This New York Times article: Study Suggests Math Teachers Scrap Balls and Slices reports on a study on whether college students learned new math concepts best by being taught with manipulatives or through abstract equations. It poses some interesting questions about how to teach math to younger students.
Modern elementary math instruction utilizes a lot of hands on manipulatives to teach math concepts. Even the Saxon math program that we have used at home uses pattern blocks, linking cubes, and coins to introduce math concepts.
My gut reaction is that there is a balance. Manipulatives are a good way to introduce concepts like place value and large numbers and fractions. But it is important for the student to take those ideas and move well beyond what is easy to represent in plastic and foam. They need to realize that the algorithm that is true for 1/2 x 1/4 is just as true for 9/16 x 17/132. Then those alorithms need to be internalized so that they can be utilized in problems that are still more abstract. Manipulatives may be quite appropriate at younger levels. But they may also become a stumbling block if the manipulative doesn't represent the problem set.
Math manipulatives represent some serious money in classroom equiping. One of my kids' favorites, the Foam Fraction Burger (R) from Delta Education costs $124 for a class set of 15 (and that assumes the kids are working in teams or taking turns). Delta's Getting Started With Manipulatives Kit for grade 2 runs $475. Then there are the manipulatives designed to teach algebraic concepts: four pan balances, early algebra dominoes, or Alge-Tiles (tm) (designed to help "students make the transition from concrete to symbolic thinking through cooperative investigations." A fully stocked classroom cupboard could easily represent thousands of dollars of equipment.
Even our homeschool closet holds numerous jars and bags of brightly colored bits and pieces that we've used over the years. And each new catalog presents new temptations in Backyard Bug or Mini Motors shaped counters, Rainbow Fractions or Measuring Worms (most with their associated activity cards or resource books). I have to be especially stern with myself over the Rainbow Fractions, which come in a dizzying array of circles, squares, and towers. The number of manipulatives for early childhood use that are shaped like cookies, candy and doughnuts could be another entire post.
It is interesting to note that the article begins with a word problem designed to strike fear into students: "One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other?"
It then concludes by giving the answer: "As for the answer to the math problem at the top of this article, the two trains pass each other at 11 p.m. at the midway point between Stations A and B. Or, using the abstract approach, t = 4."
Except 11 pm and t=4 aren't the same thing. t must be related to one of the starting times in order to have meaning. I can't for example, tell my family that I'll be at the station at t=4 and they should pick me up then. I would tell my kids that they solved the equation but didn't answer the actual question.
I'm hoping to read more of the report from the research group. They published an article about the study in Science (Volume 320, Number 5875, Issue of 25 April 2008). Unfortunately the magazine only has free archives for articles over one year old, so I'll either have to wait or try to find a copy at a library. I did manage to register for free access to their older stuff, so I can have some future fun with that.
Saturday, April 26, 2008
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3 comments:
We used Math U See for algebra and it was the best thing we did for my very visual learner. He finally "saw" algebra concepts.
I think it depends on the child.
Barb-Harmony Art Mom
Barb,
I think that you're very right that it depends on the child. I can only recall manipulatives being used once in my elementary years and that was when a substitute let us take out some place value sets.
I think that the fun value alone has been an enourmous boost for us. Math is not something to be feared, it is something that permeates our life.
I am still trying to get a copy of the actual Science article. I think that there may be greater relevance at the upper levels where it was tested. But it is interesting that the researchers found little in the way of previous studies.
On the other hand, even teachers who do informal educational research in the classroom may not bother to (or have time to) write up their findings.
I scanned the article. This was key for me, "Though the experiment tested college students, the researchers suggested that their findings might also be true for math education in elementary through high school..."
College students are operating at a different type of thinking ability -- abstract. But we know (Piaget, anyone?) that young children are concrete thinkers. To abandon manipulatives in early grades is foolish.
Shoot, I still like to draw little diagrams to help me grasp real life math problems.
If a child finds them distracting, then by all means, get rid of the manipulatives. But I think those children are rare.
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