Last time, I introduced a problem to illustrate the basic differences between additive reasoning and multiplicative reasoning used to solve a problem. I also defined levels of units and what it means to “coordinate units”. In addition, I said that “in activity” means that a level can only be accessed while working on the problem and perhaps using other physical aids. This time, I want to extend the problem to talk about students working at even higher cognitive levels…
A student can assimilate with 2 levels in activity, with 2 levels, with 3 levels in activity, with 3 levels, etc. I will give an example to show the difference between assimilating with 3 levels and assimilating with 3 levels in activity. Let’s extend the vacation problem as follows:
I am away for three weeks. At the end of the three weeks, I decide to stay away for five more weeks. How many total days is that?
Assimilating with 3 levels in activity, the student could reason this way:
3 weeks x 7 days/week = 21 days
5 weeks x 7 days/week = 35 days
21 days + 35 days = 56 days
But if you ask this student how many weeks, that would be a new problem to solve, because the student wouldn’t be able to hold all the information.
Assimilating with 3 levels, the students could reason this way:
3 weeks and 5 weeks are two “like composite units”, so they can be added:
3 weeks + 5 weeks is 8 weeks.
Then, 8 weeks x 7 days/week = 56 days
This student can operate on a week as a unit, as well as a day as a unit. The more levels that a student can assimilate with, the more problem solving power that student brings into the situation.Following is a problem for you to consider. Think about how students assimilating with 1, 2, or 3 levels might approach this problem:
In a classroom, there are six desks in each of four rows. The teacher wants to add seats until there are 36 in the class. How many more rows will be added?