"In a word, flexibility."

In What’s Math Got to Do With It?, Jo Boaler¹ summarizes the influential research conducted by two British researchers, Eddie Gray and David Tall.² They identified the reasons why so many children have to exert great effort to do simple mathematics.  They found that students who are low achievers are often learning a more difficult form of mathematics than their high achieving classmates.

For example, in addition, there are four common strategies:

1. Counting all (4 + 13 is simply counted: 1, 2, 3, 4, 5, 6...15, 16, 17)
2. Counting on (4 + 13 can be counted by starting at 13 and counting on four more ...14, 15, 16, 17, or less efficiently by starting at 4 and counting on 13 more... 5, 6, 7... 15, 16, 17)
3. Using known facts
4. Using derived facts (decomposing and recomposing numbers to make easier number combinations)  For example, 4 + 13 can be thought of as 4 + [10 +3] = 10 + [4 + 3].

Gray and Tall found a striking difference in the strategies that low achieving students used, compared to those used by high achieving students.  The low achievers used tedious counting strategies for addition and the difficult counting backwards strategy for subtraction throughout the elementary grades, sometimes relying on complex finger-counting systems as the problems got more complex.  In contrast, the high achieving students consistently used derived facts from the early grades.  That flexibility of thinking led to simply knowing facts or calculating them so quickly that they appeared to be known facts.  As students moved into the intermediate and upper elementary grades, the low achieving students still heavily relied on counting strategies in more and more complex situations.  Meanwhile, the high achieving students rarely went back to using counting methods.

Gray and Tall explain that the flexible facility to carry out a procedure or decompose/recompose and manipulate numbers, "...acts as an autonomous knowledge generator." (Gray and Tall, p. 18.) They found that high achieving students amalgamated both process and concept. These students connected:

• The process of counting with the concept of number
• The process of counting on with the concept of sum
• The process of multiplication with the concept of product

Thurston (1990, p. 847) described the "compression" that the connection of mathematical process and concept can allow:

Mathematics is amazingly compressible:  you may struggle a long time, step by step, or work through the same process or idea from several approaches.  But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression.  You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process.  The insight that goes with this compression is one of the real joys of mathematics.³

Unfortunately, as Jo Boaler points out, for low-achieving students, the learning of mathematics does not involve flexibility or compression, but is a never ending series of rules.  "What typically happens is that they get labeled as low-achieving students and people decide they need more drill, putting them into classes where they repeat methods over and over again.  This is the last thing these students need..." (Boaler, 2008, p. 152)

Watch for our March newsletter on ways to help students learn to use numbers flexibly.  Please note that helping students learn to use numbers flexibly does not preclude helping them create useful algorithms or emphasizing the connection between physical models of mathematical processes and standard algorithms.  A discussion on computational fluency will be continued in our blog during February and March.

¹ Boaler, Jo. 2008. What’s Math Got to Do With It? Viking Press, NY, NY
² Gray, E. & Tall, D. (1994), Duality, Ambiguity and Flexibility: A “Proceptual” View of Simple Arithmetic. Journal for Research in Mathematics Education, 25(2), pp. 116-140.
³ Thurston, W.P. (1990). Mathematical Education.  Notices of the American Mathematical Society, 37, pp. 844-850.

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