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 wasnâ€™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 donâ€™t think Iâ€™d do this activity â€“ either pre- or post-fix â€“ in my classroom. No matter how itâ€™s presented (and your presentation is really very good), itâ€™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. Itâ€™s a real beauty â€“ 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 â€“ a pot is a cylinder, a meatball is a sphere and thus have volumes given by the relevant formulae â€“ donâ€™t need much (if any) time.
Thanks for the comment, Franklin. I suppose Iâ€™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.
1. I wouldnâ€™t make the strong claim that geometry done inductively must result in a disjointed curriculum. But I do believe that thatâ€™s the tendency.
Hereâ€™s an example. If you were to present students with many examples of parallelograms, theyâ€™d very soon conclude that opposite sides and angles are congruent. Thatâ€™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 itâ€™s true? Thatâ€™s where deduction comes in. A proof of these two properties (and â€˜proofâ€™ just is another word for â€˜deductionâ€™) 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 isnâ€™t just a sequence of facts, loosely connected if at all. Itâ€™s a set of results that can be traced back via deduction to a small set of fundamental, intuitively obvious assumptions. Moreover, itâ€™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, itâ€™s quite possible to teach the great majority of students how to put together a mathematical proof. It isnâ€™t something thatâ€™s out of reach, as you seem to suggest. Iâ€™ve had great success when I make it the focus of my geometry courses. (I quite happy that success has been possible. If it wasnâ€™t, Iâ€™d be forced to conclude that the majority of human beings are not really capable of genuine mathematics.)
Iâ€™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?
â€œHey team. Today weâ€™re going to look at this shape called a â€˜parallelogram.â€™ Itâ€™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 donâ€™t take my word for it. We can prove it.â€
Iâ€™m just speculating so set me straight. But what my anecdote lacks is any intellectual need from the student for that proof. Theyâ€™re doing the proof because the teacher suggested it.
An inductive approach would let the students notice that, â€œHey, waitaminit, all these sides and angles are the same.â€ 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 donâ€™t mean to trivialize geometry when I call its problems â€˜puzzlesâ€™. 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 isnâ€™t that students walked into the classroom with some concrete question in mind (â€œThe wheels on my car are circles. I wonder how many times they had to rotate to get me to school?â€) and that we piggyback on that. Instead itâ€™s that they like good puzzles. I donâ€™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? Iâ€™ll be the first to admit that I donâ€™t always achieve that goal. Sometimes a bit of drill is necessary, and that isnâ€™t much fun. But still, that always the goal.
Thatâ€™s why Iâ€™m also a little suspicious when someone continually demands context or application. Taht makes it seem as if students canâ€™t really enjoy the mathematics for its own sake. Now, Iâ€™m fine with context and application. I donâ€™t mean to say that we shouldnâ€™t seek them out. Instead I mean to say that for the most part theyâ€™re not necessary. The mathematics itself suffices.
If this makes me seem unabashedly Lockhartean in my views, thatâ€™s fine. The description of Lockhartâ€™s new book â€“ Measurement â€“ at Harvard University Press says it about as well as it can be said:
â€œMeasurement 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 â€œdo the mathâ€ 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.â€
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 â€œgenuine mathematicsâ€ is. With the mathematicians Iâ€™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 (thereâ€™s an exception Iâ€™ll mention in a moment). Even when I was getting a math degree where I faced pre-cooked â€œprove Xâ€ problems the process did not resemble my high school geometry proof days at all. (In fact, Iâ€™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 â€œorganizing principlesâ€ for a math class. Iâ€™d say â€œmake math real worldâ€ is as self-defeating an organizing principle as one can find. â€œPrioritize perplexity,â€ 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 itâ€™s clear to me you prize pure mathematics, itâ€™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 isnâ€™t a trivial matter.
In the example you chose of deducing congruent opposite angles and sides from a parallelogramâ€™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 Childâ€™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.
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 theyâ€™re typically taught (and as you seem to have in mind). We donâ€™t do two-column proof. (I shudder to even write those words. I suspect that Lockhart is right about it: itâ€™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 donâ€™t care a whit for the way in which a result is proven; certainly I donâ€™t insist upon a certain method or (even worse) a certain sequence of propositions. Indeed discovery of a new method, especially when itâ€™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 thatâ€™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 â€“ thatâ€™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 doesnâ€™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 â€“ 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) â€“ that I do admit is a real challenge in the mathematics classroom. Given how much Iâ€™m required to cover, I find it difficult, indeed sometimes impossible, to give much time to it. In this yearâ€™s redesign of my course (and it seems to get redesigned every year) Iâ€™ll attempt to carve out more time for it. This means, I suspect, that Iâ€™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. Soâ€¦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 â€œopposite sides are congruentâ€ and â€œopposite sides are parallelâ€ and have them choose two to combine into and â€œIf-Thenâ€ statement like â€œIf opposite sides are congruent, then opposite sides are parallel.â€ 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 â€œDid you PLAN for us to discover this today!?â€ 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 Jasonâ€™s apology even though â€œmehâ€ 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 â€“students can identify rectangles by sight but squares are squares and are not seen as special rectangles.
Level 1: Analysis (descriptive) â€“ students can identify properties of rectangles (by drawing, measuring, and making models) but cannot yet derive other properties from given.
Level 2: Informal Deduction â€“ students can give an informal argument to justify that the figure is a rectangle from given properties.
Level 3: Formal Deduction â€“ students are capable of creating original logical arguments.
Level 4: Rigor â€“ 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 â€œtraditionalâ€ 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 Mathematicianâ€™s Delight by English Mathematician W.W. Sawyer (1911-2008)
â€œIt 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.â€
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 â€œimitation geometry.â€ 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 studentsâ€™ 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 studentâ€™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 Villiersâ€™ research on the role and function of proof identifies six basic roles for proof:
â€¢ Verification â€“to remove doubt, to convince someone of the truth of a statement
â€¢ Systematization â€“organize known results into a deductive system of postulates, definitions, and theorems
â€¢ Explanation â€“insight into why something is true
â€¢ Discovery â€“proof can occasionally lead to new unexpected results
â€¢ Communication â€“proof can create a forum for critical debate
â€¢ Intellectual Challenge â€“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 â€“verification and to establish geometry as a mathematical system â€“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 â€œtruthâ€ of a proposition within a mathematical system, especially when coming across surprising (non-intuitive) results. However this view does not stand up to actual mathematicianâ€™s experiences. Professor George Polya wrote, â€œWhen you have satisfied yourself that the theorem is true, you start proving it.â€ 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 180Â°. 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 180Â°. 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 180Â°. 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 â€œmaking sure,â€ but rather a question of â€œexplaining why.â€
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 studentâ€™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, â€œcan you explain why?â€ For example, after their very first two investigations leading to geometric conjectures, the Linear Pair Conjecture and the Vertical Angles Conjecture, students are asked:
â€œDeveloping 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?â€
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:
â€œDeveloping 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?â€
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 â€œhow is the resulting line related to the original triangle.â€ 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 isnâ€™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 Mathematicianâ€™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: Euclidâ€™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 â€“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 Euclidâ€™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.
Iâ€™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.