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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.

Marbleslides Madness!

Watch some students play our new Marbleslides activity.

In the first example, Maggie and Claire exemplify basically all of the mathematical practices and then some as they try and fail and try and succeed to set up their marbleslides.

In the second example, Mr. Bondley recorded his students’ free play levels, some of which were quite elaborate.


As I mentioned on Twitter, I don’t want to overstate the matter, but I think we’re seeing a level of human creativity unknown since the time of da Vinci.

Start your own class!

2016 Jan 19. One Marbleslide, Five Function Families.

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.

Thoughtful elementary math educator Tracy Zager offers app developers some best practices for their fact fluency games:

I’ve been looking around since, and the big money math fact app world is enough to send me into despair. It’s almost all awful. As I looked at them, I noticed I use three baseline criteria, and I’m unwilling to compromise on any of them.

She later awards special merits to DreamBox Learning and Bunny Times.

2015 Remainders

Let’s close out 2015. In this remainders edition:

  • Eight new blog subscriptions from November & December.
  • Five essential 2015 posts from this blog.
  • Three bloggers I envy.
  • Seventeen Great Classroom Action posts I never got around to posting.


  • We successfully goaded Brett Gilland into tweeting and blogging. His writing features art, wit, and insight for days. Best follow of my fall quarter.
  • Jason D’Arcangelo is an elementary math coach, making him rare company online.
  • Kendra Lomax does interesting work in elementary math education also, most recently with the University of Washington’s Teacher Education by Design project.
  • Damian Watson just came off a two-year blogging hiatus with a post featuring Malcolm Swan, Andrew Stadel, and cognitive conflict, which pushes all three of my buttons.
  • Meryl Polak likewise came off a maternity leave to post about her experience designing and implementing a 3 Act Math task.
  • Geoff Wake was one of my colleagues at the Shell Centre when I set up a tent in their offices several years ago. Great guy. Interesting thinker. I’m excited to see him maintaining a blog.
  • Jenn Vadnais does consistently interesting work with the Desmos Activity Builder. I’m tuned in, hoping to learn how she works.
  • Glen Lewis blogs thoughtfully about technology, learning, and engagement in math education.

These blogs are each low volume, producing maybe one post per month. There is zero risk of getting overwhelmed here. Just toss them in Feedly or some other RSS reader and enjoy their insight whenever they find the time to share it.

Honorable Mentions

I don’t have a lot of envy in me for other Internet math ed types – their followers, retweets, subscribers, etc. Just keep working. What does turn me green, what I do covet, though, is another blogger’s ability to stir up conversation, to mobilize and collect the intellect of his or her readers. In 2015, that was Dylan Kane, the blogger whose posts invariably had me clicking through to the comments to see what he managed to provoke from his readers, then scratching my head trying to figure out how he did it.

If your heart belongs to elementary math education, the best moderators I have found there are Tracy Zager and Joe Schwartz.

My Year in Review

If you’ve come to this blog only recently, here are five posts that received a lot of traffic and commentary this year:

Looking for favorites from the wider online math education community? Check out the #MTBOS2015 hashtag. If I had to award my own MVP, it’d be Elizabeth Statmore’s “How People Learn” and how people learn where she turns essential research into manageable practice.

Great Classroom Action

And now, shamefully presented without commentary, seventeen posts I read in 2015 that had me check myself and think, “That classroom action is great!” I haven’t shared these yet and it’s time to clean the cabinet.

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