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Christopher Danielson:

School geometry seems to me one of the most lifeless topics in all of mathematics.

Paul Lockhart [pdf]:

All metaphor aside, geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum.

Proof is part of the problem. There’s no mathematical practice with a greater difference between how mathematicians practice it and how it’s practiced in schools, between how exhilarating it can be and how inert it is in schools, than proof.

Here’s Christopher Danielson offering us a way forward:

… eventually we reach a question that sort of requires proof; it seems true, but is non-obvious, and it has arisen from the questions we have been asking about how properties relate to each other.

Then they prove.

Questions that require proof are hard to create, hard to package in a textbook, and probably impossible to crowdsource. You’re trying to nail that point where the seemingly-true hasn’t yet turned into the obviously-true and that spot varies by the class and the student.

For example, “Square matrices are always invertible” might strike that enticing balance for one student while for another its truth is too obvious-seeming to be worth the effort of a proof and for others it’s too foreign for them to have an opinion on its truth one way or the other.

This is tricky, right? And Danielson offers us a description but not a prescription. He describes the satisfying proof process in his classroom but he doesn’t prescribe how to make it happen in ours.

Here’s one possible prescription:

  • Ask students to produce something given some simple, loose constraints. Draw any rectangle you want and then draw the diagonals. Choose any three consecutive whole numbers and add them up. Draw a triangle with three side lengths that the class chooses. Add up two odd numbers.
  • Publicly display their productions and ask your students what they notice. The diagonals seem like they’re the same length. The sums are always multiples of three. Our triangles all look the same. Our sums are all even.
  • Ask students to tell you why that should be true given what we already know.
  • Ask students what other questions we can ask given our newly proven knowledge.

“You people want students to recreate 10,000 years of mathematical knowledge,” says the math reform-critic.

No one I respect thinks students should discover all of geometry deductively. But as Harel, et al, say in a paper that has fast become the most meaningful to my current work:

It is useful for individuals to experience intellectual perturbations that are similar to those that resulted in the discovery of new knowledge.

To motivate a proof, students need to experience that “Wait. What?!” moment of perplexity, the moment where the seemingly-true has revealed itself, a perturbing moment experienced by so many mathematicians before them.

That’s more useful and more fun than the alternative:

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The problem here isn’t just the coffin-like two-column stricture. The proof doesn’t arise from “a question that requires proof” but from a statement that has been assigned. That statement makes no attempt to nail the gray, truthy area Danielson describes. It informs you in advance of its truth. It’s obviously true! You just have to say why. Tell me anything more lifeless than that.

BTW: Ben Orlin is great here also.

39 Responses to “Lifeless School Geometry & Questions That Require Proof”

  1. on 22 Oct 2013 at 10:52 amDan Meyer

    Except corpses, obv.

  2. on 22 Oct 2013 at 11:07 amJoshua

    Then there are those things that seem obviously true, but aren’t, like “Square matrices are always invertible.” For a student who is likely to be enticed, those can be the best of all. Because the next question is, “Well, if that isn’t true, then what is?”

  3. on 22 Oct 2013 at 11:18 amChris Lusto

    I realize that I’m poking at what has certainly become a central tenet of your work and thinking when I say this, but I take issue with the statement that a student needs to experience a wait-what moment to motivate proof. Of course that’s a motivation, but I can’t buy its necessity for a few reasons.

    (1) Lots of proofs aren’t going to do anything in the neighborhood of satisfying that perplexity, and it’s tough to claim those proofs aren’t motivated. Non-constructive proofs are (by and large) also non-enlightening. I will certainly believe anything you prove to me by induction or contradiction, but I will also probably be about as perplexed as I was before our conversation.

    (2) The moment of perplexity and the approach of provability might be so temporally distant as to be essentially unrelated. I learned very early on that the sum of two odd numbers is even. I’m sure it gave me a moment of wait-what. But by the time I could think about proving that fact, its perplexity had long since sublimated. In other words, the perplexity might very well precede the proof, but not in any meaningful way. But, when I become mature enough to think in terms of proof, I can certainly be motivated to shore up my “given” knowledge to my own satisfaction.

    (3) If you look at my second point writ large, sometimes proof is motivated by the somebody taking a deep breath and saying, “Let’s measure twice and cut once, here. Can I be confident that these true things are true? They are? Awesome. Moving on…” They say it in papers and stuff, but that’s what they say. And that type of motivation is more external; it’s more about persuading other people that what I take to be true is a good thing to take to be true. It may never have perplexed me at all. I’m sure Gauss felt that way a lot.

    Just some thoughts. Wait-what is a powerful thing, but I’m not sure it’s necessary.

  4. on 22 Oct 2013 at 11:21 amKate Nowak

    Extra wrinkle, esp as it pertains to geometry: Dynamic Geometry Software (DGS) makes this both easier and harder. Easier to notice something that seems true. Harder to motivate why a proof should be required.

    Example: Draw an isosceles triangle. Extend one leg beyond the vertex. Bisect that exterior angle you just made. Turns out, that bisector is always parallel to the triangle’s base.

    Fun! Not obvious, authentic “Wait, what?”, not too easy nor too hard to prove. The problem is, if you construct it in DGS first and ask for a conjecture, you can easily look at countless examples and see that (to inductive-brained humans) it’s always true. So why bother proving it?

    I don’t know what the answer is. Draw examples with (fickle) rulers and compasses, I guess, and skip the Geogebra for conjecture-forming?

    I suppose this doesn’t fit “simple, loose constraints” but they’d suffer from the same consequence if conjectures were made in DGS. Also, you’d run into the same thing with numbers (odd+odd) if kids were adding a whole bunch of examples with a calculator.

    I guess my point is, proof is still hard to motivate, even when the need for it arises authentically.

  5. on 22 Oct 2013 at 11:48 amChristopher Danielson

    Thanks for the plug.

    That post seems to have resonated with a number of folks; I’m glad it spoke to you too.

    I’ll be doing the hierarchy of hexagons in my NCTM New Orleans session in the spring. It will be a fun challenge to get a group of teachers to find and prove some new geometric truths. If you’re tired of those talks with prepared PowerPoints and talking points and want to watch a presenter squirm, come join us!

  6. on 22 Oct 2013 at 12:10 pmJoshua

    I like Kate’s comment, because this “motivation through observation on a computer” has become more challenging in calculus classes for years; I’m not surprised it comes up in geometry, too. If the computer (or, in past years, graphing calculator) can just find the maximum and minimum values of a function, why should we have to do it analytically? And I heartily agree. I hated finding extreme values before I could do it on a computer, and I am loath to put my students through it when the process can be automated. I get the sense that lots of teachers hated doing two column proofs, like I did. Now we have words for why, drawn from our own experience and the experiences of our students.

    One way to deal with a reliance on computers is to explore a new sense of “elusive”—namely, when does the program break? Or when does the “fact” we want to know become difficult to demonstrate with technology? Is it still true? This point of elusiveness varies with the choice of topic and software, of course, but anything that can get students to ask why something happened (like the program failing to demonstrate the conjecture) could become motivation for a deeper understanding.

  7. on 22 Oct 2013 at 12:15 pmLynn S.

    I really like how you make this proof “real.” Here’s my question, though: How would a English language learner access this problem. Can you use any of the new resources on the Understanding Language Web site to get at the language demands of these problems? http://ell.stanford.edu/teaching_resources/math. The “Language of Math” Task Templates are interesting — but I would love to hear your analysis on it.
    Thanks.
    I enjoy getting your blog in my inbox!

  8. on 22 Oct 2013 at 12:55 pmChristian

    I was quite impressed by some of the Japanese work based on Lakatos’ idea of ‘deductive guessing’ (see Proofs and Refutations). It could provide a ‘tentative’ idea for what Chris and Kate said (if I understand correctly), in the examples I saw there was a deliberate attempt to recreate the necessity for a proof or at least a generalization of conjecture.

  9. on 22 Oct 2013 at 1:37 pmColin Matheson

    I guess my “wait-what” moment is “We are teaching geometric proofs?” Do we really think this is a transferable skill? It certainly isn’t a skill required by our adult lives (except for the minute percentage of people involved in academic math). I think the approach of starting from observations, asking questions, and gathering data to find answers is a great approach to many subjects and I applaud it (since it is essentially teaching the scientific method). However, at the end of the day is our real educational goal that 100% of our high school students need to learn the content knowledge of geometry. If it isn’t, why not teach problem solving divorced from academic math (ie would we accept geometry being replaced by a debate/philosophy class). It just seems like creating a more difficult problem for educators if we have to ask our students to learn how to problem solve, but we are stuck in a domain which is difficult for students to connect to and will do little to benefit them.

  10. on 22 Oct 2013 at 1:52 pmDan Meyer

    Chris Lusto:

    The moment of perplexity and the approach of provability might be so temporally distant as to be essentially unrelated.

    Just a general reaction that need is continuous, not binary. It’s subjective too, to add to the squishiness. Certainly, different people can experience need under lots of different circumstances. I’m trying to identify high-probability circumstances. I believe one of those circumstances is connected to this sense of wait-what.

    Kate Nowak:

    The problem is, if you construct it in DGS first and ask for a conjecture, you can easily look at countless examples and see that (to inductive-brained humans) it’s always true. So why bother proving it?

    This is a big issue.

    I’m not saying we cracked it or anything – me and Dave Major – but when we did Triangles, we had students create and share only one instance of the phenomenon we wanted to problematize (SSS congruency). I wonder if the same approach would work generally.

  11. on 22 Oct 2013 at 4:18 pmMichael Pershan

    It’s obviously true! You just have to say why. Tell me anything more lifeless than that.

    This is a really good post, and really thought-provoking. I want to raise two points here, both relating to the lifelessness of explaining stuff that you already know to be true:

    1. One of my assumptions as a teacher is that if a lot of humans are choosing to to do something, it’s probably interesting. A lot of mathematicians choose to spend their time explaining stuff that they know to be true. So its probably interesting.

    Whether we can help students see what’s cool about explaining known stuff is something worth talking about.

    2. Let’s look at that “wait-what” structure very, very carefully. You do some experiment. You notice: the sums are all the same.

    OK, pause.

    At that very moment, do the kids believe that the sums are all the same? I’d say that they do.

    Unpause, and now you’re asking “Why that should be true?”

    There’s certainly a difference between this routine and the Given/Prove shtick, but it’s not the difference between explaining something known versus explaining something unknown. That moment that you’re giving your students is very much in line with “It’s obviously true. You just have to say why.”

  12. on 22 Oct 2013 at 4:55 pmRose

    I agree that the textbook proofs are generally lifeless and uninspiring, but I struggle to find great alternatives. Maybe I’ve just not found the right set-ups, but I have found it to be very challenging managing too many conjectures. I don’t wan’t to turn the experience into “guess-which-property-the-teacher-is-thinking-about-or-is-already-in-the-textbook.” Which conjectures do you validate without turning it into this awful game? It’s really tricky, especially if you are using DGS and they can get really far afield adding midpoints or parallel lines in unexpected places, etc. On the one hand it’s lovely, on the other hand it’s unruly.

    I think I found that sweet spot you are talking about with my students when I had my students build a dynamic set of vertical angles with patty paper and pushpins (pic.twitter.com/PeuzEtcSUV). I think the dynamic piece was important for them to move from thinking “Oh, look, those angles are the same size. So what?” to “Hey, look, those angles are _always_ the same size. Why is _that_ true?” I’m not 100% confident that the same amazement would have happened with a computer screen as did with this real, manipulable object. Anyhow, the students were generally motivated to prove that vertical angles are congruent after that experience. I’ve never had that happen after having them look at a bunch of static drawings. I have a sense that dynamic objects like this one and DGS can be helpful in moving students from one VH level to the next. (By making that rectangle a dynamic one, they can see that “square” overlaps with the set of rectangles. etc.)

    I’m also a huge fan of Michael de Villiers work on proof – see http://mzone.mweb.co.za/residents/profmd/proof.pdf if you’re not familiar with it. I think that tacit in your posting here is the assumption that the role of proof is verification, and de Villiers suggests that it can be much more than that.

  13. on 22 Oct 2013 at 4:57 pmJim Doherty

    I was just reading through the Park Math curriculum today and they do a lovely job setting up some of the ‘What do you notice? What do you wonder?’ type of reasoning. They present some radically different looking triangles and ask if you think that a circle can circumscribe them. The student is then asked to draw a variety of triangles to test out the theory at hand. I recognize that the text pushes the idea of circumscribing, but it feels like an interesting question. I hope that a willing group of students can be next convinced that it might be interesting to confirm what they think is true. Then, I’d whip out the GeoGebra and try to seal the deal.

  14. on 22 Oct 2013 at 5:58 pmClara Maxcy

    I love having a geometry class! I am one of those that teach the students to use compasses to draw images, have them draw triangles and draw lines connecting midpoints, have them draw parallel lines with transversals…. The visual-ness of the lessons sets the stage for students to draw conclusions and make connections to those pesky theorems. Proof takes the form of talking about which theorems make things true- and then trying to make them (the theorems, not the students) out to be liars! All in good fun, of course!

  15. on 22 Oct 2013 at 6:58 pmJustin Lanier

    Sense-making, arguing, and conjecturing are essential mathematical activities. However, it’s important to note that what’s distinctive about proof–as the term is used in the context of high school geometry courses–is not about argument or explanation. What makes “proof” different here is that the kinds of reasons we’re allowed to give is circumscribed in a formal way.

    Now, the axiomatics of Euclidean plane geometry are thorny. It’s not a pleasant system to work in as a first-time proof writer. There are many axioms of many different kinds, and the statements that are taken as axioms and the statements that are derived as first propositions are easily interchangeable. Not to mention that this all comes at students fast and furious. No wonder that it’s hard to perceive the subtle structure that lies before them.

    And then we have the audacity to ask them “why” things are true and won’t accept any of their perfectly reasonable but nonformal attempts at doing so–if they can muster anything to say at all. We mean for them to play a certain kind of unusual game, but then our verbal prompts make us appear to be asking for something perfectly natural and common sense: “why?”

    Here is a direction I’ve taken to making the notion of a proof–embedded within a proof-system–graspable, with some success: http://ichoosemath.com/2011/11/22/introducing-proof-using-formal-systems/

    In short: Proof here is not about finding truth or convincing. It’s about a certain kind of game we’re playing.

    And a P.S.: I think it would be interesting and revealing to hear many math teachers’ responses to the question, “Why is the Pythagorean theorem true?”

  16. on 23 Oct 2013 at 5:17 amRoy

    For what it’s worth, I teach geometry at the college junior/senior level (which in my case means basically high school geometry but completely focused on proofs/axioms/foundations). I think the most natural approach to axiomatic geometry would be to start with some interesting but non-obvious property (e.g.: why the heck do you always seem to get a parallelogram when you connect the midpoints of a quadrilateral’s sides?!) and work “backwards.” That is, try to figure out why that property is true, the explanation of which will rely on simpler and somewhat more obvious properties (e.g. the “midpoint connector theorem”), which will eventually lead you to a point where you have to ask “Can we prove this geometric fact, or is it blatantly obvious?” In short, work backwards from interesting things, to more obvious things, to things that are so obvious that they don’t need to (and in fact can’t) be proved. Then and only then, in my opinion, do you introduce the word “axiom.”

  17. on 23 Oct 2013 at 6:10 amjkern

    @Colin Matheson
    “Do we really think this is a transferable skill? It certainly isn’t a skill required by our adult lives”

    I LOVED proofs in high school. The way I saw it, the task wasn’t about figuring out geometry. We already knew the geometry from experience, and most of it just makes visual sense. I loved the challenge of being given a set of rules and not taking them for granted, but actually picking out the ones that apply in the given situation, and connecting previously unconnected rules to each other. The transferable skill is in connecting discreet areas through logical arguments that link one accepted fact to another. For scienists, lawyers, anyone who wants to win an argument, geometric proofs teach something beyond geometric facts. Maybe this underlying outcome needs to be made more explicitly by teachers in order for students to look beyond the simple angle congruences that appear on the surface of the task.

    Maybe that is your point, that we could teach this logical argument skill in some other way with less confusing content, but the superficial simplicity of geometric proofs is such a good vehicle for it. Lots of complex thinking goes into just 3 written lines of a logical, research-based student answer.

  18. on 23 Oct 2013 at 6:28 amBelinda Thompson

    I also loved two column proofs in high school. Not for the geometry part so much, but the power it gave me. It was the first time I realized that I as a student could make mathematical arguments and back them up. For me, this conversation is more about “what do students think they’re ALLOWED to do?”

  19. on 23 Oct 2013 at 8:38 amDan Meyer

    Michael Pershan:

    A lot of mathematicians choose to spend their time explaining stuff that they know to be true. So its probably interesting. Whether we can help students see what’s cool about explaining known stuff is something worth talking about.

    I’m not sure the interests of career mathematicians (all of whom have opted into mathematics) tell me all that much about what students (none of whom can opt out of mathematics) find interesting. I mean, in that open-minded-to-a-fault Pershanian way, yes, their interest is worth our analysis. I just don’t know how far that will take me with a fourteen-year-old.

    Still, to jump back to Lusto’s comment above, for you, him, mathematicians, and me, proofs are concrete objects that are interesting to manipulate, as concrete to us as numbers and blocks are to our students. Once somebody reaches that point where proof logic is its own fascination, I’m not sure how much of this blog post applies.

    Michael Pershan:

    There’s certainly a difference between this routine and the Given/Prove shtick, but it’s not the difference between explaining something known versus explaining something unknown. That moment that you’re giving your students is very much in line with “It’s obviously true. You just have to say why.”

    I’m not saying “the sum of odds is even” is a needy conjecture for everybody. I’m saying every student needs a needy conjecture for the proof act is to have meaning.

    It’s completely subjective to the student. An example: I ask teachers to take three consecutive whole numbers and find their sum and product. For many, the fact that their sums all seem to be multiples of three is a needy conjecture. For others that’s totally obvious and the proof needless. For that crowd, the fact that the products are multiples of 6 is surprising and they start to tackle why.

  20. on 23 Oct 2013 at 9:45 amKevin Hall

    Been thinking a lot about this ever since your really long thread on inductive vs. deductive reasoning. I agree that having a needy conjecture can breathe life into these proofs. But what’s interested me since that earlier thread is that I think you can get that delight just from seeing a connection between things that you didn’t think were connected. Sometimes, it even works better (in terms of student delight) to have the proof come after the fact is established. That’s because if you establish the 2 things (Fact A and Fact B) in their minds as declarative knowledge that we use in class routinely, you establish their importance in our social context. Then when you show them how to connect A to B, they experience joy. For example, a couple weeks ago, I had my Precalc-by-another-name students prove the quadratic formula by completing the square on ax^2 + bx + c = 0. At first they thought it was just a tough problem, but when they realized that their answer was the quadratic formula, they had an “aw, snap!” moment. If they hadn’t learned the quadratic formula years ago, then their answer would have just seemed like an answer, but not a surprise–hard to imagine any high-fives in that scenario. So proving something they already knew to be true, but which seemed completely mysterious, was joyful.

    Maybe this is a poor man’s version of joy of learning, and I’ll take some flack for suggesting it. But I think it’s actually connected to principles of psychology, especially as it relates to humor. A lot of jokes are funny because they set up the context to make you think about one thing, and then in the punchline they draw an instant connection to something that seems conceptually distant, and its the moment in which your brain sees that connection that gives the delight.

  21. on 23 Oct 2013 at 11:05 amMichael Pershan

    Thanks for replying, Dan. And I think that your response to my first point is dead on.

    Your response to my second point makes me think that I didn’t communicate it clearly. I’m not saying that what counts as “obvious” is subjective. That’s obviously true. I was trying to argue that even in any of the proofs in your post the kids are still “given” a statement to prove.

    This is the part where you’re probably thinking I’m nuts.

    But let’s say you do these sorts of little experiments in class all the time. One day you ask kids to draw a rectangle and its diagonals, they do, and then you ask them what they notice. Sure enough, they notice that the diagonals seems to be the same length.

    At this moment, I’d argue that the kids believe that the diagonals are the same length. They think it’s true. They’ve done these experiments before, and every time you’ve done them it results in some true thing that you have them prove.

    Then you ask them “Why?”

    You’ve just given the students a statement to prove. I’d argue that if kids find proving this more engaging than a text proof, then it’s not because the statement isn’t given. It is.

    I cheated a bit by asking you to imagine a classroom where these little experiments have become a sort of learning cliche, where the kids know the drill. But I’d also argue — much more tentatively — that things aren’t much different even as a one-shot lesson. The kids still believe the proposition after the little experiment, and when you ask “Why?” you’re still giving them a proposition to prove. The proof still is an attempt to explain something that is — now — obviously true.

    Despite this, it’s obvious to me that the proof is more interesting than if you hadn’t done the little experiment. But why?

  22. on 23 Oct 2013 at 11:53 amJames Key

    Let me see if I can redeem that little problem at the end of your post.

    Give students angle 2 = 105 degrees. Ask for angle 7. Go!

    Try this problem. Time passes. Discuss what you did with a partner. Time passes. Call on students to explain their thinking: how did you get 105 for angle 7? Good chance to “construct an argument” per SMP 3.

    “Nice going team. Now let’s try another one: 2 = 110. Find 7. Go!”

    Umm, wait a minute, Mr. Key, didn’t we just do that sort of?

    “Oh, so you don’t want to have to repeat the steps just because the measure changed? Honestly, me neither. How about we just settle this issue once and for all by letting x represent some arbitrary number? You guys game for that? Go!”

    Whaddya think?

  23. on 23 Oct 2013 at 12:05 pmKevin Hall

    @Michael Pershan: Regarding your concluding “why?”, I think it’s because an isolated empirical conclusion is hard to generalize, but a line of reasoning can be extended in interesting ways.

    I don’t have a good Geometry example, but my favorite example is from physics. Since I know you work with periodic functions (and I’ll be stealing some of your lessons on that this year), this may interest you. A mass oscillating on a spring yields simple harmonic (i.e., sinusoidal) motion, and this can be derived from the fact that the spring force is directly proportional to the spring’s displacement from equilibrium. That’s cool, but not nearly as cool as how it can be extended.

    Think about any object that’s in a stable equilibrium, which means when you displace it, it feels a force back toward its starting position. It could be a pendulum, a marble rolling back and forth in the bottom of a bowl, etc. When the displacement x is 0, the force F is 0, because it’s at equilibrium. When the displacement is x, the force is F(x), but we know from calculus that in the region around x = 0, F(x) is approximately linear. So it’s approximately proportional. So for small perturbations, ANY oscillating object is guided by the exact same equations as a mass on a spring. That one principle explodes to cover a massive range of phenomena.

    You’d never get that if you only understood the mass on a spring empirically without deriving the connection to F being proportional to x. And once you’ve done the proof, you can see all sorts of surprising connections.

  24. on 23 Oct 2013 at 1:50 pmJustin Lanier

    Dan, in responding to Michael above, you say: “I’m saying every student needs a needy conjecture for the proof act to have meaning.”

    I’d note that in the Harel, et al, article you cite, there are several other categories of intellectual need given besides the need to explain. In particular, the authors include the need for connection and structure. As Kevin nicely states above, “you can get that delight just from seeing a connection between things that you didn’t think were connected.” This, in my mind, is the motivation for doing mathematics axiomatically. Otherwise we would be satisfied–in general, and in our classes–with local, not-fully-unpacked, unsystematic, “informal” proofs. These, after all, explain.

    The prescription you give in the post is a nice articulation of a good lesson structure for doing mathematics with a group. I can match it up with the way I often approach my classroom (e.g. http://bit.ly/18cPTse). It’s a way of promoting mathematical discourse and sense-making. But I don’t think it addresses formal proof.

    It’s one thing to get kids to sword fight with sticks. It’s a different thing to get them to fence. The activities are related, but kids will be confused if we start calling fencing fouls on them when they think they’re just fighting with sticks.

  25. on 23 Oct 2013 at 3:19 pmJustin Lanier

    Just to add, I think these are the most important and meaty sentences in the post: “Questions that require proof are hard to create, hard to package in a textbook, and probably impossible to crowdsource. You’re trying to nail that point where the seemingly-true hasn’t yet turned into the obviously-true and that spot varies by the class and the student.”

    I’m interested to hear more. Dan, you’ve been very successful in creating, packaging (in non-textbook fashion), and crowdsourcing other kinds of mathematical tasks. It makes me want to think about whether and how tasks involving proof are different.

    A first attempt: proofwriting requires a certain kind of established culture. It’s much harder to make happen with a single task.

    But really, I’m not sure.

  26. on 23 Oct 2013 at 10:03 pmDavid Taub

    At the risk of jumping off the bandwagon, I think there might be a deeper issue here that the proof discussion is just a symptom of.

    Dan Meyer:

    I’m not sure the interests of career mathematicians (all of whom have opted into mathematics) tell me all that much about what students (none of whom can opt out of mathematics) find interesting. I mean, in that open-minded-to-a-fault Pershanian way, yes, their interest is worth our analysis. I just don’t know how far that will take me with a fourteen-year-old.

    For me, geometry proofs changed my life – it was when math first started getting interesting. Even hard algebra felt just like “hard arithmetic” since it was all routine. But geometry was different, the idea of proofs more than anything pushed me much more towards studying more mat hand science.

    But that is just me and the small percent of the population that has an extra focus/interest in math and science. However, I can’t help feel that this is a very important percent for our entire society. Not to say that people not interested in math and science aren’t important, but to be honest, I feel a lot of these discussions completely ignores this part of the population.

    Yes, it is certainly important to figure out how to effectively teach “necessary” math (where “necessary” is currently defined legally in different countries) to all those 14-year-old’s who don’t want to be there, but “can’t opt out”. But I don’t feel we should do so at the expense of those who may fall in love with the subject.

    Going out on a limb here, I don’t personally feel it is possibly to teach these two groups in the same way. Nor do I feel it should be. I will go so far as to say that I don’t think there is much value in teaching students who have no long-term interest in math proofs, but there is very high value in teaching those who may make a career in math or science about proofs (and this has nothing to do with which exact structure the proofs take, two column or otherwise, more the idea of a constructing a proof).

    Of course there is then the thorny issue of how and when it is possible to figure out which students are which.

    These issues are common place in sport and music and not overly controversial. All students are not forced to play on varsity basketball teams, nor are all students trained as varsity basketball players. And how physically fit you are in your teenage years has been shown to have a big impact on how you stay fit and healthy throughout your life, so we can’t argue the importance of math of physical exercise “in general”.

    So why is it different for math?

  27. on 24 Oct 2013 at 4:01 amChris Hill

    Quoting Colin:
    “I guess my “wait-what” moment is “We are teaching geometric proofs?” Do we really think this is a transferable skill? It certainly isn’t a skill required by our adult lives (except for the minute percentage of people involved in academic math)”

    This is another piece that has been bothering me in teaching Geometry for the first time – what skills are transferable? Here’s what I come up with:

    Creating rigorous proofs is transferable because you’re taking your conclusions, finding facts to back them up, and laying those out in a way that someone else will find hard to ignore. I think of a proof as saying “look, it has to be this way, the argument is final.” But I can’t figure out how to show students this, or how to build students up to this level, and I don’t know what is done in other (non-math) classes that I can build from.

  28. on 24 Oct 2013 at 5:30 amKate Nowak

    “I’m not saying we cracked it or anything – me and Dave Major – but when we did Triangles, we had students create and share only one instance of the phenomenon we wanted to problematize (SSS congruency). I wonder if the same approach would work generally.”

    I suspect it would work nicely. But I don’t know anyone who knows how to build those things besides Dave Major. :-) I had an analog version in mind when I suggested making conjectures with compass and straightedge. “Everybody draw one instance of this thing. Now lets stick them on the wall and take a look. Notice anything?” Yours and Dave’s digital solutions are more elegant. And the best way, by far, I’ve seen to motivate constructions, a la the ice cream truck problem. We just need you to build a couple hundred more of them. K? K.

  29. on 24 Oct 2013 at 6:43 amDan Meyer

    Kevin Hall:

    Then when you show them how to connect A to B, they experience joy. For example, a couple weeks ago, I had my Precalc-by-another-name students prove the quadratic formula by completing the square on ax^2 + bx + c = 0. At first they thought it was just a tough problem, but when they realized that their answer was the quadratic formula, they had an “aw, snap!” moment.

    Justin Lanier analyzes what’s going on here from the POV of Harel’s framework. (Extra credit, Justin.)

    My only question is: what was the need for the quadratic formula in the first place?

    From your description, it sounds like the QF was some inert piece of knowledge, detached from the students’ neural network, until you snapped that connection into place. Could we do something about that detachment in the first place?

    Michael Pershan:

    At this moment, I’d argue that the kids believe that the diagonals are the same length. They think it’s true. They’ve done these experiments before, and every time you’ve done them it results in some true thing that you have them prove.

    Got it. One way I try to cut against that is to “notice” a bunch of stuff that’s true locally but false generally. When we add up consecutive whole numbers, for example, I might notice something right, but I’d notice a lot that’s wrong. (“They’re all less than a million!”) Students have to decide which conjectures are true, which are false, and prove both. It also gives some kids an easier ramp into the proof act to just find a counterexample.

    Justin Lanier:

    This, in my mind, is the motivation for doing mathematics axiomatically. Otherwise we would be satisfied–in general, and in our classes–with local, not-fully-unpacked, unsystematic, “informal” proofs. These, after all, explain.

    Nice application of Harel here. But the ability to think axiomatically is a prereq for it to work and we know from van Hiele that a lot of kids need a lot of time before they can think that way. Not to say thinking axiomatically isn’t important. But to say more would be to basically recapitulate this thread.

    Justin Lanier, asking why these prompts are hard to crowdsource:

    A first attempt: proofwriting requires a certain kind of established culture. It’s much harder to make happen with a single task.

    This goes a long way, I think. A culture where keen observations are shared and prized doesn’t come for free.

  30. on 24 Oct 2013 at 2:49 pmMichael Caputo

    Dan, I’ve always liked your thoughts and I’m in agreement here. My problem is that by next year, I won’t have the choice to decide about teaching lifeless proofs. It’s a Common Core Standard and, in my district, I’ll teach that or else.

    I barely have time to sneak in some If-Then statements and the converse with Pink lyrics . . .

    Where there is desire
    There is gonna be a flame
    Where there is a flame
    Someone’s bound to get burned
    But just because it burns
    Doesn’t mean you’re gonna die
    You’ve gotta get up and try, and try, and try

  31. on 26 Oct 2013 at 7:24 pmanonpls

    For what it’s worth, I think Lockhart is seriously weighing in now with his recent book ‘Measurement’ (I’ve only read maybe 20% of it so far though).

    I got the impression from ‘The Lament’ that his approach to “nailing that point where the seemingly-true hasn’t yet turned into the obviously-true” involved having a huge magazine of interesting problems to pose to students (or individuals) as needed. In ‘Measurement,’ Lockhart seems to be opening up his bag of tricks and he’s quickly becoming one of my biggest idols.

    I’d also like that add that an appreciation for history seems to be a recurring theme here with Harel (via Dan):

    “intellectual perturbations that are similar to those that resulted in the discovery of new knowledge”

    With Serra (via Dan):

    “The Great Pyramid was built in 3900 B.C. by rules based on practical experience: Euclid’s system did not appear until 3,600 years later. It is quite unfair to expect children to start studying geometry in the form that Euclid gave it. One cannot leap 3,600 years of human effort so lightly!”

    with Dan:

    “a perturbing moment experienced by so many athematicians before them”

    And with Lockhart (sorry, no quotes… But his work I’ve seen is strewn with historical annecdote and reference).

    I’ve been kicking myself for not having a stronger foundation in the history of math ever since I heard about the Bridges of Konningsburg this summer. I attended a session on math history just yesturday and learned this gem: the symbol e (maybe the value also, my notes were kinda furious) was first used by Euler in a paper called ‘Meditation on an experiment made recently on the firring of cannon.”

    …If you involve cannon balls, is it even possible to setup a boring Act 1? (and now we need another two-column proof!)

  32. on 28 Oct 2013 at 7:34 amDavid Taub

    It seems what would help are a collection of mathematical statements that seem to be true, that a few obvious examples seem to support being true, but turn out to be false which can be shown by a non-trivial and possibly hard to find counter example.

    If anyone has a few examples of these, that would be really useful the need for proofs in my opinion.

  33. on 28 Oct 2013 at 12:07 pmJustin Lanier

    @DavidTaub, you may find these posts by Ben Blum-Smith to be useful.

    http://researchinpractice.wordpress.com/2010/05/07/pattern-breaking/

    and

    http://researchinpractice.wordpress.com/2010/07/12/pattern-breaking-ii/

  34. on 28 Oct 2013 at 1:25 pmDavid Taub

    Thanks! That was really helpful.

  35. on 29 Oct 2013 at 12:04 pmJames Key

    One thing that occurs to me here is that we need to do a better job emphasizing figures that *don’t* satisfy the conditions of a theorem. That will give more “pop” to those that do.

    For instance: “The diagonals of a rectangle are congruent.” If we show students some non-rectangles, and have them notice that the diagonals aren’t congruent, then it makes sense to wonder, “Okay — what is it about having *right angles* that makes the *diagonals* congruent? That is weird and sort of unexpected.”

    Launch!

  36. on 30 Oct 2013 at 7:01 pmDavid Wees

    It seems to me that if we are trying to motivate proof (deductive reasoning), then we should probably start by looking at situations where our inductive reasoning fails, and where deductive reasoning is a more useful tool for understanding a problem.

    Some examples of these kinds of situations where deductive reasoning becomes useful to answer a question one might have:

    Example 1:
    The Bridges of Koninsberg – I introduced this problem to a group of 9th graders in inner city Brooklyn. Three weeks later, there were still pockets of kids coming up to me (from all over the school) claiming that they had found a solution to the problem. When I finally relented and showed them Euler’s clever proof that no solution was possible, they were genuinely interested in the proof. It mattered to them that we could show it was not possible so that they could lay the problem to rest. I think the hundreds of years that people attempted to solve this problem before Euler came around suggest that inductive reasoning wasn’t very useful here.

    Example 2:
    Come up with a formula that will give the maximum number of pieces with n number of straight slices of the circle (source: http://mathforum.org/library/drmath/view/55283.html). This problem is easily stated, and the first few patterns are so nice, that students are usually fooled into thinking they have the pattern, and so they will often start filling in their tables of values before they check their results carefully. It leads to a situation where inductive reasoning fails quickly, suggesting that deductive reasoning would be useful. I don’t know how interesting students find this problem though; probably not very.

    Example 3:
    There are as many fractions as whole numbers. Our intuition screams that this is crazy. Of course there are more fractions! Only a deductive proof can convince people that infinity is stranger than we give it credit.

  37. on 31 Oct 2013 at 5:23 amWilliam Carey

    > showed them Euler’s clever proof that no solution was possible, they were genuinely interested in the proof.

    This is important. In all other teaching, we present students with examples of the best: the best literature, the best art, the best physics experiments. We ought to do the same in math as well. Giving kids models of the sort of reasoning that actually goes into deductive argument can really engage them (and form them intellectually).

    > There are as many fractions as whole numbers. Our intuition screams that this is crazy. Of course there are more fractions! Only a deductive proof can convince people that infinity is stranger than we give it credit.

    There’s a lovely paper by Calkin and Wilf about this: http://www.math.upenn.edu/~wilf/website/recounting.pdf

    I’ve often used it with enterprising calculus classes, as it’s short, surprising and, written in a readable style. It includes a couple of different patterns of reasoning to support one conclusion.

    We generally spend five or six hours reading it together in class (as one might do with a tricky text in a literature class). There are vocabulary quizzes interspersed, and finally the students work in groups to construct presentations to make the argument accessible to seventh graders, which they then give.

  38. on 18 Nov 2013 at 6:40 pmJill Knaus

    I think it is a trick to find a proof that provides the right level of challenge for students. This is an area in which differentiation is very valuable. Students need to be appropriately challenged to find proofs interesting! Too hard, and they will be overwhelmed and shut down. Too easy, and they will be bored to tears.

  39. on 24 Nov 2013 at 6:24 amMichael Pershan

    I just read the Villiers paper that Rose linked to, and it’s really great. Thanks, Rose!