Recently, while working with students, we offered up a situation where nineteen counters were placed under a cloth. Seven of the counters were pulled out and the students were asked how many remained under the cloth. One child extended ten fingers, pulled them back, and then re-extended nine. He pulled back seven fingers, one by one, resulting in two still extended. Looking at those two fingers he took a moment and then said ‘twelve’. A second child began extending their fingers one by one, while counting down; 18, 17, 16, 15, 14, 13, 12, stopping when they had 7 fingers extended and also saying ‘twelve’ as his answer.
This example shows that children do not always see a situation the same way an adult interprets it, nor do they attempt to resolve it in a way that may look familiar or logical. An adult may see the situation as a simple subtraction problem and say the common subtraction algorithm is the simplest and quickest way of arriving at an answer. This does not mean the child is not mathematical or incorrect in their thinking. Actually both the strategies described above are common for young children and give us ideas into their thinking. If we are serious about the math education of children we must take the time to understand their thinking before we can begin to guide them towards more sophisticated mathematics.
During the 1960’s, the New Math program developers introduced the missing addend problem (change unknown) in the first grade, with the hope that it would connect addition and subtraction for children. If it did, then time would be saved in the learning of subtraction facts, as then subtraction could be defined in terms of addition. But teachers found the missing addend problem a source of great frustration for many children. With such feedback, program developers abandoned the introduction of these problems in the first grade.
Decisions made relative to the introduction and then abandonment of the missing addend problem were done in the absence of any data about the way children develop throughout the course of a mathematics program. That such data is desperately needed should be clear from the example given. In fact, both decisions – that of universal introduction and that of universal abandonment – were essentially incorrect. Instead, information on how we determine which children are ready for introduction of the missing addend problem and which are not would be a much more useful piece of information.
Today missing addend is once again found in first grade mathematics but we still tend to make decisions based on the mathematical thinking of adults, largely ignoring the child and their mathematics. Does the simple fact that a child is in a first grade class give them the ability to solve a missing addend problem? As long as we continue to introduce children to mathematical ideas based on the classroom they are sitting in instead of the mental operations they employ, we will continue to frustrate and fail them.