I've been thinking a lot about the role of prior knowledge in teaching. Today my CT lead an investigation around indirect variation (when one variable goes up by a certain factor, the other goes down by the same factor, or in math terms xy=k). The investigation involved balancing nickels on a ruler which had a pencil under the center point at 6 inches to act as the fulcrum. The point of the investigation was to notice a pattern that the number of nickels * distance from center was equal on the left and right sides. When I checked in with some groups to see how they were doing, more than once I heard a response of "this is stupid." When asked why, the response was that the student had done an activity like it before in middle school or elementary school. My response was, "That's great! So you have a sense of what the pattern is. if you keep stacking more nickels on the left side, what do you have to do to balance them?" These students had an awareness that more nickels meant moving them closer to the fulcrum. When I asked them if they new what the pattern was, they were not able to answer. So I encouraged the students to look for a pattern in the numbers (which they were unable to do without a lot of prodding).
If I were to teach this lesson, I would make sure to elicit from the students what experience they have with balancing activities (either formally in school or informally like on a seesaw) and ask them to make a prediction as to what will happen when they add more nickels and to explain why they made that prediction. From there I would ask them to focus on finding a pattern between the distance from center and the number of nickels. Perhaps restructuring the lesson in this minor way would 1) show respect for what students already know, 2) activate what they do already know so they can draw upon it in this lesson, and 3) focus the students on what the new part of the learning is.
Another way that I "deal" with prior knowledge in my classroom is to give pretests to see what students know. I do this before a unit and test for the specific skills that my unit will uncover. I have noticed that when I give a pretest, I get fewer comments from students like "I've done this before" or "I already know this." Whether students have done some of the math before, pretests usually reveal a lack of mastery and students seem to have a stronger desire to learn. In the case where students do have mastery, then I can structure an alterative project for them to extend their learning.
On a side note, the directions for the investigation (out of Discovering Algebra) were way too long. I think they could be rewritten in a much more concise way so that students don't get hung up on reading and interpreting steps.
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