Scaffolding Via Incrementally Complex Problems

In teaching math, you have to start somewhere. The notion of exposing young students to several procedures at once (except for, say, the standard algorithm, which is saved for a few years down the road), with minimal explanation, and allowing them to then choose the one they like has been shown, repeatedly, to be ineffective.

We all learn by imitation (called "mimicking" by someone whose name I will not mention here, and that term then mimicked by this person’s followers). The initial act of imitation serves as a foundational skill which students then link to prior knowledge.

As an example, consider this problem: "If the average weight of two boys is 80 pounds what is their total weight?" Use explicit instruction to remind them that the average is the total divided by two (in this case). “How could we work backward to get the total -- how do we reverse the steps?” I’ll ask my class. Students then figure out that by multiplying the average weight by two (for this particular problem) they will obtain the total weight. After mastery of this concept, they are then given a related problem: "The average weight of two boys is 50 pounds. One boy weighs 55 pounds. How much does the other boy weigh?" They may be given a hint such as "Can we find the total weight" which they know how to do. Once they have the total, they link to prior knowledge (subtract one number from the total to get the other number) and they solve the problem.

Problems with averages can then be given that are incrementally more complex, so that students build on what they have previously learned. This is the approach used in Singapore’s math texts. Over time, students build up a repertoire of problem solving skills related to averages by the scaffolding of problems that become more complex. Eventually, they can solve problems like the following: “John has an average of 90 on the last four tests he has taken. What score does he need on the fifth test to raise the average to 92?”

There is a pervasive belief in the math education world, however, that students will somehow grasp the rudiments of problem solving by being exposed to problems they have never seen before. Without the essential foundation instruction that relates to the structure of such problems, students will be left in the cold. I once heard a teacher admonish her eighth grade students by saying “In the seventh grade you were told how to solve certain problems. Now you have to think for yourselves. It’s no longer ‘lather, rinse, repeat’ ”. In the end, they attempt to teach generic problem solving skills, using open-ended one-off problems that do not generalize, and do not have wrong answers. (See this paper for more information on why teaching generic problem solving skills does not work.)