As a classroom teacher I worked tirelessly to create tasks, problems and questions that I thought would be good for students. I thought that the tasks I was creating were equal to what the students would be thinking. I am constantly reminded that what I perceive to be the question is not always what the student perceives as the question or task. Or, it may be that the student perceives the same question that I perceive, but does not solve the problem the way I intended. While that is perfectly alright, it can be a disconnect for teachers who believe that the task presented is related to student’s cognitive process.
Let me give you an example. I presented Elsa, Liam & Jacob (pseudonyms) with the following situation. I placed 8 little blocks on the table. I asked Jacob to count them, to verify that there were eight. I covered the eight little blocks with a cloth. I slipped a few blocks under a second cloth without specifying the amount. I asked the group to remind me how many blocks were under the first cloth. They all said, “eight.” I asked how many little blocks were under the second cloth. They paused and they said they didn’t know. I announced to the students that I had a total of 15 blocks under the two cloths and asked each of them to figure out how many I had secretly placed under the second cloth.
Liam could not access the problem. The task presented to him did not make any sense in light of his past experiences. Elsa was able to come up with an answer of seven immediately. However, Jacob said that the answer was 23. I asked Jacob how many were under the first cloth. He replied, “eight.” I asked him where the 15 were and he pointed to the second cloth. I asked him if he remembered how many I said there were in total and he remembered that I had said 15. It didn’t seem to bother Jacob that he had used 15 as an addend and as the total and had also ended up with a total ‘count’ of 23. Jacob could only think of this task as a direct addition problem. It was his goal to add the two values that he heard.
The structure of a task – as seen by the teacher – does not determine the way that a child will make sense of or solve the task.
As I prepare new experiences for engaging students in mathematical thought, I am constantly thinking about the student’s mental structures and how they may interpret the situation. As I watch for evidence of the student’s thinking, it is my goal to interpret their actions and what I think they understand through the lens of the research literature and make future moves based on their understanding of the task, not my own.