All my action is over at Substack for now.
All my action is over at Substack for now.
I have taught demo and observational classes regularly since I left full-time teaching but yesterday was the first time I taught every class for the day. Leaving myself some quick notes & impressions.
Students are receiving more feedback from computers this year than ever before. What does that feedback look like, and what does it teach students about mathematics and about themselves as mathematicians?
Here is a question we might ask math students: what is this coordinate?
Let’s say a student types in (5, 4), a very thoughtful wrong answer. (“Wrong and brilliant,” one might say.) Here are several ways a computer might react to that wrong answer.
1. “You’re wrong.”
A red x appears next to the target point.
This is the most common way computers respond to a student’s idea. But (5, 4) receives the same feedback as answers like (1000, 1000) or “idk,” even though (5, 4) arguably involves a lot more thought from the student and a lot more of their sense of themselves as a mathematician.
This feedback says all of those ideas are the same kind of wrong.
2. “You’re wrong, but it’s okay.”
The shortcoming of evaluative feedback (these binary judgments of “right” and “wrong”) isn’t just that it isn’t nice enough or that it neglects a student’s emotional state. It’s that it doesn’t attach enough meaning to the student’s thinking. The prime directive of feedback is, per Dylan Wiliam, to “cause more thinking.” Evaluative feedback fails that directive because it doesn’t attach sufficient meaning to a student’s thought to cause more thinking.
3. “You’re wrong, and here’s why.”
It’s tempting to write down a list of all possible reasons a student might have given different wrong answers, and then respond to each one conditionally. For example here, we might program the computer to say, “Did you switch your coordinates?”
Certainly, this makes an attempt at attaching meaning to a student’s thinking that the other examples so far have not. But the meaning is often an expert’s meaning and attaches only loosely to the novice’s. The student may have to work as hard to understand the feedback (the word “coordinate” may be new, for example) as to use it.
4. “Let me see if I understand you here.”
No red x or message. The student's point moves out from the origin next to the target point.
Alternately, we can ask computers to clear their throats a bit and say, “Let me see if I understand you here. Is this what you meant?”
We make no assumption that the student understands what the problem is asking, or that we understand why the student gave their answer. We just attach as much meaning as we can to the student’s thinking in a world that’s familiar to them.
“How can I attach more meaning to a student’s thought?”
This animation, for example, attaches the fact that the relationship to the origin has horizontal and vertical components. We trust students to make sense of what they’re seeing. Then we give them an an opportunity to use that new sense to try again.
The student's point moves along the horizontal axis and then vertically to the student's point.
This “interpretive” feedback is the kind we use most frequently in our Desmos curriculum, and it’s often easier to build than the evaluative feedback, which requires images, conditionality, and more programming.
Honestly, “programming” isn’t even the right word to describe what we’re doing here.
We’re building worlds. I’m not overstating the matter. Educators build worlds in the same way that game developers and storytellers build worlds.
That world here is called “the coordinate plane,” a world we built in a computer. But even more often, the world we build is a physical or a video classroom, and the question, “How can I attach more meaning to a student’s thought?” is a great question in each of those worlds. Whenever you receive a student’s thought and tell them what interests you about it, or what it makes you wonder, or you ask the class if anyone has any questions about that thought, or you connect it to another student’s thought, you are attaching meaning to that student’s thinking.
Every time you work to attach meaning to student thinking, you help students learn more math and you help them learn about themselves as mathematical thinkers. You help them understand, implicitly, that their thoughts are valuable. And if students become habituated to that feeling, they might just come to understand that they are valuable themselves, as students, as thinkers, and as people.
BTW. If you’d like to learn how to make this kind of feedback, check out this segment on last week’s #DesmosLive. it took four lines of programming using Computation Layer in Desmos Activity Builder.
BTW. I posted this in the form a question on Twitter where it started a lot of discussion. Two people made very popular suggestions for different ways to attach meaning to student thought here.
I wonder if there is option 6, that plots a diff point like, shows the coordinates, and asks if they want to revise their (4,5). This could actually be cool for Ss who plots it correctly the first time as a double check.
— Kristin Gray (@MathMinds) December 10, 2020
Unpopular opinion (apparently) from someone who’s seen many Ss start switching coordinates AFTER they’ve learned slope. Since coordinates represent location, not movement, I’d prefer #4 or better yet, "the meeting of the x&y" pic.twitter.com/mxoz8gM6Sv
— Ms. (Lauren) Beitel (@ms_beitel) December 10, 2020
“Feel free to answer like a seventh grader,” I told teachers as I led them through one of the lessons from our Middle School Math Curriculum.
A printer prints out a scaled copy of a shape on an iPad.
The question about those images was, “What stays the same? What changes?” And people did not answer like seventh graders.
A response that has a lot of formal mathematical language.
Instead, there was lots of discussion around proportionality, congruency, ratios, and other attributes of the shapes that are going to be one million miles from the minds of seventh graders in school right now.
But several teachers took me up on my offer and answered a little bit like children. I snapshotted them, paused the class, and presented them.
A response that cites the color of the scaled shape.
Things they told me that stay the same:
“I love that you folks are finding patterns, noticing similarities, deciding what varies and doesn’t vary—including color!—using your eyes, your vision, your senses. That’s math!”
I read them an excerpt from Rochelle Gutierrez which is on my mind a lot these days.
A more rehumanized mathematics would depart from a purely logical perspective and invite students to draw upon other parts of themselves (e.g., voice, vision, touch, intuition).
By naming those responses “mathematics,” I turned them into money.
As a society, we decided long ago that certain pieces of paper had value—that they’re money. In much the same way, you are the central bank of your own classroom and you decide which student ideas are money. You decide which of them have value and, by extension, you influence a student’s sense of their own value.
I’m not hypothesizing here! Watch what happened with the teachers. On the very next screen in our lesson, we ask students to describe how this printer is broken.
A printer prints out an unscaled scaled copy of a shape on an iPad.
Teachers clearly received my signal about what kind of mathematics was valuable.
A response: "My shape is drunk."
They brought metaphors, imagery, and analogies that I don’t think they would have brought if I only praised deductive, formal, and precise definitions.
The ability to decide what’s money is a lot of power! In this time of distance teaching, you have fewer ways to broadcast value to students than you would if you were in the same room together. But I’m so encouraged to see teachers using chat rooms, breakout groups, video responses, written feedback, snapshot summaries, whatever they can, to enrich as many students in their classes as possible.
I live adjacent to the Northern California wine country, which makes wine tasting a fairly affordable and semi-regular kind of outing. (Pre-quar, of course.) But wine tasting makes me anxious and sweaty in ways that help me relate to students who hate math class.
I basically only enjoy tasting with a friend of mine, Michael Kanbergs, who is the man at Mt. Tabor Fine Wines in Portland, OR, if you’re local. He has expert-level knowledge about wine and enthusiasm to match but is allergic to most ordering forces in the world, including the expert / novice distinction. So he wants to share with you his favorite wines but he’s hesitant to offer his own perception too early because that’d undermine his curiosity about how you’re perceiving the wine.
I’m grateful to Michael for modeling good teaching, and grateful to other wine experts for helping me empathize a little better with math students who might find me and my habits alienating in similar ways.
