Concrete to abstract, or abstract to concrete? Bottom up or top down? Introducing new concepts can be done either way, but which way is best? Students tend to learn better when they’re building on an existing scaffolding. However, some students are more adept at seeing the bigger picture first then analyzing the details, while others favor seeing the smaller details and constructing the bigger picture of the pieces of the puzzle.
In yesterday’s lesson introducing algebra, I started concretely with weights on the scale for variables and the fulcrum as the equal sign. But I worked with them abstractly. Instead of telling students I had 200 g, 100 g and 50 g weights, I said we had three types of weights and called them a, b and c.
The weights were set on the scales as above so we wrote out the equation:
and simplified to give:
Then we talked about balance. Whatever you do to one side you have to do to the other, so I took the c weights off and showed how it solved for a:
Now you can solve for b by dividing everything by 2 to get:
However, since you can’t exactly divide the 200 g weight into two, you can’t exactly demonstrate this, but at least you can show the math. You can also now substitute in the values for the weights. If b = 100 g.
At this point, students could look at the weights and see the numbers.
Concrete to abstract and back again, I like how the lesson turned out, although, today I had to go over how to show their work again.