(With due self-consciousness springing from the fact that I am responding to a comment on a blog 5 months later :)

I love the way that the Dan/Annie series of problems allows students using GSP to still (by the third problem) arrive at a situation in which, in spite of their technology-facilitated experimentation, they feel that they don’t have a complete story and so will want to look for one. (More below.)

And I acknowledge that my previous comments didn’t take account of GSP. To clarify my context, I have never taught a geometry class. When I wrote, “It’s the consistent lesson of my experience …,” the experience I’m referencing has all been in the context of teaching (a) high school algebra, algebra II, and calculus, (b) summer enrichment courses in number theory and group theory for middle schoolers, and (c) private tutoring at a variety of levels. In none of these contexts did students have access to a technological source of conviction parallel to GSP, so I’m not personally familiar with the set of constraints that imposes.

However. Dan’s original post is not about how to teach geometry but about how to teach proof. The point I’m at pains to make here is about how people who don’t know how to create proofs can best be supported in learning how to create them. I’m arguing that in order to learn this in the first place, students are best served by having real uncertainty and then using deduction as a tool to resolve that uncertainty. For all the reasons I’ve given above and in the linked post: the structure of deductive proof is determined by its at least formal intention to dispel doubt, so students who don’t know where to look for it or what it looks like can most easily look for it, see it for the first time, and experience it as meaning something, if it is actually dispelling their doubt.

The easiest scenario is if deduction is the most efficient available tool for resolving the uncertainty. That’s what I had in mind: when students’ desire to figure out what’s going on leads them to deduction as if by force. On the other hand if students have available another powerful tool for gaining conviction, such as GSP, then that’s great for developing geometric intuition, but from the point of view of empowering students to create proofs, it creates a new pedagogical challenge, as you describe (“… that strategy took a hit”). You’re talking about, “how does this fact change geometry class?” But I’m not focused on geometry class, I’m focused on proof: we want students to engage in deduction – how do we motivate and empower it by making it a tool they need?

What I’m arguing is that for kids new to deduction (in a mathematical context anyway), it won’t become a tool they need unless they have some sort of active, “what’s true”-typed question they are using it to try to answer. (Not just “why is this true?” That’s too abstract.) This is what my experience teaches me.

Part of my enjoyment of the Dan/Annie series of problems is the way that by the end (at least as I’m imagining it), students have a question about which there is real uncertainty in the room: for which convex quadrilaterals does this sum-of-squares relation hold and for which doesn’t it? This uncertainty may or may not motivate and empower deduction. If students are not interested to get to the bottom of it, it’s a non-starter. But even if they are, if they can mess around in GSP and come up with an if-and-only-if conjecture that holds as they slide the diagram around randomly, then conviction has been formed without deduction once again, and the students have no reason to start trying to deduce. But if it happens that students want to sort out the question and yet an if-and-only-if conjecture is not forthcoming, then deduction might be a helpful tool in looking for the right conjecture. If it works like that, then this type of sequence is a tactic we can use to motivate and empower deduction: get students interested in a question to which the answer won’t be found by randomly forming and testing conjectures, but may be able to be found through deduction. Then students have a reason to deduce. Perhaps this was what the questions were designed to accomplish?

As an example, I gave a list of the first ten [Chebyshev Polynomials of the first kind](https://brilliant.org/wiki/chebyshev-polynomials-definition-and-properties/#coefficients-of-chebyshev-polynomials-of-the-first-kind), and then had them list out their conjectures about the coefficients. We then talked about why these were true, which helped them develop a better appreciation for the theory (both from the functional equation, and from the cos polynomial).

Doubt is an excellent headache motivating the aspirin of proof. For those situations where a conjecture can be formulated that is questionably true, proof is an excellent aspirin.

Understanding is also a headache which can motivate some kinds of proof. Students ask “Why do I do that?”. They ask, “How do you know?” They ask “Why does it work?” These are the basic questions underneath most understanding. Some proofs that I think are especially appropriate for high school students are very helpful here. Studying the proofs does a lot of work toward understanding “why”. Reciting the proof or a variant of the proof gives a decent amount of evidence of the understanding. Explaining the proof gives even more evidence.

I think that proof meshes very well with maker culture. I think maker culture is showing us that people have a natural drive to dig through magic and to take things apart and to see why they work. Magic is a headache. Proof is the path in the sand left behind when one takes apart a beautiful mathematical truth to see why it works.

For over 40 years I’ve advocated and practiced a discovery classroom where students performed investigations and made their discoveries. They arrive at their own definitions and geometric properties and slowly begin to develop proofs for them. When I was a beginning teacher I believed they should discover everything by compass and straightedge thus their discoveries would be supported by Euclid. I learned from these early classroom experiences that relying on Euclidean constructions was needed for me not them. My students early on taught me that it would be easier, faster, and more convincing to use other tools such as ruler, protractor, scissors, folding, tracing, etc. Eventually I added patty paper to my set of discovery tools. All of this was based on a simple pedagogy of CONCRETE—>PICTORIAL—>ABSTRACT. Working in groups of four, students would have four similar conjectures from their investigation. With ten or more groups coming up with the same conjecture it seemed even more convincing. They felt there was enough evidence to convince themselves that their conjecture was true but I felt there was not enough evidence to have certainty. Thus I thought I had the necessary hook to ask, “How do we know for certain?” However, when Sketchpad (GSP) came on the scene, that strategy took a hit.

Now with GSP students were able to quickly construct a circle, construct a central angle and an inscribed angle intercepting the same arc. Measure the central angle and the inscribed angle. Drag the vertex of the inscribed angle and observe that the measure of the inscribed angle is always half the central angle. When students see this happen, in their eyes, “infinitely many times” not just four cases, it is more than just convincing. Students would look at me, roll their eyes and say, “how much more proof do you need?” That is certainly compelling to any classroom skeptic. The not knowing, the bit of doubt that may have existed prior is dispelled by dynamic geometry experimentation. The primary and traditional role of proof in the geometry class has changed.

The three proof exercises I offered in the last posting are three very different but linked examples of situations for exploration/explanation.

In the first I hint at a quick solution “Annie, with a twinkle in her eye, suggests to him that …”

The second one admits that the first was true (and literally a corollary to the Pythagorean theorem) but there are no right angles in parallelograms so perhaps this will not be true. Hopefully some doubt created. Exploring with dynamic geometry seems to come up with no contradictions however. Forcing them to explore “why.” “How do I recreate right triangles, like I did in the earlier cases?”

The third one announces that the first two were both true! Suggesting that there may be a bigger idea going on here and this one is true as well. But alas, a quick exploration with dynamic geometry and counterexamples are easily found.

As an aside, let me also add, I very much like Dan’s idea of “story” in our classrooms. I think this is a clever devise. It reduces the math anxiety, builds rapport and adds humor/interest to many topics including proof. Having themes such as the three exercises above adds to the opportunity of ongoing stories in your classroom. I have a character in my textbook archaeologist Ertha Diggs that appears numerous times throughout the course. I use her as an ongoing story. I share that Ertha and I were out on a dig again this past weekend and guess what happened? Then I would present a piece of broken regular polygonal plate that I claim to have found on our dig. How can we figure out the original number of sides? Or from another dig, I bring in a stone in the shape of a trapezoidal prism (actually a Chinese take-out carton covered in glue/sand). How many of these made up the arch that I suspect it came from? And on and on. I encourage classroom teachers to incorporate more and more “stories” via videos and images and use these grabbers to generate enthusiasm and perplexity. But I think the perplexity will more often be resolved by experimentation and not deductive argument.

and finally,…

People who are only good with hammers see every problem as a nail.
—Abraham Maslow

I’m certainly not advocating only using dynamic geometry software (I’m actually a bigger fan of patty paper explorations). One of my goals in a geometry course is to introduce a variety of geometry tools and eventually students can reach a comfort level with all of them and be able to decide what tool to use in any particular exploration.

I think comments #30, 31, 33, 34, and 35 establish a lot of common ground, and the conversation seems to have pretty much wrapped. So perhaps I’m out of line but I’d like to pick the thread back up for a moment, because I hear a note in Dan’s line of thinking that resonates with me and that I want to make sure gets its due. I think that MPG and Michael Serra you are likely to relate, if I can succeed in moving the conversation out of the frame given it by de Villiers’ question “Is proof about verification or other things too?”, a frame which I enjoyed very much while reading the de Villiers articles, but which in this context seems to me to drown out the note I’m hearing in Dan’s thinking and trying to amplify.

One of the things I’ve always responded to in Dan’s writing about pedagogy is the concern with what I’ll call the primal urgency of a math lesson. Dan I just spent a few minutes unsuccessfully searching for an old post of yours in which you had an image of some people whitewater rafting, with a caption something like look at the fun they’re having. You used this image as a metaphor for what you wanted math class to feel like. To me this image captures the theme in your thinking I’m trying to highlight. Whitewater rafting is a lot of fun, but another thing about it is that it’s very hard for [your body] not to go where [the river] wants you to go. You’ve always been most interested in math lessons where, not only is it a lot of fun, but it is actually very hard for [your mind] not to go where [the lesson] wants you to.

What I hear you inquiring into with this post is how can we give proof this sense of primal urgency (or as close to it as possible) for people who are completely new to it. Your proposal is that doubt has a distinguished role to play in accomplishing this goal.

I agree. I actually have a lot of conviction about this, based on the sum total both of my own classroom experience and a fair amount of experience in other people’s classrooms. I blogged about it recently (in response to this post) and less recently. I want to have due deference to the classroom experience of the people I’m conversing with, which (except for Dan) is more than mine – I’m not trying to establish any authority with this, I just wanted to say where I’m coming from. (I.e. this is not a theoretical point for me, any more than it is for any of us.) In fact, Michael Serra, based both on my sense of your corpus of work as a whole and on your “Dan and Annie” example above, I’m inclined to think that *you* might agree too, even if you’re not thinking about it this way.

De Villiers proposes that his reasons for proving have a hierarchy. He puts explanation at the beginning and systematization at the end (see the first article linked by MPG, figure 3 on page 12), with the idea that students need more supple and sophisticated skill with proving before they are ready to use and appreciate systematization as a reason for proving. I agree with his idea of a hierarchy, and I think that there is something below his bottom rung – the most fundamental and primal reason to engage in the chains of deductive reasoning we call “proof”:

Deciding what to believe.

This reason for proving has been identified in this conversation as “verification”, but that’s the wrong word. You verify something you already *mostly* believe. The more legitimate the uncertainty, the less appropriate that word, and I’m championing as much legitimate uncertainty as possible.

It’s the consistent lesson of my experience that novice provers have trouble accessing deductive trains of thought when they already have conviction, and that those same novice provers naturally gravitate to deductive trains of thought, almost like they can’t avoid them, when they honestly don’t know what to believe and want to know.

The trouble novice provers have accessing deductive chains when they already have conviction is not dispelled by formulating the question as one of explanation rather than proof, as de Villiers recommends. I believe him that one can get this going with time and enculturation, but my consistent experience is that “okay, that’s cool – why?” will always fall flat the 1st, 2nd, and 3rd times. If I’m working with students who are used to this question and what it means (e.g. when I teach professional development workshops for teachers), then explanation is a very compelling reason to prove something, but my experience is that students first encountering proof (esp. at the elem. and MS levels, but also HS often enough) don’t know how to look for that proof-type-of-explanation, and they also don’t know how to experience it as explanation when they hear it. This is the phenomenon I take de Villiers to be referring to when he writes:

Lack of initial participation in the actual activity of explaining (proving) has also been reported by teachers who have tried out some of these ideas at school level, and it appears that, in our experience, only after considerable concerted exposure to work of this kind do students become proficient in constructing their own explanations and critically comparing them.

(Bottom of p. 388 of the 2nd article MPG linked.) I describe why I think this happens in the recent post I linked above, but I have something to add here:

The thing is that the structure of deductive proof is given by the intention to dispel doubt. We’ve developed that structure over the centuries, and now (as de Villiers very convincingly argues) we use it for a wide range of purposes, and we very often find our conviction without it, but it’s still the case that how we (1) identify a proof, and (2) measure its validity, are given by the need for it to convince a skeptic. (Usually an imagined one.) We’ve chosen to use proof as a tool to understand, to give insight into reality. But the test of whether it’s a proof is not whether it gives insight, or does any of the other things de Villiers talks about. The test is whether it convinces a skeptic.

My point is that to use proof for any of the purposes de Villiers discusses, you need to know how to generate and/or evaulate proofs when you already have conviction, and that means you need to know how to serve as a skeptic even when you already have conviction. The problem is that novice provers straight-up don’t know how to do this.

On the other hand, when you really don’t know, and you really legitimately have to convince yourself, then you were actually legitimately a skeptic, so whatever you did to convince yourself is probably already pretty close to a proof. Even better if your classmate then doesn’t believe you and you have to convince them too. My experience is that students, even ones who are totally new to proving, who settle something for themselves they were legitimately uncertain about, more or less produce proofs without needing to be asked, or instructed in what a proof is or looks like. This way they can learn what a proof is and what it looks like while they are busy doing something they feel a primal urgency to do, which is figure out what the hell is going on.

To summarize, I’m arguing that legitimate uncertainty has a distinguished role to play both in giving proof primal urgency and in giving students a basic understanding of the structure of the task. And this is also what I took Dan to be pointing toward.

Postscripts:

p.s. Michael Serra, you point out that opportunities for proof as explanation abound much more than opportunities for proof to really settle a legitimate question. I’m with you on this. But I’m making a case for pushing the legit-question opportunities to the front of the queue. Imho these are the ones students should be cutting their teeth on. Once you grok the structure of proof and you learn how to be your own skeptic, then all of de Villiers’ other reasons for proving can come into play. I’m just making the case for the importance of uncertainty when you’re learning the structure.

p.p.s. To circle back to something: Michael Serra, what I meant about “you might agree based on your Dan/Annie example” was that the way you asked the question seemed designed to stimulate as much uncertainty as possible under the circumstances: not “prove this” but “Is this true?” / “Prove or give a counterexample.” So I already see you valuing the cultivation of legitimate uncertainty in how you frame the task.

p.p.p.s. One last aside: I think this is probably already clear but I just want to say that the pedagogy de Villiers is most explicitly arguing against (pp. 371-374 of the second article MPG linked) is unrelated to anything anybody in this conversation is talking about. As Michael Serra said, I don’t think anybody here thinks there’s any good to be had from telling students that they’re trying to dispel doubt when they don’t have any. de Villiers is discussing a scenario in which the teacher shows students examples of various things more or less unrelated to the work they will be doing, in an effort to bully them into not trusting their own sources of conviction. What I’m talking about is the opposite: tapping precisely into their own sources of conviction as the primal wellspring of their understanding of proof.

On the other hand, to draw on the example of inscribed vs. central angles, it seems to me that if we were teaching a more discovery-oriented (or inquiry-oriented) approach to geometry in K-12, students would be inclined already to pose the question “Why are the measures of inscribed angles half the measures of central angles that subtend the same arc?” or something along those lines (maybe just wondering why they are half the measure of the arc they subtend)?

FWIW, it doesn’t seem incompatible to me — before this move — to also ask students to draw any inscribed angle and its central angle and see if they notice anything interesting and if that interesting thing recurs in their classmates’ drawings.

I truly believe teaching is an art. The environment we set up in our classrooms, reducing the math anxiety with humor, games, puzzles, and music, inquiry-based instruction, encouraging questions, nay rewarding questions, goes a long way towards students safely engaging in the “why” process. My students knew they were the ones discovering their geometry. My kids knew they were rewarded for asking questions, and even more so for “good questions.” Whether they are in the midst of problem solving, working on the exercises, or in the midst of creating a proof with their group members, I am walking among the groups listening for explanations. I monitor and reward their explaining why to each other.

Proof as a means of verification, as I think we all agree, should happen only when there is real doubt. But I don’t believe we should avoid proof unless there is doubt. I didn’t mean to imply that explanation is the highest form of proof in the high school math classroom but I am convinced the opportunity for that role of proof is more common than verification. This is particularly true in the early portion of a geometry course as students are developing their proofing skills, the content is more straightforward, and they are still low on the van Hiele level of reasoning. For years I tried to come up with intellectually honest ways for students to discover properties inductively (and hopefully the investigation would hint at a method of proof that explains why). Try having students inductively discover the Law of Cosines or Hero’s formula. Good Luck. These are two prime examples in geometry where proof is used as a means of discovery.

I explain up front at the beginning of the year to my students what we are going to be doing in my class. Geometry, unfortunately, is usually the only math course in their K-12 curriculum where they will experience a mathematical system. Where they will be asked to explain why they think something is true. I explain that there are a number of purposes for proof and unlike other courses we will experience most of them. Many of the earlier discoveries may seem easy to accept so we will be asking why it is true. “Where did it come from? What other properties is it dependent upon?” Hopefully we will eventually come across some properties, problems, or puzzles that may not seem obvious and our goal then will be to determine whether or not it is true, and if true, hopefully show how this property fits with other properties we have previously established as true.

In fact in doing this we will be sort of recreating the historical development of geometry.

English Mathematician W.W. Sawyer (1911-2008) wrote:
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! The best way to learn geometry is to follow the road, which the human race originally followed: Do things, make things, notice things, arrange things, and only then —reason about things.

True, the CCSS claim that by high school, students will be operating at the highest level of reasoning and will have already gone through these stages of development. We shall see.

Let’s revisit the cliché example: they discover the sum of the measures of the interior angles of a triangle to be 180° usually by measuring the interior angles and calculating their sum (I still advocate this because that is what kids suggest when asked, how can we find out?). This gives something in the neighborhood of 180° and they all believe it. (However, nothing about simply adding up the measures reveals why it is true, or what properties this property is dependent upon?) Thus I insist we follow that with yet another investigation in which we cut out the triangle, tear off the angles and arrange them to get their sum. In addition to confirming their conjecture, it potentially becomes an aid in their explanation. When someone notices that some of the arrangements of the angles create a line parallel to a third sides it leads to the explanation that the triangle sum comes from the parallel properties.

Here are three different explorations/proofs for your amusement:

Dan has found the Pythagorean theorem to be such a very important property of right triangles and is wondering if there is a similar relationship with rectangles. Annie, with a twinkle in her eye, suggests to him that the sum of the squares of the four sides of a rectangle is always equal to the sum of the squares of the two diagonals. Is this true? How does she know so quickly? Prove it.

Dan was able to prove that the sum of the squares of the four sides of a rectangle is equal to the sum of the squares of the two diagonals. Then he thought, maybe it is true for any parallelogram. Is it? Either prove it is true or find a counterexample.

After seeing Dan’s proof that for any parallelogram the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals, Annie wondered if it was true for any convex quadrilateral. She has challenged Dan to prove it! Help Dan prove it or find a counterexample that proves it false.