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I posted the following three tweets yesterday, which I need to elaborate:

“Answer-getting” sounds pejorative but it doesn’t have to be. Math is full of interesting answers to get. But what Phil Daro and others have criticized is our fixation on getting answers at the expense of understanding math. Ideally those answers (right or wrong) are means to the ends of understanding math, not the ends themselves.

In the same way, “resource-finding” isn’t necessarily pejorative. Classes need resources and we shouldn’t waste time recreating good ones. But a quick scan of a teacher’s Twitter timeline reveals lots of talk about resources that worked well for students and much less discussion overall about what it means for a resource to “work well.”

My preference here may just mean grad school has finally sunk its teeth into me but I’d rather fail trying to answer the question, “What makes a good resource good?” than succeed cribbing someone else’s good resource without understanding why it’s good.


  • I felt the same way about sessions at Twitter Math Camp.
  • Kurt Lewin: “There is nothing so practical as a good theory.”
  • Without agreeing or disagreeing with these specific bullet points, everyone should have a bulleted list like this.

Featured Comment

Mr K:

This resonates strongly.

I shared a lesson with fellow teachers, and realized I had no good way to communicate what actually made the lesson powerful, and how charging in with the usual assumptions of being the explainer in chief could totally ruin it.

Really worthwhile comments from Grace Chen, Bowen Kerins, and Fawn Nguyen also.

Adrian Pumphrey:

Really, we need to literally go back to questions such as ‘Why am I teaching this?’ ‘Where does this fit into the students learning journey?’ and ‘How am I going to structure the learning so that the student wants to learn this?’ before we even think about where resources fit into our lesson. This takes a lot of time to think about and process. Time and space many teachers just don’t have.

Chris Hill:

Early on I would edit resources and end up reducing cognitive demand in the interest of making things clearer for students. Now I edit resources to remove material and increase cognitive demand. Or even more often, I’m taking bits and pieces because I have a learning goal, learning process goal and study skills goal that I have to meet with one lesson.

Kelly Stidham:

Great lessons in the context of learning around mindset and methods are the instruments we use to “do” our work. But the reflection and coaching conversations where we “learn” about our work are critical as well. Without them, we use scalpels like hammers.

But this work is much harder, much more personal, much more in the moment of the classroom. Can we harness the power of tech to share this work as well as we have to share the tools?

2014 Sep 8. Elissa Miller takes a swing at “what makes a good lesson good?” Whether or not I agree with her list is besides my point. My point is that her list is better than dozens of good resources. With a good list, she’ll find them eventually and she’ll have better odds of dodging lousy ones.

Great Classroom Action

Classrooms are back in session in the United States, which means lots of classroom action, lots of it great.


The blogger at Simplify With Me posts two interesting activities with dice, one involving blank dice, and the other involving space battles:

Once you have your ships, place one die on the engine, one on the shield, and the other two on each weapon. Which die on which part you ask. That’s the magic of this activity. Each person gets to decide for themselves.

Kathryn Belmonte posts five more uses for dice in her math classroom.

Kate Nowak set the tone for her school year with debate about a set of shapes:

Then I said, okay, so here’s a little secret: what we think of as mathematics is just the result of what everyone has agreed on. We could take our definition of “the same” and run with it. In geometry there’s a special word “congruent” where specific things, that everyone agrees to like a secret pact, are okay and not okay. Then, I erased “the same” and replaced it with “congruent,” and made any adjustments to the definition to make it correct. They had heard the word congruent before, and had the perfectly reasonable middle school understanding that congruent means “same size and shape.” I said that that was great in middle school, but in high school geometry we’re going to be more precise and formal in our language.

Hannah Schuchhardt isn’t happy with how her game of Transformation Telephone worked but I thought the premise was great:

I love this activity because it gives kids a way to practice together as a group and self-assess as they go through. Kids are competitive and want their transformations to work out in the end!

Featured Comments

Mary Dooms:

Kate does a great job connecting all the dots by focusing on the learning target at the end of the lesson. It appears all great classroom action positions the learning target there. Now to convince our administrators.

Mathspace is a startup that offers both handwriting recognition and immediate feedback on math exercises. Their handwriting recognition is extremely impressive but their immediate feedback just scares me.

My fear isn’t restricted to Mathspace, of course, which is only one website offering immediate feedback out of many. But Mathspace hosts a demo video on their homepage and I think you should watch it. Then you can come back and tell me my fears are unfounded or tell me how we’re going to fix this.

Here’s the problem in three frames.

First, the student solves the equation and finds x = -48. Mathspace gives the student immediate feedback that her answer is wrong.


The student then changes the sign with Mathspace’s scribble move.


Mathspace then gives the student immediate feedback that her answer is now right.


The student thinks she knows how to solve equations. The teacher’s dashboard says the student knows how to solve equations. But quiz the student just a little bit – as Erlwanger did a student named Benny under similar circumstances forty years ago – and you see just how superficial her knowledge of solving equations really is. She might just be swapping signs because that’s why her answers have been wrong in the past.

Everyone walks away feeling like a winner but everyone is losing and no one knows it. That’s the scary side of immediate feedback.

One possible solution.

When a student pulls a scribble move like that, throw a quick text input that asks, “Why did you change your answer?” The student who is just guessing will say something like, “Because it told me I was right.” Send that text along to the teacher to review. The solution is data that can’t be autograded, data that can’t receive immediate feedback, but better data just the same.

Related Awesome Quote

If you can both listen to children and accept their answers not as things to just be judged right or wrong but as pieces of information which may reveal what the child is thinking you will have taken a giant step towards becoming a master teacher rather than merely a disseminator of information.

JA Easley, Jr. & RE Zwoyer

Featured Comment

Justin Lanier:

I would want to emphasize that the issue is that Mathspace (and tech folks generally) tries to give immediate, “personalized” feedback in a fast, slick, cheap, low/no-labor kind of way. And, not surprising, ends up giving crappy feedback.

Daniel Tu-Hoa, a senior vice president at Mathspace responds:

[T]eachers can see every step a student writes, so they can, as you suggest, then go and ask the student: “why did you change your answer here?” For us, technology isn’t intended to replace the teacher, but to empower teachers by giving them access to better information to inform their teaching.

2014 Sep 4. I’ve illustrated here a false positive – the adaptive system incorrectly thinks the student understands mathematics. Fawn Nguyen illustrates another side of bad feedback: false negatives.

Let’s look at an example of developing the question versus rushing to the answer.

First, a video I made with the help of some workshop friends at Eanes ISD. They provided the video. I provided the tracking dots.

To develop the question you could do several things your textbook likely won’t. You could pause the video before the bicycle fades in and ask your students, “What do you think these points represent? Where are we?”

Once they see the bike you could then ask them to rank the dots from fastest to slowest.

It will likely be uncontroversial that A is the fastest. B and C are a bit of a mystery, though, loudly asking the question, “What do we mean by ‘fast’ anyway?” And D is a wild card.

I’m not looking for students to correctly invent the concepts of angular and linear velocity. They’ll likely need our help! I just need them to spend some time looking at the deep structure in these contrasting cases. That’ll prepare them for whatever explanation of linear versus angular velocity follows. The controversy will generate interest in that explanation.

Compare that to “rushing to the answer”:



How are you supposed to have a productive conversation about angular velocity without a) seeing motion or b) experiencing conflict?

See, we originally came up with these two different definitions of velocity (linear and angular) in order to resolve a conflict. We’ve lost that conflict in these textbook excerpts. They fail to develop the question and instead rush straight to the answer.

BTW. Would you do us all a favor? Show that video to your students and ask them to fill out this survey.

Let’s see what they say.

This is a series about “developing the question” in math class.

Featured Comment:

Bob Lochel, with a great activity that helps students feel the difference between angular and linear velocity:

I keep telling myself that I would love to try this activity with 50 kids on the football field, or even have kids consider the speed needed to make it happen.

Without some physical activity, some sense of the motion and what it is that is actually changing, then the problems become nothing more than plug and chug experiences.

Kate Nowak, on my recommendation that teachers ask for informal sketches before formal graphs:

I agree with everything you say here. However, I think you will get silent resistance on this because teachers don’t know what to do next if their students can’t sketch a graph. But they know their students can follow mechanical instructions, so they’ll fall back on that.

Waitaminit. Is that you? Is Kate talking about you? Let’s talk about this.

Let’s say you’re working on Barbie Bungee. You’re tempted to jump your students straight to the mechanics of collecting and graphing precise data but you decide to develop that question a little bit first. You ask them for a sketch and the results come back:


A is (basically) correct. With zero rubber bands, Barbie falls her height and no further. Every extra rubber band adds a fixed amount to the distance she falls.

So what would you do with each of these sketches? Me, I think I’d say the same thing to each student.

BTW. Kate is back in the classroom after a short hiatus so there’s never been a better time to watch her think about teaching.

Featured Comments:

Kate Nowak:

I’d need to think about it in context of the lesson and course flow. What happened before? What was done to orient them to the problem; do they have any concrete experience of the situation or is this more like just get something down, and then what kinds of things would they be basing their response on? What were your reasons for anticipating these 4? Are these kids in Algebra 2 or 8th grade? So I have more questions than answers.


I’m an engineering professor, not a math teacher, and my courses are built around design projects. What I’d tell the students is probably what I usually tell the students in the lab: “Try it and see!”

David Wees:

All four of these kids appear to have slightly different models for understanding how this graph relates to Barbie falling. I’m assuming that we are just asking for a rough sketch here, as per your previous post.

#1 seems to indicate some important understandings of the relationship between the two variables. It is hard to come up with that graph by accident. My feedback to this kid would be to ask her what else could be modeled with this graph.

#2 seems to know that the more rubber bands there are, the longer the distance is. This is a pretty key understanding. I am curious about why they chose to start their graph at the origin, and I would ask them to explain their reasoning behind their creation of this graph. Either they will notice their mistake themselves, or I will have more information with which to ask a better question. One possible response would be to ask kid #2 and kid #3 to justify their graphs and defend them.

#3 seems to be confusing the graph as a map of the actual fall itself, but there could be other explanations for their choice of graph. For example, they could be interpreting distance fallen as just distance, in which case they might be thinking that this means the distance from the ground. I need more information about their thinking, and so I would ask them to explain to me what they have done, and then depending on their response, I ask another question.

#4 did not do the question. There are many reasons why this could be true. They could not be able to read, they could not have a starting place for figuring this out, they could be unwilling to make a mistake, they could be still thinking about the problem by the time I get near them, and more. I need to know more information. Is this a typical pattern from this student? Have they produced similar graphs in the past? What socio-emotional concerns do I need to be aware of? Based on my understandings of these questions, I would ask a question like “Can you explain to me what the problem is asking?” Ideally I have already spent enough time clarifying the problem before everyone started that this particular question will not give me much information (eg. the student does know how to explain the problem) and I will likely need to ask another question. Maybe I need to ask them to describe the relationship between rubber bands and falling bands in words first.

Denis Roarty:

My second reaction, when I read a few of the Barbie PDFs is that these things are so longgg …. I was a middle school science teacher and my ideal worksheet was a one pager. We did a lot of context building by talking through the prompt, what we needed to know and the experimental design. I didn’t always pull it off well, but I also didn’t have kids mechanically following my directions.

This is a series about “developing the question” in math class.

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