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Pick A Point

Here’s my favorite moment from a workshop in Spokane last week:

It’s about the quickest and most concise illustration I can offer of Guershon Harel’s necessity principle. The moment of need is brief, but really hard to miss. It sounds a lot like laughter.

2014 Feb 19. Christine Lenghaus adapts the interaction for naming angles:

I drew a large triangle and then lots of various sized ones inside it and asked the students to pick an acute angle. I asked a student to describe the one they were thinking about and then another student to come up and mark it! This lead to discussion on how best to label so that we both agree on which angle we were talking about. Gold!

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Joel Patterson:

There’s an easy way to do this in Geogebra.
Open up blank Geogebra file, viewing only the Geometry window (no Algebra window).

Click the point tool and make a bunch of points like in Dan’s video.

Then there is a small button AA with one A in black and the other in grey. This button shows and hides labels for all points.

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Hans Freudenthal:

Youngsters need not repeat the history of mankind but they should not be expected either to start at the very point where the preceding generation stopped.

2017 Oct 16. Here are the slides.

Blue Point Rule

What is the rule that turns the red point into the blue point?


My biggest professional breakthrough this last year was to understand that every idea in mathematics can be appreciated, understood, and practiced both formally and also informally.

In this activity, students first use their informal home language to describe how the red point turns into the blue point. Then, more formally, I ask them to predict where I’ll find the blue point given an arbitrary red point. Finally, and most formally, I ask them to describe the rule in algebraic notation. Answer: (a, b) -> (a/2, b/2).

It’s always harder for me to locate the informal expression of a idea than the formal. That’s for a number of reasons. It’s because I learned the formal most recently. It’s because the formal is often easier to assess, and easier for machines to assess especially. It’s because the formal is often more powerful than the informal. Write the algebraic rule and a computer can instantly locate the blue point for any red point. Your home language can’t do that.

But the informal expressions of an idea are often more interesting to students, if for no other reason than because they diversify the work students do in math and, consequently, diversify the ways students can be good at math.

The informal expressions aren’t just interesting work but they also make the formal expressions easier to learn. I suspect the evidence will be domain specific, but I look to Moschkovich’s work on the effect of home language on the development of mathematical language and Kasmer’s work on the effect of estimation on the development of mathematical models.


  • Before I ask for a formal algebraic rule, I ask for an informal verbal rule.
  • Before I ask for a graph, I ask for a sketch.
  • Before I ask for a proof, I ask for a conjecture.
  • David Wees: Before I ask for conjectures, I ask for noticings.
  • Before I ask for a calculation, I ask for an estimate.
  • Before I ask for a solution, I ask students to guess and check.
  • Bridget Dunbar: Before I ask for algebra, I ask for arithmetic.
  • Jamie Duncan: Before I ask for formal definitions, I ask for informal descriptions.
  • Abe Hughes: Before I ask for explanations, I ask for observations.
  • Maria Reverso: Before I ask for standard algorithms, I ask for student-generated algorithms.
  • Maria Reverso: Before I ask for standard units, I ask for non-standard units.
  • Kent Haines: Before I ask for definitions, I ask for characteristics.
  • Andrew Knauft: Before I ask for answers in print, I ask for answers in gesture.
  • Avery Pickford: Before I ask for complete mathematical propositions, I ask for incomplete propositions.
  • Dan Finkel: Before I ask for the general rule, I ask for a specific instance of the rule.
  • Dan Finkel: Before I ask for the literal, I ask for an analogy.
  • Kristin Gray: Before I ask for quadrants, I ask for directional language.
  • Jim Murray: Before I ask for algorithms, I ask for patterns.
  • Nicola Vitale: Before I ask for proofs, I ask for conjectures, questions, wonderings, and noticings.
  • Natalie Cogan: Before I ask for an estimation, I ask for a really big and really small estimation.
  • Julie Conrad: Before I ask for reasoning, I ask them to play/tinker.
  • Eileen Quinn Knight: Before I ask for algorithms, I ask for shorthand.
  • Bill Thill: Before I ask for definitions, I ask for examples and non-examples.
  • Larry Peterson: Before I ask for symbols, I ask for words.
  • Andrew Gael: Before I ask for “regrouping” and “borrowing,” I ask for grouping by tens and place value.

At this point, I could use your help in three ways:

  • Offer more shades between informal and formal for the blue dot task. (I offered three.)
  • Offer more SAT-style analogies. sketch : graph :: estimate : calculation :: [your turn]. That work has begun on Twitter.
  • Or just do your usual thing where you talk amongst yourselves and let me eavesdrop on the best conversation on the Internet.

BTW. I’m grateful to Jennifer Wilson and her post which lodged the idea of a secret algebraic rule in my head.

Featured Comment

Allison Krasnow points us to Steve Phelp’s Guess My Rule activities.

David Wees reminds us that the van Hiele’s covered some of this ground already.

[3ACTS] Toothpicks

I enjoy tasks that exhaust a finite supply of things in order to make some kind of interesting structure. Here, a finite supply of toothpicks (250 of them) are exhausted to make a pyramid. (Or consider the finite fencing around Pixel Pattern.)

At some point I’d like to test out the hypothesis that removing the finiteness would make the video a lot less perplexing on the whole. In other words, we wouldn’t be as perplexed by a guy plugging away at a pyramid with an inexhaustible supply of toothpicks. We’re perplexed because we know, at the end of the video, that he’s done and we want to know what the pyramid looks like.

Here’s the task page.

Here are several interesting questions that popped up at 101questions:

  • Alison: How many rows in the end?
  • Hope Gerson: How many toothpicks does it take to make the next sized equilateral triangle?
  • Douglas Moore: How many triangles?
  • Matthew Clark: What’s the perimeter of the final triangle?
  • Scott Westwell: How many small (3 toothpick) triangles can be made?
  • Jeff de Verona: How many “total” triangles will be created (any size)?
  • Gregory Taylor: How long did it actually take to go through the entire stack?

Dissents Of The Day: Danielson, Pickford, Scammell

Christopher Danielson:

Your quest for the perfect image that will get 100% of viewers on board with the same mathematical question may be a bit quixotic …

Avery Pickford:

In [my ideal] world, I imagine spending a greater amount of time talking about the aesthetics of what makes for an interesting math problem and much less time cajoling students to ask the “right” question.

John Scammell

It’s unfortunate that we are so curriculum driven that we have to trick them into inventing the question we want them to come up with.

Here’s the thing: nobody watches Jaws and feels cajoled into wondering the question, “Won’t anybody stop that shark?!” No one watches Star Wars and feels tricked into wondering, “Will the rebels defeat the Galactic Empire?!” Those questions are irresistible, not on account of any deception on the part of the cast or crew, but because the cast and crew evoked the central conflict of their story skillfully.

This isn’t to say those questions are irresistible to everybody. Some people lack the cultural prerequisites to care about Star Wars. Some people possess the prerequisites and simply don’t care. Not everyone is interested in every movie, however skillfully it creates a narrative.

The point of the #anyqs challenge is to evoke a perplexing situation so skillfully that the majority of your students will wonder the same question (whatever that is) and the rest of the class won’t find that question unnatural or uninteresting, even if it wasn’t the first question that struck them.

Picking Up The Gauntlet

If you read nothing else, my summary judgment on stock photography closes the post.

Tom Woodard

Ten minutes after I threw down the gauntlet Tom Woodward picked it back up and whacked me with it:

Once again, help me make this better. That goes for images, argument, facts – whatever.

My response.

John Pederson

John Pederson, apropos of nothing I wrote, has developed a sudden, sloppy crush on typography, one of the artistic disciplines that hasn’t changed in several centuries, so that’s great. Because if you don’t know how to work with type, Prezi and CoolIris won’t save you.

Darren Draper

Darren Draper posted two variations on the same theme, asking would I really prefer a white background to a stock fast-food worker.


My answer is no, I prefer the stock photo, though I am glad there are a few other options besides those two. If those were your only options then go with stock photography. But carefully. The trappings of stock photos are a) exaggerated lighting, b) exaggerated framing, and c) exaggerated content, all of which give the content of your slide a lot of competition for attention.

I saw recently, for one example, a frightened kid shot under harsh lights with Scrabble tiles spelling out F-E-A-R censoring his mouth. The accompanying quote concerned Internet filtering or something. The quote was interesting and provocative but completely overwhelmed by the stock photo.

Dean Shareski

Dean perplexes me, saying I’m “stirring up trouble” with my last post. I realize this is just Dean’s usual Canadian bonhomie but, come on. Here is Dean’s commenter, Mark Kowalski. Take it away, Mark:

Even as a teacher, public critique of a person’s work is an odd experience. Maybe our social norms on feedback and politeness have gone too far one way?

If that “one way” is toward norms equating “criticism” with “insensitivity,” then I agree.

Angela Maiers

Angela has linked up a Slideshare presentation and asked for feedback.

Credit where credit is due. Y’all have taken Garr Reynold’s style and run with it for quite some distance. I have two concerns.

First, there are instances when the stock photography is so exaggerated or stylized that it distracts from the purpose of the presentation. In this example, I promise you I am not pondering the consequences of Angela’s quotation. I am scared to death of that toddler. Someone sign that kid to the Lakers but get him away from me.

Second, there are instances when the stock photography Angela has selected a) interprets the quotation for me or b) tips me to Angela’s interpretation when she’d probably rather I develop my own interpretation and add it to the discussion or presentation wiki or whateverTo cite my recent obsession, stock photography can easily be too helpful..

I didn’t mind this next one at all, an understated image that doesn’t constrain audience interpretation. The fact that I’m reduced to judging stock photography on how little it hurts a presentation oughtta concern us, however.

Alice Mercer

Alice has linked up her presentation files and asked for feedback. Take it away, people.

Summary Judgment On Stock Photography

Ditch it. Show me something real, not artificial. Serve the quotation up on a simple background with good typography and then show me some video or a photo or some audio captured naturally, in the wild, that hints at but doesn’t clonk me over the head with your point. And then let’s talk about it.

In Darren’s case, I would look for video of high school dropouts interviewed about their career paths since they left school, including, for the sake of intellectual honesty, some success stories. Find that. Or make that. Embed that. Let’s talk about that. Not about some Google Image or FlickrCC search I could have performed myself.

I realize this is several hundred times harder than typing keywords into a search engine but, as with personal hygiene, you get out of it what you put inMaybe y’all think I’m some sort of crank in these posts. But when someone uses their digital projector to curate and build conversations around interesting media they captured or aggregated themselves you really can’t imagine my enthusiasm..