What do you notice about this student’s thinking? What do you wonder about it?

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:

Response to Mistake Templates - Graphing y = x^2 - 4

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:

pic6 pic5 pic4

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.

LOS Mistake

My new favorite game is trying to classify math mistakes. (See: Classifying Math Mistakes)

Right now, I see three big categories of mistakes:

  1. Mistakes Due To Limited Applicability of Models
  2. Mistakes Due To Applying Properties of a Familiar Model in an Less Familiar Situation
  3. Mistakes Due to Quickly Associating Something In Place Of Another

I think this is pretty clearly an example of the third category. The student’s brain was working hard, and they swapped the 10 and the x.

These sorts of mistakes are interesting to me because I think a lot of teachers see these and say, “Oy, this student thinks that you can just swap out the x with the angle.” Or others would say, “Oy, this student has no conceptual understanding of trigonometry.”

Nah. This kid needs more practice with the Law of Sines so that you’ve got enough brain power available to pay attention to all the moving parts while you’re trying to solve the problem.

There’s something else that’s interesting about these associational errors, and it’s about the associations that students make. Isn’t it interesting that the x*sin(10) is more familiar to this student than 10*sin(x)? Maybe this also points to the need for more practice that mixes up missing angle and missing sides Law of Sines problems?

Photo (3)


A reflection from the submitter:

I think this 10th grader is saying .174>.34>.5.  I wonder what she would have concluded if she’d followed the directions and rounded to 3 decimal places? Many kids were tripped up by the .5, maybe she’d say they were increasing except the .5?

Do you agree with the submitter’s assessment? How do you help a student learning trigonometry nail this down?

I think it’s important to say something more subtle than “this kid doesn’t understand decimals.” One thing that this site has documented is that kids can understand something at 1:00 and then do something entirely different at 1:01. It’s best to see this not as a failure of decimal knowledge, but maybe a failure to use decimal knowledge in this situation. (Some people would say this kid’s knowledge of decimals in a certain context failed to transfer to this problem.) The difference is in how we respond. This kid probably doesn’t need the “basics” of decimals. We just need to make a connection to somewhere where she knows about decimals, I’m speculating.

We’ve been studying graphs of rational functions in Precalculus.

Me: “Take 1 minute with your group: what will the graph of y = x/(x+1) look like?”

One group, during discussion, asserted that it had to be a line, using a sort of process of elimination: it’s not a parabola, it’s not cubic, it’s not a hyperbola.

Interesting, right? Why does this seem like a linear equation? I guess that it sort of looks like one…