Friday, January 21, 2011

What Is an Authentic Problem, Part 2

What is an authentic problem? In my first blog, I mentioned that working on authentic problems leads to learning. I admit that the phrase sounded right, but that I had not thought much about how to define it. Many interpretations are possible. P.J. followed with two excellent examples of authentic problems, and there were follow up comments. I am not sure we can ever agree on what an authentic problem is, but since I intend to use the word again, I’d better define it, at least for myself.

An authentic problem is a problem that interests students in some mathematical enterprise that will teach them something important. It is a problem, which means that most, if not all, of the students do not have a canned procedure for solving it. The authentic problem will require the students to develop a strategy.

Often we call something a problem that is actually an exercise. Exercises play an important role on learning, as students probably need practice if they are to acquire proficiency. But exercises provide a different kind of learning. P.J.’s potter needs to exercise, needs to build the skills required to work with the clay, in order to be able to express a new idea when it arrives So, first an authentic problem must not be an exercise. The authentic problem needs to be a task that requires students to come up with a strategy.

I feel a need to point out that the teacher’s job here is to observe, and to work as hard as possible not to suggest, or even to hint at a strategy. Furthermore, once a strategy surfaces, a good one or a bad one, the teacher must give the student the opportunity to share this strategy with classmates. Now the teacher has an important task: the student must be induced to reveal where that strategy came from. Why did you draw the radius? What made you think of using the Pythagorean theorem? Why did you group the terms that way? Often the student will not know, but if we are going to teach our students to do mathematics, we must make this metacognition evident. Our contribution as teachers is to ask interesting questions and help students realize what moves are likely to be successful in solving problems.

And now, what makes a problem authentic? First, it must engage the students interest. Often problems from the “real world” will seem useful and engage students, and many of our problems need to make these kinds of connections, but some students are not enamored with a problem about quarterbacks, recipes, or politics. Some students respond to interesting questions about anything they do not already know the answer to.

“How many squares are there in the diagram at right?” is problem that has engaged my students for years. It is accessible, they do not have a strategy for solving it, they think they can solve it, and the questions lead to much learning, especially when the teacher asks them what a square is, or how they know that any of the shapes are squares. And that brings us to my last point for today.

There has to be learning involved. The teacher has to know why that problem is the right problem to ask students to work on that day. Something has to be learned before the next problem is started. Fun isn’t enough. Authentic means learning happens when the problem is worked on.

I agree that it would be wonderful to have a data base of problems. I have posted what I think include some authentic Geometry problems on the Illinois Council of Teachers of Mathematic Web site , http://www.ictm.org/, Geometry problems in the lower right hand corner. I hope you find them useful.