My last post on "teaching geometry" was very well received. A few people asked me about the method I had mentioned in that post for classifying geometry problems. This post is focussed on that method. Please note that I am teaching geometry to sixth graders and not advanced calculus to high school seniors.
Problem Classification Framework
I have found that classifying problems, using the framework described below, to be an effective tool in teaching kids how to think. Caution - This is a long"ish" post!
While there are additional factors that can be used to create a more sophisticated framework, here I am presenting a simple framework that requires answering three questions for each problem -
How advanced are the underlying theorem(s) that are required to solve a problem?
Every theorem used to solve a problem has a level of difficulty. For example, the theorem that sum of the angles of a triangle is 180° is an easy theorem to understand, remember and apply. Whereas, the theorem that three parallel lines divide every transversal line in the same proportion, is slightly harder to understand, remember and apply.
Similarly, the theorem that the opposite angles of a cyclic quadrilateral add up to 180° is a more advanced. It is even harder to understand that the cyclic quadrilateral theorem in the previous sentence is equivalent to the inscribed angle theorem (an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle.)
More advanced the theorem that is required to solve a problem, the harder the problem to solve.
How many theorems are involved in the solution for a problem?
Similarly, some problems require using more than one theorem to solve them. The more the number of theorems needed to solve a problem, the harder the problem. When you have many theorems that need to be used in a solution, often, the order is important and that adds to the complexity.
How obvious or hidden are the underlying theorems in the problem?
Lastly, the theorems that have to be used in the solution aren’t always obvious from the problem description. The less obvious the underlying theorem, the harder the problem. For example, in some geometry problems, you have to make changes to the figures like dividing a quadrilateral into two triangles or draw new parallel lines. These are harder to solve.
In fact, this often is the most challenging dimension to master. There are many ways in which clues to a solution can be obscured, especially as the number of possible theorems increase.
A problem that requires more advanced theorems to be used and where all these theorems are not easily visible on the surface are often the hardest ones to solve.
I use these three questions to carefully select problems for each student. Giving a student a problem that is too hard can be demotivating and giving something that is not hard enough can result in little learning. Getting this balance is critical and hence classification of problems very important.
Some Example Problems
Let’s look at a few example problems and classify them. You can also ask your high school kid to try these out with the caveat above.
Easy Problem: Find ∠y in the figure below
The theorem required for this problem is Pythagoras Theorem which is an easy theorem. It needs to be applied twice to get the answer. The fact that you should use Pythagoras theorem here is quite obvious with right angle triangle and side lengths as dead giveaways.
Harder Problem: Prove that x/y = a/b given that all three lines are parallel to each other
Here the theorem needed to solve the problem is that a line parallel to the base of a triangle divides the other two sides in the same ratio. This is a more advanced theorem that is derived from the similarity of triangles.
There are no triangles in the picture above and it takes some imagination to actually draw a line that creates two triangles that then use the theorem above (twice) to get the solution.
Much Harder Problem: Compute Angle B
In this problem, there are a few theorems used.
Sum of angles of a triangle (simple); Opposite angles of a cyclic quadrilateral add up to 180° (more advanced); Inscribed angle theorem (more advanced); Equivalence of the above two theorems (even more advanced). So we need to apply four theorems. In addition, it isn't obvious at all that there is a cyclic quadrilateral involved. See if you can figure it out!
Even Harder Problem: H is the orthocenter (intersection of the three altitudes) of triangle ABC, prove that H is the incenter (intersection of the three angle bisectors) of triangle DEF
In this problem, the theorems needed are same as the problem above viz., cyclic quadrilateral theorems (opposite angles add up to 180° and all inscribed angles are equal) and similarity theorem.
But this problem is hard even after you know that you can use these two theorems. It is hard because the cyclic quadrilaterals are hard to spot (there are 6 of them). In addition, there are a lot of lines, triangles, and quadrilaterals that can clutter your thinking. In this case, I encourage students to follow Polya's advice of focussing on the "unknown" or "what is to be proven" to de-clutter their thinking. Try proving it and observe your own thinking process to get to the solution!
While these problems may look easy to some, the classification method used can be applied to problems of arbitrary complexity.
Teaching kids "what to think" is easy, but futile. Teaching them "how to think" is hard, but essential. Hopefully, this framework will help you guide your kids to do the latter.