The Element of Surprise

I think another hallmark of an authentic problem is that it opens the possibility of surprise.  When I'm writing a problem, either for my class or for a math contest, I usually have one or two particular approaches in mind--discussing those approaches is "the" point of doing the problem.  But a characteristic of a really terrific problem is that solving it is not just a matter of finding your way through one specific set of doors in a maze: rather, it's more like navigating over rough terrain, where what seems like a detour might reveal itself to be a quicker, or more scenic, route.  Students navigating such terrain for the first time surprise us with the new paths they discover.

One example of many:  in the geometry text used at my school, CME Geometry, students learn early on about concurrence of perpendicular bisectors.  Around that time, the text asks the following question:  suppose you need to replace a broken blade from a circular saw.  You have only the fragment shown below.  How can you determine the blade's diameter?

In general, students find this problem quite difficult; in fact, in the first edition, it was a homework problem that almost nobody could solve. 

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.

What is an "Authentic" Problem?

From P.J.:

I don't have a simple definition of an "authentic" problem, but here are two examples from the last couple of weeks.  Though quite different from each other, both problems generated intense discussion and a variety of ideas.

Students entering my Geometry classes on Tuesday were confronted by a string running from the top-north-west corner of the room to the bottom-south-east corner, bearing a pink sign with the words "How long is the string?"  Even students who ordinarily sit back were up and about, measuring, arguing, and figuring out a way to determine the length of the string. They extended the Pythagorean theorem to find diagonals of three-dimensional boxes, discussed their formula's validity, and had a great time.

The week before, we started our lesson with a problem I found in a samizdat geometry book from Germany (any reference would be appreciated--my copy of a copy of a copy has no title page or publication markings of any kind). Compute the area of the trapezoid, if A is the center of the circle.

Although totally abstract, this problem was also fruitful. It took students a while to draw in the radii:

Then more questions started: what's the length of EF? Does it matter? Is EAF a right triangle, and how can we be sure? Ultimately, our discussion led us to Garfield's proof of the Pythagorean Theorem, #5 here.

These stories have two morals. First, to grab students' attention, problems must be challenging and doable. But (second), those problems don't need to be "real-world" or "applications:" kids can get excited about figuring things out, even when what they're figuring out is "just" a math problem.

What do you all think? What are your favorite problems? What made them valuable? Share links in the comments or email them to us directly.

First Steps

An essential part of good teaching is getting students interested in what you have to teach them. After that, you just need to point them in the right direction, to give some encouragement, and perhaps to help them to refine the quality of their work.

It is clear that students, then, must be involved in the learning process. They will not, for the most part, become interested in something unless they have some ownership. The best way to establish that ownership is to ask your students an interesting question, one that they perceive as being authentic and that they cannot immediately answer. Such questions will vary depending on the student, but it is the job of the teacher to think up these questions and to devote considerable class time to allow students to work on them, individually at first, and then collectively.

I do not mean to imply that thinking of these questions is easy, just that it is essential. Many of us were taught by teachers who thought that their main job was to provide clear explanations of the concepts that were in the book. We must recognize that this approach is backwards. The clear explanations need to come after the students have figured out how to solve the problem, or at least after they have worked on it long enough to care about a solution and to want to know.

It is also best if the first attempt at this explanation comes from a student, so that the class is actively thinking about the validity of that explanation. When the explanation comes from the teacher, students tend to write it in their notebook without giving it much thought. When the explanation comes from a peer, students tend to be critical and open minded, trying to find a flaw in the argument. The second is a far better way to learn.

-- John Benson

What is this blog about?

P.J.'s take:

Reading the popular press and media gives two distinct impressions about teaching.  One view--the Stand and Deliver or Dangerous Minds model--is that teaching is essentially a gift: some people just have the ability to communicate ideas and skills to other people (and some people don't).  The opposite view is that teaching is simply a set of skills that themselves can be imparted in a relatively straightforward way, such that any person who employs those skills will be a reasonably successful teacher.

I've grown into a third view:  that teaching is neither a gift nor a simple skill, but a craft.  When I was little, I was befriended by a potter who shared her next-door studio, clay, and pottery with me, and I think teaching is a lot like making pottery. 

Being a good teacher, like being a good potter, requires knowing a wide range of facts and skills, many of which can be learned easily.  Pottery knowledge includes facts: about types of clay, glaze chemistry, etc.  It includes techniques: how to knead the clay to get the bubbles out, or how to hollow out the interior, so that the sculpture doesn't explode in the kiln.  Similarly, a teacher has to know the subject and be able to anticipate common student mistakes and misconceptions.  A teacher has to have techniques for dealing with homework, for setting up different kinds of activities in class, for assessing work, and for assigning grades.

Other pottery skills require practice and patience: they aren't so easily acquired.  Actually getting the clay to make the desired shape is tough, and it can't be learned it by watching a video or reading about it in a book.  The only way is to try, and to fail, and to fail again, and to keep reflecting on the failures.  Running a discussion is like that:  the teacher starts with an idea about where the discussion could go,and what ideas could come out, but actually getting that to happen--eliciting good questions and suggestions, deciding what questions to answer, what mistakes to correct and which students to call on--is a different matter.  These skills require practice and reflection, not just factual knowledge or a quick practice session.

If you want a bunch of happy and unhappy accidents, objects that turn out as they turn out, then almost anyone can be a potter, and without much training.  Even "simple" white plates and complicated shapes can be produced by someone who spends time mastering the appropriate techniques.  But to produce great sculpture, beautiful pottery, requires vision as well as skill; the skill is what lets the potter realize his or her vision in the clay.  Great lessons and courses are like that:  they're not just expertly-run, but reflect a vision of the subject unique to the teacher and the topic.   If the desired outcomes are kids who can reproduce skills on demand, without having to think for themselves, it might be possible to make anyone a teacher in a few months or less.  But helping kids learn to think and reflect, to have, value, and evaluate their own ideas requires teachers with vision, passion, and mastery of teaching's core techniques. 

In this blog, we'll talk about the skills and ideas that we've discovered and are continuing to discover.  And we'll talk about the moments of reflection, inspiration, and frustration that keep us going and keep us asking.  We hope this will be a conversation, between us and you, the reader.  Thank you for joining us....

P.J. Karafiol & John Benson