Noticing Humans & Noticing Wonders – Guest Post by Benjamin Dickman

This is a guest post by friend of the blog Benjamin Dickman

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  • Noticing Humans refers to the importance of students being aware that they are seen by others – including, but not limited to, their math teachers – and that we, as teachers, would do well to interrogate how our external perspectives match or don’t match students’ internal perspectives.
  • Noticing Wonders refers to a specific in-class activity related to Paul Lockhart’s essay, “A Mathematician’s Lament” (pdf), that I designed – based on suggestions from my Middle School English teaching colleagues – to gain insight into how my students were thinking about mathematics.

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In noticing and wondering:

How can we follow up on our wonders about the ways in which students see mathematics?

Being Noticed

During a summer visit to my childhood home in Boston, Massachusetts, to see my family, I made a point of asking my mom[1] for a very specific suggestion: What can I say to students who voice anxiety or discomfort around an upcoming (math) assignment, especially when my prior knowledge of their quality of work and preparation suggests that they will do well? (For example, I would consider “Don’t worry about it!” or “I’m sure you’ll do fine” to be suboptimal responses.) Paraphrasing, her idea was that sharing evidence of strong past performance is fine, but that a helpful additional sentence would be one to the effect of: “As your teacher I have confidence in you, and while I know that you may not share in that confidence right now, I wish that you could see yourself through my eyes.” Less than a month later, I was reading a piece in Quanta Magazine about 2018 Fields Medal (an award sometimes called the Math Nobel) recipient Akshay Venkatesh, from which I excerpt (emphasis added):

Clearly, [Venkatesh’s] adviser must have written a glowing letter of recommendation for him — but why? Venkatesh took this question to Jordan Ellenberg, a friend and fellow mathematician. Ellenberg’s reply has stayed with Venkatesh over the years: “Sometimes, people see things in you that you don’t see.”

About a week after, I noticed on Twitter an open-request from Professor of Mathematics Education Ilana Horn about her son:

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I happen to be a math teacher who had 5 minutes, so I watched the video and left a couple of comments. I thought that my first comment got to the heart of the matter, and left a second comment even though I deemed it less relevant. Here are the comments with their respective responses:

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In retrospect, I conjecture that my first comment resonated primarily with other math educators, who can see what I see about habits of professional mathematicians, but that it would require more time and evidence to be fully believed by a child. My second comment had a different outcome – to “jump up and down with happiness” – because it comes from a different perspective: how one is seen (or even not seen) by their peers, and the ways in which one’s presence and absence are noticed, even from afar. The through line that I perceive in these three items – my mom’s suggestion to me, Ellenberg’s advice to Venkatesh, and the excerpted tweets – is that, although there is much talk in math education about noticing and wondering, I am concerned that we are (or: I am) not doing enough to ensure that I let my students know that I am noticing them for who they really are: as mathematical thinkers, as students, as humans.

I need to attend better to voicing what I notice in them, and thinking intentionally about how they share what they notice in one another. This is true for students who retain an anxiety around (math) assignments; it is true for future Fields Medalists; it is true for students who experience tracking systems in ways that are too often unjust; it is true for so many other students who cannot – or do not yet – see themselves through a caring adult’s eyes; and, I strongly suspect, it is true for so many teachers who long to be noticed and seen – knowingly or not – for who they really are: as mathematical thinkers, as teachers, as humans.

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In noticing and wondering:

How can we follow up on our wonders about the ways in which students see mathematics?

Noticing Wonders

About once a month for each of the past two years, I have met with a group of teachers at my school – some Middle School, some Upper School, some both – to talk about “Writing to Learn” strategies. One of these is called a Focused Free Write, and although it was introduced as a way for Middle School students to get a foothold on Ovid, I decided to adapt it for use with an essay that is somewhat well-known in certain math education communities: Paul Lockhart’s “A Mathematician’s Lament,” which is often referred to simply as “Lockhart’s Lament” (PDF re-link). If you’ve never read it, or even if you have, you might try to complete the activity below; the students in our Problem Solving & Problem Posing course had no prior familiarity with any of the author’s writings. To summarize (over)simplistically, I feel that this essay, or Lament, speaks in a way that resonates with many mathematics teachers. I wanted to use it to gain insight into the specific mathematics learners with whom I was working. Here is the full prompt, for which I wrote the directions in red. No additional information was provided about the source text or its author:

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We spent 10 minutes on the quiet writing part of this task, and, once we wrapped up and it came time to share out, the students organically decided that they would each read aloud one of their classmates’ focused free writes. I share this task here because I think it is a meaningful example of how strategies from another subject in another grade can be successfully transferred to the context of a math class. Moreover, I am compelled to point out explicitly that other teachers could pick a different text and/or a different excerpt and/or a different set of words to replace with ellipses. I picked this text, excerpt, and its removals as a function of what I hoped to accomplish with and learn from my students. Below are six student examples, followed by Lockhart’s original words. In each case, the portions that replaced the ellipses are in blue. Rather than analyzing the student work, I prefer to let it speak for itself; so, I will conclude with these images. There are many ways in which additional context can frame how each student-generated example reads; here, I note only that I work at an all-girls day school. Part of what made this so meaningful to me was knowing the individual students. For the reader without this shared connection, I simply suggest reading as much or as little as you’d like, and in whatever order you wish – perhaps comparing corresponding parts across the provided examples. Finally, I hope that, if you try this activity with your students – or with other teachers, or with other humans – then you will consider sharing the wonders that you notice.

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[1] I happen to think “being my mom” is a sufficient criterion for giving good advice, but my mom also happens to be a child psychiatrist whose work includes advising for the PBS show Arthur, which created a therapist cartoon character named after her!

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