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,



9 responses to “Dear Christina: Open letter to a student

  1. Christopher,

    Just a short note to say how much I appreciate the work you are doing! Great thoughts and advice for teachers old and new.

  2. Angela Kinser

    Christopher, you are one of those people for me that had the HUGE impact. (They really don’t pay you enough!) I believe that the Lord put you in my path to help me believe in myself more, and to show me that I can both do and understand math! I am the one saying, I can’t be Christopher Danielson, but I am not supposed to be you, only me. Thanks to you, I am a much more confident me though! Thanks for believing in me, and for taking all the time (and there was a lot of it) and patience with me. Every time I came by your office, you made time for me. You never rushed me, and you made sure I got it before I left. You also listened to me, and never made me feel stupid! Your feedback was always helpful too. You gave me tasks I never thought I could accomplish at first; but after your instruction and guidance, I was able to accomplish mostly every one. Thank you for all you have done for me!! Your model is one I will follow in my future classroom, and I hope to make an impact in the lives of my future students as you have made in mine. Truly, it has been an honor. Blessings, Ang

  3. Awesome. You make me miss teaching, man.

  4. I had the undergrads in my science teaching and learning seminar read Skemp this semester. Good stuff. Always enjoy your blog. Thanks.

  5. Not surprisingly, that Skemp article is very insightful. But I’m hesitant to ever use the term “Instrumental Understanding,” because the term doesn’t embody the complete fruitlessness of what it names. Also, I still can’t get past the word “understanding” being applied to that kind of mechanical indoctrination.

    I wish Skemp had gone into more explicit detail about the idea that “there are two effectively different subjects being taught under the same name, ‘mathematics’. ” A question I’ve been mulling over for the past few months is, “Just what is mathematics?”

  6. R:

    But I’m hesitant to ever use the term “Instrumental Understanding,” because the term doesn’t embody the complete fruitlessness of what it names. Also, I still can’t get past the word “understanding” being applied to that kind of mechanical indoctrination.

    Yeah, I dig what your perspective. In a sense, though, Skemp is working on how people use the term understanding. You may not mean solely to mechanical fluency when you talk about understanding something. But lots of people do mean that, and it leads to conflict. Skemp is urging us to be more clear and more precise about what we mean by understanding.

    I look forward to reading your blog for your thoughts on the nature of mathematics.

  7. Admittedly I quickly slipped from student mode back to busy mom mode as soon as classes were finished for the semester, so imagine my suprise and slight embarrassment when I went to check my grades only to find a link to your blog and my publicly answered question. All I can say is WOW! That is more than I had expected for an answer and appreciate very much you taking the time to share your response.
    In response to your hoping that we as a class felt listened to, I can say confidently that we did. There were many times that I could see you had intended the class to go one way and because you heard the students questions and concerns, the class went in a slightly different directions while still keeping within the intended lesson/concepts.
    I appreciate your challenging us to think more critically and imaginatively about mathematics and for continually guiding us to see the relationships of math and the real world.
    As well as studying math with you for two semesters I was also studying your teaching style and technique and hoping that when I become a teacher I can inspire students to become passionate about learning like you do for your students (maybe education students are more observant that way).
    I sincerely thank you for your passion for teaching and know that I as well as many of your students that have gone before me can say that you have greatly influenced us.
    Thank you again,

  8. Pingback: Questions: A poetic view | Overthinking my teaching

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