# Problem Solving versus Solving Problems

I am helping run a small professional development group at my school this year. The key parts of the proposal I wrote are:

Guiding question:

How and where in our current curricula do we explicitly and implicitly teach problem solving skills? How can we as teachers help students to become problem solvers and not simply teach them to solve problems.

Further describe your project proposal including what kind of research you will incorporate into your project:

In both mathematics and the sciences, problem solving is a crucial skill – one that forms the backbone of what it is to do professional work in these fields. Problem solving is not the same as solving problems. We believe that what most mathematics courses, and some science courses, at our school engage in is solving problems.

A student is liable to think – as even some of our most advanced seniors do – that mathematicians sit in a room all day inventing theorems and problems out of nothing, and that chemists and physicists work in laboratories producing unambiguous data which lead to the Great Discoveries. In fact, most of the work done in fields as sterile as combinatorics or as messy as molecular biology involves navigating corridors of inquiry, trying (and often failing) to draw connections, and coming up with new lenses with which to look at problems. Frustration and dead ends are part and parcel of working in these fields. Those who work in math and science based fields have honed their problem solving intuition over time. The question we have to ask ourselves is: how do we hone intuition? Problem solving is about asking questions and finding ways to answer them, and then taking the questions one step further. Solving problems, on the other hand, is applying a known method to a problem that has already been solved before. Both involve thinking, but one involves deep thought. We can’t help but hear the first line of our new mission statement whispering in the background of this proposal.

I’ve been thinking about these issues since my first year of teaching. Earlier this year, Justin Tolentino wrote a post that struck a nerve (as you can see from my comment) about my frustration about not knowing how to teach problem solving. Just today, Glenn Kenyon twittered an article he recently published on problem solving. Jim Wysocki has been teaching with a problem based curriculum and blogging about it. And if you think about it, Dan Meyer’s What Can You Do With This (WCYDWT) series is, in many senses, a concrete place to start addressing the issue of problem solving in a curriculum.

Of course the question of how to teach problem solving still remains elusive to me.

So with this post, and knowing that I have this professional development group, I’d love for anyone and everyone to throw down in the comments:

1. How you actually go about, on the ground, teach problem solving? What do those minutes look like? What are you doing? What are the kids doing? How do you decide what to say and what not to say? How regularly do you engage in this sort of activity?
2. If you do feel you teach problem solving effectively, what three pieces of advice would you give to a teacher who is starting to do it in his/her classroom so that it goes smoothly?
3. If you have tried to teach problem solving and failed, what did you do and how did it fail? (Why do you think it failed?)
4. Useful resources of any kind (books, websites, blog posts, etc.).
5. Anything else you want to say about problem solving.

PS. And yes, my friends, all of our professional development group is reading G. Polya’s How to Solve It. As I’m working through it, I am so enamored with so much of what he says that I have every fifth line underlined. Those books are rare in my life.

1. writehandrule says:

I’m a fairly new science teacher (third year) and the advisor for my school’s math team and I have much the same problem.

My own experience with students suggests that, if they have a strong grounding in all of the relevant basic concepts, they are better able to handle novel situations. I did a lab with my physics students in which I asked them to determine the acceleration of an object on inclines of various angles. Because they had already done cookbook labs relating to measuring motion and because we had spent some time working with the kinematics equations, most of the groups were able to devise a method for determining acceleration without much help. Granted, this particular scenario wasn’t intensely novel for them, but I think they successfully applied some problem solving.

Thanks for taking the time to write this entry. If you happen to figure anything out, I’d like to hear about it. I just started following you on Twitter.

2. meandthedoor says:

Wow, this is very exciting stuff! and definitely a conversation I’d like to join, albeit from a little farther away that your school PLC :)

3. TSilver says:

I’m a second year teacher (first year with math, I taught chem last year), and I really identify with this post (I just started following a couple of math/science teacher blogs recently, because I’ve felt something of a lack of professional development aimed at making me a better -math- teacher, as opposed to some vague educational theory). My school uses the Saxon Math program for all of it’s pre-IB students, and I frequently feel like I’m not teaching anything. My entire class period is dictated by the methodology of the book, so I don’t feel that I have time to try new things or teach outside of the book’s examples–which leads to the fact that all the students do is memorize a set of skills to apply to specific types of problems. They’re rarely, if ever, exposed to a problem outside the skillset set up by the text which requires them to look at a new problem, understand how it’s related to something they know, and force them to build on it. It feels like an absence of thinking.

I’m sorry if this feels more like a rant than a productive thought–I was just wondering if you might have suggestions for how to incorporate some type of problem solving into Saxon program (if you’re familiar with it).

Thanks!

4. Check out The Art and Craft of Problem Solving, by Paul Zeitz, also. Much higher level than Polya, and (rare) a different paradigm. Instead of focusing on the 4 steps, he focuses on strategies, tactics, and tools. I love Polya, too, of course.

5. I’m not going to act like I have all or even like 10% of the answers here, but. Sam I think you and I have some of the same goals and also challenges with our classes. I have had terrible results with “whole! new! kind of lesson!” The kids don’t know what is expected, or what to do, and they’re all like “It’s a fun day! I’m a just screw around until it’s over!”

TOPIC
Definition/Derivation/Formula
Example example example
Practice

Just be like:
PROBLEM.

OK so today in Algebra 2, it was Composition of Functions with Equations Day. Like, if f(x) = x-3 and g(x) is x^2, what are f(g(x)) and g(f(x)). Last year I did a very traditional lesson, and the last example was this texty problem about how if you have a % off coupon and a \$ off coupon and you can use them both, which one should you use first, and how much would you save. But it walked you through in parts and it was very boring. And it came at the end, when everyone is half asleep. So today I started the lesson by projecting images of a 20% off coupon and a \$10 off coupon from Old Navy (which I have dozens in my Inbox), and said sometimes Old Navy lets you use them both on the same purchase, so What Question Am I About To Ask You? (a trick I got from Dan) And they spent the first 5 minutes arguing about which order was better, how much you would save that way, whether the price of your purchase mattered. They didn’t get to .8x – 10 vs .8(x-10) on their own, but at least they were curious, and it was something they could hook the lesson on to. I had to leave out some of the annoying-gotcha-algebra-pain examples, but still. Definitely better than last year.

I hope that helps. And looking forward to seeing what you produce with your project.

6. MsAxthelm says:

One thing I tried to promote problem solving skills that worked pretty well was a category sort. I was trying to teach three different possible results of of rational roots: one solution, two solutions (pos & neg), or no real solutions. I made up a whole bunch of examples and asked them to sort them into three categories of their own choosing. They had to create categories so that no example could be put into more than one category. I put them in groups of three to try to give folks the benefit of collaboration.

Most groups choose one solution, a decimal solution, and complex number solution. I pointed out that one of the complex number solutions was a decimal, so those solutions could be put into two categories, which was against the rules. One group got the expected categories and shared their thinking with the rest of class. (This group did a lot of whining about how their heads hurt thinking so hard. :))

While not brain science, it got folks thinking on their own, even if reluctantly. I find the biggest challenge in these situations is motivating students to take the problem on, rather than give up and wait for the answer. Since this was a small problem that didn’t take too long, they humored me. Any advice on that front would be appreciated.

Joan

7. Problem Solving as a whole is a long process (so I have found). It’s easy to do a one shot lesson that involves students reaching higher level thinking but to do it consistently and to the point where all students are comfortable with it is where I struggle. I teach 3 sections of Algebra 2 (about 32+ students per class) and it is common for me to get numerous “I don’t know’s” when I ask questions. I find myself really being conscious about the type/quality of the questions I ask. I am trying to bring in activities to begin new units/lessons that involve students a little bit more but with 32 students in a small space it can be difficult, not to mention the lack of resources.
I also teach 2 sections of Enriched Geometry (9th grade students) and they tend to be much more motivated and eager to partake in problem solving. They tend to get bored if I don’t engage them in challenging activities. I like to use a lot of photography (similar the the Mathematical Lens from NCTM). I am an amateur photographer and take a lot of pictures. I guess it is similar to the “WCYDWT” project from dy/dan. I am still working on the problem solving thing.
To make things even more difficult my school is on a trimester system and after 12 weeks many students will have schedule changes that send them off to another math teacher for Algebra 2. I then get to “train” a new bunch of students on the problem solving ideas that I use in my class and my old students will be off to a teacher who teaches towards procedural proficiency exclusively.

8. Sam, thanks for opening up this forum. I think this is pretty much the question in math teaching. Clearly it’s a huge and ongoing conversation, so I almost feel silly throwing in 2 cents. That said:

I second Kate in that I don’t think it’s like this radical dichotomy “teach the curriculum lockstep” or “totally open-ended creative fun problem solving.” (Although I agree that there is an essential tension between the time pressure of the curriculum and the sense of expansiveness about time needed to really settle in and start cultivating resourceful and creative thinking in our students. I think the only thing to do about this is to fight for the curricula being streamlined and the big important priorities clarified. Everyone knows US state frameworks are too broad and not deep enough and yet they continue to be…) Here’s how I see it:

We all agree that over-directing our students’ efforts prevents math class from being a place to learn critical thought, creativity and resourcefulness. The flipside is that being totally disoriented is also disempowering. So, giving them a POW for which they don’t have any way in and watching them flounder for a few minutes and then give up is not the solution. (This I know from personal experience and imagine many in the math education blogosphere do too.) Students that are not used to being asked to think for themselves in math class find the experience uncomfortable. Tolerance of this discomfort, and an appreciation of the value of it, only comes from successfully solving a problem in this state. Solving a problem no one told you how to solve is infinitely more rewarding, as everybody on here knows from personal experience, but a student used to just following orders is never going to solve such a problem if just dropped into a totally disorienting situation. It has to happen in an incremental, Vygotsky sort of a way. To paraphrase Dan Meyer, we have to start by being just a little less helpful. It doesn’t help if the problem is past the point they can solve without us swooping back in to save them. They have to be able to figure it out on their own with just a quick dip into the disorienting waters of independent thought. Then, the experience of satisfaction will give them a little more tolerance for that discomfort the next time. And it builds incrementally.

I’ve had a lot of confidence with independent problem solving for a long time, but I myself had a developmental experience of this kind in the last 2 years, during which I picked up the serious study of math again. I’m now taking grad classes but when I started I was totally on my own. I was tending to get frustrated if I couldn’t solve a problem for two or three days. The book (Michael Artin’s Algebra) had starred certain problems as particularly hard and I solved two of the starred ones and felt very accomplished. The next starred problem I came to, I really struggled with, but because I had solved the last two I stuck with it. It took two weeks before I found a solution; but once I found it, I was never going to give up in a day or two again. About 5 months later, the memory of this experience of a problem yielding after 2 weeks of frustration carried me through 2 months of unsuccessful work on a problem. Now 2 months feels fairly normal. The point is, the experience of success after time spent in the frustrated, disoriented state makes it much easier to stay involved in a problem through this frustration later. So the willingness to stay engaged, and so the opportunities for creativity and resourcefulness, grow incrementally. So the problems need to grow less helpful, but they need to do it gradually.

1. I am such a fan of that second paragraph you wrote. I think the exact same way!

“we have to start by being just a little less helpful. It doesn’t help if the problem is past the point they can solve without us swooping back in to save them. They have to be able to figure it out on their own with just a quick dip into the disorienting waters of independent thought. Then, the experience of satisfaction will give them a little more tolerance for that discomfort the next time. And it builds incrementally.”

Sam

9. Carolyn Marchetti says:

Is this your quote or did you get it from somewhere. I wanted to use it in a blog and give credit where it is due. Thanks!

“Problem solving is about asking questions and finding ways to answer them, and then taking the questions one step further. Solving problems, on the other hand, is applying a known method to a problem that has already been solved before”

1. Hi Carolyn,

I think this is my quote. I might have written this proposal with a chemistry teacher, so she might have had a hand in that quotation… but it was so many years ago! I’m 99% certain it’s not from a book or another blogpost because then I would have cited it.

Always,
Sam