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Thursday, December 29, 2011

Math Lesson Formula

Okay so, seven years in, and I feel I am finally cracking this nut: how do you make any math lesson work for most kids under most circumstances? Throughout the year I have been tweaking most of my lessons to follow the same basic formula. Not that we do the same boring thing every day - there are infinity variations to make it work for me or a particular group of kids. Not that I'm saying teaching doesn't require a whole mess of skills besides knowing how to set up a lesson. Anyway.

I will illustrate with the most frustrating of topics : log laws. I can't think of a topic that seems more boring and pointless to most math teachers and students. I know their virtues as well as you, but let's be honest, 99% of your kids don't really need to know them for anything they are likely to do for the rest of their lives. I posted about it last year, but there was a piece missing, and now it really sings. To believe this works, you have to believe that the one doing the work is the one learning. Nobody gets much out of Miss Nowak doing dramatic performances of math problems and proofs other than Miss Nowak learning how to do dramatic performances of math problems and proofs under the withering attention of 24 bored and irritated teenagers. I don't want to give the impression that I'm giving them a worksheet and being all like, "You're on your own, kids! Time for me to kick back and drink coffee." Because I'm running around, scanning for common questions or points of confusion or missed connections, re-capping with the whole group every five to ten minutes, encouraging and validating, etc. But if you believe "teaching" = "lecturing" then you are not going to see the validity of this approach, and I can't help you.

Phase 1: Productive Struggle
Hook the new thing to something they already know or know how to do. Then make them do it. A few times. Let them discuss and work together. No reason this has to be done in silence. Whether calculators are allowed depends on whether the calculator will let them avoid the things you want them to remember and see. (This particular lesson is no-calculators.)

Phase 2: Generalize. Make them write whatever they have been doing with letters. This is harder for most kids than you'd probably expect, especially if they've never been asked to do it before.

Phase 3: Use it. Presumably this new thing you've discovered is good for something. Even if that something is obviated by ready access to a shmancy calculator.

Phase 4: Prove it. The hardest part for kids, and the hardest part for me to figure out how to get them to be the ones doing the work. I have had some success with this approach of setting up an organizer and basically telling them what to write. But they still need lots of hand-holding. But at least they are doing more than watching/copying a dramatic performance.

Phase 5: Lots and lots of practice. I want them to understand, but I also want mental automation of relationships and procedures. Because later they are going to use this stuff to learn something new.

I would like to say Phase 6 is apply it to a novel and interesting problem, but I'll be real, I'm not there yet with log laws. Though I am there with good projects on some other topics that lend themselves to applications. Give me another seven years.


  1. Thank you. Thank you. Thank you for putting into a flow I can apply to factoring quadratic expressions. I'll be modifying this: with the NowakLessonFlow(TM)

  2. Aw I can't open an ExamView file on my home computer. But I'm sure you'll make it awesome.

  3. Looks like you are doing some abstracting from repeated computation. I always like to ask them, "what math are you doing over and over again regardless of the input?"

  4. Thanks for making it sound so simple and doable.

    What do I do if I don't know how to 'prove' it?

  5. If you don't know how to prove it? I want to say "look it up in a textbook" but I suspect your question is deeper than that. Do you have an example?

  6. I don't mean to be That Guy, but in math if you don't know how to prove it you don't really understand it. You can use it, sure, and even be good at it, but something is missing. Yes, it works, but why?

    As for getting the kids to prove it, filling in blanks is a great first step. You might also try writing out the thought process that goes from one point to the next in complete sentences. The equations are important, but they're the outline to hang a story on.

  7. Yes, that is why I make sure I put the proving it in there, That Guy.

    And if you think 80% of them can do anything without the scaffolding, you're just ignorant of the realities on the ground populated by your average 16 year old.

  8. I was pointed to this blog by a physics modeler. As a college physics prof entering my 4th decade of "service", I have a strong opinion about what math education should be, K-14.

    All but a fraction of 1% of math students will never need to do math proofs (or understand log laws). They don't need to truly and deeply understand math: they need to understand how to use specific mathematical tools. IMHO, math education in the US has ended up being a barrier to student success rather than a help.

    When I get a student in my algebra-based physics course, they've had an average of 5 or so prior experiences with algebra, but are unable to apply it outside of "math world".

    If I were emperor, I would eliminate almost all stand-alone math courses because math doesn't stand alone. Math is a language that we can use to help us understand the world about us. Math should always be contextual and context-free exercises should be eliminated. Again, IMHO...

    "I can't think of a topic that seems more boring and pointless to most math teachers and students. I know their virtues as well as you, but let's be honest, 99% of your kids don't really need to know them for anything they are likely to do for the rest of their lives."

    Doesn't this statement argue that teaching logs to students who don't need them is equivalent to Robert Heinlein's admonition about porcine music? “Never try to teach a pig to sing; it wastes your time and it annoys the pig."

    Not saying that students are pigs, but...

  9. If what to teach were my decision, I'd get into this argument, but it's not.

  10. It looks to me like Kate has used the content to teach a larger goal: the ability for students to abstract from repeated calculations (see also the "habits of mind" and/or Mathematical Practice #8). I also think the numeric examples do a fantastic job of scaffolding the proof -- being able to do a numeric example that is "general in spirit" is very close to doing a proof with X and Y and whatnot.

    If these are the true goals of the curriculum instead of "all kids love log laws", it's a great thing, since those are the skills students will actually use in college and careers. I wish we could all have learned that way, so that it would be easier to teach that way...

    It pisses me off when an adult does something inherently mathematical but then says they "can't do math", because it means their impression of mathematics is that it's a useless pile of formulas and rules. Bleh.

    Thanks Kate!

  11. Could you describe again the purpose of Phase 5? The goal of that activity is not clear to me.

    (Caveat: I hate, hate, hate those exponent problems where it's crap like simplify x^3y^4z^-2 to the -4 power. Why do we teach that??)

  12. The purpose of Phase 5 is so they can pass the Regents exam and go to college and have a future.

    I hate those exponent problems, too. Barf.

  13. The kids must already know that "log2(8) = 3 because 2^3 = 8", which makes me wonder why you don't go straight to phase 4. Do you think they would find this approach confusing?

  14. quintic - They would just not know what the heck was going on - what you were proving, why it mattered, etc. I think it's important to immerse them in the meaning of the symbols on the page before trying to prove anything.

    Especially given that the meaning of the log function takes one whole class period, so you'd be approaching log laws on a new day.

  15. I love what you are doing to teach for understanding. The process you describe ties it all together. Except in step 3, use it, I would prefer a physical example rather than a numerical manipulation.

    For example, a model for terminal velocity of a falling object is:
    v^n = cW, where n is an unknown integer, c is an unknown constant, and W is the weight. Solving for v and using logs leads to:
    nlog(v) = log(cW) ... now you use your new relationahip to give:
    log(W) = nlog(v) - log(c).

    Now gather data for v vs W by dropping coffee filters (stacking them together to vary the weight, and measuring velocity with a stopwatch and a meter stick).

    Plotting log(W) vs log(v) will give a straight line, the slope of which will be the value for n; and the intercept yielding the value for the constant.

    This is a powerful example (actually the only one I know of) where the relationship being taught is very useful. Google "coffee filter lab AP" to view some written procedures.



  16. I don't know that a 6th phase is necessary in every lesson. However, Joe's application to terminal velocity would be an excellent 6th phase. Unfortunately, as someone who teaches both math and physics, I can tell you the math teachers have precious little time to experiment.

    Looking at this another way: need the math abstractions students prove in phase 5 always be applied in some "real" situation? What happens when we stop at the proof?

    Kate's lesson format frees me from the annoyances of applied math (not everything we must teach in high school math has a cool, engaging application at the level we've just taught) and instead lets us show students the beauty of math as a language.

    [Dang, I can't believe I wrote that last paragraph. I'm definitely an applied math person who'd rather use the math tools I'm given than go about proving them.]

  17. Also, Bowen, I learned alot of this from you. So thank YOU.

    Megan - "beauty of math as a language?" Who ARE you, and what have you done with Megan?!

  18. Time and time again, I move on without enough of Phase 5...lots and lots of practice. Given the very real time constraints of any given course, I too often feel like I've done my own version of Phases 1-4 and then feel satisfied with how my lessons have helped them really see what's happening. So I move on. And then when I give a quiz, I see that they need more practice. And I think, "But you totally saw the connections yesterday!" And once again, I am reminded that yeah, Phase 5 is just as important as the rest of the teaching/discovering. Thanks for the reminder. Maybe I won't forget as often.

  19. For those trying to follow the practical mechanics of teaching, could you share a version of the materials you used to implement this lesson?

    Did you use Keynote slides/SMART Notebook? a worksheet or packet of materials? clay tablets & cuneiform stylus on the document camera?

    I'm trying to visualize how you "launched" Phase 1 with your students in the same way I can visualize one of your Solve-Crumple-Toss or Speed Dating activities or your circumcenters Treasure Hunt. But I'm falling short (which is probably just another failure of imagination on my part, but oh well, I'm trying).

    - Elizabeth (aka @cheesemonkeysf on Twitter)

  20. Hi Elizabeth - That's a great question. I always find myself asking some version of "what does that look like live?" when I read about somebody's lesson.

    The kids get a handout with all the screenshots from this post. I either put them in pairs or quads depending on how the class works together. I also project the same handout on the smartboard.

    I've realized relatively recently how important the launch is. I have to say something like, "you're going to use the stuff about logs you learned yesterday. this is new so if you don't remember, it's ok, but I want you to look it up in your notes or talk to the people around you. For now, just do the first problem. I'll give you two minutes and randomly choose someone to share their thought process." Then, I give them two minutes and listen to what they are saying. I use a smartboard flash thingie to randomly choose a student and we make sure we all understand the easy one.

    And then, they're off, basically. Different kids/groups might work at a different pace. I continue listening carefully, and I usually stop the class every 5-10 minutes to recap. I try to make sure I only recap up to a point that everyone has finished. Ideally, the quicker kids/pairs/groups finish a little early and start their homework, and the slower kids/pairs/groups at least reach a minimally-acceptable stopping point for the day. Sometimes I put starred challenge problems throughout to slow down the faster-working kids.

  21. For the last step of just finding interesting ways to induce practice, you could try playing Logarithm Block:

  22. What'd you think of my space-whale-adding-logs video thing?

  23. Have you considered using slide rules for introducing log formulae? I gave my students "magic sticks" and let them experience how they turned multiplication into addition. Two examples:

  24. The human ear's sensitivity to sound varies on a logarithmic scale.
    I.e., the stereo volume on notches 1, 2 and 3 has leads to "sound power" 2^1=2mW, 2^2=4mW, 2^3=8mW...

    (assuming a powers-of-two amplifier setup...)

    What age are your students? Maybe the above is not the best example.....

    Oh... and maybe you can ask "If I put two speakers on on 2nd notch next to each other, do I get the equivalent of sound of having the speaker at notch 3 or notch 4?

    notch 2 = log_2(4mW)

    but you don't add notches, you should add the power right?
    So 4mW + 4mW = 8mW,
    which is equiv to notch 3.

  25. Wonderful post.
    I second everything picrust is saying, and wonder: how do you find enough time in class for all that practice? Or do you give it as homework? Do you make up exercise sets before each class?

    One thing I'd like to add is I find it important to go over the "results" (not individual calculations but rather the conclusions) with the whole class before letting them loose on practice exercises. If I don't, I find some students sometimes miss the big ideas. Maybe instead of whole-class discussion students should be encouraged to compare conclusions with other groups, but there needs to be some check that everyone is on track.

  26. I finally got 'round to reading this after it sat in my twitter favourites since you wrote it.

    I like that you've abstracted / generalised your process in an easy to follow way. Granted, timing is still an issue as to how much you can allocate for each step but that's ok; you've highlighted what works and what's important.

    I agree that many kids struggle with generalisation. I've read somewhere that it's to do with cognitive development; however, there is such a thing as neuro-plasticity so we can, as teachers, influence positively - i.e. the more you do it, the better you get at it. Sadly, I think kids don't get enough time or opportunities to practice abstraction and generalisation because teachers give them the 'rules' or 'laws' or set definitions - to save time, maybe so there's more time for step 6 (ironically, practice).

    Having worked in a different career prior to becoming a teacher, I can honestly say that the most useful thing I learned from maths is mathematical thinking which involves abstraction and justification. full disclosure: career was in software dev't - may be different in other careers....maybe.

    You're on a winner here. Thanks for sharing.

  27. Yeah, it's nice when WE may have some sort of order to our teaching madness. I'm relieved to learn that I have been doing your prescribed 5-phase without knowing it. Love the scaffolding for proof!! Yes, they need it!!

  28. Hi, this is a truly wonderful and most helpful post. I am just writing to share my thoughts on communicating the historical significance of logarithms with students. Some historical facts and maybe a piece of mathematical literature, spiced up with oddities, rivalries or plain mistakes seem to do the trick for me in most cases.
    I believe it was in "The Parrot's Theorem" by Denis Guedj, when I first read that it took John Napier nearly forty years to construct his logarithm tables. Far from trivial, he explains in this 1616 english translation of the preface of his great book "Mirifici logarithmorum canonis descriptio", what propelled his actions:
    Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter. But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three. (from the MacTutor History of Mathematics Archive).
    Thanks for the read!

  29. Very well done!! This is a brilliant method, and one that I have used (in bits and pieces with different things).

  30. One thing I'd like to add is I find it important to go over the "results" (not individual calculations but rather the conclusions) with the whole class before letting them loose on practice exercises. If I don't, I find some students sometimes miss the big ideas. Maybe instead of whole-class discussion students should be encouraged to compare conclusions with other groups, but there needs to be some check that everyone is on track.Watch World TV OnlineNepal radio Online

  31. I will be working on logs/exponentials with my students the first real time soon. I am glad to see some kind of blueprint to start this. My initial plan was to focus on exponentials and use those to introduce logs as a method to solve problems.

  32. They should really have a pretty solid understanding of and fluency with exponents before starting this.

  33. I started logs yesterday by doing the mystery function game. First with easier functions, then with P(8)=3, P(2)=1, P(1)=0, etc, I called it logarithms after we'd gotten pretty comfortable with it.

    It was the funnest start to this unit I've ever had.

    I handed our your lovely worksheet at the end of class, for them to start as homework and finish in groups.

  34. Thanks for laying out your approach to logarithms. As far as inverse operations go, there is one in the pairs that seems to be established as the backwards one and thus harder to grasp. Subtracting is harder than adding. Dividing is just so backwards of Multiplying. And Logarithms undo exponentials. You can just see the "What!?! Let me think. Oh, I got it. Wait, WHAT!?!" on the students faces. I agree there is that certain stage (stage 5) that you just need to give the students time to cope and work with it. I did do applications where students were to solve problems involving logarithms. My students almost liked the word problems. I am going to even go as far as saying some student just liked the word problems, for they saw the need for logarithms. PLEMDAS not PEMDAS anymore!As for the proofs, I really did just have to say "Part of Precalculus is to juice you up for Calculus, and you will need these properties." PowerPoint Presentation:

    Worksheet concerning Logs:


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