Your contributions to the comments have been on another level lately. I’m especially indebted to the commenters who a) get my project here, b) disagree with its premise or implementation, and then c) say something about it.

Franklin Mason dropped by last week’s Makeover Monday to take exception with a) my focus on “real-world” math, and b) an inductive approach to geometry. It was a fascinating conversation.

I’m calling that entire thread up to the top right here for future brain food.

I wasn’t familiar with the Discovering Geometry text. I went out and read a bit about it. I take it that the method of the text is primarily inductive, not deductive. Proof seems to come as an afterthought. The result, I would think, is that geometry becomes just a loose collection of facts. Deduction establishes the connections between them, and with limited deduction, there is limited connection.

I don’t think I’d do this activity – either pre- or post-fix – in my classroom. No matter how it’s presented (and your presentation is really very good), it’s still just a plug-numbers-into-a-formula kind. A much better task for a high school classroom is to systematically derive the relevant volume formulae from the volume definitions and postulates.

This would be an opportunity to present Archimedes derivation of the sphere area formula. It’s a real beauty – one of the high points of the entire year. I really look forward to it.Another point: I would think that, by the time high school students reach a geometry classroom, we should expect them to largely abstract away from the particulars of such things as pots and meatballs. The applications of the results – a pot is a cylinder, a meatball is a sphere and thus have volumes given by the relevant formulae – don’t need much (if any) time.

Thanks for the comment, Franklin. I suppose I’m confused why induction must necessarily result in a disjointed curriculum.

Also, it seems difficult for most adults to think deductively, to say nothing of building a curriculum for students around deductively.

Two points:

1. I wouldn’t make the strong claim that geometry done inductively must result in a disjointed curriculum. But I do believe that that’s the tendency.

Here’s an example. If you were to present students with many examples of parallelograms, they’d very soon conclude that opposite sides and angles are congruent. That’s induction.

But what if we wanted students to know more than that as a matter of fact opposite angles and sides are congruent? What if we wanted them to know why it’s true? That’s where deduction comes in. A proof of these two properties (and ‘proof’ just is another word for ‘deduction’) derives them from prior results about congruent triangles and parallels. The proof thus forges the connection between the new facts and the old.

My view is that the importance of geometry lies in its systematicity. It isn’t just a sequence of facts, loosely connected if at all. It’s a set of results that can be traced back via deduction to a small set of fundamental, intuitively obvious assumptions. Moreover, it’s by linking the new back to the old via deduction that we explain why the new is true. No deduction, no explanation.

2. From my own experience, it’s quite possible to teach the great majority of students how to put together a mathematical proof. It isn’t something that’s out of reach, as you seem to suggest. I’ve had great success when I make it the focus of my geometry courses. (I quite happy that success has been possible. If it wasn’t, I’d be forced to conclude that the majority of human beings are not really capable of genuine mathematics.)

I’m curious what the classroom sequence looks like here.

Because a proof begins with that which is to be proven. So we pose the given (a quadrilateral with opposite sides parallel) and the objective (opposite angles and sides are congruent) and we prove it deductively.

But how did we get that place of wanting to prove that objective? Did the teacher just write it on the board at the start of class?

“Hey team. Today we’re going to look at this shape called a ‘parallelogram.’ It’s defined by its opposite parallel sides. What else can we figure out about this shape. Well it turns out its opposite sides and angles are congruent also. But don’t take my word for it. We can prove it.”

I’m just speculating so set me straight. But what my anecdote lacks is any intellectual need from the student for that proof. They’re doing the proof because the teacher suggested it.

An inductive approach would let the students notice that, “Hey, waitaminit, all these sides and angles are the same.” Which would then provoke the need to prove why. Thus the inductive can lead to and provoke the need for the deductive.

My view of this matter is that proofs are puzzles and that those puzzles are fun. (I don’t mean to trivialize geometry when I call its problems ‘puzzles’. Some puzzles are deep and profound, their solutions surprising and beautiful. An example: can the parallel postulate be proven from the other postulates of Euclidean geometry? That puzzle kept mathematicians busy for millennia. It turned out that the solution had quite an impact on geometry specifically and mathematics generally.)

The motive to find the proof of some geometrical theorem or other is internal to the problem. It isn’t that students walked into the classroom with some concrete question in mind (“The wheels on my car are circles. I wonder how many times they had to rotate to get me to school?”) and that we piggyback on that. Instead it’s that they like good puzzles. I don’t know why we human beings are like that, but we are. This is why my first question when I consider what do to in my classroom is always this: Is it a good problem? Will it be fun to look for a solution? I’ll be the first to admit that I don’t always achieve that goal. Sometimes a bit of drill is necessary, and that isn’t much fun. But still, that always the goal.

That’s why I’m also a little suspicious when someone continually demands context or application. Taht makes it seem as if students can’t really enjoy the mathematics for its own sake. Now, I’m fine with context and application. I don’t mean to say that we shouldn’t seek them out. Instead I mean to say that for the most part they’re not necessary. The mathematics itself suffices.

If this makes me seem unabashedly Lockhartean in my views, that’s fine. The description of Lockhart’s new book – Measurement – at Harvard University Press says it about as well as it can be said:

“Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living.

In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.

Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.”

First (and apologies to Michael Serra who I know reads this blog) let me say Discovering Geometry is kind of meh for teaching deductive reasoning. I taught out of it for four years and deductive reasoning always came off as an afterthought.

However, you do seem to have a stilted idea of what “genuine mathematics” is. With the mathematicians I’ve worked with, the process has gone more or less:

1.) Gather data on the problem.

2.) Poke at the data and make inductive guesses.

3.) Test those inductive guesses and generate proofs as necessary.

4.) Build new mathematical structures and make intuitive connections as necessary.The Euclidian-pure setup of deductions leading to deductions leading to more deductions is not really what happens except for in routine solving (there’s an exception I’ll mention in a moment). Even when I was getting a math degree where I faced pre-cooked “prove X” problems the process did not resemble my high school geometry proof days at all. (In fact, I’d say my geometry proof days were not helpful at all for my later mathematics, but we did two-column proof, which is alien language compared to most real mathematical work.)

[OK, so that one exception: the folks in logic can work like the pile-of-deductions on occasion, like the recent work in inconsistent logic to rebuild set theory. Also, the people working with proof software sort of work that way — like the confirmation of the Kepler-sphere-packing proof — but they program computers to do the logical-deduction-piling.]

It is interesting you invoke Lockhart since the example of geometrical proof he gives in his Lament is one that involves intuitive rotation to prove that an angle embedded in a semicircle is 90 degrees, as opposed to using the laundry-list of deductions method. (He calls two-column proof spawn of the devil or something like that.)

Anyhow, if you want to see mathematical thinking exposed for the messy wreck that it is, try one of the Polymath projects (I recommend #5, where a large part of the early work was consumed by just getting the data in a form that looked clear enough we could start thinking of patterns).

We talked over here about “organizing principles” for a math class. I’d say “make math real world” is as self-defeating an organizing principle as one can find. “Prioritize perplexity,” on the other hand, lets us chase down curious mathematics wherever it lives, either in the world outside the classroom or in the world of numbers and shapes.

While it’s clear to me you prize pure mathematics, it’s less clear to me how you create the experiences around it. Humans like good puzzles, as you say. No disagreement there. But creating those puzzles isn’t a trivial matter.

In the example you chose of deducing congruent opposite angles and sides from a parallelogram’s parallel sides, how do you turn that proof into the sort of satisfying puzzle Lockhart writes about, rather than an exercise assigned by a teacher?

**2013 Aug 3.** Pierre van Hiele, from *The Child’s Thought and Geometry*:

The student learns by rote to operate with [mathematical] relations that he does not understand, and of which he has not seen the origin. Therefore the system of relations is an independent construction having no rapport with other experiences of the child. This means that the student knows only what has been taught to him and what has been deduced from it. He has not learned to establish connections between the system and the sensory world. He will not know how to apply what he has learned in a new situation.

**Featured Comments**

Franklin Mason, responding again to Jason Dyer:

Insofar as I understand what you say, I agree. I most certainly do not teach my geometry classes as I suspect they’re typically taught (and as you seem to have in mind). We don’t do two-column proof. (I shudder to even write those words. I suspect that Lockhart is right about it: it’s sole purpose is to so regiment proof that it can be quickly and easily graded.) Instead we write them out, sentence by sentence, paragraph by paragraph. The emphasis is always on clarity and completeness. I don’t care a whit for the way in which a result is proven; certainly I don’t insist upon a certain method or (even worse) a certain sequence of propositions. Indeed discovery of a new method, especially when it’s more intuitive that what we had before, is given very high praise. This does make it hard to grade student work, especially in those cases where a student attempts an approach that’s new to me. It takes time and real thought, and that means that I can grade only a selection of student work. But I see no way around that.

This ability to cogently map out the logical relations between a set of concepts – that’s the real importance of my course. I do know, of course, that very few if any of my students will become mathematicians. (Now I reply to Mike C.) But that doesn’t imply that the skills of a mathematician are useless to non-mathematicians. What I really teach is the ability to think well and write clearly. Geometry simply provides the opportunity to practice this. (A really good opportunity, I would say. There are lots of beautiful results.)

As for the non-deductive part of mathematics – gather data and then squeeze it until a good hypothesis pops out and then test that hypothesis however seems best (perhaps seek out a counterexample, perhaps attempt to construct a proof) – that I do admit is a real challenge in the mathematics classroom. Given how much I’m required to cover, I find it difficult, indeed sometimes impossible, to give much time to it. In this year’s redesign of my course (and it seems to get redesigned every year) I’ll attempt to carve out more time for it. This means, I suspect, that I’ll have to do less. Do less and make time for students to explore and make discoveries. The challenge is to do this and yet still make certain that we prove all of the key results.

The inductive-experimental method is great for generating hypotheses, but the deductive approach allows us to prove these hunches are universally true. Both are needed in a great mathematician.

Some of the contributors to the discussion have raised the issue of what mathematics really is or what mathematicians do. But realize that almost every kid in America is forced to take geometry and an insignificant number of them are going to be mathematicians. So…who really cares what mathematicians do? They can learn that in college.

What will keep kids engaged? That is the question. Or perhaps what can we teach that will enrich the lives of a large number of kids?

At The Urban School of San Francisco, thanks mostly to the amazing Henri Picciotto, we handle it like this: the first semester of our Geometry-ish course is mostly inductive reasoning. They determine the congruent triangle theorems, for example, by using a compass and ruler (virtual or actual), discovering that once we set a specific Angle, another Angle, and a given Side length, they can only make one possible triangle. Thus, AAS is convincing, but not actually proven.

Then in the second semester, we introduce deductive reasoning as we dive into Quadrilaterals. We give them a list of phrases like “opposite sides are congruent” and “opposite sides are parallel” and have them choose two to combine into and “If-Then” statement like “If opposite sides are congruent, then opposite sides are parallel.” Then in their groups they sort their individual lists into Definitely True, Probably True, Probably False, and Definitely False. Then they send them to the teacher and we make a big list of the four categories. Invariably, even in a small class of 4 groups of four, the same sentence will appear on both Definitely True and Defintely False as different groups place it differently. Ta-da! A need to figure out who is right! Then we spend time sorting through the list proving (with a couple different methods) or disproving (by counter-example) the different statements. Occasionally we encounter something they know to be true from middle school (where they learned properties of quadrilaterals) that they want to use to prove another property, so we add that to our list and prove it as well. The teacher also suggests useful and/or challenging properties to add to the list as necessary.

This way, we teach them that they can figure out cool things themselves, (I will never forget a 9th grader asking me “Did you PLAN for us to discover this today!?” after deriving the Inscribed Angle Theorem in the first semester). And we also teach them the importance of deductive reasoning and how to use it as we.

Michael Serra stops by to leave a treatise on how he treated deduction in Discovery Geometry:

Great discussions. The topic of inductive and deductive reasoning obviously is very dear to me. First, I accept Jason’s apology even though “meh” was a dagger to the heart.

There have been a number of commentators saying things about Discovering Geometry (DG) that are simply not true or very misleading. They appear to be speaking from no direct involvement with the book but from how others have characterized the discovery approach.

I take strong exception to people characterizing DG as the book that is inductive with little or no proof or proof that only appears in the last chapter or that proof is seen as an afterthought. Inductive and deductive reasoning appear throughout the text.

So permit me to make my case.

Whenever there is discussion about proof in geometry we should begin with the van Hiele model of geometric reasoning. Research by numerous mathematics educators including Usiskin 1982; Senk 1985; Burger and Shaughnessy 1986; Geddes and Fry 1988; Clements and Battista 1992; and more recently Battista 2007, support the accuracy of the van Hiele model. The model consists of five levels of geometric reasoning that students pass through from visualization to rigor.

Level 0: Visualization –students can identify rectangles by sight but squares are squares and are not seen as special rectangles.

Level 1: Analysis (descriptive) – students can identify properties of rectangles (by drawing, measuring, and making models) but cannot yet derive other properties from given.

Level 2: Informal Deduction – students can give an informal argument to justify that the figure is a rectangle from given properties.

Level 3: Formal Deduction – students are capable of creating original logical arguments.

Level 4: Rigor – students are capable of reasoning about mathematical structures (i.e. Euclidean -vs.- non-Euclidean geometries).

The van Hiele model asserts students cannot move to the next higher level until they have successful mastered the previous level.

But the big item is that research shows that 70% of high school students enter geometry operating at level 0 or 1 on the van Hiele measure of geometric reasoning (Shaughnessy and Burger 1985; Senk 1989).Yet “traditional” geometry textbooks that begin with establishing postulates and proving theorems are ignoring the research and expecting students to immediately begin their geometry experience at levels 3-4. When the teacher and the textbook are presenting geometry at van Hiele level 3 or higher, while the students are functioning at level 0 or 1, it should be no surprise that there is such a high failure rate in traditional geometry courses.

One of my favorite quotes that I refer to often comes from Mathematician’s Delight by English Mathematician W.W. Sawyer (1911-2008)

“It would, I suppose, be quite possible to teach a deaf and dumb child to play the piano. When it played a wrong note, it would see the frown of its teacher, and try again. But it would obviously have no idea of what it was doing, or why anyone should devote hours to such an extraordinary exercise. It would have learnt an imitation of music. And it would fear the piano exactly as most students fear what is suppose to be mathematics.”

Some geometry teachers claim that they can successfully teach all of their geometry students how to create geometry proofs. Perhaps they only have those 30% that are entering geometry and functioning above van Hiele levels 0 and 1. I suspect however, they may also be teaching an “imitation geometry.” Their students are trained to go through the motions of stating the theorem about to be proved (5 points), then stating the given (5 points), then stating the show (5 points), then drawing the diagram to the right (5 points), then drawing the big T (5 points), then putting the given information in the first few lines of the T-proof (5-15 points), then writing given to the right of each statement (5-15 points), then throwing in enough statements to garner enough points to get credit for the exercise without having any clue as to what he or she was doing or why anyone should devote hours to such an extraordinary exercise. They are doing imitation geometry.

Discovering Geometry, in its first edition, was an innovator in addressing students’ needs for gradual development of the deductive process. Discovering Geometry is the only high school geometry textbook on the market that is aligned with the van Hiele model and other research on geometric proof. We accept that the vast majority of student are entering geometry at very low van Hiele levels of geometric reasoning and our goal, with careful deliberate scaffolding, is to move them to higher and higher levels of geometric reasoning as they progress through the course.

Geometry student’s consistent difficulties with understanding proofs (an international problem BTW) should not be solely attributed to their inability to reason but perhaps our inability to recognize that there are many purposes for doing proofs and we have been stressing the wrong purposes for proof at inappropriate times. Professor Michael de Villiers’ research on the role and function of proof identifies six basic roles for proof:

• Verification –to remove doubt, to convince someone of the truth of a statement

• Systematization –organize known results into a deductive system of postulates, definitions, and theorems

• Explanation –insight into why something is true

• Discovery –proof can occasionally lead to new unexpected results

• Communication –proof can create a forum for critical debate

• Intellectual Challenge –proof can be a testing ground for intellectual stamina and ingenuity

The function of proof in a high school geometry course has been mostly two-fold: to remove doubt, to convince someone of the truth of a statement –verification and to establish geometry as a mathematical system –systematization.

Many high school mathematics teachers seem to hold this naive view that the main function of proof is to provide verification that a given statement is true. The role of proof as a means of verification is a useful method of verifying the “truth” of a proposition within a mathematical system, especially when coming across surprising (non-intuitive) results. However this view does not stand up to actual mathematician’s experiences. Professor George Polya wrote, “When you have satisfied yourself that the theorem is true, you start proving it.” If the sole or primary purpose for doing proofs in a high school geometry course is to provide verification of the truth of a statement then students functioning below level 4 on the van Hiele scale will continue to question, or worse, disregard the process of proof.

If systematization is emphasized as a primary function of proof right from the start of a geometry course the same poor results will persist. The van Hiele model tells us that systematization requires the highest van Hiele level of geometric reasoning. Geometry textbooks that begin their geometry program with lists of definitions and postulates, and then begin doing two-column proofs are ignoring the research. Only an honors course or any class in which all students are finally operating at van Hiele level 4 would have any chance of success in a course that looks at geometry from the perspective of a mathematical system. Until some magic happens and all students beginning geometry enter the course functioning at van Hiele level 3 any attempt to create a mathematical system for a regular or informal geometry course is likely to continue to have major problems.

Students can acquire a very high degree of confidence in a conjecture arrived at by inductive methods but these methods may not explain why the conjecture is true. Here is where proof can come to the rescue. An example would be the inductive discovery that the sum of the measures of the three angles of a triangle is always 180°. A good inductive first approach would be to ask students to measure the three angles of a number of triangles thus gaining reasonable confidence that the sum is indeed 180°. The same can be done with dynamic geometry software. Either investigative approach is a good first step because it is the first approach that students would take. These inquiry approaches do give students confidence in their conjecture however they gives no insight as to why the sum is always 180°. The investigation should be followed by a second investigation where they cut out the triangle and then tear off two of the angles and arrange them on both sides of the third angle to create a straight line. From this arrangement they can see that the three angles create a line parallel to the third side. This visually explains what properties this conjecture is dependent upon and why the conjecture is true. This can also be done quickly with a patty paper investigation.

From The Role and Function of Proof in Mathematics by Michael de Villiers:

It is not a question of “making sure,” but rather a question of “explaining why.”

Using proof as a means of explaining why something is true is the most meaningful role proof can play in a high school geometry course. Asking why something is true, after performing investigations that have convinced students that it is true, is a powerful 1-2 punch. Explaining why can be an effective tool regardless of a student’s van Hiele level. This is the approach we take with Discovering Geometry.

In practically every lesson in the fourth edition of Discovering Geometry (DG4) students are asked to perform geometric investigations and then make their geometry conjectures. After performing their investigation and making their conjecture they are asked, “can you explain why?” For example, after their very first two investigations leading to geometric conjectures, the Linear Pair Conjecture and the Vertical Angles Conjecture, students are asked:

“Developing Proof You used inductive reasoning to discover both the Linear Pair Conjecture and the Vertical Angles Conjecture. Are they related? If you accept the Linear Pair Conjecture as true, can you use deductive reasoning to show that the Vertical Angles Conjecture must be true?”We then ask them to work with their cooperative group members to develop a paragraph proof explaining why the conjecture is true then check their reasoning with ours.

Later after discovering the Triangle Sum Conjecture students are asked:

“Developing Proof The investigation may have convinced you that the Triangle Sum Conjecture is true, but can you explain why it is true for every triangle?”We then direct them to look back at the arrangement of the three angles that they tore off and reassembled forming a line. We ask “how is the resulting line related to the original triangle.” This is their lead-in to creating a paragraph proof explaining why their conjecture fits with what they have already discovered/proved about parallel lines.

The inductive and deductive reasoning in DG4 continues with investigating, conjecturing, and explaining why from Chapter 2 through Chapter 12. It isn’t until the last chapter, when there may be students ready for van Hiele level 4 reasoning, that we introduce geometry as a formal mathematical system.

Again from Mathematician’s Delight by English Mathematician W.W. Sawyer (1911-2008)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.

Once deductive arguments began to sprout up around Ancient Greece, it still took a while for the process to be accepted. From the first deductive arguments by Thales to Euclid’s Elements took over 300 years. We owe it to our students to give them time to move up through the van Hiele levels so that they come to understand the role proof plays in mathematics.

I’ll post these comments on my blog as a pdf that you may download.

Another great resource for proof in geometry is the supplement Tracing Proof in Discovering Geometry. My publisher Kendall Hunt has promised to place excerpts from it on their website .

Thanks for the opportunity to get these thoughts off my chest.

## 38 Comments

## Grant Wiggins

August 1, 2013 - 11:24 am -I have a longstanding interest in this issue and wrote part of my dissertation on an alternative view of geometry teaching that would focus on the ideal sequence for ‘understanding’ regardless of the logical order of Euclid’s proofs.

First of all, there is a long history to the critique of doing it deductively. Descartes blamed the Greeks for hiding their methods of proof hehind fancy deductions. (See Rules for the Direction of the Mind). Hegel’s critique in the Phenomenology is the most insightful and withering; it had great influence on Dewey’s views about reflective thinking. Hegel noted that there is no inherent obvious rational strand to proofs and their order. If you didn’t know which little ‘trick’ to do to add a line here or extend the figure there you would not have a clue as to what would happen next. He suggested it was akin to a card or magic trick.

Then there is arguably the greatest inductive geometry course of them all: Harold Fawcett’s The Nature of Proof in the 1930’s. What was especially great about that was the transfer goal: use your newfound reasoning and problem-solving powers and apply them to issues of propaganda and political rhetoric.

My own great epiphany was to learn about non-Euclidean geometries in college. I think it is unconscionable that most people do not know about this move, what mathematician Morris Kline called as important an idea in human thought as Darwin’s theory of evolution. I often taught mini-lessons on it when I was a HS teacher, and most kids were fascinated. It leads to deep conversation about what we should assume as given, a vital conversation with great transfer value. Arguably the most absurd aspect of deductively-taught geometry is that you have to take all the axioms on faith and close off discussion when the questions fly.

Try this experiment: half way into a conventionally taught deductive geometry course, ask kids this question: Why are some statements axioms and others theorems? What is the difference between an axiom and a theorem? You will find that most kids haven’t a clue and will often conflate the two or get the logic all wrong.

Good issue; worthy discussion. PS: Morris Kline’s chapter in Mathematics and Western Culture is a great read; I used it with my students for years.

## pete capewell

August 1, 2013 - 2:20 pm -The inductive-experimental method is great for generating hypotheses, but the deductive approach allows us to prove these hunches are universally true. Both are needed in a great mathematician… or mathematical partnership:

I’ve recently enjoyed the fictionalised account of the relationship between Ramanujan and G.H. Hardy in David Leavitt’s ‘The Indian Clerk.’ I daresay it takes some liberties with the historical truth, but it illustrates the two-sided coin of hypothesis and proof rather nicely, and there was enough narrative drive to keep the pages turning. A holiday read perhaps?

In the context of your debate, Ramanujan was an inductive genius with an ability to draw on a vast and intimate familiarity with patterns in number to generate inductive rules and formulae (mostly in number theory rather than geometry) whilst Hardy brought the rigour of proof to the party. Now, before you flame me, I know this is a caricature of what they did, but it’s a useful distinction in the generative versus the skeptical parts of the mathematical mind.

Now you might reasonably argue that the truly revolutionary geniuses such as Newton, Euler or Gauss had both, and these are the guys who give us the Kuhnian paradigm shifts in mathematical thought, but Andrew Wiles has made a name for himself proving a well established hypothesis of someone-else’s making (Fermat’s Last) and I’d sell my grandmother to have his place in mathematical history, so it seems to me that proof is king to inductive hypothesis’s queen in the chess game of math: the queen has reach, but it’s the king that matters in the end-game.

## Mike C.

August 1, 2013 - 3:46 pm -Some of the contributors to the discussion have raised the issue of what mathematics really is or what mathematicians do. But realize that almost every kid in America is forced to take geometry and an insignificant number of them are going to be mathematicians. So…who really cares what mathematicians do? They can learn that in college.

What will keep kids engaged? That is the question. Or perhaps what can we teach that will enrich the lives of a large number of kids?

My view is that we need options. Why not have a class on that focuses on induction and one that focuses on deduction? Make them both half year classes and make them optional.

## Brian

August 1, 2013 - 6:37 pm -High school geometry (as I took it, but textbooks now seem less clear) is where you focus on deductive reasoning and logic and its power and limits. You take all the real world information you have and hunches from induction, and then use pure logic and deduction to see if it really holds. Or to extend what you started with in ways that would not be intuitive if you left it in a real-world representation, limited only to what repetition lets you see through induction. Those translations and matching things up to the real world (by just adopting arbitrary axioms, for instance) are important, but ideally something that happens in other math classes and science, too. The purely logical steps, represented by the two-column proof, are supposed to be your extended practice in that skill, an experience provided in high school mainly through geometry class. Now, should that skill be diffused over more of the math curriculum? Probably. But should we be uncomfortable that it’s often abstract puzzles? Probably not. Should teachers apply strict standards to what an acceptably precise proof is? Probably. Should that obscure the value of practice in deductive thinking to the point where students who are practicing it come to think that they are no good at it just because they skip an occasional step or write the wrong buzzword in the second column? Probably not. It’s one thing to revere the pristine logic of a correct proof, but very misleading to suggest that producing that on the first try every time says anything about a student’s ability to think deductively, logically, inductively, creatively, clearly, etc…

## Franklin Mason

August 2, 2013 - 7:59 am -I’ll respond to Jason’s comment.

Insofar as I understand what you say, I agree. I most certainly do not teach my geometry classes as I suspect they’re typically taught (and as you seem to have in mind). We don’t do two-column proof. (I shudder to even write those words. I suspect that Lockhart is right about it: it’s sole purpose is to so regiment proof that it can be quickly and easily graded.) Instead we write them out, sentence by sentence, paragraph by paragraph. The emphasis is always on clarity and completeness. I don’t care a whit for the way in which a result is proven; certainly I don’t insist upon a certain method or (even worse) a certain sequence of propositions. Indeed discovery of a new method, especially when it’s more intuitive that what we had before, is given very high praise. This does make it hard to grade student work, especially in those cases where a student attempts an approach that’s new to me. It takes time and real thought, and that means that I can grade only a selection of student work. But I see no way around that.

This ability to cogently map out the logical relations between a set of concepts – that’s the real importance of my course. I do know, of course, that very few if any of my students will become mathematicians. (Now I reply to Mike C.) But that doesn’t imply that the skills of a mathematician are useless to non-mathematicians. What I really teach is the ability to think well and write clearly. Geometry simply provides the opportunity to practice this. (A really good opportunity, I would say. There are lots of beautiful results.)

As for the non-deductive part of mathematics – gather data and then squeeze it until a good hypothesis pops out and then test that hypothesis however seems best (perhaps seek out a counterexample, perhaps attempt to construct a proof) – that I do admit is a real challenge in the mathematics classroom. Given how much I’m required to cover, I find it difficult, indeed sometimes impossible, to give much time to it. In this year’s redesign of my course (and it seems to get redesigned every year) I’ll attempt to carve out more time for it. This means, I suspect, that I’ll have to do less. Do less and make time for students to explore and make discoveries. The challenge is to do this and yet still make certain that we prove all of the key results.

## David Scott

August 2, 2013 - 8:41 am -As I try to move my instruction towards more a problem-based-learning approach, I have been wondering about how I would do that well with Geometry, especially since I love teaching proofs (I avoid column-proofs and teach paragraph proofs) and I do believe that students get great satisfaction from understanding a proof or even being able to come up with one on their own. When it comes to learning or discovering geometric proofs, I personally have the desire to be able to figure things out on my own, but I also am proud of myself when I can understand what someone else has proven. For this reason, I required that my students memorize a particular proof of the Pythagorean Theorem this year. Once they understand the flow of the argument, I think there is a lot of value in their being able to reproduce it.

When I think about teaching a unit on circles to my students, I abhor the endless problems in textbooks where students are forced to apply relationships of chords and intercepted arcs and secants and all the rest to solve a host of problems. Instead of memorizing rules that the students do not understand (and that the teachers may not even be able to prove), I want to spend more time in class looking at the proofs of relationships between chords and various angles in a circle.

I have been feeling quite strongly recently that a deductive approach to Geometry is the best way to teach it, since I believe the proofs are the greatest beauty of the subject, and most applications I have seen seem quite contrived.

However, in the spirit of Dan Meyer’s tasks, I have begun to notice math in the world around me and build problems based on these situations.

Two examples:

1) The San Antonio airport has an interesting “suitcase sculpture” –> see link: http://tinyurl.com/kshftp2

When I was flying through San Antonio recently, I realized that there are some very interesting geometry questions related to building this structure.

2) Carpeted stairs – How much carpet is needed? What will the shape of the carpet look like laid out flat – especially if it is a curved staircase, or one that has triangular stairs interspersed.

These types of problems make me think it may be possible to teach many geometry topics inductively, though I know I will still give my students a hearty introduction to paragraph proofs, first principles/axioms and deductive reasoning!

The biggest challenge I see at present as an educator is the amount of time needed to find/create and sequence appropriate problem-based-learning tasks so that I have a curriculum that addresses all the standards.

Geoff Krall has encouraged me with his efforts in this direction.

Thanks again for all the insightful comments on this thread.

David Scott

## James Key

August 2, 2013 - 9:24 am -Great discussion here. I’m getting a lot out of this.

1. Franklin on the Meatball task from Makeover Monday: “No matter how it’s presented (and your presentation is really very good), it’s still just a plug-numbers-into-a-formula kind.” I think your criticism stems from the fact that you have a mind-blowing sphere-task that you want your students to appreciate, and by comparison, the Meatball task falls short (for you). I think what Dan is trying to do here is to take *an existing task* (from a published textbook) and show how we, as professional teachers of mathematics, can “put our stamp on it” in a way that better serves our students. So you guys are kind of approaching things from two different angles. The value of the made-over Meatball task is that, like other 3-Act Tasks, students get to practice lots of modeling-type skills. Ask questions, generate guesses, ask for appropriate info, perform calculation, ask if the answer makes sense, extend the question, etc etc. I’m sure Franklin’s logic-based sphere-task is mind-blowing, but it’s kind of an apples-to-oranges comparison.

2. Franklin on the importance of deductive reasoning: “But what if we wanted students to know more than that as a matter of fact opposite angles and sides are congruent? What if we wanted them to know why it’s true?” I think you make a compelling argument here, and I enjoyed reading your brief description of the value of deductive reasoning. Dan seems to be railing against the “teacher writes parallelogram fact on board and assigns/teaches proof” method of instruction, and to be asking math teachers to consider how to get students to see that fact for themselves, in order to stimulate the intellectual need to prove it. He is not suggesting that we *not* prove it. My own view is that we should be doing both things — put the contruction paper and the scissors and the rulers and the compasses and the geometry software into kids’ hands, and let them play. Let them “discover” all kinds of things about shapes. Then explore the logical connections between ideas that Franklin writes about. Win-win. Good quote summarizing Dan’s reply: “Thus the inductive can lead to and provoke the need for the deductive.”

3. Franklin on what drives learners of geometry: “The motive to find the proof of some geometrical theorem … is internal to the problem. Students…like good puzzles.” I think you and Dan are 100% on the same page here. Dan’s core philosophy is all about perplexity. My only comment here is that there are really *two levels* to the puzzle, and again, I think we should do both. Level 1) Hey guys — make a parallelogram. Now play around for a while and get back to me when you’ve learned a few things. Level 2) Okay, great job. So you all noticed that the opposite sides are congruent, whereas trapezoids (for instance) are not like that. Anyone care to explain that? Like what is it about the nature of a parallelogram that makes is have that special congruency? Let’s spend some time exploring the *reasoning* together. Some teachers do a great job with Level 1; others are outstanding at Level 2. A well-rounded professional should work to become great at both — and you guys can quote me on that. *wink*

4. Franklin on applied math: “That’s why I’m also a little suspicious when someone continually demands context or application.” I don’t think Dan’s purpose here was like, “Okay, let’s teach students about spheres and cylinders. Anyone have any concrete student-relatable ideas we can work with? Oh hey, I know — meatballs!” It’s more, “Okay, so this (crappy) task is out there, how can we make it better?” In other words, it was already an applied problem, and we just tried to improve it so that it meets our instructional goals better.

5. Dan on intellectual need: “I’m just speculating so set me straight. But what my anecdote lacks is any intellectual need from the student for that proof. They’re doing the proof because the teacher suggested it.” Dan, you have taught me so much, in so many ways, on so many levels. But here’s a challenge for you: help me teach proofs better. You blog a lot about 3-Act math and so forth, and all that stuff is awesome. I would love to see “the Dan Meyer treatment” of, say, the proof of the Alternate Interior Angles Theorem. Because I would contend that teaching proofs to students is both really, really hard, and also really, really important. I agree with Franklin that almost all students can, and should, attain a degree of mastery at writing proofs. I would love to see you start a conversation on your blog about how to teach deduction well. [Maybe you think that math teachers are already doing this well, or maybe it’s just not where your interest lies, but I will observe that it does not appear to get much attention in your writing.]

6. Jason on 2-column proofs: “two-column proof … is an alien language compared to most real mathematical work.” This was in response to Franklin having quoted Lockhart’s Lament, in which Lockhart gives a withering critique of formal proofs, as construed by (I would guess) many/most math teachers in the traditional classroom. Lockhart prizes the intuitive and denounces the formal. Can the formal approach be rescued? I believe it can, and I would love to engage in a conversation about how, and to hear what everyone else thinks. Please! I’ll tip my hand just a little here: I think Dan’s premise applies in this area as well, that *students need to approach abstraction in degrees, and to do it from the bottom up.* So “prove that vertical angles are congruent in general* is a horrible first task. Instead, give students one angle, say 50 degrees, and ask them find the opposite angle. It’s a puzzle! And they can solve it. Then ask 6 students to go to the board, make up their own angle, and figure out the opposite angles. *Only then* ask students to generalize the results they are getting, and to attempt to prove that result using variables. If we approach instruction this way, then doing a proof should feel *really natural* — we’re just doing *exactly the same reasoning we did with specific numbers,* but we’re doing it with variables/generalized numbers.

## James Key

August 2, 2013 - 9:25 am -So I ended my previous comment with this claim: “Doing a proof should feel *really natural* — we’re just doing *exactly the same reasoning we did with specific numbers,* but we’re doing it with variables/generalized numbers.”

Here’s a task for the readers of this blog. Warning: I give my own response to the task below, so do me a favor and wheel out your own paper so you can get your own response down before you read mine.

#1. Draw a pair of vertical angles, label one 70 degrees, and deduce the opposite angle.

#2. Now write a proof using “x and y” instead of 70 degrees.

#3. Now analyze the two approaches — what are the similarities? what are the differences? Pretend you are sitting in that desk in row 2 of your geometry class. What aspects of your proof in #2 would feel strange to the student who has just completed task #1 (perhaps using several angles)?

I think the ideal proof gets as close as possible to capturing the thinking in #1, but – for me at least – there are significant differences in the two approaches, and these may contribute to students’ difficulties. Anyway, here’s my contribution. I look forward to seeing yours!

Case 1: 70 degrees.

1. The adjacent angle is 110, cuz it’s on a straight line. So 180 – 70 is 110.

2. The angle adjacent to that angle (i.e. the *other* adjacent angle) is therefore 70 degrees, cuz they are *also* on a straight line (albeit a different line). So 180 – 110 = 70.

3. So my answer is 70, and I observe that it is congruent to the angle I started with.

4. Conclusion: *this pair* of vertical angles is congruent.

Case 2: x degrees. [Note the differences here, if you please.]

0) Let’s label the angles x, y, and w, with x and y adjacent, y and w adjacent, so that x and w are opposite each other.

1) x and y are a linear pair of angles, and so x + y = 180.

2) y and w are also a linear pair of angles, so we also have y + w = 180. 3) Since x+y and y+w are both equal to 180, they must equal each other: x+y = y+w.

3) The y-term is common to both expressions, so we cancel it out using subtraction, leaving x = w.

4. So *every pair* of vertical angles is congruent.

[Note: does anyone talk about this last inference? We called it “universal generalization” in my college logic class. I never mention it in the classroom, but it goes like this: since we chose x and w to represent *arbitrary values,* and then proved that x = w, it follows that *all pairs* of vertical angles are congruent.]

The “Case 2” proof above is what I would normally do in the classroom, but does anyone want to convince me that a third proof is better? Compare:

Case 2B: a revision on the above proof.

0) Let’s have x, y, and w as before. But now let’s try to organize our “generalized thinking” in exactly the same way as our “specific number thinking.”

1) Since x and y are a linear pair, we can figure out y by doing 180 – x. So y = 180 – x.

2) Now that we have an expression for y, we can figure out w (which makes a line with y) by doing 180 – y. So w = 180 – y.

3) But since y = 180 – w, we get that w is equal to 180 – y = 180 – (180 – x). Conclusion: w = 180 – (180 – x). Now I know the relationship between x and x! I just wish I understood it better. Hmmm…

4) Hold on — I know that subtraction distributes, so I can write 180 – (180 – x) as 180 – 180 – (-x), or 180 – 180 + x. But that’s 0 + x, which is just x! So w = x!

Analysis: the proof for Case 1 (70 degrees) involved subtraction. We used actual numbers, so everything was nice and easy. The proof for Case 2 (x degrees) involved the Transitive Property — two expressions that both equal 180. That did not appear to come up in the first proof. That is a major difference, or at least I think students would perceive it as such. The remix of the proof for Case 2 involved an algebraic observation, namely that when you *subtract* 180 – x from 180, you get x. I would point out that this is more similar in spirit to the first approach, the one I used for Case 1/70 degrees. Is this important? What do people think?

Question #2: how important is it to organize either proof — Case 2 or Case 2B — into 2-column form? What is gained/lost by writing the proof in full sentences, as opposed to citing reasons one nifty phrase at a time (Linear Pair Postulate, Transitive Property, and so forth)?

## Kevin Martz

August 2, 2013 - 10:31 am -This is fascinating to read the different sides go back and forth. You should share this when you can.

## Isaac D

August 2, 2013 - 11:32 am -@ James Key

Point 2 in Case 2 involves what we call the Transitive Property in logic and classical geometry, but in algebraic terms it is usually taught as “substitution”.

I’m not sure that Case 3 is necessary. I would instead rephrase Case 1 to include the unstated assumptions, in which case it starts to look more like Case 2.

1) The original angle (70°) plus the adjacent angle (either one) measure 180° because they are on a straight line. 70° + y = 180°. To solve this I subtract 70° from both sides, so y = 180° – 70° = 110°.

2) The angle which is adjacent to that one is also on a straight line with y, so y + w = 180°. Since y= 110°, that means 110° + w = 180° and therefore w = 70° which is the same as the original angle.

The only difference between this and Case 2 is that we solved for y halfway through the problem and substituted that value (110°) into the second equation instead of substituting y directly.

The difference between this and your version of Case 1 is that I am starting with 70° + y = 180° instead of jumping to the next step in the solution, y = 180° – 70°

## James Key

August 2, 2013 - 12:03 pm -@Issac D:

Great point. And I especially like the fact that your “teacher move” is something the students will understand readily: “So, you’re saying that y = 110 b/c you did 180 – 70. Can we back up a step here? Let’s put this puppy in reverse. Where did this number ‘180’ come from? Oh, yeah — it’s really 70 + ? = 180, from which ? = 110.”

But hold on a minute — I re-wrote your argument, which was about the case where x = 70, in terms of a general x, and got this:

1) The original angle (x) plus the adjacent angle (y) measure 180° because they are on a straight line. x + y = 180°. To solve this I subtract x from both sides, so y = 180° – x.

2) The angle which is adjacent to that one is also on a straight line with y, so y + w = 180°. Since y = 180 – x, that means (180 – x) + w = 180° and therefore w = x [by adding ((x – 180)) to both sides] which is the same as the original angle.

Isaac says, “The only difference between this and Case 2 is that we solved for y halfway through the problem and substituted that value (110°) into the second equation instead of substituting y directly.”

Based on the above, I’m gonna argue that your work is basically more like my argument 2B than my argument 2A. The main difference, as you point out, is the use of the Transitive Property. That is, the logic of 2A hinges on the inference that x+y = y+w b/c “two things that both equal 180 must be equal to one another.” But this “move” is nowhere to be found in your argument above (as I understood it), and is likely to be perceived as “some strange new thing” by students.

Maybe we should actually go a bit further here, to [DAN ARE YOU READING THIS] provoke the intellectual need for the step in the proof where the Transitive Property got used. Perhaps something like this:

Summary of argument where x = 70:

1. x = 70

2. x+y = 180

3. 70+y = 180

4. y = 180-70 = 110

5. y+w = 180

6. 110+w = 180

7. w = 180-110 = 70

8. x = w

Here’s my classroom dialogue to follow up this hypothetical scenario: “Okay, guys — great work. Now I want you to try something a little weird. Bear with me for a second — in steps 2 and 5, the number 180 appears. Can someone remind me where that number came from? Oh, right — we just assumed in the beginning of this class that angles on a straight line make 180 degrees. Okay, here’s the weird part I promised — I want you to try this problem again, but this time use the postulate that everyone uses on Venus — that the angles on a straight line make TWO HUNDRED degrees. Does that sound strange to you? Well, like I said, bear with me and try it, I’m really curious what you might learn from this…”

1. x = 70

2. x+y = 200

3. 70+y = 200

4. y = 200-70 = 130

5. y+w = 200

6. 130+w = 200

7. w = 200-130 = 70

8. x = w

“Hmm…that’s weird — we still got 70 degrees for w.”

Me: “That’s right! Now let’s try to wrap I minds around this: what just happened?”

1) In both instances, we started with a certain value, 70 degrees.

2) In both instances, the sum of the angles on a line was some certain number, 180 in the first case, 200 in the second.

3) In both instances, we got an intermediate value (110 or 130), which we then used to get the final value.

4) In both instances, the final value was the same.

“Yeah, great! So here’s the really cool part — it didn’t matter that the sum was 180, or 200, or whatever. It just mattered that the sums were the same!!”

So maybe it doesn’t matter whether we figured out that middle value after all.

1. x + y = 180

2. y + w = 180

3. x + y = y + w

4. y = w

Okay, I gave it my best effort, but I still don’t feel I’ve provided a great link between steps 1/2 and step 3. Anyone care to improve on my presentation? (And is this worth all the effort? The Transitive Property is not all that hard to teach…or is it? It’s the kind of thing that is easy to apply once you’ve practiced it a bunch of times. What do you guys think? Worth the time and effort or no?)

## James Key

August 2, 2013 - 12:10 pm -@Dan:

Okay, Dan, this one’s for you: here are some tasks for your next Makeover Monday! I’m dying to see what you and the math teacher community come up with. Feel free to choose a (proper) subset of these tasks: ;-)

1. Prove that vertical angles are congruent.

2. Prove that alternate interior angles are congruent when you start with parallel lines.

3. Prove a triangle has 180 degrees.

4. Prove that an exterior angle in a triangle is equal to the sum of the remote interior angles.

5. Prove that the base angles of an isosceles triangle are congruent.

6. Prove that a parallelogram has two pairs of congruent sides and two pairs of congruent angles.

I could list more theorems, but that’s a good start right there! Seriously, readers — how many of your freshmen/sophomores could prove every single one of those theorems? Or even prove one of them? I’m gonna wager that there are a great many students out there who could prove NONE of these theorems the year after they complete geometry, and that is a great travesty.

P.S. I think the above list of 6 tasks is a good final exam for first semester geometry. :-)

## James Key

August 2, 2013 - 12:35 pm -More thoughts inspired by Isaac D:

I’ve been trying to refine my thinking on that last argument.

So we did it with x+y = 180, and we did it with x+y = 200. In both cases we got w = 70, the same as x.

How about this:

1. x+y = CONSTANT SUM

2. y+w = CONSTANT SUM

Maybe this is the next move — present the pair of equations above to the student, and just see what they come up with. “I’m not telling you what the sum is, just that it’s the same in both cases. Can you still get to x = w? Let me know what you come up with.”

I realize now that the “problem” here is motivating the application of the Transitive Property, which is what this discussion has become about. We are so far away from two lines that criss-cross, it’s not even funny. If students already had “Transitive Property” in their Algebraic Toolkit, then they’d be good to go. But in my experience, proving the Vertical Angles Theorem is the first time they apply that particular property. (I guess in Algebra 1 they might do it in the “linear systems” unit, i.e. y = 2x + 4 and y = 3x – 5, hence 2x + 4 = 3x – 5.)

## Lizzy

August 3, 2013 - 9:15 am -First of all, James Key, if you don’t have a blog, please make one. If you do, please give me the link. I love how you took just one simple problem and used it to explore the nature of proof, the abstractions involved in trying to generalize any relationship, and the transitive property to boot.

I don’t know how pertinent this comment is, but I’ve been thinking about Dan’s blog post for the last two days straight because I guess this issue has been on my mind for a long time. I really agree with Pete Capewell in that induction and deduction are partners and I’ve also been thinking about the cliche “math is a language”. I’m not sure how much I agree with this statement, but thinking back on the 10 years I spend studying Japanese I think at least in terms of the learning process there are a lot of similarities. I had teachers who focused on linguistics and teachers who focused on conversation practice. Some of the teachers who focused on conversation did it through repeated use of drill (say the same sentence 10 times but alternate the adjective used) some focused on creating real-world scenarios in which we had to use our limited Japanese creatively (but usually with incorrect grammar) to get our point across. I learned a lot from each type of teacher. The teacher who focused on linguistics helped me see grammatical patterns and learn to recognize and adapt them for myself. The teacher who focused on drill helped me perfect my accent and also helped me come up with grammatically correct sentences automatically without having to think or stress about it. The teacher who threw us into situations and had us use halting, broken Japanese to convey meaning helped me learn to think on my feet and to not be ashamed when mistakes were made.

I don’t think that all math teachers who like drilling are inherently lazy, or are not creative, or don’t care. I think that some at least must have achieved some kind of success and they see their students moving beyond the drill, interpreting patterns and extending their knowledge. Or they see that kids need practice to make things stick. Or they may see that some kids find repetition comforting. Teachers who focus on presenting neat, clear proofs without giving their students a stimulus for wanting to know the proofs are still showing their students mathematical beauty. I don’t want to go to an art museum to have blank canvases with jars of paint waiting to be used. I want to go to an art museum and appreciate what others have wrought. So a math teacher who is passionate in the presentation can inspire wonder in students even if there’s no “hook”. Conversely though, I don’t want to go into an art class and have the teacher spend the whole time showing off his portfolio and so teachers who allow students to create mathematics for themselves are also serving students’ needs. All these teaching styles can have value. A good teacher needs to be able to experiment with style and use appropriate methods at appropriate times, but I think we also need to teach from who we are. A linguist will teach linguistics. If he tries to teach street Japanese he won’t be successful because he’s not passionate about it. And the poor grammar makes him cringe.

Regardless of the method in which I learned Japanese, I only really learned it when I moved to Japan. Only when I was forced to use it on a daily basis and each new word I learned made the world make more sense did I really pull together all the disparate knowledge I’d learned from teachers over many years.

Dan seems to be arguing for making math class like living in Japan- make each piece of knowledge precious because each new idea makes the world make more sense. This is a beautiful and admirable goal, but I would have been really scared and lost if I’d gone to Japan without the framework that my teachers had given me.

The big difference between math and language though is that linguistics is weaved through every level of language learning. It’s ok if a teacher focuses more on conversation one year. They still do cover grammar. In math, we seem to leave all the formalization of proof for geometry and because it’s all packed into one year with one teacher, the student doesn’t get to see multiple teachers over many years playing with different styles. This is also why presenting proof formally and consistently is both important and also really hard on students because it’s all so new and there’s so much of it and we have so little time to get through it. There’s a lot of pressure on geometry teachers because they know this is their students’ first shot (and usually only shot) at formalizing mathematical reasoning. I like the inductive-> deductive approach, but I think it’s ok if teachers teach formal Euclidian geometry. I found the linguist Japanese teacher’s class just as valuable as the street Japanese teacher’s class.

## James Key

August 3, 2013 - 5:01 pm -@Lizzy:

Thanks! You can find my (woefully underdeveloped) blog here: http://iheartgeo.wordpress.com/

Also, thanks for sharing about your experiences learning Japanese, and how different teaching styles contributed to your learning in different ways. That was good to hear.

More refinements on vertical angles:

It occurred to me that it might be interesting to start with the “middle angle,” since everything else sort of hinges on what happens in relation to that angle.

Let x and y be a linear pair, and y and w be a linear pair. Suppose y = 100. Find x and w, and observe their relationship to one another. [Note how this differs from my original formulation of the task, in which x was given to start with.]

Here are some “high-level observations” about what really goes on as you do this problem/proof:

1. x and y are supplementary.

2. y and w are supplementary.

3. 2 angles which are supplementary to the same angle must be congruent to each other, hence x = w (both are supp to y per #1 and #2).

Fact 3 is established via algebra, but it should be clear intuitively once a student sees that it comes back to this:

1. ? + y = 180

2. y + ? = 180

Clearly there is only room for one value of ?, and that value gets assigned to both x and w. The rest is details.

So my 1-sentence summary of the vertical angles theorem is this: “Any two vertical angles must be congruent, because they are both supplementary to the same angle.”

In keeping with Dan’s theme around the “Ladder of Abstraction,” I feel that this last step is particularly important, but I admit that (for me at least) it happens all too rarely in the classroom. Call it “sense-making” at the highest possible level. In other words — it’s one thing to solve the problem when x = 70. It’s another thing to prove it in general for an arbitrary x. But it’s still another thing to get up above that argument and notice what makes it tick.

Challenge: come up with a 1-sentence summary of the proofs of some theorems from elementary geometry.

One further example — the alternate interior angles theorem. Label the angles 1234 on the first line and 5678 on the second line, so that 3 and 6 are alternate interior angles.

1-sentence summary of argument: “When you have parallel lines, any pair of alternate interior angles (such as 3 and 6) must be congruent b/c they are both congruent to the same angle (namely, angle 2).” The details of the proof will flesh that out — 2 and 3 are a vertical pair and 2 and 6 are a corresponding pair — but that one sentence is the *essence* of the proof.

Dear math teachers, how often do you bring out this essence in your lessons, and how do you go about it? Have you succeeded in *getting your students* to be able to describe the essence of an argument? How did you do that?

## Laura Hawkins

August 3, 2013 - 8:19 pm -At The Urban School of San Francisco, thanks mostly to the amazing Henri Picciotto, we handle it like this: the first semester of our Geometry-ish course is mostly inductive reasoning. They determine the congruent triangle theorems, for example, by using a compass and ruler (virtual or actual), discovering that once we set a specific Angle, another Angle, and a given Side length, they can only make one possible triangle. Thus, AAS is convincing, but not actually proven.

Then in the second semester, we introduce deductive reasoning as we dive into Quadrilaterals. We give them a list of phrases like “opposite sides are congruent” and “opposite sides are parallel” and have them choose two to combine into and “If-Then” statement like “If opposite sides are congruent, then opposite sides are parallel.” Then in their groups they sort their individual lists into Definitely True, Probably True, Probably False, and Definitely False. Then they send them to the teacher and we make a big list of the four categories. Invariably, even in a small class of 4 groups of four, the same sentence will appear on both Definitely True and Defintely False as different groups place it differently. Ta-da! A need to figure out who is right! Then we spend time sorting through the list proving (with a couple different methods) or disproving (by counter-example) the different statements. Occasionally we encounter something they know to be true from middle school (where they learned properties of quadrilaterals) that they want to use to prove another property, so we add that to our list and prove it as well. The teacher also suggests useful and/or challenging properties to add to the list as necessary.

This way, we teach them that they can figure out cool things themselves, (I will never forget a 9th grader asking me “Did you PLAN for us to discover this today!?” after deriving the Inscribed Angle Theorem in the first semester). And we also teach them the importance of deductive reasoning and how to use it as we.

## Michael Serra

August 4, 2013 - 8:55 pm -Great discussions. The topic of inductive and deductive reasoning obviously is very dear to me. First, I accept Jason’s apology even though “meh” was a dagger to the heart.

There have been a number of commentator saying things about Discovering Geometry (DG) that are simply not true or very misleading. They appear to be speaking from no direct involvement with the book but from how others have characterized the discovery approach.

I take strong exception to people characterizing DG as the book that is inductive with little or no proof or proof that only appears in the last chapter or that proof is seen as an afterthought. Inductive and deductive reasoning appear throughout the text.

So permit me to make my case.

Whenever there is discussion about proof in geometry we should begin with the van Hiele model of geometric reasoning. Research by numerous mathematics educators including Usiskin 1982; Senk 1985; Burger and Shaughnessy 1986; Geddes and Fry 1988; Clements and Battista 1992; and more recently Battista 2007, support the accuracy of the van Hiele model. The model consists of five levels of geometric reasoning that students pass through from visualization to rigor.

Level 0: Visualization –students can identify rectangles by sight but squares are squares and are not seen as special rectangles.

Level 1: Analysis (descriptive) – students can identify properties of rectangles (by drawing, measuring, and making models) but cannot yet derive other properties from given.

Level 2: Informal Deduction – students can give an informal argument to justify that the figure is a rectangle from given properties.

Level 3: Formal Deduction – students are capable of creating original logical arguments.

Level 4: Rigor – students are capable of reasoning about mathematical structures (i.e. Euclidean -vs.- non-Euclidean geometries).

The van Hiele model asserts students cannot move to the next higher level until they have successful mastered the previous level.

But the big item is that research shows that 70% of high school students enter geometry operating at level 0 or 1 on the van Hiele measure of geometric reasoning (Shaughnessy and Burger 1985; Senk 1989).

Yet “traditional” geometry textbooks that begin with establishing postulates and proving theorems are ignoring the research and expecting students to immediately begin their geometry experience at levels 3-4. When the teacher and the textbook are presenting geometry at van Hiele level 3 or higher, while the students are functioning at level 0 or 1, it should be no surprise that there is such a high failure rate in traditional geometry courses.

More to come.

## Michael Serra

August 4, 2013 - 9:03 pm -One of my favorite quotes that I refer to often comes from Mathematician’s Delight by English Mathematician W.W. Sawyer (1911-2008)

“It would, I suppose, be quite possible to teach a deaf and dumb child to play the piano. When it played a wrong note, it would see the frown of its teacher, and try again. But it would obviously have no idea of what it was doing, or why anyone should devote hours to such an extraordinary exercise. It would have learnt an imitation of music. And it would fear the piano exactly as most students fear what is suppose to be mathematics.”

Some geometry teachers claim that they can successfully teach all of their geometry students how to create geometry proofs. Perhaps they only have those 30% that are entering geometry and functioning above van Hiele levels 0 and 1. I suspect however, they may also be teaching an “imitation geometry.” Their students are trained to go through the motions of stating the theorem about to be proved (5 points), then stating the given (5 points), then stating the show (5 points), then drawing the diagram to the right (5 points), then drawing the big T (5 points), then putting the given information in the first few lines of the T-proof (5-15 points), then writing given to the right of each statement (5-15 points), then throwing in enough statements to garner enough points to get credit for the exercise without having any clue as to what he or she was doing or why anyone should devote hours to such an extraordinary exercise. They are doing imitation geometry.

Discovering Geometry, in its first edition, was an innovator in addressing students’ needs for gradual development of the deductive process. Discovering Geometry is the only high school geometry textbook on the market that is aligned with the van Hiele model and other research on geometric proof. We accept that the vast majority of student are entering geometry at very low van Hiele levels of geometric reasoning and our goal, with careful deliberate scaffolding, is to move them to higher and higher levels of geometric reasoning as they progress through the course.

Geometry student’s consistent difficulties with understanding proofs (an international problem BTW) should not be solely attributed to their inability to reason but perhaps our inability to recognize that there are many purposes for doing proofs and we have been stressing the wrong purposes for proof at inappropriate times. Professor Michael de Villiers’ research on the role and function of proof identifies six basic roles for proof:

• Verification –to remove doubt, to convince someone of the truth of a statement

• Systematization –organize known results into a deductive system of postulates, definitions, and theorems

• Explanation –insight into why something is true

• Discovery –proof can occasionally lead to new unexpected results

• Communication –proof can create a forum for critical debate

• Intellectual Challenge –proof can be a testing ground for intellectual stamina and ingenuity

The function of proof in a high school geometry course has been mostly two-fold: to remove doubt, to convince someone of the truth of a statement –verification and to establish geometry as a mathematical system –systematization.

Many high school mathematics teachers seem to hold this naive view that the main function of proof is to provide verification that a given statement is true. The role of proof as a means of verification is a useful method of verifying the “truth” of a proposition within a mathematical system, especially when coming across surprising (non-intuitive) results. However this view does not stand up to actual mathematician’s experiences. Professor George Polya wrote, “When you have satisfied yourself that the theorem is true, you start proving it.” If the sole or primary purpose for doing proofs in a high school geometry course is to provide verification of the truth of a statement then students functioning below level 4 on the van Hiele scale will continue to question, or worse, disregard the process of proof.

If systematization is emphasized as a primary function of proof right from the start of a geometry course the same poor results will persist. The van Hiele model tells us that systematization requires the highest van Hiele level of geometric reasoning. Geometry textbooks that begin their geometry program with lists of definitions and postulates, and then begin doing two-column proofs are ignoring the research. Only an honors course or any class in which all students are finally operating at van Hiele level 4 would have any chance of success in a course that looks at geometry from the perspective of a mathematical system. Until some magic happens and all students beginning geometry enter the course functioning at van Hiele level 3 any attempt to create a mathematical system for a regular or informal geometry course is likely to continue to have major problems.

Students can acquire a very high degree of confidence in a conjecture arrived at by inductive methods but these methods may not explain why the conjecture is true. Here is where proof can come to the rescue. An example would be the inductive discovery that the sum of the measures of the three angles of a triangle is always 180°. A good inductive first approach would be to ask students to measure the three angles of a number of triangles thus gaining reasonable confidence that the sum is indeed 180°. The same can be done with dynamic geometry software. Either investigative approach is a good first step because it is the first approach that students would take. These inquiry approaches do give students confidence in their conjecture however they gives no insight as to why the sum is always 180°. The investigation should be followed by a second investigation where they cut out the triangle and then tear off two of the angles and arrange them on both sides of the third angle to create a straight line. From this arrangement they can see that the three angles create a line parallel to the third side. This visually explains what properties this conjecture is dependent upon and why the conjecture is true. This can also be done quickly with a patty paper investigation.

More to come.

## Michael Serra

August 4, 2013 - 10:10 pm -From The Role and Function of Proof in Mathematics by Michael de Villiers:

It is not a question of “making sure,” but rather a question of “explaining why.”

Using proof as a means of explaining why something is true is the most meaningful role proof can play in a high school geometry course. Asking why something is true, after performing investigations that have convinced students that it is true, is a powerful 1-2 punch. Explaining why can be an effective tool regardless of a student’s van Hiele level. This is the approach we take with Discovering Geometry.

In practically every lesson in the fourth edition of Discovering Geometry (DG4) students are asked to perform geometric investigations and then make their geometry conjectures. After performing their investigation and making their conjecture they are asked, “can you explain why?” For example, after their very first two investigations leading to geometric conjectures, the Linear Pair Conjecture and the Vertical Angles Conjecture, students are asked:

“Developing Proof You used inductive reasoning to discover both the Linear Pair Conjecture and the Vertical Angles Conjecture. Are they related? If you accept the Linear Pair Conjecture as true, can you use deductive reasoning to show that the Vertical Angles Conjecture must be true?”

We then ask them to work with their cooperative group members to develop a paragraph proof explaining why the conjecture is true then check their reasoning with ours.

Later after discovering the Triangle Sum Conjecture students are asked:

“Developing Proof The investigation may have convinced you that the Triangle Sum Conjecture is true, but can you explain why it is true for every triangle?”

We then direct them to look back at the arrangement of the three angles that they tore off and reassembled forming a line. We ask “how is the resulting line related to the original triangle.” This is their lead-in to creating a paragraph proof explaining why their conjecture fits with what they have already discovered/proved about parallel lines.

The inductive and deductive reasoning in DG4 continues with investigating, conjecturing, and explaining why from Chapter 2 through Chapter 12. It isn’t until the last chapter, when there may be students ready for van Hiele level 4 reasoning, that we introduce geometry as a formal mathematical system.

Again from Mathematician’s Delight by English Mathematician W.W. Sawyer (1911-2008)

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.

Once deductive arguments began to sprout up around Ancient Greece, it still took a while for the process to be accepted. From the first deductive arguments by Thales to Euclid’s Elements took over 300 years. We owe it to our students to give them time to move up through the van Hiele levels so that they come to understand the role proof plays in mathematics.

I’ll post these comments on my blog as a pdf that you may download.

Another great resource for proof in geometry is the supplement Tracing Proof in Discovering Geometry. My publisher Kendall Hunt has promised to place excerpts from it on their website .

Thanks for the opportunity to get these thoughts off my chest.

Regards,

Michael

## Jason Dyer

August 5, 2013 - 4:53 am -@Michael: If you want a more detailed review than “meh” (I know, call me Ebert) I could try tracking down a copy (it’s been a couple years) and do some page-by-page comments; I don’t have one accessible right now.

I know part of the issue was with the practical exigencies of the school year making “the last chapter” a blur of distraction (prep for the state test, prom, a general student desire to get out the door). I favor a model that at the least samples level 4 earlier in the year, because it feels too new and different from the prior levels for them to get an mastery in the time they have.

## Franklin Mason

August 5, 2013 - 6:11 am -I have three comments for Mr. Serra, two specific, one general:

1. Perhaps we agree about two-column proof. It gives rise to the illusion that proof construction is mechanical, and that all correct proofs must take the same form. I find two-column proof profoundly unnatural. It’s better to simply write out your proofs (mostly I call them ‘explanations’) in simple, unadorned language supplemented by whatever symbolism seems appropriate. The goal is not to adhere to some external, teacher-imposed standard of what constitutes a good proof. Rather it’s to satisfy the desire to understand the why of a thing. If a proof does that, it’s good.

2. You say that it’s often inappropriate to pursue systematization from the start in a geometry classroom. Perhaps we mean different things by ‘systematic’. I mean the use of a prior result or postulate (or both) to explain why some new proposition is true. To systematize is to explain. Without it, new facts float free, ungrounded and so unexplained. Thus I think it exactly right to pursue systematization from the start in a high school classroom. I suspect that when a teacher has little success with it, it’s because she begins with the formal apparatus of a typical geometry class. All that – the T, the Given and Conclusion, the postulate and theorem names, etc. – erects a barrier than some students cannot clear. Moreover (and more importantly) it tends to obscure what real mathematics is – the search for explanations that compel.

I do think it important to spend quite a bit of time with the postulates. My experience is that students find them quite intuitive – there is only one line through a pair of points, there is only one parallel to a given line through a given point not on it, etc. Thus when we can trace a proposition back to those postulates, their curiosity is satisfied. Students must be led to recognize that postulates are not some set of arbitrary rules imposed by the teacher but are instead just clear, clean statements of what they already believe.

3. I take issue with your example of how to explain that the sum of the angles of a triangle is 180 degrees. You say:

“The investigation should be followed by a second investigation where they cut out the triangle and then tear off two of the angles and arrange them on both sides of the third angle to create a straight line. From this arrangement they can see that the three angles create a line parallel to the third side. This visually explains what properties this conjecture is dependent upon and why the conjecture is true. This can also be done quickly with a patty paper investigation.”

No one can ‘see’ – not with the eyes anyway – that a straight line is created. It will look straightish if students have been careful. But from the appearance nothing can be inferred. For all students know, the sum of the three angles is a bit more or a bit less than 180 degrees. Moreover, to say that a straight line is created is to assume that the sum of three angles is 180 degrees. Thus if we argue in this way – cut out two angles, place them around the third, a line is formed, thus the angles sum to 180 degrees – our argument is circular.

The only way to really explain why the sum of the angles of a triangle is 180 degrees is to invoke the Parallel Postulate (the second of the postulate examples given above). Through that third vertex, there’s a line parallel to the opposite side. Draw it in. Students will already understand that when lines are parallel, alternate interior angles are congruent. This leads almost immediately to the desired result.

To reiterate a point made above: if all of this is presented simply and naturally – as a set of spoken or written sentences – with no unnecessary formal apparatus, almost all students see how it works immediately.

## James Key

August 5, 2013 - 7:40 am -@Michael and Franklin, regarding the proof that a triangle has 180 degrees:

I find that students in a high school geometry course are generally familiar with the “180 degrees property” as a fact, although they will not be able to give the reason for it. My own approach is this:

1. Draw a triangle and specify two of the angles. Let’s say they are 60 and 70 degrees. Ask students to find the third angle — they will get 50 degrees. Ask how they knew that. They will say “because a triangle has 180 degrees.” When asked why that is the case, they won’t furnish an explanation. So my next question is this:

2. “Okay, so you guys got 50 degrees for that third angle, and you said that’s because a triangle has 180 degrees. On a scale of 1 to 10, how sure are you that you’re correct about that?” High degrees of certainty are expressed. “Okay, great. Now in order to understand *why* that is the case, let’s spend some time thinking together. Can you think of another situation involving angles that add up to 180?” Someone will say “angles on a straight line.”

3. So now I’m like, “Great. So when we say the angles in a triangle make 180 degrees, we’re really saying we can arrange them along a straight line. Let’s explore that.” Then I do the activity that Michael mentions — tear off the corners and arrange them on a line.

Franklin counters that this does not provide a *logical basis* for the theorem, since we don’t know *for certain* that the angles are in fact collinear. That is a valid point, but the good news is that the two approaches combine beautifully: once you do Michael’s tear-off-the-corners activity, you can do Franklin’s draw-a-parallel deductive proof. Why do they work together so well? Dan, this part’s for you:

*Because tearing off the corners gives us a reason (read: provides intellectual need) for drawing that line that is parallel to the base.*

In other words, imagine the class where the teacher just said, “Okay, class, go ahead and draw a triangle. Now pick one of the sides, and draw a line parallel to the side that goes through the opposite vertex.” Once you draw that line, the game is *already over.* The rest is just making some inferences about alternate interior angles. I think that, from a mathematician’s standpoint, the really crucial thing was *drawing that particular line to begin with.* So the nice thing about Michael’s approach is that it gives us a reason to *want* to draw that line.

In George Polya’s book “How to Solve It,” he analyzes the problem of calculating the length of the diagonal of a 3D box. He argues that it is pointless to draw in two right triangles, because in so doing *you have essentially already solved the problem for the students.* Recognizing that this problem can be solved by the Pythagorean theorem, recognizing that this problem is merely an extension of the 2D box problem (find the diagonal a of a rectangle, knowing its sides) — this is the “puzzle” part of the problem, the part worth figuring out for oneself. Applying the PT twice to get the answer? That is merely an exercise.

To summarize: drawing a line parallel to the base of the triangle? That is the part worth figuring out for oneself. Doing the rest of the proof is a mere exercise — although it is clearly worth doing.

## Todd Curtis

August 5, 2013 - 8:45 am -Great discussion.

Instead of tearing up or de-constructing a triangle to show students why it has 180 degrees, and perhaps or perhaps not getting something like a straight line, why not construct the triangle *from* a straight line?

Easy enough with a long strip of heavy-duty paper laid flat (and obligatory discussion about how many degrees are there), then folded at two places (vertices 1 and 2), with the ends rising to meet each other to create the third vertex. Lay on its edge and measure the angles if you’d like, though the lack of perfectly stiff paper will make that tricky.

## Michael Serra

August 5, 2013 - 10:05 am -@Jason & Franklin

Jason it seems more like you have an issue with planning your school term than an issue with DG4. I contend that if you have a regular geometry class (one like most of us where 70% of them are entering geometry at vH level 0 or 1) then it is not likely that you will get very many of them to vH level 4 in order for them to appreciate/understand the distinctions between postulates, definitions, and theorems, i.e. establishing a mathematical system, before the end of the term. But that is OK. For the vast majority of students taking geometry using proof as a means of explaining why something is true is extremely important and valuable in itself and all students should experience that. There are also a lot of other very valuable opportunities left in geometry. I use to plan out my entire semester on a calendar that I gave to my students the first week. I would actually schedule “catchup” days because I knew I’d need them, just not know when there would be massive flu outbreaks or when the school would schedule field trips. I also saw myself as a conductor of a geometry symphony and so when making out the semester/year schedule I’d pick a couple different geometry melodies to weave through the term such as problem solving, applications, reasoning, or art and culture connections.

Franklin, I think we are pretty much in agreement on the role of proof –that of “explaining why” rather than establishing “truth” or as a means to teaching within a mathematical system. I also think we are pretty much in agreement on proofing techniques. The two-column proof is a device whose time has passed. I’m sure it was created to make proof grading easier for the teacher without regard to good pedagogy.

I’d like to elaborate. I prefer to move from oral arguments early on (explaining to each other in their cooperative groups) then move to taking those verbal explanations and turning them into paragraph proofs. When the “explaining why” gets to be too complex we turn the paragraph proofs into flowchart proofs. This approach I feel is the most successful because my students (inner city kids) had severe limitations in their writing skills. A flowchart proof I found the most successful because it is very visual (the primary learning mode of most students). A cooperative group of four students working on a large 3′ by 4′ sheet of white board talking, planning and explaining to each other the necessary details of a proof is a sight to behold.

The discovery and “explaining why” that goes on throughout the DG course is not random and disjointed but follows a very logical sequence. In fact it sort of follows the history of the development of deductive reasoning.

Franklin, I was not saying that lining up the two angles adjacent to the third angle of the triangle constituted a proof that the “line” created was a line or was a parallel line. I’m saying that rather than telling them the method of the proof, which is not going to come to them naturally, this image hopefully leads them to the idea of a line through the vertex parallel to the third side. The more we create opportunities for the student to feel they came up with the idea the more ownership they take. This is where good teaching comes in (“be less helpful”).

Let me also add, that this venue is very exciting. My communication with fellow teachers in my day was limited to teachers at my school, and at local and state math conferences. I am really happy to see this development and I confess I am quite envious of all of you working in the midst of this revolution.

Regards michael

## Michael Serra

August 5, 2013 - 3:21 pm -You can now download the pdf of my comments about proof at my website under Articles and Handouts.

Here is a direct link to the intro to Tracing Proof in Discovering Geometry on Kendall Hunt’s website:

Tracing Proof Introduction

## Kevin Hall

August 5, 2013 - 3:54 pm -Michael, I don’t think you pasted the link correctly. At least, nothing happens when I try to click on it.

## Jason Dyer

August 6, 2013 - 4:41 am -@Thomas, Michael:

I found a copy of the textbook lurking about, so I’ll be able to dig through specifics soon.

It sounds like we don’t disagree that much. We all dislike two-column proof.

Franklin starts with deductive proofs and would like to incorporate more induction.

Michael starts with inductive proofs and puts deductive proofs in the last chapter of his book.

I would like to mix both types of proof in relatively equal proportions. My issue isn’t just with arranging the year in a timely fashion; if one goes by the book there is only a limited amount of things to be done in the deductive chapter, and if I had a quarter or a similar chunk of time where I could do actual justice to deduction I would need to supplement so much I might as well be using a different textbook.

I have some issues with van Hiele that take a long time to unpack (if I have time maybe I’ll try later, but I am setting up for students coming to my room starting tomorrow) but one of the relevant ones is: deduction is not just a level of abstraction but a process. I feel saving deduction for the end is analogous to spending all school year working on reading exercises but not writing anything until the end. (There are even strong links between building an argument in an essay and building a mathematical deductive structure.)

## Kevin Hall

August 6, 2013 - 5:51 am -Michael’s book uses flow chart proofs much earlier than the last chapter.

## Ryan Brown

August 6, 2013 - 7:27 pm -Okay, I feel like I’ve gotta chime in here.

I’ve been teaching from the Discovering Geometry curriculum for the past 7 years. It is hands down the best curriculum I have taught from. From a pedagogical standpoint, the progression of this course is brilliant. As Mr. Serra pointed out in previous comments, the teacher must understand where his or her students fall in the van Hiele levels of geometric understanding. Most high school students are not able to understand the role of deductive proof as a systematization of geometry. Many students do not appreciate or see the need for formal deductive proof at all. Many will be convinced, for example, that triangles have an angle sum of 180 degrees based on 2-3 measured examples or the fact that their 6th grade math teacher told them so. Prove it? “Why do I need to?” they will ask. They most certainly are not ready for formal deductive proof at the beginning of a Geometry course.

That being said, I have to say that deductive proof is woven throughout the whole of Discovering Geometry. Serra’s “Developing Proof” exercises are an integral part of the process. Are they formal proofs? Of course not. But they lay the groundwork for explanation in a deductive way that is less intimidating to the average geometry student. Students working though DG are experiencing informal deductive proofs as early as Chapter 2, and the students that catch on are doing formal deductive proofs by Chapter 4. This is most certainly not a curriculum that saves deductive proof for the last chapter of the book.

@Jason, you mention that you’d like to mix both types of proofs equally. Discovering Geometry is for you then. However, you have to look at a typical lesson as more than just the steps of an investigation followed by work time on exercises. The opportunities for deductive reasoning exist in nearly every lesson starting at Lesson 2.4. Early on, they are intended for group and class discussions. Have students share arguments in groups and share on large whiteboards. Encourage debate. Challenge students with “devil’s advocate” scenarios. Help students formulate more reasoned explanations. By chapter 4, your middle to high end students will be grasping more formal deductive reasoning via flow chart proofs. By the end of chapter 6, most of your students, including the lower kids, will be able to at least reason deductively. The improvement in quality of students’ arguments over the first 6 chapters is a cool thing to see.

This has been a great discussion to follow. I appreciate the perspectives that have been shared. Having taught Geometry from other curriculum in the past, I have found that students are turned off by a more “traditional” approach to geometry. They resort to procedural steps and memorization of postulates and theorems without developing a conceptual understanding of deductive argument. I remember my high school geometry teacher who had us write reasons in our two column proofs like “Theorem 3.4.2” which I assume made his correcting job easier. I learned nothing from this method. I really feel that unless you are teaching high end honors students, you’ve got to meet them where they are at – van Hiele level 1 and then build deductive understanding from there.

## Andy

August 7, 2013 - 6:32 am -Great conversation folks! When I read Ryan’s response to Jason’s regarding how much deductive reasoning is “covered” in Serra’s DG, I tend to agree with Ryan above. I’ve taught from this book for a handful of years and would argue that deduction is emphasized throughout the text as Ryan notes in his post. So I keep going back to Jason’s post wondering how one who’s taught from this text really can’t see deduction woven throughout the text. There’s a ton more there than just in the final chapter. But I don’t think this is what’s happening. What I do think is going on here is a difference in opinion when defining “deduction” and “deductive reasoning”.

Jason: I would agree with Ryan above that deduction is presented way before the final chapter. In fact, I’ve never taught the final chapter due to time constraints and (more importantly) because I’ve never taught students who were ready for this.

But if I ask my students to complete an angle chase (or play Mastermind, or complete a Sudoku puzzle) and *justify their series of steps*, that is deduction in practice. If you’re familiar with the “What’s Wrong with this Picture?” theme Serra threads throughout the text, these problems are absolutely brilliant. These little tasks opened the door to some of my struggling students. I’ve had students verbally creating beautiful deductive arguments–the same students who probably couldn’t write this up formally in whatever format they choose.

Jason: my impression is that you’re *not* considering a student’s 2 step argument in a little task like I just outlined as a deductive proof? I totally do. If I’m wrong here, then I’m missing something in my understanding of what you wrote (sorry!) Other than adding formality and more rigor, I’m not sure what the book’s final chapter does for *most* kids (I’ve rarely taught a 9th or 10th grader ready for this formality . . . maybe this is more a difference in the populations we all teach?).

Serra has said more than once above to consider the “Tracing Proof” document that goes with his text. Even if you haven’t taught (or don’t teach) from DG, this is an important document. It helped me better understand how I present deduction to students.

## Jason Dyer

August 7, 2013 - 9:29 am -@All: Keep in mind I generally like the book, it just wouldn’t be my go-to textbook if my goal was to get the students to do deductive proof. (If your goal is not to teach proof but rather the act of deduction, you don’t even need a math class for that.)

So I went through the text; there’s proof earlier than I remember, but it still strikes me as pretty scant. For example, in the review section for each chapter, here’s how many problems involve making a proof:

1. 0 out of 57

2. 0 out of 26

3. 0 out of 32

4. 3 out of 37

5. 1 out of 27

6. 0 out of 33

7. 0 out of 28

8. 0 out of 47

9. 0 out of 29

10. 0 out of 28

11. 0 out of 21

12. 0 out of 28

13. 14 out of 31

## Isaac D

August 7, 2013 - 9:37 am -@All This is a great discussion, and very helpful. Although we seem to be agreeing more than disagreeing, the disagreements are very important and I’m learning a lot from all sides in this.

@Jason I’d be extremely interested in hearing your concerns with van Hiele especially as it relates to deduction and the current discussion. Please do share your thoughts when you have time to do so.

## Kris Boulton

August 8, 2013 - 12:23 am -I couldn’t possibly read everything here, but having skimmed the first few comments, I find myself agreeing with Franklin in at least one important way, and I find myself with a different use or interpretation of inductive learning, which I would like to share:

1) I agree with Franklin that, it seems to me, no-one is any more or less inspired for having spotted for themselves that, say, the opposite angles in a parallelogram are always equal. In fact, taking the time to spot that is arguably time wasted. A teacher can exposit that fact, and if students then either worked on proving the fact, or using it to solve angle problems, then that process can be very enjoyable – here is where I agree with Franklin, and with Daniel Willingham, we are naturally wired to enjoy problem solving, provided the problem doesn’t feel impossible.

2) I’ve used induction successfully (by my own measure!) in a way prescribed by Engelmann to teach conceptual ideas such as what a parallelogram actually is. In my case, I’ve used it for Prisms and Surds. On both occasions, if I tried to define what they were I figured I would only overload the working memory of many students – the explanation could never be sufficiently clear and simple, nor concrete, for a novice. On the other hand, by just showing lots of examples and non-examples in a carefully designed sequence, almost everyone seemed to get it, and many (not all) were able to articulate some kind of definition for themselves. I would then often have to provide them a more correct definition, however.

## Andy

August 8, 2013 - 6:38 am -@Jason: First, I agree with our comment about teaching the art of deduction versus deductive proof. Playing a game of Mastermind involves deductive reasoning (which is not proof). Having a student who just played said game justify their reasoning and arguing their choice of position based on given information: that’s deductive argument/proof IMO. I believe this is where we are parting ways in opinion.

But I think that there is also a bit of difference of opinion here about what deductive proof needs to look like. Even though you don’t seem to be a fan of the 2 column, it sounds like you’re looking for formality in the write up (format and language used). I do see the value in that, esp for a student on path for a university degree in math related fields; but since the vast majority of my students are no where near that, I don’t emphasize that until they’ve been convinced that there is even a need to justify their thinking. Convincing them takes time (months).

Second, using the review sections as an assessment of this book’s (or any book for that matter!) handling of proof is misleading. There are entire sections of a few chapters in the first half of the book *devoted to writing* proofs. Yes, the review sections don’t necessarily reflect that, but I’ve never taught a book (or evaluated–formally or informally–a book for that matter) using the chapter reviews.

As stated many times so far throughout this thread, a better assessment of how this text handles deductive reasoning can be found in the “Tracing Proof” document Serra links to above.

## Ryan Brown

August 8, 2013 - 10:07 am -Here is the link to the “Tracing Proof” document: http://math.kendallhunt.com/documents/ALookInside/DiscoveringGeometryFourthEd/TracingProof.pdf

This is well worth the read by anyone in this discussion thread, even if you have not seen the Discovering Geometry text. Great information about building toward formal deductive proof and how the van Hiele levels relate to student understanding of proof.

In regard to our recent conversations about the instances of proof in the DG text, I encourage you to read about the “Developing Proof” problems as well as “What’s Wrong with this Picture” exercises that are spread throughout the book, beginning in Chapter 2. For students at any level on the van Hiele scale, building toward a solid understanding of proof begins with practice putting their arguments into writing. The What’s Wrong with this picture problems are excellent spots for this development of proof to start.

I would echo @Andy’s sentiments about judging a book by it’s review section. The review sections tend to be focused on individual skills and recognition of stand alone properties (with a small handful of proof based problems worked in). A closer look at each chapter would reveal a much deeper examination and progression toward student understanding of proof. Starting with Chapter 2 (which half of is devoted to the development of inductive reasoning for problem solving), I count 15 instances of proof based, deductive argument problems and examples. In subsequent chapters, the opportunity for deductive argument and eventually formal proofs continues to ramp up.

## Jason Dyer

August 8, 2013 - 12:06 pm -If you don’t like the review-problem method then: counting somewhat liberally, I get 90/723 or about 12% of pages devoted to deductive proof.

Let me note where I think some difference in our opinions is occuring—

Chapter 2.6 Special Angles on Parallel Lines

There are some really nice patty-paper inductive explorations here, where students fill in blanks to make conjectures like “If two parallel lines are cut by a transversal, then corresponding angles are ___.”

Where deductive reasoning hits is on page 129:

You used inductive reasoning to discover all three parts of the Parallel Lines Conjecture. However, if you accept any one of them as true, you can use deductive reasoning to show that the others are true.EXAMPLE: Suppose we assume that the Vertical Angles Conjecture is true. Write a paragraph proof showing that if corresponding angles are congruent, then the Alternate Interior Angles Conjecture is true.

Solution: (then is given immediately after)I could imagine in a class having the students stop reading and derive the thing for themselves, but the big chunk is devoted to inductive exploration, there’s no faciliation of creating the deductive proof in the text, and no followup problems where students create deductive proof independently.

Also, the “if you accept any one of them as true” bit is a little mysterious and completely hiding the momentous occasion of creating a new axiom. This is, I know, Intentional, but it is hard for me to see how the section puts equal value on deductive and inductive proof.

I’m not even saying it’s impossible to have a class with a fairly equal mix using the text (plenty of counterexamples in this very thread); but the text as written (where, let’s face it, most teachers will plow through at face value) does not seem to encourage such an approach.