Jason Dyer’s Redesign Of My Redesign Of Darren Kuropatwa’s Design

This is great. This is the culture of criticism we need.

I don’t know if Jason has redesigned my slides (since they’re all intact) so much as he has extended them in a particularly meaningful way. We motivated the concept of range by fixing the means, telling the kids implicitly, “you can’t use that anymore.” Now, Jason inserts this slide, which fixes the mean and the range, asking the kids, “now what can you do with this?”

This approach to skill development works both with fancy visual application problems and with skill acquisition. Rather, than 1) defining some concept like “range,” and then 2) using it in example problems, we instead 1) discover the limitations of our current tools and then 2) invent new ones. These mathematical operations didn’t arise just to employ degenerates like me. They arose because we needed them.

To the extent that Jason seems to think we should skip range altogether, I disagree. (Why not talk about it?) To the extent that he thinks we should engineer a situation where range is no longer useful, where the students must develop stronger tools like variation and standard deviation, I say nice job.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. I dunno. This goes further in the direction of “what concepts can I cram on this slide.” None of you seem to have noticed that there isn’t enough context for the question. I don’t know enough about the situation to know whether I should take the consistent or riskier but possibly quicker route.

    This part of the lesson does a poor job with the “why” of statistics.

  2. “Which route is better?”

    At this point, with the equalized range, it’s a matter of where the bank is, or the dry cleaners, or if I love the five minutes of the commute that goes past the lake enough to take it regardless of how much longer it may be.

    That was my initial reaction, before reading Tom’s comment, and I think I’m agreeing with him? The mathematical concepts have lost their link to the original question with this latest redesign.

  3. Agreed though that the way y’all have volleyed this idea / concept around is awesome.

    And I’d say that my input / Tom’s at this point is another part of the cool process – can the new design withstand further critique?

  4. @Tom, w/r/t information clutter, personally, I’d add a new slide replacing the old, replacing the means-fixed numbers with the new, range-also-fixed numbers. That’s easy if you have the source slide, which Jason didn’t.

    w/r/t justifying the stats here, is your objection that this visual will reasonably support “mean” but “range” less so and “variation” not well at all. I agree, to the extent that stat problems are notoriously difficult to contrive in any meaningful way. If I taught the course, I reckon I’d keep these slide until I found a better approach.

  5. My objection is that which route I will take will depend on the situation. Do I have a date across town in 20 minutes? Maybe I take a chance on getting there in 15 taking the short cut.

  6. I originally tried to be all smooth and make the new slide look like a natural extension of the old one, but I couldn’t even figure out which font was in the original.

    And I wouldn’t skip teaching statistical range altogether. Mostly I was just grumping, because the of the holy quadfecta of standardized test statistical terms (mean, median, mode, range) one of them I have never seen in the wild, ever (mode) and another shows up far less often than other things the students could be learning instead (plus or minus error, for example).

  7. @Tom, Hm. I guess it wasn’t sufficiently clear that this is a workday commute. The situation, along with the mean and the range, is held constant.

  8. If you did have students talk about this, it seems that the notion of “tolerance” (in an engineering sense) would naturally come up. How much tolerance does the system have for failure and what conditions could you invent that change the answer. A nicer neighborhood to drive through or your favorite donut shop, perhaps, but also if you work a job where you get fired for being late vs. a cushy job where if you get there a few minutes late it doesn’t matter. Tolerance is how much the punishment for error hurts.

    Because if there is no tolerance for being late, there is no reason to do any of the math. If one route has the potential to take 45 minutes, you would always have to leave 45 minutes and sit out your extra time in the parking lot. So your time spent commuting is always the highest number. Who cares what the mean or range is?

    You’ve chosen a known situation, but a fairly risk free one.

    You could imagine that the raw numbers could be a measure of how close you came to a perfect landing in a helicopter, for example. If “30” is a perfect landing, with every “5” measures of error meaning some additional injury. In that case, looking at the mean is not just an inconvenience, it could be deadly. In a riskier scenario, the ideas of margin of error and accuracy come in nicely.

  9. What if students were presented with a map of a new place of living and work in a city that they no nothing about. Assume that there are no freeways to conveniently merge onto.

    Would the main street(s) be better to take or would side streets be better to take?

    What if you don’t have the average travel time for five days?

    What if it was left out until students asked for it?

    What if all you had was a paper map?

    What if you had to go one route to work and a different route to home?

    What if you had to make errands to and from work?

    What about travel time to go to lunch during your hour lunch?

    What if . . .

    What if not . . .

  10. Dan/Jason/Darren, moving away from the mathematical content, I’m really impressed with the process of review that evolved here.

    Dan, you’ve been intimating that this is what you’d like to see more of in the edutechyblogosphere, rather than look at my *insert hip web2.0 thingy here*.

    How could we encourage it?

    Is bouncing from blog to blog a good enough method (it could be), or do we need a more central place for this to happen?


  11. One problem with scaling this up is that you can’t do it lesson by lesson — you have to start with “What is math?” and work your way down.

  12. @Tom Couldn’t you simply have a site set up like a help forum, where people come to have their lessons critiqued? I don’t think Dan was thinking of a curriki-like lesson plan depository. Maybe I shouldn’t speak for him…

    On the other hand, Dy/Dan could become Bourbaki 2.0.

  13. At Sylvia,

    You could. I just think that if you’re getting informed feedback, it is quickly going to spiral back down to “What are you really trying to do with this unit?” “Why are you teaching statistics?” etc.

    This is not to say that the project is impossible, but there has to be a clear point of view, and you have to be able to say to a contributor, “That’s an interesting way to take this, but we’re trying to do something else. Perhaps you’d like to start your own project to explore your ideas. If you need any advice getting started, let us know!”