At #TMC13, my session is about designing problem-based courses. But all math courses involve solving math problems; most do little else.

So what makes a course “problem-based”? R.A.C.S.V.P. (Répondez aux commentaires, s’il vous plaît.)

I’d love to get this fleshed out before we all get to Philadelphia!

Merci!

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When I think of problem-based courses, I think of courses where the math is primarily *learned* THROUGH the problems being solved. So instead of (lecture on math -> example problem together -> students do problems alone), it is more (students do problems -> develop a math idea).

In other words, instead of math being *taught* from the top down, where the ideas/concepts are being presented and applied (passive, little independent thought, few connections made), math is being *learned* from the bottom up, where the ideas/concepts are being discovered and then solidified/generalized together as a class (active, a lot of independent thought, many connections made).

x,

Sam.

Let’s put forward that communication with other human brain things is very nearly impossible, because we have no idea if, when we speak, our words mean the same thing.

I mean, we can assume that our words mean the same thing, and that’s very often a fair guess with simple words like “apple” or “pelvic muscles.” But when you get to stuff like “fair”, “shape” and “blue”, all of the sudden things get wooly.

And you asked: what does “problem-based” mean? (“Based” is a whole lot like “apple,” in that people spend very little time disagreeing with each other over its meaning. So really we’re wondering about the meaning of “problem.”) Are we talking about an awesome worksheet where the correct answers spell out the punchline of some animal pun? Are we talking about teaching via explanations of difficult problems up at the board? Are the kids solving problems relating to material they’ve never seen before? Or are they practicing?

I see no way out of this clusterbrunch. So here’s my thought: don’t stress out too much about what words mean, because that way lies madness. Instead, just describe what it is that you care about, using as many probably-neutral words as possible, such as “piano” or “spongebath.”

After all, it’s not the words “problem-based” that you’re pedagogically committed to. You might be committed to students struggling over math, or having opportunities to do math in the way that mathematicians do it, or having fun associations with learning. As long as your commitments are made clear, it matters little what you consider to be a “problem” and what counts as an “exercise” or “drill.”

(I’m not pretending that we can completely escape talk of the meaning of words, or that such discussions have no value. But if we’re just trying to pin down how another person thinks, I’m all for ditching fuzzy terminology.)

I’m not entirely clear on my own personal pedagogical commitments, otherwise I’d share them here. There are some things that I can say, for example: “Understanding difficult ideas usually requires lots of effort, and so the difficulty of an exercise is decent first-pass heuristic for whether an activity is worth a student’s time.” Or: “I want students to learn how to answer difficult questions in math, and the best way to do this is to give them opportunities to solve difficult math questions.” And so on.

That’s my proposal: don’t worry about the meaning of the phrases student-centered, problem-based, project, full immersion or lecture. Punt on all of those (interesting) questions, and instead just restate our thoughts and commitments in simpler language.

What Sam wrote is more substantive than what I came to say, but I would like to add that we often make a distinction between problems and exercises. In a conventional class, students do exercises to solidify the procedures they’ve learned. Problems take problem-solving skills. There is no obvious procedure when you start.

Sue’s remark is an important one, I think. I had a student who complained that in my AP Calculus class AP stood for ‘All Problems’ The idea that there is an important distinction to be made between problems (don’t know what to do when I see it) and exercises (I know what to do but I may still get it wrong) and I think that we need to be aware of it when we assign HW. I am beginning to think that HW is a more meaningful and productive endeavor if I restrict what I ask my students to do on their own to exercises and save the problems for when we are working as a team. I think I’ll get more productive work on the HW front AND more productive work when we are together.

I lead a session at an EdCamp yesterday where we talked about problem based learning in math (as opposed to “project”). Some people said problem based meant that there was an actual PROBLEM – like an issue or fight or conflict. Whereas “project” was more of a made up thing that just had lots of pieces. So in problem based learning you’re actually tackling an issue that real live people have.

I didn’t take this viewpoint when I did it. My thought was that “project based learning” means long and including many smaller problems and tons of objectives. And “problem based learning” means a single situation or thing that needs one or two pieces of learning to achieve the goal.

When I taught the PrBL Finite Math class in the first summer session, I tackled it from the standpoint of this:

1. Each week there were different groups (or consulting companies) formed by me.

2. They all had 3 problems to do (from 3 different clients with different situations, etc.)

3. They had to solve the problems using any sources (google, text, phone-a-friend, each other, etc.)

4. They had to present the solution and options to the client at the end of the week.

5. They had to act as clients for the other companies (asking questions, etc.)

6. They had to give feedback on their co-workers.

It was my job to:

1. Create ill formed problems that needed more information.

2. Be prepared to supply the other information or coach them on how to find it.

3. Create problems that really needed the math in order to be solved.

4. Be open to novel solutions that avoided the math (and be prepared, on the front end, to tell students how much they would be docked if they avoided the math)

5. Have daily milestone checks (I didn’t discover this part until after it was over – someone suggested it at EdCamp).

It was the hardest class I’ve ever taught (and I’ve been teaching for almost 20 years – with about 40 Finite Math classes under my belt!)

Good luck with your presentation. Please share online whatever you give to those lucky soles at #TMC13!

(P.S. Your question has prompted me to do this reflection, so I’m using the above as the outline for my long-awaited post on how this class went. Thanks bunches!)

>But all math courses involve solving math problems; most do little else.

For us, the truth of that statement is not evident; it depends how you define “math problem”.

“Solve the following quadratic equation” (CC A-REI.4) is practicing a procedure, even when the numbers change. We don’t deign to call that a math problem.

“Bisect the angle with a compass and straightedge” (CC G-CO.12) is not solving a math problem, either; it’s repeating a practiced construction.

Michael Pershan asks, “Are the kids solving problems relating to material they’ve never seen before? Or are they practicing?”

Michael’s two questions represent two extremes. The former may be problem solving, but it’s likely to prove ineffective as a learning tool only because the leap is too big; it’s asking students to do things like derive theorems on their own, which is beyond the ken of the average student.

Even Exeter’s problem sets are a meandering hodgepodge that run the gamut between simplistic (some Math 3 problems belong in elementary school) and out-of-reach (students will invariably seek out assistance—what’s the value in that?).

***

We think the ground for real problem solving is found in between those two extremes:

● A math problem should require solving something new by requiring students to apply in insightful ways material (e.g., theorems) that they are already familiar with.

That’s the fundamental rationale for each and every problem we pose on our blog, http://fivetriangles.blogspot.com

For instance, both NYS Regents and Common Core have constructions, as we summarize here: https://docs.google.com/document/d/1s6l8_r4sWMuJhAHun94E1jTxc-a752UHoQfU67IH-Sg/edit

The shortcoming remains that every construction that’s listed is rote, there’s no “math problem”.

We take the standard construction skills repertoire and pose construction “math problems” in which students KNOW they will have to use one or more of those standard constructions. But they don’t know yet which ones are needed and how they will be applied. (See near the bottom of the above Google Doc for links.)

These non-obvious constructions, where the solution to each “math problem” is just out of reach and requires students to figure out how to apply something they already know, are situated at the middle ground we seek.

Hopefully, students won’t be bored (too routine) and they won’t be frustrated (too difficult). It’s like Goldilocks: they’ll solve real math problems and it be “just right”.

For all of the other “math problems” on our blog, there are basic skills underlying the non-obvious solutions. If a teacher wants to use any of them, it’s up to a teacher to study the questions and know when to effectively interpose them.

PBL, PrBL, IBL, etc all require the student to have a need to learn or a desire for inquiry. In PrBL, as in PBL, we need to have a “problem” that must be addressed. Students must take time to reflect on what they already know and what they will have to learn to be successful. They also need time to reflect at the end on what it is they just completed. It is through reflection, discussion, and inquiry that students go deeper in their learning and connect with other life events so that retention is maximized. So, for example, if you take a standard word problem that requires more than a direct path to complete, you can PrBL-ize it. Make it more open-ended, spend purposeful class time exploring what math concepts they already know or can find out. Then explore what concepts they might not have learned yet. You might even identify class experts who understand certain concepts well and could hold small group workshops with those students who struggle. Then, and only then, will you have the students attempt to answer the problem. The answering can start on day two or even day three if the material is extremely foreign to their present knowledge level. Teachers should also have planned for multiple ways to “solve” the problem and be encouraging (demanding) of having groups work in multiple directions. Finally, groups should have the opportunity to explain, both orally and in writing, how they worked through the problem. I would even suggest that the answer or range of answers be given to the students on the last work day so that every group arrives at the same (or similar) end state. Then is when I would add an assessment that shows that each student has learned the requisite knowledge.

Another thought:

You’re saying that the sort of learning that happens in a problem-based course depends a great deal on the structure of that course.

But you could just as easily say that those different courses are just solving different sorts of problems, so that there is no shared meaning of “problem” between these courses. Exeter means a certain thing by a problem, Goldner means another, and CPM another still.

It seems to me that you can either fix the meaning of “problem” and talk about how various structures succeed or fail to live up to kids solving those sort of “problems”, or you can allow various meanings of “problem”. But since, no matter what you do, different course structures lead to different sorts of learning, I just want to know about those different sorts of learning.