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I spent an hour on The Kathleen Dunn Show on Wisconsin Public Radio earlier this week. I was disappointed we didn’t get around to my thoughts on #PackerNation and Steve Avery, but I enjoyed our conversation about math, education, and technology just the same. Even though Dunn admitted she’s uncomfortable with math, she was gracious enough to let me assign her a math problem. It was also my first time on a call-in show, and the callers did not disappoint. [Show link.]

2016 Jan 27. As long as I’m wearing my public relations fedora, EdSurge just posted my interview with Blake Montgomery.

Geoff Krall:

The crux of Problem-Based Learning is to elicit the right question from students that you, the teacher, are equipped to answer. This requires the teacher posing just the right problem to elicit just the right question that points to the right standard.

Our existing knowledge and schema determine what we wonder so kids wonder kid questions and math teachers wonder math teacher questions. Sometimes those sets of questions intersect, but they’re often dramatically disjoint.

Which makes Geoff’s “crux” a form of mind control, or maybe inception, which is impossible. Kids wonder so many wonderful and weird things. And even if that practice were possible, I don’t think it’s desirable, since it seems to deny student agency while pretending to grant it. And even if it were desirable, I wouldn’t have the first idea how to help myself or other teachers replicate it.

If PBL is to survive, it needs a different crux. Here are two possibilities, one bloggy and one researchy.

First, Brett Gilland:

[The point of math class is to] generate critical thought and discussion about mathematical schema that exist in the students minds. Draw out the contradictions, draw attention to the gaps in the structures, and you will help students to build sturdier, creatively connected, anti-fragile conceptual schema.

Second, Schwartz & Martin:

Production seems to help people let go of old interpretations and see new structures. We believe this early appreciation of new structure helps set the stage for understanding the explanations of experts and teachers – explanations that often presuppose the learner will transfer in the right interpretations to make sense of what they have to say. Of course, not just any productive experience will achieve this goal. It is important to shape children’s activities to help them discern the relevant mathematical features and to attempt to account these features (2004, p. 134).

Notice all the teacher moves in those last two quotes. They’re possible, desirable, and, importantly, replicable.

2016 Jan 12. Logan Mannix asks if I’m contradicting myself:

As a science teacher follower of your blog, I’m not sure I follow. Isn’t that what you are trying to do with many of your 3 act problems? Get a kid to ask questions like “is there an easier way to do this” or “what information do I need to know to solve this”?

I am interested in question-rich material that elicits lots of unstructured, informal mathematics that I can help students structure and formalize. But I never go into a classroom hoping that students will ask a certain question. I often ask them for their questions and at the end of lesson we’ll try to answer them, but there will come a moment when I pose a productive question.

The possibility of student learning needs to rely on something sturdier than “hope,” is what I’m saying.

2016 Jan 13. Geoff Krall writes a post in response, throwing my beloved Harel back at me. (My Kryptonite!) It’s helpful.

Show the following five sentences to one group of students:

  1. A newly-wed bride had made clam chowder soup for dinner and was waiting for her husband to come home.
  2. Although she was not an experienced cook she had put everything into making the soup.
  3. Finally, her husband came home, sat down to dinner and tried some of the soup.
  4. He was totally unappreciative of her efforts and even lost his temper about how bad it tasted.
  5. The poor woman swore she would never cook for her husband again.

Then show all those sentences except the fourth, italicized sentence to another identical group of students.

Which group of students will rate their passage as more interesting?

For Greg Ashman, advocate of explicit instruction, the question is either a) moot, because learning matters more than interest, or b) answered in favor of the explicit version. Greg has claimed that knowledge breeds competence and competence breeds interest.

I don’t disagree with that first claim, that disinterested learning is better than interested ignorance. (Mercifully, that’s a false choice.) But that second claim is too strong. It fails to imagine a student who is competent and disinterested simultaneously. It fails to imagine that the very process of generating competence could be the cause of disinterest. It fails to imagine PISA where some of the highest achieving countries look forward to math the least.

That second claim is also belied by the participants in Sung-Il Kim’s 1999 study who rated the implicit passage as more interesting than the explicit one and who fared no worse in a test of recall. Kim performed two follow-up experiments to determine why the implicit version was more interesting. Kim’s determination: incongruity and causal bridging inferences.

That fifth sentence surprises you without the context of the fourth (incongruity) and your brain starts working to understand its cause and connect the third sentence to the fifth (casual bridging inference).

Kim concludes that “stories are interesting to the extent that they force challenging but resolvable inferences on the reader” (p. 67).

So consider a design principle for your math classes or math curriculum:

“Ask students to make challenging but resolvable inferences before offering them those resolutions.”

Start with estimation and invention, both of which offer cognitive benefits over and above interest.

[via Daniel Willingham’s article on the brain’s bias towards stories, which you should read]

2015 Jan 11. John Golden attempts to map Willingham’s research summary onto mathematics instruction.


The New York Times looks at the dismal testimony of an “accident reconstructionist”:

The “expert witness” in this case would not answer questions without his “formula sheets,” which were computer models used to reconstruct accidents. When asked to back up his work with basic calculations, he deflected, repeatedly derailing the proceedings.

Watch the video. It’s well worth your time and I promise you’ll see it in somebody’s professional development or conference session soon. It offers so much to so many.

And then help us all understand what went wrong here. What’s your theory? Does your theory explain this catastrophe? Does it recommend a course of action? If you could go back in time and drop down next to this expert as he was learning how to make and analyze scale drawings, how would you intervene?

My own answer starts off the comments.

BTW. Can anyone help us understand how the expert came to the incorrect answer of 68 feet?

BTW. Hot fire:

The motorcyclist’s lawyer filed a counter-motion to refuse payment to the expert witness. It contained the math standards for Wichita middle schools.

[via Christopher D. Long]

2016 Jan 2. The post hit the top of Hacker News overnight.

2016 Jan 2. One of the Hacker News commenters notes that the actual deposition video is available on YouTube.

Featured Comments:

gasstationwithoutpumps offers one explanation of the error:

3 3/8″ at a 240:1 scale gives 67.5′ which rounds to 68′

It is easy to mix up 3/8″ and 3/16″, which is one reason I prefer doing measurements in metric units.

katenerdypoo offers another:

It’s quite possible he accidentally keyed in 6/16, which when multiplied by 20 gives 7.5, therefore giving 68 feet. This is also a reasonable error, since the 6 is directly above the 3 on the calculator.

Jo illustrates a fourth grader’s process of solving the scale problem.

Robert Kaplinsky chalks this up to pride:

Lastly, it’s worth noting that eventually the heated conversation shifts from the actual math to whether or not he will do it or can do it. At that point it seems to become a pride issue.

Alex blames those awful office calculators:

The reconstructionist is given an office calculator, which doesn’t even have brackets. He needs to enter a counter-intuitive sequence of “3/16+3” to even get the starting point. When I was at school I remember being aware that most people wouldn’t be able to handle that kind of mental contortion. They’d never been asked to.
So what’s the problem, and how might we solve it? Well, the man’s been given the wrong tool for the job. He’s never been asked to use the wrong tool before & so this throws him. This makes him defensive and he latches onto an excuse about formula sheets.

Jeff Nielso:

The motorcyclist’s lawyer is the unrelenting classroom didactic whose motivation is based on making his student look and feel stupid. I was waiting for Act 2 where the lawyer would jump up, grab his felt marker, and demonstrate just how easy he can show the procedure.


Interesting note: my grade 7 math class is in the middle of our unit on fractions, decimals, and percents, so I showed them this video so we could work on the problem. I thought they’d get a chuckle out of it and feel good about solving a problem that the expert on TV couldn’t solve.

Their reaction was unanimous. They identified with the guy and wanted them to give him his formula sheets. Some of them were pretty riled up about it!

They’re quite accustomed to me showing them videos and doing activities that are designed to build up their understanding that everyone approaches things differently, and we’ll all get there even if we take different paths. This guy wasn’t allowed to follow his path and do it his own way, and they were unfairly putting him on the spot and forcing him to do it their way.

It’s a rich problem, so I’ll use it again, but I think I’ll set it up and frame it a little differently next time!


Malcolm Swan:

When we’ve done analyses of the results of [our professional development efforts], we’ve found that teachers often move from a transmission approach where they tell the class everything and the students have been fairly passive, they’ve usually moved in two directions.

One is retrograde. They’ve moved towards individual discovery. They say “I’ve been saying everything to these students for so long. What I’ll do now is withdraw and let them play with the ideas. I’ve been saying too much. I’ll withdraw and let them discover stuff.” That’s worse than the place where they started.

The other place is where they move in and they challenge students and work with them on their knowledge together. That’s a better place. That’ smore effective.

And so in professional development, people take a path. Over time they might move from transmission to discovery to collaborative connectionist. So they might actually get worse before they get better.

That’s one of the problems with evaluating whether its been successful by looking at student outcomes. People take awhile to learn new things.

Earlier in the talk, he describes counterproductive designs for professional development:

Most of the time [in teacher professional development] we inform people of something and then we say “go and do it.” That’s not the way people learn. Usually they learn by doing something and then reflecting upon it.

So when you start with a professional development, you say, “Try this out in your classroom. It doesn’t matter if it doesn’t work. Then observe your students and then as a result you might change your beliefs and attitudes.”

You don’t set out by changing beliefs and attitudes. People only change themselves as they reflect on their own experiences.

And then productive ones:

If you’re designing a course, we usually start by recognizing and valuing the context the teacher is working in and trying to get them to explain and explore their existing values, beliefs, and practices.

Then we will provide them with something vividly challenging. It might be through video or it might be through reading something. And this is really different to what they currently do.

And through this challenge we ask them to suspend their belief and try and act in new ways as if they believed differently.

And as they do this we offer support and mentoring as they go back into the classroom to try something out.

And then they come back together again and it’s taken over then by the teachers who reflect on the experiences they’ve had, the implications that come out of their experiences, and recognize and talk about where they’ve changed in their understandings, beliefs, and practices.

What’s great about Malcolm Swan and the Shell Centre is their designs for teacher learning line up exactly with their designs for student learning. It all coheres.

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