Problem solving

Too long for a tweet; not really worthy of a blog post. Here goes anyway…

I just walked past a computer science classroom, where the instructor was drawing an elaborate set of diagrams—presumably representing subroutines and whatnot.

This is the sort of thing I mean.

Students were dutifully taking notes.

I thought quickly to myself, How effective can this possibly be? Don’t you have to learn programming by solving problems?

And then I thought to myself, a bit more slowly, And how is mathematics any different? 

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Minutiae: Five practices artifact

I had the pleasure of promoting one of my favorite books on Twitter last night, which got me thinking about my own use of the Five Practices for Orchestrating Mathematics Discussions.

This week, we were working on problem solving in the first of three courses for future elementary and special ed teachers. We were discussing solutions to this problem:

Roberto is shopping in a large department store with many floors. He enters the store on the middle floor from a skyway, and he immediately goes to the credit department. After making sure that his credit is good, he goes up three floors to the housewares department. Then he goes down five floors to the children’s department. Then he goes up six floors to the TV department. Finally, he goes down ten floors to the main entrance of the store, which is on the first floor, and leaves to go to another store down the street. How many floors does the department store have?

It’s not a deep, rich problem. But everyone can get a start on it, no matter their math background; it doesn’t feel to students like there is a particular piece of mathematics content that they are supposed to apply, etc. For these reasons and more, it’s a good first task for these students in this course. I did not create it.

Here is my artifact of selecting and sequencing.

I wanted students to see a variety of diagrams (going from most concrete in representing the problem to most abstract so that we could notice this as a difference).

I wanted to identify two key issues (both come from experience with this problem and this student population): (1) That we need to account for the precise meaning of the phrase the middle floor, (2) That there are assumptions about the problem that remain unstated (e.g. that there are no subterranean floors), and (2a) That identifying these assumptions, stating them and dealing with their consequences is a mathematical task.

As for connecting? Classroom Discussions by Chapin, O’Connor and Anderson is the ticket for making that happen.

It’s about understanding

Four textbooks, each with some variation of the title Mathematics for Elementary Teachers. Consider the first chapter of each one:

God bless George Polya, but this is the wrong way to work with future elementary teachers.

Wrong.

Wrong.

Wrong.

When we’re working with future elementary teachers, it’s not about problem solving. It’s about understanding.

Canvas allowed me to have meaningful online discussions for the first time this semester. The first discussion question:

What does it mean to understand in mathematics?

Consider giving examples of something you understand and something you do not, or to compare understanding in math to some other discipline or area of life (is it like understanding cooking? driving through your hometown? something else?)

Some typical responses:

1. Understanding math is to know the concepts and how they apply, to have a good framework of math and be able to build using that framework.
2. It is easy to understand math if you know the steps you need to solve the problem, especially when there’s a lot of options in the process taken to get to the solution.
3. Understanding in mathematics is the ability to cognitively understand numbers and symbols, to be able to use them in functions and equations, and to be able to see various relationships between those numbers, functions, and equations.
4. To understand math is to know why you are doing what you are doing. You should be able to describe every step that you are going through and why you are doing it. Going through the math problems I can do them but explaining why is something that i would struggle with.

Then we came back to it in the final discussion:

We began the semester with a discussion of what it means to understand in mathematics, and whether this might be different from other subject areas.

A major goal of the course has been not just to remind you of the procedures of elementary matheamtics, but to deepen your understanding of it. Often this involves a period of disorientation in which you feel that you may not understand things you previously did.

For this final discussion, reflect on your understanding of one or more topics in this course. How has your understanding of a mathematical idea been changed this semester? Maybe you understand something you previously did not; maybe you thought you understood something, and now know that you do not. Maybe your understanding of what it means to understand has changed.

Some typical responses:

1.  I started this semester assuming to understand mathematics you simply needed to know the question being asked, different ways to solve it and how to explain to others your reasoning behind your answers. Since then, I have looked at understanding math differently. Understanding math is knowing WHY you do certain things a certain way, what could happen if those rules weren’t followed and what to do to dig deeper into presented arguments or algorithms.
2. My understanding of mathematical ideas has been changed in a good way. In the beginning of this class I described understanding math as being able to do a problem or being able to do the procedural steps. Now, I see that in order to fully understand the problem in addition to being able to solve the problem I should be able to ask why did I just do that, how is this relevant, and what is the meaning of this problem.
3. When we had to do the one’s task, I didn’t know why in addition the 1 means one when we carried it over but in subtraction the 1 means ten.  It almost made me think that I didn’t understand addition and subtraction, which is crazy because it is something that you learn at an early age.   It wasn’t until we really talked about it in class that I finally understood the concept.
4. One of the topics that I come back to when completing the various activities we’ve done this semester, and when thinking about what it means to understand mathematics is the Lesh model.  I feel like, in order for children to understand math for any given exercise, they’re able to go back and forth across the model. There are pictures, spoken symbols, written symbols, manipulative aids and real world situations that all lend to an understanding in the subject.  It seems that whether we, children, or anyone, is really coming to an understanding in math, we can start with one and cross into any other.
5. I honestly thought I had most of the answers for understanding basic elementary math, but clearly I was wrong. Going back to learning place value was definitely a realization that I really didn’t understand everything I was taught. For example when it came to the lattice method for multiplication I knew how to do it but I never really put a thought as to why each number went in each place. After coming up with different solutions and finding new discoveries for place value and decimal points it was like a whole new world of math had come my way. Also this last section with dividing fractions and drawing pictures to match them has really thrown me for a loop.

At the beginning of the semester, understanding in mathematics means “being able to do it”. If I can get a correct answer using the standard addition algorithm then I understand addition. Procedural fluency is good enough for them, and by extension it’ll be good enough for their students when they become teachers.

Here at the end of the course, they are writing about meaning and about connections and about how ideas are represented. They are writing about bumps along the way-they know that understanding doesn’t come quickly from a perfect explanation, but that it involves struggles and wondering and getting thrown for a loop.

The fact that these students equate doing correctly with understanding at the beginning of the course is not their failing. It’s ours. Ours is a system of teaching mathematics that emphasizes procedural talk:

Let’s imagine we would like to change this.

In this case, it is not teachers’ problem solving skills that should have first priority (although I’m all in favor of improving these).

No, it’s teachers’ ideas about what it means to understand.

They cannot leave their content courses feeling like procedural fluency equates with understanding.

I am in no way claiming that all of my future elementary teachers have come through the semester with the kind of understanding of place value, fractions, etc. that I want. But they have certainly expanded their idea of what it means to understand these things. They know why it matters. And they’re hungry to question things that they used to accept.

Diagrams and decorations

I recall a poster in math classrooms of my youth that implored me to “draw a picture” as part of the problem-solving process. A useful strategy, to be sure. But it turns out that it’s a learned skill.

Pictures that are useful for demonstrating or examining the mathematical structure of a problem are special. We don’t all make them intuitively. I have asked the future elementary teachers in my courses to draw a picture that might be helpful in solving the following word problem:

Tabitha has five baskets of apples. Each basket has eight apples. How many apples does Tabitha have altogether?

I frequently get back something of this form:

I refer to this as a decoration, and I contrast it with diagram. A diagram demonstrates mathematical structure; it represents mathematical ideas differently from a symbolic form. A decoration makes the symbols look prettier or more contextual, but does not on its own represent the underlying mathematical relationships.

We can decorate our diagrams. The inclusion of an apple in the picture does not preclude it being a diagram. But it doesn’t necessarily make the picture useful for solving problems either.

With that in mind, which of the following are diagrams and which are purely decorations?

Exhibit A

Exhibit B

Exhibit C

P.S. Extra credit to anyone who can find or take a photograph of the “draw a picture” poster. I seem to recall it being one in a set of five or so problem-solving posters.