The value of manipulatives

(Note: I'm "resurrecting" an old post, with added information and a video. The topic is still very much valid.)

Manipulatives are IN, in math education. But do they TRULY facilitate learning to such an extent as people promoting them claim?

The entry at Text Savvy, Hands-on, Brains-Off, explains some of the pitfalls in manipulative use. Very enlightening!

Quoting from an article at Education Week (Studies Find That Use of Learning Toys Can Backfire):
In a similar series of experiments at the elementary-school level, the researchers found that children taught to do two-digit subtraction by the traditional written method performed just as well as children who used a commercially available set of manipulatives made up of individual blocks that could be interlocked to form units of 10.

Later on, though, the children who used the toys had trouble transferring their knowledge to paper-and-pencil representations. Mr. Uttal and his colleagues also found that the hands-on lessons took three times as long as the traditional teaching methods did.

The video below also illustrates how manipulative use can lead to problems.

The girl solves an addition problem involving thousands by drawing thousand-blocks, hundred-sheets, etc. on the board, taking 8 minutes. Then she says that at home she has been taught to stack the numbers and add. She solves it that way, too, taking 1 minute. But she gets two different answers.

(Hat tip to Denise)



Clearly, manipulatives aren't the best way to solve such problems, and children shouldn't be led to believe so. Not that this girl believed that way... she seems to understand which way is easier and quicker.

My take is that manipulatives aren't the ultimate answer to math teaching.

They're a reasonable starting point. But children should NOT be taught to rely on them. Children need to be taught and shown how to transfer all of that concrete play into the abstract.

In my books, I often instruct concepts with pictures, which essentially take the place of manipulatives. Then the student does a bunch of problems that have the same pictorial representation in them, including having to draw  those same pictorial models to illustrate the math problems. THEN after that, the student goes on to totally abstract representation.

Here's one example from my book Add & Subtract 2-B: Adding with Whole Tens. The student first adds using the pictorial model, and then with numbers only.

I don't know if that approach suffers from the drawbacks mentioned in the article above. I hope not; from the feedback I get, it seems to work well.

Comments

Christina in GA said…
I don't think my son has relied too much on the pictures shown in your multiplication book. It helped him "see" the concepts and the logical way it's laid out is finally helping multiplication make sense.
kriswiss said…
I believe that students need to learn the concpt first in math and then we can give them manipulatives to enhance the learning process.
alpatel said…
student learn the concept of maths. but easily understand through Maths' 3D Animated Video and new stereoscopic econtent
Cool math games said…
I think the best way to learn is by playing games; math games that is. Better yet- cool math games :-)
The most fascinating notion revealed by the video is in the comments left on YouTube, indicating that regardless of ostensible politics, educational conservatives can't see the point of letting students think, possible "inefficiently" or explore various models.

Further, the notion that there's some "best" way to teach math that trumps all others is intriguing, but I don't find it very convincing, regardless of what way is being touted as "best."

I have no doubt there are some ineffective ways, confusing ways, ways that suppress student thinking, etc. And I don't recommend doing those, but they tend to dominate US math teaching in school.

Models, with or without manipulatives, are part of how good teachers try to open up possibilities for students and how students construct their own understanding. If they make mistakes or construct 'inefficient' models on the way to grasping key mathematical ideas, I'm not sure how that's a bad thing. But then, I really AM a progressive educator.
NickDMax said…
As Christina in GA pointed out, these are tools to help one visualize and experience numbers, quantities, and arithmetic operations to aid in building mathematical and numerical intuition. They are (as far as I know) not supposed to be a new "method" for asthmatic. You can't go drawing boxes/bins/bags every time you want to add a few numbers, but you can help someone "see" how the paper-and-pencil algorithm is working, what "carry the one" is doing. We NEED the representational system and its associated algorithms to effectively do math, but we also need ways of helping students understand what is happening in that very abstract system. Any calculator can add numbers, humans can see patterns, make logical leaps, make intuitive guesses, understand, grasp, make use of the deeper nature of numbers -- that is the goal here. The goal is more algebraic thinking then mechanical arithmetic skill. You still need to teach the skill, but you have to try to help someone understand what is going on and why things work because eventually they will cease to use the skill. When was the last time you added numbers like 1568+1423+680?

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