I got an email from a fellow edu-blogger a couple of days ago asking for my input on the subject of academic rigor. Specifically this person asked:

Is the quest for more rigor an issue for you? Is it good, bad, meaningless? What does rigorous teaching look like in your classroom?

I hope she doesn’t mind my sharing the answer, because after writing it I thought it’d make a good blog post. I said:

For me, “rigor” in the context of intellectual work refers to thoroughness, carefulness, and right understanding of the material being learned. Rigor is to academic work what careful practice and nuanced performance is to musical performance, and what intense and committed play is to athletic performance. When we talk about a “rigorous course” in something, it’s a course that examines details, insists on diligent and scrupulous study and performance, and doesn’t settle for a mild or  informal contact with the key ideas.

Example: A rigorous course in geometry goes beyond just memorization of formulas, applications to simple geometric exercises, and “hand-waving” attempts at proof. Instead, such a course treats details as important, the ability to explain on a deep level the truth of formulas and results as a key goal for students, and sets a high bar for the exactitude of mathematical arguments. Euclid’s “Elements” for example is the prototype of the rigorous treatment of geometry. It’s not a difficult work to understand, necessarily; in fact one of the enduring qualities of the Elements is the clarity and precision of not only each individual proposition but also in how the overall collection of propositions fits together. By contrast, many modern books on geometry are highly non-rigorous, omitting details, putting theorems out of order, and defining a proof as a “reasonable explanation” only.

Is rigor good? It depends on the audience and the goals of the class. When I teach a geometry course for junior and senior Math Education majors, rigor is of the utmost importance because I want those pre-service teachers to go into their classrooms with tough, precise minds for the sake of their students. If I were to teach a geometry class for fifth-graders, on the other hand, I think rigor would obscure the subject, and I would depend a lot more on intuitive explanations and perhaps constructivist techniques for discovering key ideas in geometry and save rigorous proofs for another day. Similarly, when I teach calculus at my college, the audience is about 50% business majors, and so we designed the course not to cover much theory. This is not a rigorous treatment of calculus, but it is more effective for the students than if we included the epsilon-delta proofs and what not.

The quest for more rigor is most important in the post-calculus courses I teach (geometry, abstract algebra, and introduction to proof). These are subject areas where precision and detail-orientation are essential for a complete understanding of the material. Students are not allowed to give examples when a proof is called for, and I nitpick every little thing in their proofs up to and including the choice of punctuation and prepositions. [If any of the five who took this course from me this past semester are reading this, feel free to chime in with an "Amen." - RT.] At the calculus level and below, I lay off on the theory but the rigor in the course comes from getting details of mechanical calculations right. And this is a big issue, because students in high school are generally taught only to produce a correct answer, not a clear and detailed solution. I am on a mission to make sure students can not only get right answers but also communicate their methods in a clear and audience-appropriate way, and that’s what “rigor” looks like there.

[After-the-fact note: To clarify, in calculus I insist on details in mechanical calculations but also on the details of processes and in paying attention to nuances in solving application-style problems. For example, students know that if you just set and solve for , that this doesn't give you an inflection point; and in an optimization problem you can't just find the critical number of the model function, you must also test it with the First or Second Derivative Test to see if it really yields a maximum. Or at least, they don't complain when they forget to do it and I take off points!]

I have two kids, ages 3 and 5. (There’s a third one on the way in three weeks, but that’s another story!) I’m pretty rigorous with them, too — when the 5-year old says “Mimi comed to our house this weekend” I correct her grammar, and she gets it right the next time. You have to do it in a gentle way, but getting details right now will help them get the more complicated things right later. If I were to project myself out of higher ed and into the K-12 sphere I could see my teaching being “rigorous” in that kind of way — insisting that kids get the details right and not gloss over things, but doing it in a lovingly persistent way. I wish more K-12 teachers would do this, though, because it’s obvious from my freshmen in the last 4-5 years that this isn’t happening (or at least it’s not sticking).

[Final note: That last sentence isn't a slam on either my freshmen, who were really quite excellent this year in calculus, or their teachers. It's an observation, and I stand by it. I can show you their work at the beginning of the semester if you don't believe me. Why this kind of "rigor" is not sticking with them is something I can't fully explain because I don't know what was going on with them in high school. Is it them? Is it their teachers? Is it the system? Is it the preponderance of standardized testing, which makes rigor more or less irrelevant? Comment!]

Hat tip to Darren at Right on the Left Coast for this article, which starts off saying in a plainspoken way:

Here are two of the clues to America’s current mathematics problem:
1.”Student-centered” learning (or “constructivism”)
2.Insufficient practice of basic skills

The article then goes on to say, of constructivism:

In small doses, constructivism can provide flavor to classrooms, but some math professors have told me the approach seems to work better in subjects other than math. That sounds reasonable. The learning of mathematics depends on a logical progression of basic skills. Sixth-graders are not Pythagorus [sic], nor are they math teachers.

That’s right. Constructivism, when used with the right kinds of students and in the right ways, can be quite effective. But it’s important to remember that not all students are ready for this, and not all material is taught effectively this way. When I teach geometry to junior and senior math majors, it’s almost entirely constructivist, because the process of mathematical investigation and discovery is precisely what I am trying to teach them (through the medium of Euclidean and non-Euclidean geometry). But I’d be crazy to try constructivism at that level on, say, a precalculus class full of students who have little skill in and absolutely no taste for math at all. Those students aren’t dumb, but they need structure and guidance a lot more than they need the supposed thrill of mathematical discovery.

And then, about drill and practice:

Another problem in math classrooms is the lack of practice. Instead of insisting that students practice math skills until they’re second nature, educators have labeled this practice “drill and kill” and thrown it under a bus.

I wish I had a dollar for every time I heard that phrase. It’s a strange, flippant way to dismiss a logical process for learning. Drilling is how anyone learns a skill. [...] Everyone drills – athletes, pianists, soldiers, plumbers and doctors. Drilling is necessary.

It isn’t good or bad – it’s simply what must be done.

I’ve said it before here: No human being can do meaningful creative work until they are completely fluent in the rudiments of what they are working with. Musicians, athletes, and skilled workers all know this. For some reason, there’s no outcry among music educators that we need to just hand new musicians a saxophone and try to get them to discover how to play it all by themselves. This fact — that drill and mastery precede creative work — is so painfully obvious that I feel a little embarrassed for my colleagues in math instruction who don’t seem to get it.

Constructivism and drill/practice are pedagogical tools, not religions. You look at your class, your students, and the material to teach, and then choose the right combination of tools for the job. To hear some proponents, and opponents, of constructivism, you’d think that you’re supposed to choose sides and swear undying allegiances instead.

We interrupt this blogging hiatus to throw out a question that came up while I was grading today. The item being graded was a homework set in the intro-to-proof course that I teach. One paper brought up two instances of the same issue.

• The student was writing a proof that hinged on arguing that both sin(t) and cos(t) are positive on the interval 0 < t < π/2. The “normal” way to argue this is just to appeal to the unit circle and note that in this interval, you’re remaining in the first quadrant and so both sin(t) and cos(t) are positive. But what the student did was to draw graphs of sin(t) and cos(t) in Maple, using the plot options to restrict the domain; the student then just said something to the effect of “The graph shows that both sin(t) and cos(t) are positive.”
• Another proof was of a proposition claiming that there cannot exist three consecutive natural numbers such that the cube of the largest is equal to the sum of the cubes of the other two. The “normal” way to prove this is by contradiction, assuming that there are three consecutive natural numbers with the stated property. Setting up the equation representing that property leads to a certain third-degree polynomial P(x), and the problem boils down to showing that this polynomial has no roots in the natural numbers. In the contradiction proof, you’d assume P(x) does have a natural number root, and then proceed to plug that root into P(x) and chug until a contradiction is reached. (Often a proof like that would proceed by cases, one case being that the root is even and the other that the root is odd.) The student set up the contradiction correctly and made it to the polynomial. But then, rather than proceeding in cases or making use of some other logical deduction method, the student just used the solver on a graphing calculator to get only one root for the polynomial, that root being something like 4.7702 (clearly non-integer) and so there was the contradiction.

So what the student did was to substitute “normal” methods of proof — meaning, methods of proof that go straight from logic — with machine calculations. Those calculations are convincing and there were no errors made in performing them, and there seemed to be no hidden “gotchas” in what the student did (such as, “That graph looks like it’s positive, but how do you know it’s positive?”). So I gave full credit, but put a note asking the student not to depend on technology when writing (otherwise exemplary) proofs.

But it raises an important question in today’s tech-saturated mathematics curriculum: Just how much technology is acceptable in a mathematical proof? This question has its apotheosis in the controversy surrounding the machine proof of the Four-Color Theorem but I’m finding a central use of (a reliance upon?) technology to be more and more common in undergraduate proof-centered classes. What do you think? (This gives me an opportunity to show off WordPress’ nifty new polling feature.)

BusinessWeek’s TechBeat blog has this article about the federal panel report on K-8 mathematics instruction that I blogged about here. It’s good to see this report getting attention in the blogosphere and MSM. It needs more. One thing from the BusinessWeek article that needs a slight bit of correction, though — it says:

The sad thing about the report that despite the unanimity on a panel that represents a broad spectrum of the mathematics and math education communities, it will take a decade or more for its recommendations to be implemented. It simply takes that long for curriculum guidelines to be recast, textbooks to be rewritten, and teachers to be trained or retrained. And in that time, a lot more damage can be done.

That may be true of traditional public schools, where red tape and opposing political forces must be overcome at every turn, but it does not have to be true of private or charter schools where reaction times to changing pedagogical climates can be much faster. I think this report, and the dire consequences of ignoring it, create a prime situation for charter schools and private schools to lead the way to better education for our kids. If governments would open up schools to market forces to a greater degree, we might even get the traditional public schools on board once parents choose to send their kids to places that are getting on the ball faster.

A federal panel examining K-8 mathematics education in the USA has made some forthright recommendations, according to this article in the NYT today. Unlike many federal panels, this one has an uncommon amount of common sense in its conclusions. For example, this finding that is striking in the way it refrains from choosing sides in the math wars:

Parents and teachers in school districts across the country have fought passionately over the relative merits of traditional, or teacher-directed, instruction, in which students are told how to solve problems and then are drilled on them, as opposed to reform or child-centered instruction, which emphasizes student exploration and conceptual understanding. The panel said both methods have a role.

“There is no basis in research for favoring teacher-based or student-centered instruction,” said Dr. Larry R. Faulkner, the chairman of the panel, at a briefing for reporters on Wednesday. “People may retain their strongly held philosophical inclinations, but the research does not show that either is better than the other.” [...]

“To prepare students for algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency and problem-solving skills,” the report said. “Debates regarding the relative importance of these aspects of mathematical knowledge are misguided. These capabilities are mutually supportive.”

Say what? An appeal to actual research rather than anecdotes and personal biases when thinking about effective math teaching? Amazing. And this shocking discovery:

[T]he panel found that it is important for students to master their basic math facts by heart.

“For all content areas, practice allows students to achieve automaticity of basic skills — the fast, accurate, and effortless processing of content information — which frees up working memory for more complex aspects of problem solving,” the report said.

Dr. Faulkner, a former president of the University of Texas at Austin, said the panel “buys the notion from cognitive science that kids have to know the facts.”

“In the language of cognitive science, working memory needs to be predominately dedicated to new material in order to have a learning progression, and previously addressed material needs to be in long-term memory,” he said.

Why, it’s almost as if they think that mastery precedes creativity or something. And finally:

The report makes a plea for shorter and more accurate math textbooks. Given the shortage of elementary teachers with a solid grounding in math, the report recommends further research on the use of math specialists to teach several different elementary grades, as is done in many top-performing nations.

The article goes on to give some of the panel’s recommended benchmarks for mathematical skills in grades 3-7. There’s also a link to the panel’s report.

All I can say is that I hope math educators, prospective teachers (especially prospective elementary school teachers), curriculum designers, ed schools, school boards, and everybody else who is a stakeholder with some influence in this process are listening. We’ve got 2 years until our oldest starts kindergarten and she needs teachers and curricula who get math right.

[h/t God Plays Dice]

Following up on his three posts on classical education yesterday, Gene Veith weighs in on mathematics instruction: 

I admit that classical education may be lagging in the math department. The new classical schools are doing little with the Quadrivium, the other four liberal arts (arithmetic, geometry, astronomy, and music). The Trivium, which is being implemented to great effect (grammar, logic, and rhetoric), has to do with mastering language and what you can do with it. The Quadrivium has to do with mathematics (yes, even in the way music was taught).

This, I think, is the new frontier for classical educators. Yes, there is Saxon math, but it seems traditional (which is better than the contemporary), rather than classical, as such.

Prof. Veith ends with a call for ideas about how mathematics instruction would look like in a classical education setting. I left this comment:

I think a “classical” approach to teaching math would, going along with the spirit of the other classical education posts yesterday, teach the hypostatic union of content and process — the facts and the methods, yes (and without cutesy gimmicks), but also the processes of logical deduction, analytic problem-solving heuristics, and argumentation. Geometry is a very good place to start and an essential to include in any such approach. But I’d also throw in more esoteric topics as number theory and discrete math (counting and graph theory) — in whatever dosage and level is age-appropriate.

At the university level, and maybe at the high school level for kids with a good basic arithmetic background, I’d love to be able to use the book “Essential College Mathematics” by Zwier and Nyhoff as a standard one-year course in mathematics (and in place of the usual year of calculus most such students take). It’s out of print, but the chapters are on sets; cardinal numbers; the integers; logic; axiomatic systems and the mathematical method; groups; rational numbers, real numbers, and fields; analytic geometry of the line and plane; and finally functions, derivatives, and applications. You have to see how the text is written to see why it does a good job with both content and process.

(I took out the mini-rant against the gosh-awful Saxon method.)

Any thoughts from the audience here?

One of my linear algebra students is an education major doing student teaching. Today he showed me this method of simplifying radicals which he learned from his supervising teacher. Apparently it’s called the “Illini method”. Googling this term returns nothing math-related, so I think that term was probably invented by his supervisor, who went to college in Illinois.

The procedure goes as follows. Start with a radical to simplify, say . Look under the radical and find a prime that divides it, say 5. Then form a two-column array with the original radical in the top-left, the divisor prime in the adjacent row in the right column, and the result you get from dividing the radicand by that prime number in the left column below the radical. In this case, it’s:

Now look for a prime that divides the lower-left term, say another 5. Again, put the dividing prime across from the dividend, and the quotient below the dividend. With our example, the array at this stage looks like:

In general, continue this process of dividing prime numbers into the lower-left entry in the array, writing the prime across from that entry, and writing the quotient beneath that entry, until you end up with a 1 in the lower-left entry. So the final state of our example would be:

Now, look at the left-hand column of the array. Group off any pairs of numbers you see. Multiply together all numbers which are representative of a pair. In our case, there is only one such pair, a pair of 5’s. Any numbers that occur singly are placed under a radical and multiplied. In our case, that’s the single 2. Then multiply the product of numbers which are in pairs times the radical which contains the singleton numbers. So we end up in our example with .

Here’s another example with a larger number, :

There are three groups of 2’s, so outside the final radical we’ll put . And the 3 and 11 are by themselves, so under the radical we put 33. Hence .

Pretty clearly, all this method is doing is presenting a different way to do the bookkeeping for doing the prime factorization of the number under the radical. The final step of grouping off the prime pairs and leaving the un-paired primes under the radical is analogous to finding all the squared primes in the prime factorization.

This method is nice and systematic, and I can see why students (and student-teachers) might like it. But it seems to be obscuring some important concepts that students ought to know. With the method of factoring, looking for squared primes, and then removing them from the square root, at least you are dealing directly with the inverse relationship between squares and square roots. The Illini method, on the other hand, uses an approach of “put this here and then put that over there” with minimal contact with actual math. It does work, and it does keep things in order. But do students really understand why it works?

Your thoughts?  What does this method make clearer, and what does it obscure? Should high school algebra teachers be teaching it?

Here’s a problem I have with the way most calculus textbooks are written, and therefore by default the way most calculus courses end up being taught. Tell me if I am crazy or missing something.

We teach calculus from a depth-first viewpoint. That means that whenever we encounter a concept, we go as deeply as possible in that concept before moving on to the next one. There are some subjects where this makes sense, but in calculus we have a small number of main ideas that are made out of several concepts, and if we stop to attain maximal depth on every single thing, there’s a good chance that we never arrive at the main idea with any degree of understanding.

The big ideas of calculus — the rate of change (derivative) and accumulated change (integral) — are actually really simple if you consider them simply for what they are and what they were invented to do. Derivatives, for instance: You have a function, and it is changing in all kinds of ill-behaved ways. The object is to find out exactly how quickly it is changing at a given point. We quantify that rate of change by sticking a tangent line on the graph of the function at that point and measuring its slope. Really, that’s it. Slopes of lines. The rest are technical details on how to calculate this slope with some degree of accuracy, and those details range from graphical estimation to interpolation tricks to algebraic techniques.

But in Stewart’s Calculus book, the coin of the realm of calculus texts, here’s what students have to study before the derivative is defined: an entire chapter of precalculus review (a mind-numbing section 1.1 on functions and notation, mathematical models, families of functions, exponential functions, inverse functions and logarithms), then a chapter on limits in which students have to master finding limits from graphs, calculating limits using the Limit Laws, the epsilon-delta definition of a limit (mostly untaught these days), continuity, and limits at infinity.

Then there’s a section on “Tangents, Velocities, and Other Rates of Change” followed by two sections on the Derivative.*

This approach plays directly in to the greatest weakness of the average calculus student, which is algebra/precalculus content mastery and the ability to master technical details of calculations and theory. How likely is it, for the student who struggles to read mathematics or use algebra correctly, that this student will be in any shape to learn what a derivative is, and what one is for, by the time they get there?

You want students to master those technical calculations and theory, of course. But you also want those to be mastered in context, not just as mathematical tricks to be learned as parlor games. The few students who survive the onslaught of detail mastery and are still psychologically around to learn what a derivative is, often find it extremely hard to know what f’(3) = 2 actually means. All they know is that you bring the power down and subtract one, and maybe the Product Rule.

I’d prefer some kind of approach to calculus that is not depth-first but more like breadth-first, where students get a good grounding in the overall ideas of calculus and do some basic work before mining into the really deep details. Not all students really need those deep details, after all.

* OK, there is a section (2.1) where the ideas of tangent lines and velocities are briefly introduced. And then summarily ignored until the end of that chapter. The students typically ignore that material right along with the book.

Here’s a promotional video for a new math curriculum from Pearson called enVisionMATH. (It must be a sign of the times that grade school math curricula have promotional videos.) Watch carefully.

Four questions about this:

1. Should it be a requirement of parenthood that you must remember enough 5th grade math to teach it halfway decently to your kids?
2. Does the smartboard come included with the textbooks?
3. Did anybody else have the overwhelming urge to yell “Bingo!” after about 2 minutes in?
4. When will textbook companies stop drawing the conclusion that because kids today like to play video games, talk on cell phones, and listen to MP3 players, that they are therefore learning in a fundamentally different way than anybody else in history?

The last question is all about the research-free digital nativist assumption that is the source of many lucrative curriculum deals these days. Data, please?

[ht Teaching College Math Technology Blog]

The Everyday Math curriculum has been rejected in the state of Texas. I’ve blogged about Everyday Math and how it attempts to teach multiplication before. But I didn’t know that it had activities like this:

A. If math were a color, it would be –, because –.

B. If it were a food, it would be –, because –.

C. If it were weather, it would be –, because –.

I’m not sure exactly what the point of an exercise like this is — perhaps the curriculum is just trying very studiously not to get too deep into mathematics itself, thereby teaching math without the social stigma of being very enthusiastic about it. Or maybe the idea is to get kids to see math from a different point of view, as a sort of oblique path through math anxiety.

Either way, it’s the wrong approach. The only way to come to terms with math, conquer math anxiety, and appreciate (and learn) the subject is… to get good at it. And that only comes by doing, lots and lots of doing. You replace practice with long division for this stuff, you’re not doing what you ought to be doing. To paraphrase what somebody said a couple of thousand years ago to a similarly math-disaffected person, there is no royal road to understanding arithmetic or algebra, no cutesy affective end-arounds to get out of the hard work of learning.

I think there could be an enormous market in coming years for “alternative” at-home math curricula to counteract the sloppy mess of “modern”, usually NSF-funded, math curricula — and those “alternatives” would look awfully similar to the math texts of the 50’s and early 60’s.