A thing that complicates your project, I think, is that most math *teachers* have never gotten to do mathematics with good interpretive feedback.

100%. Encountering a productive teaching practice is one challenge. Supporting teachers in adopting it is another.

“Quickly and fitfully and artfully” is a wonderful turn of phrase – can I steal that to use with parents? It really describes the dynamic I’m shooting for in my classes.

A thing that complicates your project, I think, is that most math *teachers* have never gotten to do mathematics with good interpretive feedback. So I think an adjacent project is inculturation of math teachers to *do* the sort of math we want to teach our students to do which means putting teachers face to face with unfamiliar mathematical ideas (here not being a formally trained mathematician is helpful to me!).

The snapshot thingy looks really interesting. Is there a way with desmos to have multiple people working on the same answer in that framework? The mathematical discussions among teachers I’ve been having this year use zoom as our communication tool, and the face to face is awesome, but boy does the whiteboard suck as a vehicle for communicating mathematical ideas to one another. We usually end up working on paper and taking photos to share with one another, which we then annotate and argue about. I wish there were a really freeform converse.desmos.com that was like Zoom, only with tools to communicate about math that didn’t suck.

When you say, “we just attach as much meaning as we can to the student’s thinking in a world that’s familiar to them,” that’s *kind of* true. You’re attaching a particular meaning that is not the one that the student intended.

Yeah, I think this is dead on and gets at the different capacities of machines and humans.

My least favorite version of feedback for learning and identity formation is where the student gives significant thought to a question and is told (by machine or human) “you’re wrong, purely and totally, and also here is an adult who will explain how to be right without any reference to anything right in your own thinking.”

My favorite version is where a teacher seeks first to understand the student’s thinking, points out its value, and offers feedback that references the parts that were right as much it references the parts that need more development.

Computers … can’t do that.

As you point out, our feedback in the example in the OP doesn’t speak as specifically as a human can to the student’s thinking. So we don’t try. We don’t say, “Here’s what I think you were thinking about.” Instead, we say, “Here’s what your thinking makes me think about. Is this useful for you?”

Or, to lean on summation notation, how would interpretive feedback help students craft an argument that the sum of the first n numbers is half the product of the nth number and the (n+1)th number?

Yeah, this is a useful contrast. I think something important about the model we’re building for collaboration between humans and computers is that our computers offer interpretive feedback whenever we can imagine it, and everywhere else – especially for answers to questions like yours above, where student thinking evolves so quickly and fitfully and artfully that computers can’t and don’t deserve to parse it – we make that thinking visible to teachers and support them in giving interpretive feedback. The centaur model for human computer interaction is conceptually messier than lots of ways people use computers in math class but I think we’re seeing how effective it can be.

Thanks for all your comments and questions here—all very effective interpretive feedback on my thinking about interpretive feedback.

what is this coordinate? (1) It’s where red dot is , or( 2) its the intersection of the lines, 4 = x and 5 = y, under the red dot.

I don’t react well to convention and notation questions, especially from computers who can’t understand the irony.

Let’s all get the new year we need.

The idea of reflecting a student’s thoughts back at them is really important. The more students engage with their thoughts and the thoughts of others, the more they’re *humans* as opposed to machines. The example you give is subtle. Presumably (?) what the student meant when they wrote (5,4) was to correctly plot the indicated point . So what the computer’s doing is not actually reflecting their thoughts back at them, or even interpreting their thoughts. The computer is interpreting their thoughts through the lens of a mathematician who has a particular understanding of the coordinate plane that *is different from that of the student*. When the computer metaphorically asks the student “is this what you meant?” the computer is willfully *misunderstanding* the student so as to change the way the student constructs the meaning of coordinate points. That might be one strategy in a human teacher’s toolbox, but they’d also be able to converse at the level of meaning with their students.

When you say, “we just attach as much meaning as we can to the student’s thinking in a world that’s familiar to them,” that’s *kind of* true. You’re attaching a particular meaning that is not the one that the student intended. That’s a little weird. Absent some much more direct instruction, it moves the task of syllogizing the order of coordinate pairs from the teacher to the student. If the students aren’t articulating the result of that inference to a human being who can interpret it, my experience has been that all sorts of weird stuff happens. From your example here, a student might infer that the greater coordinate is the horizontal one. So you give them more examples. But young people are, as you rightly point out, very creative. Students will infer complex and wrong rules in ways that computers can’t detect.

Beyond that, I wonder if this sort of feedback is effective at teaching grammar, but not logic or rhetoric. For example, I can imagine this sort of feedback being pretty aces at teaching a student how to translate between summation notation and arithmetic notation. And that’s an important first step. But the really interesting mathematical questions tend to be about argumentation and proof, not about notation. I’m reminded of Paul Lockhart’s example of the creative arguments young people produce to prove Thales’s theorem. Or, to lean on summation notation, how would interpretive feedback help students craft an argument that the sum of the first n numbers is half the product of the nth number and the (n+1)th number? I genuinely don’t know. And would teaching the grammar in this way also make it more difficult for students to play with and reason about those more rhetorical questions? I also don’t know!

I think, Dan, you’re working the right project — it’s all about getting students to cogently articulate meaning while making mathematical arguments. I’m really curious to see where the ceiling is!

Karim! Nice of you to interrupt your sojourn from the world of math ed to offer us this interesting question:

The question I have is, What types of problems will this method not work for? It seems that for the feedback to work, there has to be some pre-existing answer to evaluate against, e.g. a dot that’s already plotted at (4,5) or a giant whose general look is already established.

I think interpretive feedback only works for questions that are asked inside a world that is familiar to the student. That’s one reason why we give students early experiences with Marcellus (“write in words what a scale giant means”) and Land the Plane (“first drag the point”). Those early experiences make the world familiar enough to students that our interpretive feedback is useful.

And I can’t think of a reason why that category of questions would be any smaller than “all of math itself.”

The challenging work is to make mathworld familiar to students, to ask questions that invite student thinking, and to find ways to interpret that feedback in mathworld again.

A toy example – a student solves 2x – 4 = 10 with x = 3. If the idea of equivalency is familiar to that student – ie. what an equation represents – then the teacher can say, “Okay, what you’re telling me here is that 2*3 -4 should be 10. But I’m getting 2. Let me know what you want to do next.”

Computers and people giving students interpretive feedback all over mathworld. I think that’s the project.

The question I have is, What types of problems will this method not work for? It seems that for the feedback to work, there has to be some pre-existing answer to evaluate against, e.g. a dot that’s already plotted at (4,5) or a giant whose general look is already established. But what if the coordinate plane were blank? Or what if the task involved no context at all? In situations like this, do you default to a different kind of feedback, e.g. aggregating lots of responses and allowing the class conversation to serve as the nudge? Or do you try to only write tasks that lend themselves to some kind of automated feedback (in which case how do you avoid allowing the evaluation tail to wag the inquiry dog)?

But to the larger point, this is cool. In addition to being humane — as in, not treating students like pegs to be hammered into holes the way that many tech tools do — the approach also does a nice job of demonstrating that mathematics is conventional, i.e. something that humans built rather than something that fell from the sky. That’s helpful.

https://teacher.desmos.com/activitybuilder/custom/5f4266941c1a7e2f01197cc0?collections=5cfa65e024858f111b713941#preview/9dbc3c60-558e-42dd-8c03-e1b237b4bec9