one post with all the stuff on unit multipliers we've put together over the last year and a half...

archive of posts on math manipulatives

original post about Doug's fraction tiles
original post about Doug's number lines

-- CatherineJohnson - 14 Sep 2006

LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson

I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

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