The Rule Of Least Power: An Initial Approach

This is why I only use ten percent of my textbook’s printed pages:

The text has already imposed a rigid, powerful framework around an interesting drawing of a ski-lift. It has labeled the points, scaled the axes, and written the questions. The textbook has told my students how to care. The student can interpret this drawing only as the textbook intends.

To a certain extent, I have no problem with this. I want my students to interpret this drawing in a particular way. I want to use it to learn slope. But by applying this powerful framework in advance, the textbook has told my students exactly how they should be curious, which isn’t any kind of curiosity at all. It doesn’t train my students to draw these strong, interesting connections on their own and it presumes their engagement with the problem.

For example, if a textbook were to repurpose my last What Can You Do With This? prompt, it would run like this:

Just a guess.

The textbook would apply the most powerful framework to the problem, imposing a definite line of inquiry on the student before she even gets around to asking herself, “why does the tennis ball blur like that?”

By contrast, an application of the Rule of Least Power to the problem looks like this:

I put this picture up, just a picture, totally absent any mathematical framework, the least possible power I can apply here, and I ask, “What do you guys notice about this photo?”

The moment any student mentions the blur I drive the conversation her direction. The student has given me permission to apply more power to the situation. I ask, “Does anyone know why cameras do that?”

Several students take photography as an elective and mention shutter speed. I have the students take out their cell phone cameras and take a picture. I ask them to explain the camera’s pausePerhaps we digress with these images..

Having been given permission now to talk about shutter speed, I apply more power:

We talk about “1/25” and what it means to photographers. I might draw another blurred tennis ball on the board, one with a longer blur, and ask them to describe the differences. (A: a longer blur would mean it was dropped from a greater height.)

Finally, after this careful, deliberate application of power, I ask, “Can anyone tell me how high up off the ground this tennis ball was dropped?” No one can, not without measurements, and once someone mentions that, I project the last picture.

And we take on the problem. We have voluntarily committed ourselves to a mathematical framework. That commitment wasn’t forced upon us by an external agent. (Again: the involuntary commitment.)

The Rule of Least Power, as I have applied it to my classroom, means:

  1. Tell no student to care.
  2. Tell no student how to care.
  3. Apply increasingly powerful frameworks to mathematical objects only as the class cares about them.

Please don’t confuse this with hardcore, Waldorfian constructivism. I have an agenda, a standard to meet, a lesson objective. But I don’t fence my students onto a narrow path to my objective. I instead pave the ground beneath them so that the path to my objective is the easiest and the most satisfying to walk.

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. Thanks!

    Those 3 rules (for a first attempt) distill pretty nicely what I shoot for. I know when I get it, and when I don’t, but I’ve been lacking a framework in which to analyze my approaches. This (more than the w3 reference) crystalizes that.

    In other news, we going through a textbook adoption. I’m wondering if there is an inverse proportion between teachers who argue adamantly for one or the other, and teachers who are actually good at teaching math?

  2. What if someone mentions the shadow? Is that the wrong thing to notice? Seems like a perfectly good thing to notice to me. But if I mention the shadow, and you don’t really go anywhere with that, but you then pick up on blur when someone mentions it, then I am going to get the clear impression that I failed to notice the right thing.

    I wanted to explore the shadow. Feels to me like you imposed a line of enquiry on me, just like the textbook would.

  3. I was going to mention exactly the same thing that Robert did. It seems to me that you’re still sort of herding your students toward a “right” answer, but you’re making it seem a little more surprising/organic for them. I love the graphics, but setting it up as a series of slides getting to one specific and correct point means that you’re not doing a whole lot more than your textbook does, at least in this example.

  4. @Robert and Jeff- I see what you guys are saying and I would definitely let the student talk and dive into the questions being raised. I think I’d have to bring it back to the objective of the lesson though and point out the blur. Right? At least Dan’s pictures allows that conversation to happen. The textbook picture doesn’t even open that door.

  5. I agree with MrTeach, this textbook doesn’t allow for any student questions/insights/inquiries. Dan has a choice as to when he advances the next slide. He has control over when (and if) more information is provided to the students. There is no such option with the text pictured.

    Robert, if one of my students mentioned the shadow, I wouldn’t dismiss it. We’d pursue it for as long as it took to realize that it wouldn’t get us anywhere. It doesn’t really take as long as you think it might – once you’ve created an environment where it’s okay to make a conjecture and explore it, students are pretty darn good at working things out.

  6. Dan, you should have been a science teacher. Science brings meaning and ownership to those empty math problems.

    I have no problem with the textbook’s example except for the word “investigation” – it should read “practice” instead.

    Quite frankly, I don’t want the book to do anymore than it is doing- that’s what the teacher is for. Even if the book tried, I think it would fail since this process has to match the style and finesse of each teacher.

    There was a time when I dismissed the textbook too. I would proudly boast that I seldom or never use it. But I later realized that even though I didn’t need it to teach, they needed it to learn. I wasn’t happy with the hidden message I was giving students. I was downplaying the importance of reading. Reading the sections in the book really helped them learn the material. The book’s ability to offer student’s time to review and quietly reflect on the concepts we investigate in class is important.

    The bottom line is this: Reading is a necessary part of life- even textbook reading. Textbooks may not seem like real reading material; It’s not always the type of reading that you can get lost in with a blanket by the fireplace. Nevertheless, reading the textbook is a valuable skill. It will help students navigate their way through the next textbook, manual, sales report, city council minutes or the myriad of other sources they will face as citizens.

  7. Dale, I think you’re on to something . . . integrated units! Combining science and math (or reading and science) always makes instruction more relevant–creating connections for students. Not sure how easy that is to accomplish at the high school level, but it’s something I constantly strive for in elementary school.
    Which leads to my mantra for education (and life!) “balance”… textbooks and other resources have their place (practice, reinforcement, review, reflection) but should not replace the zeal of a creative, teacher-generated lesson plan.

  8. What Jackie and MrTeach said.

    The fundamental criticism here seems to be “Waitaminit, you still have an agenda,” to which I can only reply, “Yes, I do.” though only after several seconds of stupefied silence.

    But the journey is the difference. It no longer makes sense to me – psychologically, pedagogically, or otherwise – to deploy the entire scenario at once, pushing every nuance at the kid in advance, buying the kid out of any ownership stake in the process.

  9. I’m very curious how you steer the subsequent conversation around to the fact that the ball is accelerating and your questions can not be answered with simple linear equations. Acceleration is a concept that I find my best 9th grade students struggle with mightily to actually explain and understand and I don’t have any clear idea how to teach.

  10. Nothing wrong with agendas as long as you don’t let them hamstring you or your students. A sufficiently-sophisticated and knowledgeable teacher can give pretty open structures and, having thought through in advance a lot of the likely responses that might be elicited, make decisions about which ones are the ones most worthy of pursuit. Ooh, a JUDGMENT??? Yes, that’s part of the teacher’s job. But that’s not enough. The best teachers can make on-the-fly choices when the unexpected arises to: (a) pursue a promising line of inquiry despite the prior game plan; (b) stick with the game-plan but clearly acknowledge to students that something of interest has arisen. Such teaching requires that the teacher briefly address this fact by inviting students to pursue the question on their own (and ensuring that it is likely someone will, perhaps by allowing someone to volunteer to take responsibility for doing so), or promising to look into it and bring it back for future investigation, or promising to raise the question for everyone to investigate in a future (possibly the next class or shortly thereafter).

    What ISN’T an option is ignoring things presented in good faith. (Goofy, wise-guy stuff one does get to ignore, though even there it’s possible that something heuristic has been raised; I just don’t believe in rewarding rudeness very often). Some sort of acknowledgment must be made for each good question.

    No teacher can always anticipate every good or provocative question, insight, or comment, and so being prepared and being poised to react are vital. And we can be expected to “misplay” opportunities, going too far with ones that aren’t productive, not far enough with ones that later prove or could have proved deep and fascinating.

    But all that said, you’ve got a hell of a better chance if you aren’t constantly using stuff where the “discovery” game is rigged to produce “one right response.” Kids aren’t so dumb that they won’t notice the game is fixed.

    Only one question: why the fear of “constructivism”? I don’t even see what this has to do with constructivism as I understand it: just a theory of how people learn, not how to teach. Whether you have loaded, limiting frameworks or not, kids construct their own understanding. It simply may be hard for them to construct anything you haven’t utterly planned them to, which would be boring as it gets, at least for me and most kids.

  11. That textbook page is actually pretty good. I’m tired of the “real-world” questions that bare no resemblance to the real world like: “Agriculture: Farmer Smith has two fields, one field is twice the size of 3 acres more than the other…”

    However, wouldn’t it be interesting to ask the students which part was steeper, then ask them to find someway to quantify steepness. Could our students invent a measure of slope on their own? I like what you’re saying Mr. Meyer and I’m almost curious enough to try it.

  12. I object to the “science gives meaning to those empty math problems” sentiment. Math IS abstract. It’s free, puzzly, pattern-seeking play. It’s truth with or without context. It’s consistent relationships and patterns in quantity or shape that hold no matter the size or materials. That’s precisely why it’s so useful. Sometimes it’s beneficial and great to bring some physics or biology into my math class, but I’d lose much richness and freedom if I was required to do it all the time.

  13. Kate’s comment brought to mind a study done recently in Ohio State (published in Science, April 5 2008 – entitled “The Advantage of Abstract Examples in Learning Math”.

    The study found that students may benefit more from abstract math than through concrete examples, especially in being able to apply those concepts to other situations. It seems as if the students who learned the math through a ‘real-world’ example were not as easily able to transfer that math to other examples.

    Will this completely change the way I teach? Of course not. Aside from the fact that it reflects results from one study, it highlights an underlying message that most students have different learning styles. To completely teach from the textbook or to completely teach a constructivist approach will benefit some students and alienate others.

    I believe a good teacher uses a multifaceted approach throughout the school year, so that on a global level, he is able to connect with all of the students some of the time, and not just some of the students all of the time.

  14. The study found that students may benefit more from abstract math than through concrete examples, especially in being able to apply those concepts to other situations. It seems as if the students who learned the math through a ‘real-world’ example were not as easily able to transfer that math to other examples.

    This study was questionable in a number of ways, but I will simply point out the primary one that the real-world example was horribly confusing.

    More applicable would be an explanation of the applicability of e (to Algebra II students) as lim n->inf(1+1/n)^n versus something that falls naturally out of an interest rate problem.

  15. I loved this example of helping students build their knowledge. I am a currently a student teacher and have been exploring different methods for engaging students. I agree that text books can box students in and reduce the amount of positive exploration and curiosity.

    Your example of the tennis ball was engaging. I loved breaking the slides up and using student questions to guide the lesson. It is lessons where students feel like they are part of the process that true learning happens. The students had to discover the information, which made it more like a puzzle or a mystery, and what child doesn’t like to solve a mystery!

    In some of the early comments to this blog, the question of the shadow was brought to attention. I had also thought about this, but I believe that it is something that can be explored with the students the next day. It is fine to redirect students back to the objective, but their questions should be addressed later on. If our desire is for students to really question and become engaged, we have to indulge their questions and acknowledge the importance of their thoughts.

  16. Dan
    the only use I can find for my textbook anymore is to provide some practice and the occasional ski lift problem that inspires me to go and find or create the digital media that matches what they are trying to do. It’s almost impossible to sit here and think, hey I’m teaching adding polynomials tomorrow I need a picture of a “widget”.
    I’m not sure that science teachers get that it’s built in to most of their objectives from the start and it’s just not for math. I hope that you and the open source guys can build that repository for us because my LCD projector is way under used and we just don’t have the time and resources to do it alone.
    By the way, implementing your assessment system this year and I love it so far. Thanks for all you do!!

  17. the only use I can find for my textbook anymore is to provide some practice and the occasional ski lift problem that inspires me to go and find or create the digital media that matches what they are trying to do.

    Right. I find inspiring contextual stuff in my textbook but the textbook kills it by providing the necessary inputs and only the necessary inputs, by prescribing a route through the problem and then pointing frantically at that route in steps a, b, c, and d. Nothing to do but go out and recreate the scene properly yourself.