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Thinking like mathematicians

This post was written by Mary Gage Davis, Opal School Curriculum Specialist 

Our work at Opal School is driven by the desire to support children in making meaning; math is no different.  Much of our math curriculum is based upon teaching for number sense.  With good number sense, children can think flexibly, compose and decompose numbers and have the opportunity to truly think like mathematicians.  This may look and feel very different from the mathematics that many of us grew up with.  You may be wondering why your children are not being taught traditional algorithms (how to "carry" or "borrow", for instance) in math workshop.  You may wonder if children are expected to do arithmetic calculations.  The answer is “yes”, we expect children to be able to compute accurately and efficiently, but we want children to do this as active mathematical thinkers.  Students are encouraged to look at the numbers before they calculate, to think rather than follow rote steps, to base their procedures upon strong number sense.  

For example, when given the problem 368 + 208, some children may:

  • Split the numbers by place value and add them together:  200 + 300 + 60 + 16
  • Recognize that it is more efficient to keep the 368 whole and add 200 then 8.
  • Turn 368 into a friendly number by taking 2 from 208 and adding it to 368, making 370 + 206.

In order to do this kind of work, children need to develop a mental model of the relationship between numbers.  Many of these strategies and big mathematical ideas are developed through mental math mini-lessons at the beginning of math workshop.  Children are given a string of related problems and asked to solve them mentally.  As children share their thinking, it is then modeled on the open number line or with arrays depending upon the operation.  As the number string unfolds, so do relationships between numbers, operations and problems that highlight specific strategies.  (Future blog posts will address the specifics of the models: open number line and array.)

At the heart of mathematics is the process of setting up relationships and trying to prove these relationships mathematically in order to communicate them to others.  Creativity is at the core of what mathematicians do.  (Fosnot and Dolk 2001)

In the 1980’s as researchers began to explore whether or not algorithms should be the goal of arithmetic thinking.  In a study conducted by Kamii and Dominick, researchers compared three groups of children: those that had been taught only traditional algorithms, those that had been taught none and those that had been taught both.  These children were asked to calculate 7 + 52 + 186.  The results are described on this chart:

  Download Algorithm chart

You may notice that the greatest percentage of children to get the correct answer were those who had not been taught algorithms.  Even more interesting, you may notice the range of answers given.  Those children with no algorithm experience were in the closest proximity to the answer.  “It appears that most of the errors in the first group were place value errors; in the latter group, they were calculation errors.  This is strong evidence that the algorithm actually works against the development of children’s understanding of place value and number sense.  As they focus on doing the procedures correctly, they sacrifice their own meaning making; they sacrifice an understanding of the quantity of the numbers they are dealing with.”

The mathematics curriculum at Opal School is greatly influenced by this and other research that supports teaching for number sense.  Learning to compose and decompose numbers, building a strong number sense and a repertoire of efficient strategies takes time.  At Opal School, we believe this time is an important investment in children’s learning.  Upcoming blog posts will give a closer look at what this work might look like in the classroom across the grades.