I made a little Desmos activity to see if it’s possible to use their activity builder to share and comment on student work. I asked people to circle something they noticed in this student’s work. Here is the overlay showing everything that everyone circled.

I’m not sure what to make of all that overlaid, but I’m definitely interested. The written answers people offered were also really interesting. Here is a sampling:

Guest: “This is a common error for my students as well. They  do not recognize that this is a quadratic function and try to get a straight line.”

Kevin: “These don’t seem to be in any particular order.”

Lane: “Does he have an eraser? Does he get confused calculating with zero?  Does he know the shape of a parabola?  Does he know that a function cannot possible have one point on top of another? Does he sometimes get confused with x^2 and 2x?  Could he have analyzed his own mistakes with a calculator?  By not checking with a calculator will some of his errors snowball and cause further confusion?  Is the student feeling frustrated? I think it is good this student understands the choice of input does not have to be in a particular order.”

Jonathan: “I notice that there is a disconnect in the student’s knowledge of linear vs. quadratic equations. I wonder how come the student did not use any negative values in her table.”

While there were a lot of great observations, the one that stood out to me was that this student could probably learn to recognize that this sort of equation will produce a U shape. Knowing that this sort of equation produces a U will make it more likely that they will test negative x-values, or at least more reliably guess the rest of the shape. I agree: it seems as if this student is trying to fit a U-shaped function into a line-shaped paradigm.

What activity could we design that would help students like this one develop their thinking?

Inspired by Bridget, I put this together:

In the Desmos activity, I asked if people could think of a way to improve my rough draft. Here were three responses that represent some of the variations people had:

On twitter, Bridget had a second idea for an activity that would help students like this one develop their thoughts.

To wrap things up, I shared a mockup of Bridget’s alternative activity and asked people what they thought about it.

Some selected responses:

Max: I prefer my version of the previous activity — this activity doesn’t invite students to consider why the parabola is symmetric — it’s easy enough to connect the linear and nonlinear representations and not confront the whys of the symmetry of the nonlinear representation. Maybe including y = x^3 + 4 as a third example (with both graph and equation provided) would support that sense-making?

Brian: I like the idea of having one more equation than graph. I’m also wondering about the choice to have two linear functions vs. one quadratic. This activity provides less structure than the previous since, to determine what the function’s graph looks like he would need to do it himself. The other provided the Desmos graph. On the other hand, this activity does provide the student with a more possibilities of visualizing the function, which could yield insight into how he’s thinking about the quadratic function.

Bridget: I wonder if the graphs should be discrete points instead of continuous. Not sure if it would make a difference or not. I also wonder if the missing representation should be another quadratic. I’m trying to consider if the connecting representations should include tables. I’m not sure…

Also, I think the previous slides connect more to the issue at hand. On an assessment-do you think this student could be given the equation y=x^2-3 and choose the correct graph from four multiple choice? I’m not sure…

Overall, this was fun! I’m excited to try it again.

OK OK OK I think I’ve got where 1024 comes from but what is going on with that 11?

Update: I think banderson2 nails it in the comments. “It comes from the power of 2. 2 = 8/4 so 8/4 + 3/4 is 11/4.”

The submitter directs us to 2a:

This student has gotten something very right, no? What does she know, and how would you build on it to help her with this sort of problem?

(Thanks Zach P!)

Did this kid just get excited by a coincidence? Or is there something deeper going on here?

(Thanks Tina!)

Tina says: “Two students have done this so far. Not a mistake, but still curious what these kids are thinking:”

She’s talking about the 4/6 thingy. Any ideas, people?

(Thanks to Pam for the submission.)

Matt submits the above, and Matt writes, “I think it’s especially interesting that this student left the mistake on the board even though she had found the correct solutions by graphing in Desmos.  I’m not really sure if she did half of forty, or sqrt 4 and then stuck a zero on it (she wasn’t sure either).”

I vote for “half of 40.” You?

What do you notice in this student’s work?

Thanks to Matt Owen for the mistake.

There’s a ton to comment on here. I doubt you’ll need much in the way of a prompt, but here goes: what mistakes are missing? You grade this test on Sunday; what does Monday’s class look like?

Thanks to Tina Cardone, who is not-so-slowly taking over this blog, for the submission.

I want to share a theory on this mistake:

The student had an association between negative exponents and reciprocals and “half-powers” and square roots. As the student was parsing the problem he “fulfilled his obligation” to use that association on the number. I guess what I’m positing is that the mind works by making a connection, and then remaining in tension until that connection is used in a problem. (I’ve often had the experience of feeling as if there’s an insight that I haven’t used yet in solving a problem, and it’s like having a small weight on my back.) The mind comes to relief at the moment that the insight is used.

The student’s mind made the connection between negative powers and reciprocals and was in tension. He then used this insight at the first opportunity he saw, to relieve himself from the burden of his insight.

Some of you might disagree. For instance, you might think that the student had just memorized some rule poorly, had no conceptual understanding of powers, and gave the answer that he did.

But I think that the answer felt right because he used the fact that he knew. I’d predict that this student would be able to answer $x^{1/2}$ correctly.

If you think that the student just memorized a rule, is there any reason to think that a student would get a question such as $x^{1/2}$ correct?