First Principles

1) All students are sense-makers.  All students have ways of thinking (mathematically).

2) There is little to no value in a teacher demonstrating a procedure and a student then practicing it, no matter how creatively the teacher demonstrates the procedure nor how important we think the procedure is.

3) We learn the most when we think deeply and then communicate our thinking.  We all learn more when we have the opportunity to receive feedback on our thinking and to revise.

I think this covers it.  It’s funny, despite having spent upwards of 7 years thinking about math education specifically, I haven’t ever written down my first principles.  These might be controversial.  What thoughts do you have about them?



15 thoughts on “First Principles

  1. Pingback: Sarah Langer and Joy Kogut | Work in Pencil

  2. I think we should push on the second one a bit. “Little to no value” begs the question, doesn’t it? What is the value in a teacher demonstrating a procedure? When is it valuable? Under what circumstances? Deborah Ball, who’s more often than not on the side of the angels, identifies “explaining ideas and processes” as a high-leverage practice. Is she right? Under what circumstances? Etc.

    PS. My name is Dan and I love classroom blogging. I’m really glad the two of you have decided to make this project public.

  3. I like points 1 and 3. Social constructivist learning theory has always resonated the most with me. I’m not as big a fan of point 2. I think I understand you have a desire to move away from direct instruction, but I don’t believe that equates with never demonstrating anything to students. Not everything people learn requires exploration and deep thinking. For example, if I want to learn how to change a tire on my car, I want to watch someone do it and then try it out myself. I don’t want to be handed a set of tools and be told to have at it.

    I also think that people often wrongly assume that listening to a lecture or watching a demonstration cannot possibly be a good learning experience. If you believe that each person is taking in information and making sense of it, then they can do that during a lecture or demonstration as well. They are taking in words and visuals. Of course we can’t control what each brain does with that information, but they all have the capacity to make some sense of it. Now, is it the most engaging and motivating way to teach? It depends on your goals, your students, and your presentation style. The act of demonstrating itself is not a problem, it’s whether it lines up with your ultimate goals for your learners.

    One day you might use some demonstration and practice to develop procedural fluency with a particular skill, but on another day you might drop the students in the deep end hoping they’ll see how this tool they developed earlier can assist them.

    By the way, I applaud your efforts at rethinking your teaching and then implementing it. It takes a lot of courage and I wish you great success!

  4. Thank you both for your thoughts. I want to continue to think for myself whether I agree with the extreme version of #2 and what I actually mean by it.

    First, perhaps I should have edited this to say, (like Dan’s questions) every time we want to demonstrate a procedure and have students practice it, we should examine …
    A) Why do we need students to learn that procedure? Is it actually necessary as a procedure?
    B) If we did a well-chosen problem, would a few students figure out the procedure and then be able to teach the class the “procedural” method that I want to teach?
    C) Will a teacher demonstration actually be effective?
    D) Do students actually need to practice it repeatedly?

    If the answers to all of these questions lead me to having a teacher demonstrate and students practice, then I will go for it, but that should happen rarely.

    I think that’s where I am currently with #2.

  5. Brilliant First Principles. Its an interesting challenge to consider what my list might be. Very very similar to yours.

    Regarding concerns about the second, mine seem to be slightly differently oriented. First, I would suggest that rather than saying “There is little to no value in a teacher demonstrating a procedure and a student then practicing it,” I would argue there is some sort of negative value in this teacher action. My argument resides in robbing children of their sense of self as knowledge creators. We teach them to mimic, rather than think.

    My understanding of social constructivist frameworks, which basically emerged under the fascist -oversight of Communist Russian psychologists such as Vygotsky, Davydov, and Leont’ev, had precisely that goal in mind–how to understand the learning of *something in particular.* I prefer to study mathematical learning in such a way that does not put some particular mathematics first, adhering to a Piagetian constructivist approach.

    So, I suggest a discussion about what it means to learn, and what is to be the goal of learning, is necessary to make more meaning of #2.

    And then back to my point in the second paragraph, a discussion abut what is the nature of learning in the powered structures of an institution like school, or the school classroom. Here, a solid sociological framework can maybe help understand implications for student identity, personal epistemology, and sense of self that the second point brings to the fore.

    It is nice to be introduced to your blog. I truly look forward to following, and sharing!

  6. Lawler: “I would argue there is some sort of negative value in this teacher action. My argument resides in robbing children of their sense of self as knowledge creators.”

    Just so I have this straight, Brian, under no circumstances could a teacher demonstration result in value for the student? Does that extend to teacher explanation, more generally? Can you speak to your personal experience? Have you never experienced value from a demonstrated procedure?

    • Hello Dan, and kind authors of this post. Dan presses me by trying to create an either/or caveat which would reduce a complex and very serious concern about the goal of education. I don’t think it comes from a need for an ego to be stroked, but rather to say “the act of a teacher saying something to students, if disallowed, makes our job impossible.” Yes, I agree with that.

      I think my point was to say, “I concur their is little positive value in the act of a teacher demonstrating a procedure, followed by children repeating it. But further, that sort of action is quite dangerous in its potential for its negative value. Children are taught that math is stuff to be learned (or worse, discovered). Children come to learn, after repeated correcting, that they are incorrect, full of misconceptions, not possessing of interesting, creative, or mathematical thought. Each of these are dangerous for an Educational System that seeks to create intellectually and morally autonomous people (this being a restatement of Piaget’s goal for education).

      I can certainly be caught often saying something like, “Here is my way of thinking…”, or “Here is how I saw someone in Per. 3 solve this problem” or “Here is the way mathematicians have agreed to define [fill in blank, e.g. matrix multiplication].” I work hard to level the various ways of thinking, and keep it clear that all mathematics is a product of human ideas.

      A friend shared a paper with me recently that re-engaged me with thinking about the role of curriculum and of (mathematical) knowledge in the classroom. And what the teacher is to do; how to act. My read on this paper matches my own definition of what the teacher is to do: they must interact (they cannot not), they must be terribly interested in their students thoughts and ideas, allowing the child’s ideas to direct the learning environment–of course coupled with one’s own way of knowing–interpreting and responding as a result. In this way knowledge, and curriculum, co-emerge among the student & teacher (& students & …). Diversity is the goal, rather than the closed and singular ways of knowing that is abundant in US classrooms today, and expected by policy such as CCSS-M. The paper is in my Dropbox here.

  7. With #3 you’ve described one scenario by which a student can develop a hypothesis (think deeply), test that hypothesis (communicate it to an expert), receive feedback on that hypothesis and revise/iterate on it.

    I would just expand this principle to include receiving feedback from an environment, such as a chemistry lab allowing a student to test/revise/iterate on a hypothesis.

  8. I’m having a hard time with #2 because “demonstrating a procedure” is such a broad context for me. Are we demonstrating a procedure on how to do the math (like factoring and deriving equations) or how to carry out an experiment/task? There are so many math tasks that I do in my classes where I must give the kids procedures — for efficiency and safety.

    Saying, “Here are two ropes. Go!” simply will not work for the recent Conway’s Rational Tangles that I did with them. Same thing with Barbie Bungee, “Here’s Barbie and some rubber bands. Go!”

    I think when “demonstrating a procedure” is perfectly timed and well delivered, in small doses, it becomes a critical role (and goal) of the teacher.

    If we’re talking about doing mathematics strictly, then some sliver of procedure would still be necessary. I don’t want “procedure” to become a bad word, synonymous with what has happened to the word “lecture” — I’d be delighted to sit through a lecture from anyone of you here.

  9. Just to submit one more thought to the “procedure” thread — it seems that at least two different types of procedures are being spoken of.

    Procedures of how to use tools/materials – I’m generally in favor these. I’ve done Barbie bungee as well, and the point is the experiment, so yes, of course, demonstrate how to do the experiment!

    Procedures of how to do math skills (for lack of a better term) – I’m still skeptical of these needing to be teacher-demonstrated except as a last resort. For one example, this week I demonstrated how to calculate the z-score as a quick side-note as my seniors worked on a Stats project. On the one hand, it worked well because I taught it in the moment when students were interested in it. On the other hand, in retrospect, I could have assigned it as a question to one of the more-ahead groups, who could have then taught it to the class, probably better than I did. And then they would have been experts! Or I could have put it as a mini-challenge to the class.

  10. So glad you’re blogging. I realize this is an older post, but I wanted to say that I’m interested in exploring #2 more. Do you think that this idea is specific to mathematics? Do you believe that mathematics at the university level would ideally be taught with this principle in mind?

    I have more to say, but there’s a three year old demanding I do other things.

  11. Absolutely brilliant post and inspires me (as a new assistant principal working with the math department) to really refine and articulate what I stand for. It’s currently just all in my head swirling around like crazy.

    I’m going to go out on a limb and say that I agree 100% with your most ‘controversial’ #2. It is clear to me that you know what real learning looks like.

    Once my blog gets up and running this will be one of the first posts I analyze further, quote, and beg people to read.

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