Tag Archives: understanding

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.




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.

Some hopeful words

Keita writes (in the context of the lattice algorithm, but not just on the subject of the lattice algorithm):

Evidence and lots of evidence is coming out supporting this and things are going to change. The old way of confusing people is dying.  Understanding is coming.

I needed that.

Dear Christina: Open letter to a student

Dear Christina,

At the end of two semesters of the math for elementary teachers courses, you were assigned to read the Richard Skemp article “Relational Understanding and Instrumental Understanding,” and to apply the ideas of the article to your own learning in these courses.

You wrote elegantly about your understanding of fractions, of decimals, and of how the courses helped you to make sense of relationships between these ideas; relationships that had eluded you through many years of schooling.

And you finished your paper with a question for me:

Throughout [these courses] you have followed through with intentional teaching as well as not letting us get away with instrumental … understanding. You have challenged me to think beyond what I think I know and to confront the math concepts that I thought I would never comprehend. That said, I have already taken some education courses and many of the texts express the importance of teaching…that accommodates the different learning styles in the classroom…My experience within many levels of schools is that there is no accommodation for different learning styles and that most curricula are still set up for instrumental understanding and that leaves a lot of students slipping through the cracks when they are highly capable and bright children. How can a teacher in today’s schools find the time to teach to all, especially with all the standardized testing, and curricula that don’t fit the … learner?

These are important questions. No matter the grade level, no matter the subject, no matter the political environment, these are questions that thoughtful caring teachers (and most of us are!) struggle with.

The present testing environment in public schools and the recently developing politics around public school teaching change the pressures around these questions, but the questions have always been there.

Many people have written passionately on this topic. Rather than give a comprehensive answer, I’ll focus on what I find to be the most important piece of answering this question.

In a word: listen.

Of all the things you can do in the classroom that take up time in class, the time spent listening to your students will pay off the most.

Instead of assuming you know what your students will struggle with, listen to them to find out.

Instead of planning tomorrow’s lesson based on what comes next in the textbook, listen to your students’ ideas and use them to help you decide what makes sense to do next.

Instead of telling them how they should see the world, listen to how they do see it.

And if you listen, you’ll say less. But what you say will relate to your students’ ideas.

And if you listen, you’ll cover less. But your students will hear more.

Let me tell you about two people who have had a profound influence on my own teaching.


Anne Bartel taught me to listen to students. Anyone who has been involved in mathematics teaching and learning at the state level in Minnesota or in the Minneapolis Public Schools has met Anne, and odds are good they consider her a mentor.

I met Anne in my second year of teaching (spring, 1996) and had the opportunity to work closely with her for a number of years. She is an incredible listener. She puts people at ease. She makes them feel that they are being heard and understood. Anne, at about 5 foot 2, can hold the attention of a large group of fussy teachers in large part because they know she’s listening.

For a couple of summers of professional development work, I marveled at her ability to work with adults. But I didn’t strive to emulate her model. I figured I would have to develop other skills in my own work. I figured I could never do what she does. I figured she could do it because She’s Anne Bartel. She had a natural gift that most don’t have, right?


Over time, I came to realize that she could do what she does because she learned to do it. She studied it. She practiced it. And she could teach it.

I realized that if studying makes you that good, I’d better get to work. I’ve been working on it ever since (for about 14 years now).

I hope it has shown in our two semesters together. I hope you and your classmates have felt listened to, have felt that your ideas have been valued and incorporated into the courses.

I know we studied fewer topics than another version of each course might have covered. But in reading your paper, and the papers of your classmates, I have some strong evidence that you have made connections among the big ideas of these courses; that you have retained what you mastered and that you continue to think about these ideas and to wonder long after we have moved on.

I would gladly pit you all against a group of students coming through a course that emphasized coverage in a lesson-planning smackdown (that is why we are studying this stuff, right? so that we can teach children someday?)

This isn’t pie-in-the-sky idealistic stuff. I can point to any number of practices in these courses, any number of topics we spend extra time on, any number of ways of approaching ideas that result from my listening to my students.

I have learned by listening what future elementary and special education teachers know and what they struggle with. I have learned what attitudes about mathematics and about themselves as learners of mathematics they bring to my classroom on the first day of class. And I continue to listen because there is always more to learn.

Which brings me to the second person I want to write to you about.


Gary Knowles was the head of my teacher education program at the University of Michigan (he has since changed institutions at least once and I don’t know where he is now).

Gary made a huge impression on me because his practice was consistent with his professed beliefs about teaching and learning. As teachers, we often say to ourselves and others some form of the following, “I know my students don’t get it, but I don’t have time…” or “I know I should, but…”

These words would never pass from Gary’s lips. He knew that teachers need to think about their practice, to discuss with others, to problem-solve, to reflect. In effect, Gary knew that teachers improve their practice in large part through listening to themselves.

Everything else in our program followed from that core belief. We were trained to write reflectively, to question critically and to talk intelligently about our teaching.

Now, at the college level, our constraints are quite different. But the same principle applies to Gary and to me as it does to public school teachers. You need to identify your core values and you need to teach within the constraints of the system in which you choose to work.

As you know, I have tried to have an impact on your core values around teaching-not least by having you read Skemp. But for me, Skemp’s article is just a summary of a discussion we have had for a full academic year now. I have enjoyed it; I hope it has been meaningful for you.

I know you’re going to be a great teacher, Christina. Keep in touch.

Your colleague,