A method related to Usiskin’s Problem Analysis for looking and enriching math topics.
I saw some elements of this somewhere (probably on the web, but what a vague reference!) and have developed and used it in a couple of classes with high school and community college teachers. Recently I saw an article in Mathematics Teacher about using Problem Analysis from Usiskin and noticed some similar ideas and so I thought I’d put the idea of the question tree out here. It is also related but at a different scale to a Wu story, and it might remind you of a Frayer model, just to throw out another buzzword.
The fundamental purpose of a Question Tree (or q-tree) is to provide a structure for developing an in-depth understanding of a specific topic. It also provides insight into innovative ways to approach the topic, and can be used to extend ideas across a curriculum. It is primarily a tool for instructors, but might have some use with students in developing higher level thinking skills.
I’ll describe it from the inside and working outward, but the usual way that I’ve used it is by starting at one of the leaves and working both in and out.
First the q-tree lives in some environment. This is the context for the q-tree. It can include grade-level and content information. An example: 3rd grade, geometry, proportionality.
The trunk of the q-tree is an instance of the context and is the topic. As we form the branches, you’ll see how the topic should be formed, but for now think of it as defined term, e.g. similarity of geometric figures.
There are three branches on the q-tree and each branch has a leaf. The leaf is an instance or example of the branch, and is usually formed as a question or activity:
- theory branch – a theorem or statement revealing some abstract property of the topic; the corresponding leaf is a proof, as a problem that requires the theory to resolve
- application branch – a class of applications that use the topic; the corresponding leaf is a problem, a word problem connected to the application
- process branch – a specific method related to the topic; the corresponding leaf is an exercise, a ‘textbook’ problem that requires the process to solve
Although it would be easy to imagine a topic having many different branches with multiples of each type, but for the q-tree, there are only three – one of each. And those three branches should be related, so the application should depend upon the process, which should be a consequence of the theory, etc.
The q-tree also has two roots, which are connections or instances outside the context or topic:
- generalization root – an extension or abstraction of the topic
- X-problem root – a motivational or summary problem related to the topic, usually more of an open task or a longer-term project
As I said starting out, the usual approach to forming a q-tree is to start with a leaf then work to the trunk and then form the other branches and roots. So let’s see it in action:
Starting point: A typical calculus problem about finding the extrema of a given function: e.g. Find the extrema of on the interval . This is the exercise. And the corresponding process is the usual procedure of finding the critical points and testing them and the end points for extrema status. The topic could be ‘application of derivatives to graph properties’ in a context of a first-year calculus course, derivatives and applications. From here we can form the theory as the extreme value theorem, or a conceptual statement relating derivatives to extrema, and the proof could be either a proof or a consequence of the theorem, or an abstract problem that requires a careful use of the theorem to resolve. The application is constrained max-min problems, with a problem as one of the classic maximize the volume of a box under some limits. The generalization could be extrema in higher dimensions or characterizing extrema with the 2nd derivative when you have an unbounded domain (moving from global to local extrema). For an X-problem we could start with a practical problem, say of taking a load of something in a truck and needed to make a box to hold as much as possible, so that the constraints either have to be assumed or discovered separately.
The end result is a collection of related topics that give us a deeper understanding of the basic ideas. The roots give us links that we can use to present the material in class or to connect to other ideas. In a classroom driven by standards and/or learning objectives, the individual q-tree could be grouped around the class objectives. The process of forming the q-tree is a nice exercise in thinking about a topic from multiple perspectives, and is valuable for both teachers and students.