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Bob Lochel, responding to commenter Jenni who wondered how, when, and where to integrate tasks into a unit:

In my years as math coach, the most efficient piece of advice I would give to teachers is this: think about your favorite problem from a unit, the problem you look forward to, or that problem which is number 158 in the last section which you know will generate all kinds of discussion. Without fail, this problem is often done last, as the summary of all ideas in the unit. Okay, why not do it first? Keep it simmering in the background, flesh it out as ideas are developed and pratice occurs. It often doesn't take a sledgehammer to make a good unit great.

I took machine-graded learning to task earlier this week for obscuring interesting student misconceptions. Kristen DiCerbo at Pearson's Research and Innovation Network picked up my post and argued I was too pessimistic about machine-graded systems, posing this scenario:

Students in the class are sitting at individual computers working through a game that introduces basic algebra courses. Ms. Reynolds looks at the alert on her tablet and sees four students with the “letters misconception” sign. She taps “work sample” and the tablet brings up their work on a problem. She notes that all four seem to be thinking that there are rules for determining which number a letter stands for in an algebraic expression. She taps the four of them on the shoulder and brings them over to a small table while bringing up a discussion prompt. She proceeds to walk them through discussion of examples that lead them to conclude the value of the letters change across problems and are not determined by rules like “c = 3 because c is the third letter of the alphabet.”

My guess is we're decades, not years, away from this kind of classroom. If it's possible at all. Three items in this scenario seem implausible:

  • That four students in a classroom might assume "c = 3 because c is the third letter of the alphabet." I taught Algebra for six years and never saw this conception of variables. (Okay, this isn't a big deal.)
  • That a teacher has the mental bandwidth to manage a classroom of thirty students and keep an eye on her iPad's Misconception Monitor. Not long ago I begged people on Twitter to tell me how they were using learning dashboards in the classroom. Everyone said they were too demanding. They used them at home for planning purposes. This isn't because teachers are incapable but because the job demands too much attention.
  • That the machine grading is that good. The system DiCerbo proposes is scanning and analyzing handwritten student work in real-time, weighing them against a database of misconceptions, and pairing those up with a scripted discussion. Like I said: decades, if ever.

This also means you have to anticipate all the misconceptions in advance, which is tough under the best of circumstances. Take Pennies. Even though I've taught it several times, I still couldn't anticipate all the interesting misconceptions.

The Desmos crew and I had students using smaller circles full of pennies to predict how many pennies fit in a 22-inch circle.

131003_1

But I can see now we messed that up. We sent students straight from filling circles with pennies to plotting them and fitting a graph. We closed off some very interesting right and wrong ways to think about those circles of pennies.

Some examples from reader Karlene Steelman via e-mail:

They tried finding a pattern with the smaller circles that were given, they added up the 1 inch circle 22 times, they combine the 6, 5, 4, 3, 2, 1, and 1 circles to equal 22 inches, they figured out the area of several circles and set up proportions between the area and the number of pennies, etc. It was wonderful for them to discuss the merits and drawbacks of the different methods.

Adding the 1-inch circle 22 times! I never saw that coming. Our system closed off that path before students had the chance even to express their preference for it.

So everyone has a different, difficult job to do here, with different criteria for success. The measure of the machine-graded system is whether it makes those student ideas invisible or visible. The measure of the teacher is whether she knows what to do with them or not. Only the teacher's job is possible now.

Featured Comments

Sue Hellman:

This doesn’t even touch the students who get questions RIGHT for the wrong reasons.

Dave Major:

Dashboards of the traditional 'spawn of Satan & Clippy the Excel assistant' sort throw way too much extremely specific information straight to the surface for my liking (and brain). That information is almost always things that are easy for machines (read. programmers) to work out, and likely hard or time consuming yet dubiously useful for humans to do. I wonder how many teachers, when frozen in time mid-lesson and placed in the brain deli slicer would be thinking "Jimmy has 89% of this task correct and Sally has only highlighted four sentences on this page."

Let me highlight another conversation from the comments, this time between Kevin Hall, Don Byrd, and myself, on the merits of direct instruction, worked examples, inquiry learning, and some blend of the three.

Some biography: Kevin Hall is a teacher as well as a student of cognitive psychology research. His questions and criticisms around here tend to tug me in a useful direction, away from the motivational factors that usually obsess me and closer towards cognitive concerns. The fact that both he and Don Byrd have some experience in the classroom keep them from the worst excesses of cognitive science, which is to see cognition as completely divorced from motivation and the classroom as different only by degrees from a research laboratory.

Kevin Hall:

While people tend to debate which is better, inquiry learning or direct instruction, the research says sometimes it’s one and sometimes the other. A recent meta study found that inquiry is on average better, but only when “enhanced” to provide students with assistance [1]. Worked examples actually can be one such form if assistance (e.g., showing examples and prompting students for explanations of why each step was taken).

One difficulty with just discussing this topics that people tend to disagree about what constitutes inquiry-based learning. I heard David Klahr, a main researcher in this field, speak at a conference once, and he said lots of people considered his “direct instruction” conditions to be inquiry. He wished he had just labelled his conditions as Condition 1, 2, and 3 because it would have avoided lots of controversy.

Here’s where Cognitive Load Theory comes in: effectiveness with inquiry (minimal guidance) depends in the net impact of at least 3 competing factors: (a) motivation, (b) the generation effect, and (c) working memory limitations. Regarding (a), Dan often makes the good point that if teachers use worked examples in a boring way, learning will be poor even if students cognitive needs are being met very well.

The generation effect says that you remember better the facts, names, rules, etc that you are asked to come up with on your own. It can be very difficult to control for this effect in a study, mainly because its always possible that if you let students come up with their own explanations in one group while providing explanations to a control group, the groups will be exposed to different explanations, and then you’re testing the quality of the explanations and not the generation effect itself. However, a pretty brilliant (in my opinion) study controlled for this and verified the effect [2]. We need more studies to confirm. Here is a really portent paragraph from the second page of the paper: “Because examples are often addressed in Cognitive Load Theory (Paas, Renkl, & Sweller, 2003), it is worth a moment to discuss the theory’s predictions. The theory defines three types of cognitive load: intrinsic cognitive load is due to the content itself; extraneous cognitive load is due to the instruction and harms learning; germane cognitive load is due to the instruction and helps learning. Renkl and Atkinson (2003) note that self-explaining increases measurable cognitive load and also increases learning, so it must be a source of germane cognitive load. This is consistent with both of our hypotheses. The Coverage hypothesis suggests that the students are attending to more content, and this extra content increases both load and learning. The Generation hypothesis suggests that load and learning are higher when generating content than when comprehending it. In short, Cognitive Load Theory is consistent with both hypotheses and does not help us discriminate between them.”

Factor (c) is working memory load. The main idea is found in this quote from the Sweller paper Dan linked to above, Why Minimal Instruction During Instruction Does Not Work [3]: “Inquiry-based instruction requires the learner to search a problem space for problem-relevant information. All problem-based searching makes heavy demands on working memory. Furthermore, that working memory load does not contribute to the accumulation of knowledge in long-term memory because while working memory is being used to search for problem solutions, it is not available and cannot be used to learn.” The key here is that when your working memory is being used to figure something out, it’s not actually being used to to learn it. Even after figuring it out, the student may not be quite sure what they figured out and may not be able to repeat it.

Does this mean asking students to figure stuff out for themselves is a bad idea? No. But it does mean you have to pay attention to working memory limitations by giving students lots of drill practice applying a concept right after they discover it. If you don’t give the drill practice after inquiry, students do worse than if you just provided direct instruction. If you do provide the drill practice, they do better than with direct instruction. This is not a firmly-established result in the literature, but it’s what the data seems to show right now. I’ve linked below to a classroom study [4] and a really rigorously-controlled lab study study [5] showing this. They’re both pretty fascinating reads… though the “methods” section of [5] can be a little tedious, the first and last parts are pretty cool. The title of [5] sums it up: “Practice Enables Successful Learning Under Minimal Guidance.” The draft version of that paper was actually subtitled “Drill and kill makes discovery learning a success”!

As I mentioned in the other thread Dan linked to, worked examples have been shown in year-long classroom studies to speed up student learning dramatically. See the section called “Recent Research on Worked Examples in Tutored Problem Solving” in [6]. This result is not provisional, but is one of the best-established results in the learning sciences.

So, in summary, the answer to whether to use inquiry learning is not “yes” or “no”, and people shouldn’t divide into camps based on ideology. Still unanswered question is the question when to be “less helpful” as Dan’s motto says and when to be more helpful.

One of the best researchers in the area is Ken Koedinger, who calls this the Assistance Dilemma and discusses it in this article [7]. His synthesis of his and others’ work on the question seems to say that more complex concepts benefit from inquiry-type methods, but simple rules and skills are better learned from direct instruction [8]. See especially the chart on p. 780 of [8]. There may also be an expertise reversal effect in which support that benefits novice learners of a skill actually ends up being detrimental for students with greater proficiency in that skill.

Okay, before I go, one caveat: I’m just a math teacher in Northern Virginia, so while I follow this literature avidly, I’m not as expert as an actual scientist in this field. Perhaps we could invite some real experts to chime in?

Dan Meyer:

Thanks a mil, Kevin. While we’re digesting this, if you get a free second, I’d appreciate hearing how your understanding of this CLT research informs your teaching.

Kevin Hall:

The short version is that CLT research has made me faster in teaching skills, because cognitive principles like worked examples, spacing, and the testing effect do work. For a summary of the principles, see this link.

But it’s also made me persistent in trying 3-Acts and other creative methods, because it gives me more levers to adjust if students seem engaged but the learning doesn’t seem to “stick”.

Here’s a depressing example from my own classroom:

Two years ago I was videotaping my lessons for my masters thesis on Accountable Talk, a discourse technique. I needed to kick off the topic of inverse functions, and I thought I had a good plan. I wrote down the formula A = s^2 for the area of a square and asked students what the “inverse” of that might mean (just intuitively, before we had actually defined what an inverse function is). Student opinions converged on the S = SqRt(A). I had a few students summarize and paraphrase, making sure they specifically hit on the concept of switching input and output, and everyone seemed to be on board. We even did an analogous problem on whiteboards, which most students got correct. Then I switched the representations and drew the point (2, 4) point on a coordinate plane. I said, “This is a function. What would its inverse be?” I expected it to be easy, but it was surprisingly difficult. Most students thought it would be (-2, -4) or (2, -4), because inverse meant ‘opposite’. Eventually a student, James (not his real name), explained that it would be (4, 2) because that represents switching inputs and outputs. Eventually everyone agreed. Multiple students paraphrased and summarized, and I thought things were good.

Class ended, but I felt good. The next class, I put up an similar problem to restart the conversation. If a function is given by the point (3, 7), what’s the inverse of that function? Dead silence for a while. Then one student (the top student in the class) piped up: “I don’t remember the answer, but I remember that this is where James ‘schooled’ us last class.” Watching the video of that as I wrote up my thesis was pretty tough.

But at least I had something to fall back on. I decided it was a case of too much cognitive load–they were processing the first discussion as we were having it, but they didn’t have the additional working memory needed to consolidate it. If I had attended to cognitive needs better, the question about (2, 4) would have been easier, and I should NOT have switched representations from equations to points until it seemed like the switch would be a piece of cake.

I also think knowing the CLT research has made me realize how much more work I need to do to spiral in my classroom.

Then in another thread on adaptive math programs:

Kevin Hall:

My intention was to respond to your critique that a computer can’t figure out what mistake you’re making, because it only checks your final answer. Programs with inner-loop adaptivity do, in fact, check each step of your work. Before too long, I they might even be better than a teacher at helping individual students identify their mistakes and correct them, because as as teacher I can’t even sit with each student for 5 min per day.

Don Byrd:

I have only a modest amount of experience as a math teacher; I lasted less than two years — less than one year, if you exclude student teaching — before scurrying back to academic informatics/software research. But I scurried back with a deep interest in math education, and my academic work has always been close to the boundary between engineering and cognitive science. Anyway, I think Kevin H. is way too optimistic about the promise of computer-based individualized instruction. He says “It seems to me that if IBM can make Watson win Jeopardy, then effective personalization is also possible.” Possible, yes, but as Dan says, the computer “struggles to capture conceptual nuance.” Success at Jeopardy simply requires coming up with a series of facts; that’s highly data based and procedural. The distance from winning Jeopardy to “capturing conceptual nuance” is much, much greater than the distance from adding 2 and 2 to winning Jeopardy.

Kevin also says that “before too long, [programs with inner-loop adaptivity] might even be better than a teacher at helping individual students identify their mistakes and correct them, because as as teacher I can’t even sit with each student for 5 min per day.” I’d say it’s likely programs might be better than teachers at that “before too long” only if you think of “identifying a mistake” as telling Joanie that in _this_ step, she didn’t convert a decimal to a fraction correctly. It’ll be a very long time before a computer will be able to say why she made that mistake, and thereby help her correct her thinking.

2013 Aug 14. Christian Bokhove passes along an interesting link summarizing criticisms of CLT.

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.

Franklin Mason:

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.

Dan Meyer:

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.

Franklin Mason:

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.)

Dan Meyer:

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.

Franklin Mason:

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.”

Jason Dyer:

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).

Dan Meyer:

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.

Pete Capewell:

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.

Mike C.:

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?

Laura Hawkins:

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.

Tweet-Sized Tasks

Here is one of my favorite quotes on task design from one of my favorite math educators:

A good problem seems natural. A good problem reveals its constraints quickly and clearly.

Is it possible to pose a task so quickly and clearly that it would fit in a tweet?

I asked and lots of you gave it a shot. Here's mine as well as a few of my favorites:

Extra merits for roping in your personal life:

Extra demerits for trolling:

I'll depart from Sallee briefly and say that it's nice, sometimes, when the constraints aren't fully revealed. I'd like the task to be clear, but in life the constraints often require clarification. When you ask yourself, "What extra information do I need here?" you're doing the work of mathematical modeling.

Feel free to play along in the comments, but you'll have to constrain yourself to 140 characters.

Featured Tasks

Jonah:

Two points A and B on a paper, 13″ apart. You have a pencil and a 12″ ruler. Construct the line segment AB.

Caitlin Browne:

How many squares are on a standard checkerboard?

Bowen Kerins:

Is it really possible for Steven Seagal to have “millions of hours” of weapons training?

2013 Sep 22. From Nat Banting on Twitter:

Give students the sums when rolling two irregular dice. Ask them to design the dice based on data.

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