### Montessori Math Manipulatives

*by Anne Zelenka*

Another school year is about to start and I'm again thinking how much I love the Montessori approach. I like that classrooms are multi-age. Older kids can act as leaders and younger kids can learn from them. Cliques are less likely to develop than in a single grade classroom. Age is less important than developmental stage in determining what the kids work on. The Montessori approach allows for kids to go as fast or as slow as they need. I love it.

I am not, however, a huge fan of the Montessori equipment. It seems too fussy somehow. Now an article in Scientific American has given shape to my doubts. Judy DeLoache, a specialist in early cognitive development at the University of Virginia, writes about how children develop symbolic thinking. In her research, she's found that children learn to think symbolically over several years and that they make many errors during that development. Thinking symbolically requires understanding dual reprentation: seeing an object "both as itself and as depicting something else."

Here's a couple examples of disconnects I've seen between the materials and the math they represent:

Example 1. The binomial cube. This cube is intended to represent the equation (a+b)^3. My husband and I were introduced to it at parents' night last year at the elementary school. The linkage between the wooden cube and the algebraic equation is a huge leap, one the elementary teacher wasn't prepared to take. To me, it was much more complicated to understand the algebra with the manipulative than with written mathematical symbols. [If you're interested, here's an explanation of the binomial cube.]

Example 2. Square root with a peg board. Henry and I did a square root together using a board and pegs one day at school. While I understood immediately how to physically manipulate the material, the connection between the material and the math behind it was not easy for me to grasp. This may be because I have spent my entire mathematical education using symbols and not using manipulatives, other than a stint at age three with the Montessori chains for skip-counting. But I wonder how many kids quickly learn to manipulate the materials but don't make any sort of leap to the math behind it.

Disclaimer. I'm no expert! Just reporting on what I've seen of math manipulatives in Montessori. I'm also not arguing for doing pencil-and-paper work in preschool. The kids have fun with the Montessori materials. I'm skeptical about how much they really prepare them to do symbolic math but I'm in favor of the approach as a whole.

I am not, however, a huge fan of the Montessori equipment. It seems too fussy somehow. Now an article in Scientific American has given shape to my doubts. Judy DeLoache, a specialist in early cognitive development at the University of Virginia, writes about how children develop symbolic thinking. In her research, she's found that children learn to think symbolically over several years and that they make many errors during that development. Thinking symbolically requires understanding dual reprentation: seeing an object "both as itself and as depicting something else."

DeLoache discusses the educational ramifications:

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."At the primary level (preschool through kindergarten, roughly ages 3 to 6), math manipulatives are used with great enthusiasm in Montessori classrooms. In elementary (ages 7 to 12) Montessori, manipulatives are used less often while paper and pencil work increases.

Here's a couple examples of disconnects I've seen between the materials and the math they represent:

Example 1. The binomial cube. This cube is intended to represent the equation (a+b)^3. My husband and I were introduced to it at parents' night last year at the elementary school. The linkage between the wooden cube and the algebraic equation is a huge leap, one the elementary teacher wasn't prepared to take. To me, it was much more complicated to understand the algebra with the manipulative than with written mathematical symbols. [If you're interested, here's an explanation of the binomial cube.]

Example 2. Square root with a peg board. Henry and I did a square root together using a board and pegs one day at school. While I understood immediately how to physically manipulate the material, the connection between the material and the math behind it was not easy for me to grasp. This may be because I have spent my entire mathematical education using symbols and not using manipulatives, other than a stint at age three with the Montessori chains for skip-counting. But I wonder how many kids quickly learn to manipulate the materials but don't make any sort of leap to the math behind it.

Disclaimer. I'm no expert! Just reporting on what I've seen of math manipulatives in Montessori. I'm also not arguing for doing pencil-and-paper work in preschool. The kids have fun with the Montessori materials. I'm skeptical about how much they really prepare them to do symbolic math but I'm in favor of the approach as a whole.

## 7 Comments:

My husband nearly had a fit when he was visiting a local Montessori preschool. One of the kids had made the correct leap that if you slice a cylinder, you get an oval, and was trying to explain this to the teacher, who blew him off with something like "we don't cut our toys." T. whispered "you're right" to the kid when he got a chance.

Yeah, John Holt talks about this in How Children Learn. As an adult, we may 'see' how a manipulative demonstrates something, but we're looking back at it. It may not work for the kids -- and, as with everything, people learn in different ways, so what works for some may not work for others.

Elizabeth - what an interesting story. It's kind of annoying to me in Montessori that all the material has to be used in particular ways, not in whatever ways the children want. While the overall discipline makes for an orderly environment, it doesn't promote creative inspiration like what you're talking about.

Marjorie - But wasn't it John Holt who was all into the Cuisinaire rods or something like that? I thought he was a big proponent of manipulatives.

Good memory, Anne -- thats right, in his revisions in the updated version of How Children Learn, he referred to how he used to be very pro-manipulative and then he explained why he was no longer.

I need to review my Holt, what I'm thinking about appeared in the updates of the revised version of How Children Fail

Please do not clump all Montessori schools together. There is an abundance of amazing work happening in many montessori schools that use the materials in many different facets. What's interesting to me is that mainstream education is marketing "manipulatives" that are exactly what is used in Montessori schools. You do need to touch and manipulate before you can abstract. Dr. Jane Healey writes many books about how the brain learns best and at the preschool age, its with your hands, not Paper and pencil. As with any educational system there is always narrow mindedness, but believe me in my classroom those connections made by exploring materials are spectacular and important.

As a montessori parent, I thought this blog was very one sided. My 5 year old daughter was having trouble sleeping last night, and instead of saying she was thinking of butterflies and fairies to help her go to sleep, she said she was thinking of numbers! It was pretty empowering to know that she is really comprehending the concept of numbers. All schools have there pros and cons what works for some doesn't for others.

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