When kids memorize times tables, we blithely believe that they understand multiplication. Too often, that is not the case. In multiplying, we are looking at coordinating units. What does that mean? I will use an example to introduce the idea:
If I go on a trip for 5 days, but then I have a 2-day delay before returning, I am gone for 7 days.
This addition (two of the same units – days) is within 1 level, so I am not coordinating units.
If I go away for three weeks, how many days is that?
Now, we have two levels of units to coordinate, days and weeks, in order to answer the question. There are 7 days in a week, distributed across each of 3 weeks, that’s 21 days. This is the difference between additive and multiplicative reasoning.
On my team here at the AIMS Center for Math and Science Education, our task is to try to determine if a student has developed the mental structures necessary to carry 2, 3, or more levels of units.
Considering the three weeks problem… At one end of the spectrum, a student might quickly respond with 3 x 7 = 21, that’s 21 days, with little cognitive energy used. Another student might count 1, 2, …, 7, that’s 1 week, 8,9, …, 14, that’s 2 weeks, 15, 16, …, 21, that’s 3 weeks… so, 21 (while using fingers or something else to monitor). This second student correctly answers the question, but there is a great expenditure of cognitive energy. The second student might not even be able to remember that the 21 means “21 days”.
The second student above is doing the most basic coordination of 2 levels of units. This coordination, accessing the second level, is called “in activity”, because it can only be accessed while working on the problem and perhaps using other physical aids. The first student assimilates with 2 levels, that is, he/she can carry out work in 2 levels easily and quickly because the mental structures needed are in place.
Our team is trying to identify indicators that teachers can look for that will allow them to see the number of levels a student can assimilate with. We are also working to find assessments that can help gauge those indicators. Finally, we hope to identify experiences that teachers will be able to facilitate that will enhance the development of more levels of assimilation.
Our primary sources for information will be Professor/Researcher Les Steffe and some of his former students (i.e. Dr. Anderson Norton, Dr. Catherine Ulrich, Dr. Amy Hackenberg).