Students are receiving more feedback from computers this year than ever earlier. 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 enquire math students: what is this coordinate?

A target point at (4,5).

Permit's say a student types in (5, 4), a very thoughtful wrong reply. ("Wrong and brilliant," one might say.) Hither are several ways a computer might react to that wrong answer.

1. "You lot're wrong."

A red x appears next to the target point.

This is the near common way computers respond to a student's idea. Only (5, iv) receives the same feedback equally answers like (1000, one thousand) 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.

two. "You're wrong, only it's okay."

A red x and the message

The shortcoming of evaluative feedback (these binary judgments of "right" and "wrong") isn't simply that it isn't nice plenty or that it neglects a student'due south emotional state. It's that information technology doesn't attach enough significant to the pupil'due south thinking. The prime directive of feedback is, per Dylan Wiliam, to "crusade more than thinking." Evaluative feedback fails that directive because it doesn't attach sufficient meaning to a pupil'south thought to crusade more thinking.

3. "You're incorrect, and hither's why."

A red x and a message that the student might have switched the coordinates appears next to the target point.

Information technology's tempting to write down a list of all possible reasons a student might have given different incorrect answers, and then respond to each ane conditionally. For example here, we might program the computer to say, "Did yous switch your coordinates?"

Certainly, this makes an effort at attaching meaning to a pupil's thinking that the other examples so far have not. But the pregnant is often an adept's significant and attaches only loosely to the novice's. The educatee may accept to work as difficult to empathize the feedback (the give-and-take "coordinate" may be new, for instance) as to use it.

4. "Let me see if I sympathize yous hither."

No red x or message. The student's point moves out from the origin next to the target point.

Alternately, we tin enquire computers to articulate their throats a scrap and say, "Permit me see if I sympathize you here. Is this what you lot meant?"

We brand no assumption that the educatee understands what the trouble is asking, or that we understand why the student gave their answer. We just attach equally much pregnant as we tin to the student'south thinking in a world that'due south familiar to them.

"How tin can I attach more meaning to a student'south thought?"

This animation, for example, attaches the fact that the human relationship to the origin has horizontal and vertical components. We trust students to make sense of what they're seeing. Then we requite 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 nosotros use well-nigh frequently in our Desmos curriculum, and information technology'south oftentimes easier to build than the evaluative feedback, which requires images, conditionality, and more programming.

Honestly, "programming" isn't fifty-fifty the right give-and-take to draw what we're doing here.

We're building worlds. I'm not overstating the affair. Educators build worlds in the aforementioned manner that game developers and storytellers build worlds.

That globe here is called "the coordinate plane," a world we built in a computer. But even more ofttimes, 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 lot about it, or what information technology makes you wonder, or y'all ask the grade if anyone has any questions about that thought, or yous connect it to another student's thought, y'all are attaching meaning to that student'southward thinking.

Every fourth dimension you work to attach meaning to student thinking, y'all assist students learn more math and yous help them learn nearly themselves as mathematical thinkers. Yous help them understand, implicitly, that their thoughts are valuable. And if students become habituated to that feeling, they might but come to understand that they are valuable themselves, as students, as thinkers, and every bit people.

BTW. If yous'd similar to learn how to brand this kind of feedback, check out this segment on last week's #DesmosLive. information technology took 4 lines of programming using Computation Layer in Desmos Action Builder.

BTW. I posted this in the grade a question on Twitter where it started a lot of discussion. 2 people made very popular suggestions for dissimilar ways to attach meaning to student thought here.