Today marks the beginning of Girls’ Angle’s 9th session and fifth year of existence. To celebrate the occasion, I’m blogging about one of the core philosophical principles of Girls’ Angle: Math Accommodates Many Ways of Thinking.
In fact, I am often amazed by how differently mathematicians think. Richard Feynman expressed this beautifully:
(Please let us know if this link goes bad.)
Although Professor Feynman was a theoretical physicist, his comments seem to me equally applicable to mathematicians.
I am concerned that the way math is typically taught in school in the US often fails to acknowledge the extraordinary differences in the way we think – too often, schools demand that students solve their assignments in very specific, overly rigid ways.
As another example from my experience, I sometimes bake a yummy chocolate brownie for the class. The students must figure out how to divvy up the brownie into equal portions for all before anybody gets some. The catch is that the brownie isn’t a rectangle or circle; it’s some unusual shape.
In one class, I made the trapezoidal brownie shown at right for 16 students. I did design this brownie to have a particularly elegant dissection, which one of the students did succeed in finding.
The cutting scheme at left shows the solution that the brownie was designed for. For the goal of divvying up the brownie for 16 people, it’s a particularly elegant solution because with 16 congruent triangles, one can see at a glance that the problem is solved. Many students’ first instinct is to compute the area of the individual pieces right away, but this solution renders the area computation unnecessary. Also, one can cut the pattern with long broad slices that run clear across the brownie.
However, the students came up with a variety of alternative solutions representing different ideas that each leads to important concepts in mathematics. Rather than push the solution above as the “right one,” it’s quite wonderful to discuss each solution’s advantages and disadvantages. I’ll describe two other solutions that students found.
Divide and Conquer
In this approach, the student invoked a powerful problem-solving strategy: divide and conquer. The maxim is: When faced with a difficult problem, try to break it down into simpler ones. Here, the student turned the trapezoid into a rectangle and two triangles.
The student divided the rectangle with ease and noted that the triangle on the right was already the necessary area. The triangle on the left, however, presented a problem. How can a triangle be divided equally into thirds? Instinctively, many students draw the solution shown but reject it because they feel that the areas of the triangles get smaller and smaller. Ask the student “What is the area of a triangle?” Typically, the student will respond, “One half base times height!” Then ask, “What are the bases and heights of the three triangles you’ve drawn?”
When a student who had been doubtful about the equality of the areas of the triangles suddenly convinces him or herself about the equality, it is a special moment in their education. They will have seen that what they think to be true on the surface may not be the case, and that with reasoning one can find the truth.
So, although not as much of a “Gestalt” solution as the congruent triangles dissection, this solution illustrates the powerful divide and conquer technique and unveils an area property that is essentially equivalent to the preservation of the determinant of a matrix under the operation of adding a multiple of any column to any other column (or adding a multiple of any row to any other row). When the student later encounters this fact about determinants, the brownie cutting experience may very well make things go easier.
One Piece at a Time
Two students working together came up with the solution shown at left. First, they computed that the area of the individual pieces had to be 4.5 square inches. Next, they made a scale model of the brownie on a piece of graph paper so that each grid square represented one square inch. By clumping together 4 and ½ squares over and over, they knew that they would have to end up with 16 equal, albeit strange-looking, portions.
At first blush, this might seem a clumsy solution because of the awkward shapes produced. The pieces are awfully hard to cut out and some might complain about the shape of the piece they get. But, remember the principle: Mathematics accommodates many ways of thinking. We should always strive to find the merits of a student’s approach.
What is one of the merits of this approach?
By understanding that every solution will result in pieces 4.5 square inches in size and that any dissection into pieces of size 4.5 inches is a solution, these two students no longer had to concern themselves with the specific shape of the brownie. They could focus locally, tracing out little 4.5 square inch pieces, and know that in the end, they would achieve a solution. The technique will conquer a brownie of any shape! It’s nice to know a method that will always work. There are shapes where neither of the previous approaches would apply. There may be no way to dissect the brownie into congruent pieces and no convenient way to reduce the problem to simpler dissection problems.
Even more, the technique is a delicate clue for the Calculus concept of integration.
In school, we sometimes hear about the “best” solution or “good” and “bad” solutions. In truth, there are many “good” solutions, and the measure of “goodness” depends on the goal. And, if this goal is greater understanding, virtually all solutions are worth examination. Rather than dismiss the solutions that don’t conform to some predetermined “right way,” let’s instead ask “What mysterious, possibly new, point of view is hidden in the mind of the child?” and help them articulate it, develop it, and refine it.
Math accommodates many ways of thinking and it will take new ways of thinking to enrich society. I can’t wait to learn the thoughts of our Session 9 members in just a couple hours!