I remember preparing for an interview for my first teaching position in the 90’s. I was told that I would likely be asked to explain my approach to teaching problem-solving. I jumped on the Internet to research problem-solving and craft my response. What I found was that problem-solving in math basically meant teaching students to solve word problems. I ended up getting the job and, for a number of years, taught what I thought was problem-solving. What I’ve come to find out, however, is that while we certainly need to teach students strategies for solving word problems, problem-solving is so much more than solving word problems.
Problem-Solving > Word Problems
Think for a minute about a problem you’ve solved recently. I’ll give you a personal example. My current car lease ends next month, and I have to decide what to do. Usually, I just turn in my old car and lease another one. This year, however, is different. We are in the midst of an unprecedented shortage of new cars, driving new car prices way up. Not the best time to buy or lease a new car. At the same time, used car prices are surging and many used cars are selling at close to their original MSRP. Once again I jumped on the Internet to research the situation. I found out that I might be able to purchase my car at lease-end and turn around and sell it at a higher price! But that would leave me without a car. So, I have decided to purchase my car at lease-end and hold onto it until new car prices start to come back down. I should still get a trade-in value on my current car higher than what it will cost me to purchase it at lease-end. Of course, all this sounds great in theory and seems to be the right decision based on the data, but I won’t really know if I made the best decision until sometime next year.
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I think you’d agree that what I described was some heavy-duty problem-solving with pretty significant consequences. Yet not a word problem in sight. You see, true problem-solving is messy and goes way beyond solving word problems.
George Polya is often called the Father of Problem-Solving. In 1945, he outlined a 4-step process for solving problems in his ground-breaking book How to Solve It. You can see the four steps pictured below.
Now think about the process I went through while solving my car problem. Don’t you see the four steps in what I did?
The problem is that well-intentioned teachers have tried to turn the problem-solving process, which is inherently messy, into an algorithm—if you do these steps, then you can easily solve problems. This is why we see students boxing numbers, underlining questions, and looking for “key” words, all shortcuts that basically give students permission to not read and understand word problems.
So how do we teach students to become problem solvers? Well, it might sound simplistic, but we give them rich problems to solve and get out of the way. Again, with the best of intentions, teachers often provide too much support and students come to depend on it.
I recently facilitated a book study on the book Productive Math Struggle: A 6-Point Action Plan for Fostering Perseverance, by John J. SanGiovanni, Susie Katt, and
One way to increase productive struggle and thinking in our classroom is to flip the sequence of our instruction. Rather than the traditional direct teaching approach of I do, We do, and You do, we flip the process so students are given a problem to solve before direct instruction.
Here’s an example. Say students have been using a part/whole diagram to represent join/result unknown word problems. So they have been practicing identifying if each number in a word problem represents one of the parts or the whole and creating part/whole diagrams, such as this one. By looking at the diagram, you can probably construct the word problem they were solving, right?
Now you would like to introduce a new structure—join/change unknown. It’s a more complicated type of problem. Here’s an example of this type of problem.
Mariana had $20. Her grandmother gave her some money for her birthday. Now Mariana has $28. How much money did Mariana’s grandmother give her for her birthday?
I could proceed to teach this new structure with a scripted lesson: Boys and girls, you have been using a diagram to solve join/result unknown problems. Today, I’m going to show you how to use the diagram to solve a new structure—Join/change unknown. I would model a problem or two, we would work a couple together, and then they could practice more on their own. A typical I do, We do, You do lesson.
But instead, what if we read the new problem out loud together, and then I commented, Huh. This problem sounds a little different. Work with a partner to see if you can solve it using your part/whole diagram. In other words, the You do comes first! Would every pair of students be successful in solving the problem? Probably not. But after all students have the opportunity to struggle with it, think of the rich discussions we can have. It’s likely that at least some students will determine that it’s a new structure and then I can come along behind and put a name to it.
So I hope you will commit to thinking of problem-solving as something beyond just solving word problems and give students the opportunity to productively struggle in your classroom. I think you’ll see engagement soar!
If you want more information on addition/subtraction structures check out this post.
Cy Fair does model drawing!! It works
Howdy, neighbor! Yes, model drawing is so powerful. I’m looking forward to next year when this whole group comes up knowing it!
I’ve been using model drawing with my first graders and it’s been powerful to see how they understand what’s happening in the problem, especially those tricky missing addend problems. This is the most confident I’ve felt with first graders and problems in a very long time!
I totally agree, Nilda! It really helps them to visualize the math that is happening in the story!
Looks like a great lesson!
The Math Maniac
I am a math coach in Massachusetts and want institute bar models in all grade levels, K-5, for the 2017-2018 school year. Can you recommend a resource for me to use for research prior to professional development?
A book that I have used and think is a great resource is Char Forsten’s book on model drawing.