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28 Jul 2005 - 17:51 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?
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
Quick Thought about Fraction Manipulatives
Fraction Manipulatives Part 2
New Study on Manipulatives Part 2
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This supports what I thought, after reading that study I wrote about -- that very young children might actually relate to manipulatives as toys, and no more. I think you might need to have a certain comfort level with abstraction in general, and the concept being 'manipulated' in particular, before you can relate to manipulatives as anything other than toys. -- CarolynJohnston - 28 Jul 2005
You nailed it EXACTLY. I'm impressed. I'm going to go read the article in full now... -- CatherineJohnson - 28 Jul 2005
I don't think it is that they relate to them as toys per se, but yes, I agree about the ideas that small children might not understand the abstraction. All sorts of child development courses discuss issues like object permanence and concepts of volume. You know that famous Seymour Papert example? You have two transparent cups, one short, and one tall. The short one is filled with juice. You pour the juice out of the short cup and into the tall cup (in its entirety.) You ask the child "which cup has more" , and children under a certain age say "the taller cup", even though it's the same amount of liquid. Papert or Minsky tells a famous story about some grad student trying this test on Minsky's kids--and one of them tells the grad student "oh, you should try that on my brother; I don't understand the concept of volumes(or whatever it was called) yet" and the grad student is dumbfounded. (the details of the story are probably apocryphal...) Another reason might simply be that abstraction is possible--but not this kind of open ended abstraction. Usually, with manipulatives, kids are supposed to play on their own undirectly, and somehow derive these rules. But that's a really big leap--even if they weren't seen as toys, they might be seen as building materials, or abstractions representing people, etc. Imagine telling a high school student: "here's a curve; here's a bunch of very thin tiles; derive something" and expecting them to derive integration is the area under the curve. Bit of a leap, no? -- KtmGuest - 29 Jul 2005
I don't think it is that they relate to them as toys per se, but yes, I agree about the ideas that small children might not understand the abstraction. But I think preschoolers relate to almost everything as toys. It's either food, or it's a toy. (Everything else you said, I agree with). But here's another angle on it: manipulatives, played with independently, form the basis of almost all learning in Montessori schools, or so I've heard (no personal experience here). Presumably it works to some degree, or we couldn't have Montessori public schools, as they do in some places, including here. When do Montessori-schooled kids make the move to pencil and paper? -- CarolynJohnston - 29 Jul 2005
Well...I think in the study they used direct instruction with the manipulatives. That's the impression they gave (and given that it was a peer-reviewed study I would imagine they kept the teaching method consistent across the two conditions). One thing that interests me about this is the fact that parents find skinnies & bits confusing. We're told that that's because 'we aren't used to the way math is taught now' or 'we expect to see children sitting in rows doing worksheets' or 'parents don't like change' & blah, blah, blah. But what if the answer is much simpler? What if the reason we see skinnies & bits as confusing is that: skinnies & bits are confusing? -- CatherineJohnson - 29 Jul 2005
Presumably it works to some degree, or we couldn't have Montessori public schools, as they do in some places, including here. When do Montessori-schooled kids make the move to pencil and paper? Good question. I still want to know why paper and pencil is easier. I absolutely agree that it IS easier, based on personal experience. But why????? -- CatherineJohnson - 29 Jul 2005