No-Drop Zones

From the #iPhone-game-as-metaphor-for-curriculum-design hashtag, we have Geared, which I purchased because I’m almost completely obsessed with little spinny things, a purchase which I almost immediately regretted.

Two reasons:

  1. The early levels are ridiculously easy. Not a serious problem in and of itself. The same is true of Flight Plan, which you’ll recall I rather liked.
  2. But game play gets harder only over a series of completely nonsensical contrivances. You’re dropping gears into a system, blitzing your way through easy. Then on level 21, as the game flips to medium, you’re confronted with “no-drop zones.” That’s really it. Everything else is the same. You’re arbitrarily excluded from routes you know would otherwise work for reasons that have nothing to do with the function of gears.

There’s no good reason to criticize an iPhone game from this forum except for the robust metaphor it offers for conceptual growth in math. Few textbooks get this right – and I include here the ones that do a pretty good job of being less helpful:

whenever possible, introduce new skills and new knowledge as the solution to the limitations of old skills and old knowledge.



Please argue with me here but I don’t think my freshmen really care if career professionals use math in their jobs. This “career” concept is supremely abstract to most and therefore mostly useless to me as a motivator. I’ve found a much stronger motivator in a palpable sense of forward momentum, in a coherent skill set, in real, uncontrived challenges.

I’m teaching remedial Algebra for a fourth year now and the change I make to my curriculum far more than any other is to add this connective tissue.

You’re comfortable with a dot plot? Fine. Let’s put you in a place where a dot plot is tough to execute – say, a large data set with no mode and a huge range. That’s annoying. Then bring in the box-and-whiskers, the histogram, or whatever. I try not to introduce the next concept simply because it’s the next chapter in the book or the next bullet point on a list of standards or because it’s “what we’re learning today.” In other words, I try to stay away from the no-drop zones.

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. Do students ask you “Why do we need to know this? What is this good for?” I think it’s clear the textbook authors *expect* them to ask that. But maybe that’s not the right question. Because the truth is, many of them are not going to use, say, the FOIL method in the course of their everyday lives.

    Maybe if you provide intriguing enough questions, they won’t feel compelled to ask things like that, anymore than they ask what the “point” of a crossword puzzle or sudoku is. There’s no “point” to it per se, it’s just a fun little puzzle. And of course it strengthens the problem-solving part of your brain.

    Is it naive of me to think that if you pose interesting enough problems, the students won’t ask why, they’ll just be excited to figure out the technique for solving them?

  2. I cut this note out of my blog draft but wondered if it would come up here:

    This doesn’t mean they won’t exploit their abstract future careers to put their teacher on the defensive, asking, “when will we ever use this?”

    And I agree that if you reframe the class around the ability to solve problems, especially if you equip students with the ability to confront new, unknown problems without fear, that’s powerful stuff, and mitigates a lot of the usual posturing.

  3. “Please argue with me here but I don’t think my freshmen really care if career professionals use math in their jobs. This “career” concept is supremely abstract to most and therefore mostly useless to me as a motivator.”

    I won’t argue with you, but I find it rather sad that there are really no long-term motivators available to kids at this age. Actually, quite a few parents in local homeschool groups where I belong make it a top priority to help kids find such motivators. Career is only one possibility. While I don’t like them for several reasons, math competitions provide another. Then long-term involved projects, such as robotics or software design, can do the trick. For some, it is college entrance tests.

    Learning tools because old tools are weak is good; it’s beautiful really! I love the pedagogical device. I think it’s necessary. Not sufficient, though, and something long-term – career, project, team work, something – has to be in place as a motivator, as well.

  4. Isn’t the problem in your teaching in the first line of your game plan? “Quick summary of relevant prior skills”?

    In what remedial class do all students have those relevant prior skills? Only if the teacher starts on the first day building from the very basics, includes everyone, and avoids no drop zones will the class and the kids succeed.

    A textbook can’t do that. But textbooks are developed by phds. They are “research based” and “peer reviewed.” Teaching from the text covers a teacher’s ass.

    You are among the few who can say that the texts suck because you’re operating from an understanding of math and how it works that most teachers can’t grok.

    You not only teach the class but you develop your curriculum as you go. That requires a much higher understanding of the process, a higher order of brain development. The vast majority of teachers, regardless of education level, can’t do this. If asked to do it, they will be in over their heads. Read Robert Kegan’s In Over Our Heads: The Mental Demands of Modern Life.

    There aren’t enough of you and there never will be.

    The problem is the system. School as factory assembly line, where the standards march on. A place where the teacher is the sole source of knowledge and, if the teacher is junk, you lose. A system that creates Behind, a miserable place of no drop zones.

    I’m working on something that can help. Meanwhile, thank God for teachers like you. And thanks for the great metaphor.

  5. Maria, I totally agree with “I find it rather sad that there are really no long-term motivators available to kids at this age.” I don’t know why this is true except for developmental reasons. I remember caring what my future would be, but not caring enough about it for it to affect me in a specific way. Just generally.

    I don’t think kids say to themselves, “I need to focus on this lesson about properties of chords because I will use it in life some day,” but rather “I need to focus on this lesson because I need to get good grades because I need to get into college.” And that’s if they don’t first think “I don’t give a shit about this”.

    So, from a motivation standpoint, I don’t think it motivates most kids to do something because they might use it some day, as much as it demotivates them to not do boring bookwork because it is contrived and uninspiring.

  6. Touzel, re-reading my comment after your reply, I realize I should have said, “… there are really no long-term motivators available to SOME kids at this age.” I personally know many kids who do have such motivators, and I see some patterns in what their adults do to provide such motivators.

    For example, most kids these days are motivated to produce social media, so any class that helps them make better YouTube videos, become better bloggers, or figure out networking opportunities of FaceBook is hugely motivating. Right now, I am co-teaching an unclass about physics, modeling and math in the context of programming and computer game design. All kids in the group are gamers, and the opportunity to develop their own games and share with the world (we use Scratch for now) is motivating them rather long-term. Math and physics enters powerfully.

    Then particular kids have particular long-term interests. For example, kids from my daughter’s “writing circles” – little clubs and online groups for writing – are very interested in metaphors, linguistics (even as an area of math – look at Eastern European linguistic Olympiads for a sample), visual literacy and diagramming work (, logic and definition-making and anything else that can help them become better writers – now, not when they graduate college.

  7. @Dan – first of all, I love this post. Lack of contrivance or arbitrariness is one good candidate for First Law of Math Curriculum Design.

    Secondly, your little spinny things gave me a flashback to an art exhibit I went to about 15 years ago, which I have inexplicably not thought about since becoming a math teacher. The artist was Arthur Ganson. He makes machines. One of them is a series of twelve gear setups that each reduce the revolution rate to 1/50 of the previous. A motor turns the first gear at 200rpm; the last gear is embedded in concrete. All Ganson’s machines are charming, whimsical, and sort of mathematically provocative. (For example.)

    @MatrixFrog, Maria and Touzel – a propos of the motivation question I read a compelling post by Jesse Johnson on how the question “what’s it for?” is actually usually a code for “I’m bored” or “I’m lost.” MatrixFrog I think you’re right that it’s the wrong question to focus on, or at least those of us teachers who are not engaged by the question should feel free to not focus on it. It is enough to focus on keeping the problems provocative, the level of challenge appropriate, and making sure the students are getting opportunities to be creative.