Dan Meyer’s recent TEDx video has spawned some confusion characterizing the method as entirely constructivist (wherein as many mathematical techniques as possible are “discovered” by the students rather than given to them).
I don’t believe this to be the case, but I don’t think the pitch has been clear as to why. I consider the main point to be a problem solving method I’ll refer to (via Dan) as Be Less Helpful, in which:
a.) The problem is given with minimal information. The students decide what additional pieces are required.
b.) The students are encouraged to bring as much of their own experience as possible, of a concrete nature when relevant but also of a mathematical variety.
Concrete question, no artifact:
How fast do you have to be driving to outrun a speed camera?
Concrete question with an artifact (aka What Can You Do With This):
How large are the two types of DS-i?
Abstract question:
Call any natural number with two or more digits where each successive digit is smaller than the previous a decreasing number. (For example, 531, 98642, and 60 are decreasing numbers, while 94105 is not.) How many decreasing numbers are there?
Engineering challenge:
Here is a plastic cup. Design a box to hold 50 of these cups.
Physical challenge:
Throw a ball to a receiver twenty feet away such that the receiver catches the ball without slowing down.
These problems are compatible with a constructivist method of teaching but no particular method is required. The students may already have all the mathematical tools they need and are practicing their application.
Additionally, while problem solving can generate a constructivist curriculum
(Suppose students who only know the angles of a triangle add to 180º.) Solve for x.
Not all constructivist tendencies have to do with problem solving:
List everything you know about triangles.
In essence, the technique provide an “open” environment akin to how real-world problem solving (both concrete and mathematical) occurs. When creating a new device, no textbook awaits the engineer giving the exact data needed. When solving a new theorem, no creature sits at the mathematician’s shoulder whispering the lemmas.
Filed under: Education, Mathematics |
The Number Warrior 
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I use that “abstract question” every year in my senior elective. In fact, I give tests where the main parts are worth 90, and one question of this type is worth 20, where the first 10 points are available for making progress.
I like that the “cases” approach leads recognizable sub-answers, which lead in turn to a surprisingly simple answer, which suggests an alternate approach… but I’m usually pulling large parts of this out of them (not on the test)
Jonathan