Every elementary school teacher has seen children struggle with subtraction. From these struggles, attitudes of “I’m not good at math” emerge. Our team recently worked with students on the concept of subtraction. We presented situations in which students would count out 23 cubes, hide them, and then remove some of the cubes. The students were then asked to think about how many were left. Many of them struggled, unable to accurately find a solution. There appeared to be a general inability to make sense of the problem. But we did have students that were able to solve the problem. Some in ways that surprised us.
In my last blog post, I introduced Dan, a third grade student, who was asked how many cubes were left after removing 5 from a pile of 23. He counted backwards, ”23, 22, 21, 20, 19,” while lifting five fingers, and then stated there were 18 left. Why would he do it that way? For us (adults) it seemed easier to count backwards 22, 21, 20, 19, 18 with the last number word spoken being the answer. His description gave us insight into his thinking. Dan showed an understanding of 19 to 23 as a set of counting acts that can be removed from the larger set of 1 to 23. He showed an ability to think of 23 as a number composed of 18 and 5. Notably, he could do this without actually counting the remaining cubes. He was able to think about the cubes in his mind and mentally operate on them to achieve the answer.
Jean Piaget, an esteemed researcher in childhood development, described mental operations as the ability to accurately imagine the consequences of something happening without it actually happening. During mental operations, children imagine “what if” situations. These mental operations were developed through Dan’s experiences of counting forward and backward. In the course of his activity, Dan is representing his past experiences of counting, uniting the counting acts 23 through 19 into a composite of 5 by monitoring his counting with fingers; 23 is 1, 22 is 2, 21 is 3…19 is 5. With the development of these mental operations, Dan can reason through these and other problems. Strategies of subtraction, often taught in classrooms (examples: subtract by breaking the subtrahend into two parts, use known subtraction facts, subtract in tens and ones, and add up) can be constructed in ways that continue to make sense for Dan. In doing so, the math becomes his, and for him his math makes sense.
During a recent visit to our center, Dr. Les Steffe emphasized that teaching children subtraction should never be the goal of a teacher unless there is good reason to believe that their students have the ability to count forwards and backwards from at least twenty along with an ability to monitor their counting. Without these necessary mental operations, children will continue to struggle with subtraction, despite the teachers best efforts. Join us in future blog posts as we continue our work exploring children’s thinking in subtraction situations and how teachers can help them construct meaning in their mathematics.