Unnatural Currents

This is basically perfect, a single panel that goes 80% of the way to explain a) why your students roll their eyes when you assert the value of math, and b) the uncreative state of curriculum development in 2010.

In the second hour of my St. Louis workshop last Saturday I assigned an application problem to each group and asked them to locate the problem’s “hook,” which is to say, the point of the problem. Invariably, we found the hook in the middle or at the end of the problem, buried beneath all kinds of exposition and set-up. This is something like pressing the gas for ten minutes and then turning the key in the ignition.

One teacher said, “My students do all this work and they don’t know why they’re doing it. They’re doing it because the textbook is telling them to do it but when they get the last answer, they forget how they got it, they don’t know what it means, and they don’t care.”

We also noticed that the publisher sometimes chose an unnatural hook for the problem. For instance:

Let me put a situation to you: a high-rise is on fire. It is surrounded by marshy grass and shrubbery. Everyone has evacuated except for a man and his son who are waving and coughing from a window on the eighth floor. A firetruck speeds up to the building and parks as close as it can, some twenty feet away. The rescue ladder starts extending …

… and is anybody wondering anything besides “will the ladder reach?!

That is the natural current of this evocative situation. The unnatural current is to ask, “how high is the top of the ladder above the ground?” or, to reduce the example to absurdity, “what is the circumference of the firetruck’s tires?”

This leads me to three recommendations:

  1. Don’t force it. I’m not saying that the only math that matters is math that applies to some real-world context. Like Deborah Loewenberg Ball said, “On the question of context, it’s worth remembering that mathematics itself is a context.” I’m saying that if you can’t find a natural current for the Pythagorean theorem, don’t wrap a neckerchief around a dog and tell your students that the natural current is to apply the Pythagorean theorem. That is an unnatural current. Unnatural currents posit mathematical investigation as an unnatural act, which hurts all of us.
  2. Stop using clip art and stock photography. Clip art and stock photography are still water. They lack current of any kind, which is their whole point. They exist so that as many interested parties can purchase them and apply them to as many different purposes as possible. On the upside, still water can lead you anywhere you want, to any question or content standard. On the downside, you and your paddle are doing all the work, and your students will notice that you’re sweating when you assert the value of math.
  3. Find the natural current in some evocative photography or video. Paddle with that current. The boat in the river video is effective because, at its conclusion, no one is wondering anything but, “how long will it take him to go up the down escalator?” I found the natural current of the video and paddled with it. If that current doesn’t lead to the skill you hoped, try remixing the problem.

    This is difficult. A good example is this video from commenter Scott Haluck. He put a lot of care and craft into it.

    I asked him what question he’d ask and he replied:

    • “if everyone started at the same time, how far apart were they when they started?”
    • “what time will each person be passed by the fastest racer?”
    • “how long is the race?”

    And I’ll suggest here that there is a strong natural current to a video that shows four people racing across a screen. That video begs one very specific question, especially if you pause the video in the middle of the race. Ask a different question and you might hit more or different content standards, but you are pushing your students along an unnatural current and they will notice.

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. Thank you so much for this post! I feel this way every time I’m at a PD session and someone says, “We have to find the real world application in math to engage students.” I mean, we want to engage students, but we also don’t want to make up silly scenarios just for the sake of it. This post is one I will keep on my own list of “must reads.”

  2. I was thinking about stock photography as soon as you showed that dog.

    It leaves me imagining a textbook writer, faced with the task of pushing out a new text that aligns with the newest batch of semi-arbitrary mandatory mathematical standards.

    He sits there in a cubicle, skimming over thousands of clipart thumbnails, thinking to himself, “Triangles … triangles … triangles …” His desperate mantra triggers hopefully when he catches a glimpse of the high-contrast, roughly three-sided red shape. Okay, a triangle … a triangle … how do I use this triangle? *scribble scribble type type* Quickly, he resumes skimming for the next usable stock photo so he can keep on pace to push this book out before summer hits.

    The moral of the story is … is it humanly *possible* to do better than this when textbooks are reorganized and rewritten every 2-5 years due to a new round of math standards? Not to let them off the hook for, wow, that horrible dog-bandana problem, but I’m left wondering if the blame should be pointed higher up than at the textbook publisher.

    And maybe crowdsourcing (ie. teachers sharing good stuff) is really the only way to pool together a full course’s worth of truly inspired curriculum.

  3. Another wonderful post!
    @josh: good points all, but let me kvetch further on the issues. We get new editions not just because of new standards and frameworks, but because the publisher has a payroll and needs to employ all its people, especially in marketing, and they need new stuff to sell.

    Then, regarding clip art, the most egregious is probably not chosen by the author, but by various page designers whose job seems to be to make EVERY SPREAD SHOUT!!! so that the book passes the “flip test,” and make a prospective buyer (i.e., district curriculum rep) stop long enough to consider it. The dog and the bandana? Maybe the author thought that one up, I admit. Shame!

  4. This is great! I like engagement driving a need to know: “That is the natural current of this evocative situation.

    Sorry, on another topic but I don’t know how to reach you:
    I work with teachers interested in project-based learning and I’ve been figuring out what we can learn from the practices of mathematicians through Polya, Schoenfeld, and others so we can help kids as they approach real world problems. Any project-based learning experiences (messy, real-world, sustained) you can share with me would be great.

    In return… I know you like your infographics so I thought I’d point you to a pair I find compelling. They show amounts of time spent at different stages of math problem solving for a pair of students and a university mathematician. They are on pages 61 and 62 of this pdf LEARNING TO THINK MATHEMATICALLY: PROBLEM SOLVING,METACOGNITION, AND SENSE-MAKING IN MATHEMATICS Alan H. Schoenfeld, et al: http://citeseerx.ist.psu.edu/viewdoc/download?doi=
    As you did in Ukiah with the water usage graph there’s a lot of story here, esp. in the second one.

  5. Thank you so much for this. I stirve to make as much relevant as possible but I don’t want to force anything in class that is not natural. I actually stood up in front of my Pre Calculus Honors class yesterday and started the lesson by apologizing for it not being entertaining or ‘fun’ like some of my Algebra II lessons the students had me for last year. One of them laughed and said, “Mrs. C it’s ok. It’s Pre Calc. We knew what we signed up for and it wasn’t ‘fun’ activities. We want to be ready for Calculus next year. That’s our goal with this class. You teach, we will learn.” I almost cried. I feel like my students lifted a weight off of my shoulders. I will still strive to be creative and relevant, but I refuse to fill my curriculum with contrived math.

  6. You called it, Dan. Often, trying to force the video fit a standard makes it lose it’s initial appeal. Answer the question that everyone wants to know, “Who will win?”, and the rest will take care of itself. The students will push further if they want.

  7. @Scott Haluck – I was just about to create a vid similar to this one. If you check this comment thread again,I’d appreciate an email letting me know how you made it. jybuell at gmail


  8. That dog poster goes 99.19992% of the way towards explaining why I want to stab myself in the eye with a sharpened broom handle any time somebody asks me why I could possibly want to become a math teacher in this day and age.

    It’s because the people who extruded this poster are the one who will be 100% in charge of math education if I don’t.

  9. OK, actually I have something more constructive to contribute than just frustrated snark.

    As a recovering serial entrepreneur (6 start-ups in 20 years) moving into math teaching this year, I find it absolutely galling (verging on insulted) when mathematicians insist that ‘MATH SKILL/TECHNIQUE A’ [say, exponential functions] is exactly how real business people model ‘BUSINESS ISSUE A’ [say, the elusive “marginal cost”].

    This makes me roll my eyes because the marginal cost function is something only economists, executives at major corporations, and newly-minted MBAs care about.

    That’s because the marginal cost function doesn’t really exist as a phenomenon in the business world. Sorry, kids. No Santa Claus and no marginal cost function. Not any more. It only models anything in the great celestial fantasy world of writing a business plan or projections. I feel bad about saying this, but it hits me in the gut the same way the dog bandana stock photo did.

    The reality is that the marginal cost function doesn’t even begin to take into account in all the weird-ass corner-case factors that go into modern R&D and manufacturing — things like one-time and repeating license fees, government fees, R&D tax credits on some percentage of the initial investment, and exceptions that have to be built into every cost/expense model. The reality is that the modern “manufacturing” environment does not even EXIST in a single country any more, much less in a single factory owned by a single company.

    Which means that we’re actively misleading kids when we tell them that things like the marginal cost function will be essential when they enter the “work world.”

    IMHO, it would be better to advise them that models like these are most useful when you are working on pitching a project you’re trying to get funded or approved. At least then everybody is in agreement that you are operating in the great theoretical fantasy land of business models.

    If you present a compelling enough business model and concept, the best you can hope for is that some smart venture capitalist will poke holes in your assumptions and challenge you to create an even more complex model that takes into account the things you didn’t think of but that they (in their Yoda-like wisdom and experience) have seen before. This is the value they add to the project that makes your model and plan more realistic, though by no means a sure thing. And once the iterative process has gone back and forth enough times, and your reasoning and tenacity convince them that you are either crazy enough or smart enough to want to partner with, then the real work of growing a business can begin.

    But we need to stop trying to convince students to take their wide-eyed theoretical models into the business world. It either gets their asses kicked or it convinces people of greater power and lesser intelligence to (God forbid) put them in charge of things.

    And we end up with situations like Enron and AIG.

  10. Hi,

    just nitpicking here–it doesn’t “beg the question”, which is a kind of logical fallacy where you “beg the question to give you the answer” as it were (in other words, an argument with the logical fallacy of “begging the question” tries to use the question as the answer).

    What you mean here is that the video “prompts one very specific question”.

    [Example of begging the question: Say you are trying to argue that blue eyed people are evil. You go through a long convoluted argument and show something that a blue eyed person has done, and point out that it could have been done for either benevolent or malevolent motives. Then you say that the benevolent motives are inconsistent with the nature of blue eyed people, so the motive must have been the evil ones. Therefore, blue eyed people are evil.

    When you said that the nature of blue eyed people precluded them from having good motives, you were essentially using the idea that blue eyed people are evil as part of the proof that blue eyed people are evil. That’s what begging the question really means.

  11. I was referred to this blog on “contrived problems” after I created this playlist about how I teach such a problem. http://www.youtube.com/my_playlists?p=392F0B88B1F1B1A8 (apps of quadratic functions to areas, clipart pic and all). I disagree with the blog’s premise that real problems are valid and agree with Elizabeth. There is no problem solvable with the skills from high school that represents a real life solution. Does playing Wii golf at pro level make me a pro golfer? “Real problems” aren’t real even in college – ask any engineer in the field. Let’s get our kiddies understanding the math we are teaching. Make the problem clear, show ALL the aspects of the math being used in every possible representation and relate them – making sure the problems require them to repeat old skills, learn new skills and to think logically about how to apply them. Use direct instruction, practice and practice. Apologize for mathematics – NEVER. Even after 35 years teaching, mathematics never ceases to amaze me. Want to teach it better? Read this book: Teach like a Champion by Doug Lemov.

  12. I just realized something about this problem: the “You” in “You can use properties of special right triangles…” is, in fact, directed at the *dog*, who has recently acquired a dog of his/her own (and consequently needs to share his/her bandana).

    – Elizabeth (aka @cheesemonkeysf on Twitter)