You have probably all heard the statement, “If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck.” Generally this statement is used to convince ourselves that we can trust what we see, or that we can make sense of our world by gathering evidence and comparing what we notice against previous experiences. At first glance, what did you see in the picture above?
But isn’t it true that after further investigation, we sometimes adapt our views? Are our original ideas misconceptions? The notion of a misconception implies that there is one, authoritative ‘Truth’ out there and that what we understand is either a match and therefore true, or not. Since individuals process events via their personal experience, whose experience should be taken as the authority? We create our reality as we live it. We form conjectures, ideas, or theories and adapt them as we go. For some young children, anything that can be pushed is called ‘car.’ And not everyone agrees on what color the square is or what color the dress below is.
Askew and William (1995) explain that all children “constantly ‘invent’ rules to explain the patterns they see around them.” Many of the rules that children invent are true in one way or another but only apply in a limited context. Many children observe that when a whole number is multiplied by 10, the original number gets a zero at the end. However, this does not hold true for all decimals. Young counters may feel comfortable with the idea that four coins is more than three coins – and we would all agree – but cannot fathom the idea that one coin is valued more than another. I encourage you to read Shel Silverstein’s poem “Smart.”
Children begin to develop number sense through their counting experiences. At first it may be “one, two, lots.” Then the values from three onward start to have meaning. They become values that the child can conceive of and build meaning around. As you have read in my colleagues’ blogs, there are many stages in a child’s construction of number. Adults may become impatient and think that children are sometimes wrong or have built up some misconception. Remember, the child’s understanding may be correct, but only in limited contexts.
This is not an argument for “touchy-feely” math, but rather an argument for more mathematical dialogue between child and adult. Ask good questions of the child to elicit explanations of their thinking. Present similar situations in which you believe that their current understanding may break down in order to provide opportunities for the child to adapt their understanding to accommodate the new situation. Let them ponder, reflect, and wrestle with new ideas for a while rather than jumping in with your understanding right away. Challenge yourself to see the situation like the child sees it.