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

understandin 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:

- 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.
- 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.
- 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.
- 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

understandin 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

thoughtyou understood something, and now know that you do not. Maybe your understanding of what it means to understand has changed.

Some typical responses:

- 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.
- 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.
- 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.
- 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. - 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.

This is an invaluable column that I plan to share widely. Thanks so much for posting it, Chris.

I agree that starting with Polya is not helpful for this course, but I don’t think problem solving and understanding are in opposition. If you ask the students at the beginning what they think problems solving means, I think they are likely to groan and talk about “word problems” and following procedures to solve them — basically the same answers as you get now. I see problem solving as working on problems that you don’t already know how to do. Using different representations, justifying reasoning, and communicating are intertwined.

It sounds like your course is only one semester, and that’s not nearly enough. Ours is a three semester sequence, which is also not enough to have them leave with the level of understanding that I’d like, but it does give us a lot more room. I love the first few weeks of class where students solve big, unfamiliar problems without my telling them how to do so (although I don’t use the book much here; it’s not that helpful). These weeks completely disrupt their notions of what a math class can be, and also build their confidence that they can be smart in math, all of which is helpful when we get to number and other topics focused on understanding elementary math.

Would you say that problem solving is not that important in elementary school either? If it is important in elementary school, then teachers need to have experienced it themselves if they are to have a hope of teaching it, and sadly, with twelve or so years of math classes, many of our students have not done any meaningful problem solving.

Would your thinking be different if you had three semesters? Do you teach the part that’s about patterns, functions, and algebra (or whatever you might call it) or geometry? Do you feel differently about problem solving and those topics?

I appreciate this post, which is making me think (or overthink — love your blog title too).

Thanks for stopping by,

dborkovitz. I look forward to future conversations.I don’t see problem-solving and understanding in opposition. I do think that where we begin our courses sets a tone for the semester. While problem-solving is an important mathematical practice, it’s not really the heart of what I want these courses to do.

You wrote on your blog about people in STEM careers transitioning to teaching. Those people are probably pretty good problem solvers. But they probably don’t know a whole lot about understanding in mathematics.

Everyonewho goes into teaching wants their students to “understand”. But they haven’t considered the meaning of the term, and they are likely to think that it’s more clear cut in mathematics than in other content areas.In short, I feel like an emphasis on problem-solving sets students up to participate in the practice of mathematics. This is important. But an emphasis on understanding sets students up to participate in the community of teaching.

Furthermore, you know what Polya’s first problem-solving step is, right?

Understand the problem. There it is, first word of the first step of the problem-solving process. My students thinkunderstand the problemmeansread the problem carefully several times, highlighting key words. What if we worked on Polyaafterwe developed productive ways of talking about understanding? Wouldn’t students be better positioned to make sense of Polya’s teachings?Also, for the record…As a community college, we are at the mercy of the four-year institutions to which our students transfer. Most of our intending elementary teachers hope to attend one nearby state school. That school now requires three math content courses, so most of our students take those three. They are (1) number and operation, (2) geometry and measurement, and (3) algebra and statistics. Technology is infused across the three courses, especially in that latter two. We are working hard to increase the algebraic content across all three courses so that the third one can focus on summarizing the algebraic ideas. But we’re not there yet in terms of curriculum. I am ashamed to inform you that our state requires only one math course for future special ed. teachers. These people may go on to do high school math pullouts, but they have only been required to take the first course in our sequence.

Christopher, thanks for reading my blog and for the conversation.

I think we probably agree on more than we disagree on, but I do disagree with this paragraph:

“In short, I feel like an emphasis on problem-solving sets students up to participate in the practice of mathematics. This is important. But an emphasis on understanding sets students up to participate in the community of teaching.”

I just don’t see them as either or, and I do think elementary teachers need to know something about the practice of mathematics, as their future classrooms should be places where mathematics is practiced (yikes, that sounded so good in its parallelism, but it also sounds like students doing rote practice instead of doing mathematics, like I meant…..). At my college, our math for teachers classes are their only gen ed math classes (they are in the math department, not the ed department), and so I’m also conscious of having that gen ed goal in there – I want my students to have at least a glimpse of why some of us love mathematics. We “convert” some folks to math in these classes, and they decide to be math majors, and it’s not the unit on fractions that does that for them. Also, realistically, a lot of them don’t end up being teachers.

I’m also not sure that we mean the same things when we say, “understanding in math,” “problem solving,” or “practice of mathematics.” (sigh, as I tell my students, communication is hard). So, I’m not even sure how much we in fact agree or disagree.

I would not say that I emphasize “problem solving” in the first unit. It’s called “Introduction to Processes,” and a lot of what goes on there is about representations, reasoning, etc. I don’t even remember Polya’s steps, and I don’t find them helpful in this context. What I love most about teaching this unit is that it’s the one place where I make my pedagogical decisions based primarily on what is best in terms of process; content and process are always both there, but most of the time we are tethered by having to “cover” the content. So, we are similar here in that our most important goals to start the class are process goals – you are not starting by asking your students to describe everything they know about multiplication, after all.

I don’t use the phrase “understanding” as a process though, and I would use different words for some of the ways you are using it. I don’t think Polya’s “understand the problem” is the same as other ways you’re using “understanding,” where I would be more likely to use the phrases “conceptual understanding” or “profound understanding of fundamental mathematics” or “specialized math content understanding for teaching.” I spend a lot of time on this kind of understanding. I think you may be saying that we should start with this because it’s most important. I might go the other way and say that if we start with breaking their mold of what math class looks like with less familiar problems, then when they get to these topics that they have seen before, they are primed to do them in a different way.

But you know, this is not an argument that I care about winning, in the sense that I think it’s a valid point of view to start with conceptual understanding of arithmetic, and I have no desire to try to convince you to change. You haven’t convinced me either, but I think you’d probably be OK with what I do (btw, here are a few of my favorite activities from the first unit http://debraborkovitz.com/category/pedagogy/fav-prob/). We also run the first two courses in our sequence as a yearlong course, so I have two semesters to play around with, which is a big difference, and I am not constrained by transfer requirments.

In Massachusetts, the sped teachers have the same math content requirement as for elementary ed, which is good. But yikes, the early childhood (up to grade 2) only require one class (our faculty agreed to 2, but not all 3 in our sequence, which is a pain as the students haven’t sorted themselves out, so they just miss 1/3 of the material). For grad students and people who go through alternative certification, well you read about that, no requirements at all.

Oh, we have a lot of work to do…..

-Debbie

Although focus on mathematical understanding over rote problem-solving is more the focus now, and is something I’m mostly in agreement with, I think the pendulum could swing too far this way as well. Can anyone here honestly say they understood every math concept from their university-level math courses perfectly before applying them? That they never took shortcuts and reverted to formulas or algorithms? For the sake of efficiency, for most people some level of doing this is necessary. I’ve also sometimes found it easier to focus on understanding more once I had the mechanics down-pat, especially when I was working with new notation.

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