Knobless Cylinders and the mathematical mind

Bee Pape is the expert here, and a search for her documents is well worth the time. She has extensive extensions with the knobless cylinders. She's been teaching for a long time in a Montessori school, done research with early number development. She also knows the work of Piaget and Constance Kamii. Many of my own thoughts are generated from a workshop of hers that I attended combined with my own research. I found her delightful, because I've read and understood so much of the same material.

Jean Piaget was 26 years younger than Maria Montessori. His developmental theories were also based on the observation of children and many interesting and experiments. These included conversations with the children about what they were thinking. He was familiar with the ideas of Maria Montessori and was at one point president of the Swiss Montessori society. One of his fundamental beliefs is that children construct their own knowledge through interaction with the world. In many ways, this is similar to Maria Montessori (and many current educational theories) that young children learn best through concrete materials and actual experiences.

Jean Piaget observed an interesting phenomenon called "conservation." There is "Conservation of liquid," "conservation of number" and several more. These are a developmental stage that children go through where they can only hold one attribute of a physical material in their mind at a time. For example, in "conservation of liquid" once a child agrees that the liquid in two equal shaped glasses is equal, if one glass is poured into either a wide/short glass or a taller/thinner glass the amount of liquid is no longer the same. The same thing is true with "conservation of number." If a child agrees that two rows of similar materials (poker chips) have exactly the same amount, if you stretch out one line – then they are no longer the same. There are many reasons for this. Sometimes the child will say that the longer line has more, sometimes the shorter, sometimes they will provide an answer that is difficult to explain. The result is consistent and has been shown many times. Young children do not recognize that quantity is consistent regardless of spatial arrangement.

This developmental understanding of number only comes with actual experience (in many forms) with things to manipulate. Experience with objects to count, adding or subtracting one, matching, all promote learning about equality, greater than and less than. What do you do if there is one more doll than you have a bed for?

If you look at current educational research and activities for children, you see seriation, ordering, pattering, matching, and measurement. Maria Montessori was brilliant. How many people remember their training that a child should truly spend time with and master many of the sensorial materials before being introduced to mathematics? In my training, the knobless cylinders were one of the last materials that were introduced to a child because there is no control of error except visual. The knobless cylinders are a set of 40 with 4 subsets in different colors. You have a few pieces that are actually identical. A child can sort them by color. A child can sort them by size. A child can sort them by height. If one is missing, because they are sitting on it – they have a problem to solve and a interesting learning lesson about quantity. They can explore relationships. You can even set up logic puzzles that are designed for the youngest child and visually based. Find me a piece that is this height and this color (or not this color…)