My classroom uses many Math fun activities, or games, during the day. Typically, activities are preceded by an introduction to the concept -- which includes student participation and soliciting their observations -- and modeling of the activity by the teacher.
I have noticed that the structure of the activity is often at the symbolic level, occasionally beyond the level of understanding of many students.
For example:
Today, the class did an activity focusing on "greater than/less than/equal to", in which each student received numbered cards to flip. The student wrote the number he or she flipped n the line. Then they added the numbers and put the sums on the final line. They did this twice -- once for each side of the circle.
Students were instructed to look at the sum and to decide on the symbol to place in the circle:
__ + __ = __ O __ + __ = __
Many students picked it up right away, and got started before the teacher finished passing out the cards.
Other students seemed somewhat baffled. They would fill the blanks in with a plus sign, or fill the circle in with a sum. One student asked "what's the point in writing a sign, if you already know which one is bigger because of the number it is?"
The teacher had used the alligator analogy to explain how to use the symbols (the alligator is hungrier for the bigger number, so the mouth faces towards the bigger number). This analogy helped students figure out how to show greater than or less than, but not why.
The activity was engaging for all the students, once they figured out the "rules" of the game. I wonder, however, if the game might have been made higher-level by asking students to consider why we use symbols to show the relationships between numbers. This kind of activity ensures students get ample practice, but does not address their deeper conceptual understandings about math. In effect, it asks them to reproduce existing structures, rather than asking them to construct their own understanding about the structures.
If the activity aimed to do the latter, the teacher would get the added benefit of seeing student thinking more clearly -- far beyond the immediate concept, and perhaps see misconceptions that may prove problematic in future.
I imagine the discussion needed to make the activity a higher-level one would require careful design on part of the teacher before it took place. Here is an instance in which much classroom practice would be needed to get students ready to talk about their mathematical thinking. The benefits, however, would be far-reaching, and -- I think -- would be well worth the initial time investment.
You do a great job describing this important case. Your analysis is very sophisticated, but I am more impressed with your initial noticing, which is perhaps the most important revelation for a math teacher to have - when you present material in a pre-determined way (especially when that way is abstract) some students will catch on, but others will simply be lost and really have no way of catching on. As you state, even once "the rules of the game" are explained, the students have little understanding of the deeper concept that underlies the game, which clearly you and your teacher understand. But without a deep conceptual understanding to begin with, a game like this can be played, but still make no sense. You may not learn any math from it.
ReplyDeleteSo, the game still has its place; but something abstract like that (or anything procedural or based on "rules") should not be the first thing students see. Instead, students should be ask to make sense of the "big idea" for themselves. Wouldn't it be important and convenient for us to be able to signal which set of stuff has more than another? What are some ways we might do that? Students can think on their own and them compare, student-to-student. This is more valuable than starting with an abstract game.
I can say this until I am blue in the face, but it means nothing until the intern herself notices this phenomenon, which is exactly what you have done. So you are on the path to success for certain!