What does academic rigor look like?

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!]

Filed under: Calculus, Education, Geometry, High school, Teaching , academics, Calculus, Geometry, Math, math education, rigor

Try this at your school’s next holiday play

If you’re wondering why India is earning a reputation for outpacing the rest of the world in math and science, here’s a data point:

Students of the St Michaels Primary School in Mahim celebrated Children’s Day in a unique way. Instead of reciting poems and participating in fancy dress competitions, these students stumped their parents with their expertise with numbers and logical reasoning.

Intelligence enhancement programme, the brainchild of school’s manager Fr Hugh Fonseca, saw children babble out Vedic mathematic formulas and do complex calculations in a flip second with ease.

“It was wonderful to see my child go up onto the stage and fearlessly rattle out those numbers in front of a huge audience without hesitation,” said Naseem Sheikh, whose son Maqdoom recited skip numbers both forwards and backwards. Echoing her sentiments was Uday Babu, an engineer. “My son Ganesh is a lot more confident now.”

The programme, which began two months ago, entailed training the students in Chess and Vedic mathematics. “The emphasis was on all-round development of the child and not just imparting bookish knowledge. We want to mould the young minds as early as possible, hence the programme is for the Class I students for now,” said C Raji, a teacher training the students in Chess and Vedic mathematics.

Can you imagine this kind of thing being tried in an American school?

Actually, I could see this kind of thing happening, and I think many kids would like it, especially the chess part. But I think a vocal plurality of parents and administrators would freak out — the parents because math and chess are nerd things and therefore unnatural and sinister; the administrators because they want to make parents happy. If that semi-cynical assessment is correct, then it’s a sad statement about our culture, and remember that it’s the culture, stupid.

Filed under: Early education, Education, Math, Student culture , Student culture, Math, mathematics, culture, india, children's day, vedic mathematics

Shortages in SMET fields: Not just for Americans

The Australians are also facing critical shortages of students choosing to study science, math, engineering, and technology (SMET) fields:

“It is no exaggeration to say that the relative decline in the science, technology, engineering and mathematics capability and literacy of South Australian school students is a very serious situation that requires decisive remedial action by the government,” said Engineers Australia state president Bill Filmer.

“There is an urgent need for reprioritisation in schools, staffing and curricula to overcome this problem to enable South Australia to be more competitive in the knowledge-based economy.”

The report also identified the lack of training in science given to primary school teachers as a key issue and questioned their commitment to teaching science. [Emphasis added]

As to that last sentence above, insofar as I can understand the teacher education curriculum in Australia from a little bit of Googling, the curriculum for primary teachers does seem awfully lightweight on the math and science end. The curriculum at the University of South Australia has students take a three-course sequence in “Studies in Science, Mathematics, and Society and Environment Education”, and the course descriptions go like this:

[For the first course] This course engages students with constructivist perspectives of student learning; social constructivist pedagogies including interactive approaches to teaching; thinking and working mathematically scientifically, environmentally and socially from socially inclusive and critical perspectives; planning for learning in mathematics, science and society and environment; key concepts embedded in sorting and classifying, pattern, number, living things, interdependence and ecologically sustainable components of Years 3 to 9 curriculum.

[Second course] This course engages students with constructivist perspectives of student learning and focuses on interactive approaches to teaching and student questions; thinking and working mathematically, scientifically, socially and environmentally from socially inclusive and critical perspectives; planning for learning in Mathematics, Science and SOSE; student centred inquiry; equity (fair trade) governance (political, social and economic systems); key concepts embedded in spatial sense and geometric reasoning, energy systems, matter and fair tests, personal footprints, democratic participation and poverty as aspects of the Years 3 – 9 curriculum.

[Third course] This course engages students with constructivist perspectives of student learning and focuses on interactive approaches to teaching and student questions; thinking and working mathematically, scientifically, socially and environmentally from socially inclusive and critical perspectives; planning for learning in Mathematics, Science and SOSE; student centred inquiry; the SOSE value of social justice and equity through refugees and Indigenous Australians; key concepts embedded in measurement, earth systems (soils and weather), plant and animal relationships as aspects of the Years 3 – 9 curriculum.

Like I said, it seems light on the actual science and math content, but the students will certain get lots of social justice issues and a bias towards constructivism as the religion pedagogy of choice. Perhaps I don’t understand Australian culture as I should, but if I were a student being taught by someone thoroughly drilled in this kind of thing, I probably wouldn’t like math or science either. And if I were a teacher who wanted to teach math and science because, well, I really liked math and science, I would be a little put off by the back seat that the actual disciplines take to all this constructivism and social justice stuff.

It would be interesting to take the countries who are having these kinds of problems in one column, and the countries that are eating our lunch in SMET fields in the other column, and compare how science and math teachers are trained in each column.

Filed under: Education, Math, Teaching , australia, Math, Science, smet, Teaching

Straight talk on constructivism

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.

Filed under: Math, Teaching , constructivism, direct instruction, Math, math education

It’s Tricki

Tim Gowers gives a lengthy report here on the development of the Mathematical Tricks Wiki, which he is now calling the Tricki. The Tricki will be a wiki/database of mathematical problem solving techniques that will, if development proceeds, eventually be something like an expert system that mimics how a human mathematician’s brain works when solving problems. In Gowers’ words:

The main content of the Tricki will be a (large, if all goes according to plan) body of articles about methods for solving mathematical problems. Associated with these articles will be many qualities that will vary substantially from article to article. For example, some will be about very general problem-solving tips such as, “If you can’t solve the problem, then try to invent an easier problem that sheds light on it,” whereas others will be much more specific tips such as, “If you want to solve a linear differential equation, you can convert it into a polynomial equation by taking the Fourier transform.” Some articles will be written at a very elementary level, and some will be quite advanced… Some will concern particular areas of mathematics, such as algebraic geometry or probability, whereas others will concern techniques that are relevant to many different areas. And so on.

Gowers thinks that the Tricki could revolutionize the way mathematics is done in much the same way that or ArXiV have done, and I think he’s right. Read the whole thing for some fascinating ideas on how he envisions the navigation and human-computer interaction of this system.

Filed under: Math, Problem Solving , gowers, Math, mathematics, timothy gowers, tricki

Technology in proofs?

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.)

Filed under: Computer algebra systems, Education, Grading, Math, Problem Solving, Teaching , Math, math education, mathematics, proof, Technology

It’s official: They’re prime

The numbers believed to be the 45th and 46th Mersenne primes have been proven to be prime. The 45th Mersenne prime is and the 46th is .Full text of these numbers is here and here.

Of course what you are really wanting to know is how my spreadsheet models worked out for predicting the number of digits in these primes. First, the data:

• Number of digits actually in : 11,185,272
• Number of digits actually in : 12,978,189

My exponential model () was, unsurprisingly, way off — predicting a digit count of over 24.2 million for and over 35.8 million for . But the sixth-degree polynomial — printed on the scatterplot at the post linked to above — was… well, see for yourself:

• Number of digits predicted by 6th-degree polynomial model for : 11,819,349
• Number of digits predicted by 6th-degree polynomial model for : 13,056,236

So my model was off by 634,077 digits — about 6% error — for . But the difference was only  78,047 digits for , which is only about 0.6% error. That’s not too bad, if you asked me.

There’s only one piece of bad news that prevents me from publishing this amazing digit-count predicting device, and you can spot it in the graph of the model:

So evidently the number of digits in will max out around and then the digit count will begin to decrease, until somebody discovers , which will actually have no digits whatsoever. Um… no.

Filed under: Crypto, Geekhood, Math , Math, mathematics, mersenne, mersenne prime, prime, prime numbers

Estimating the digits in a Mersenne prime — for dummies

At the end of this post, I made a totally naive guess that the recently discovered candidate to be the , the 45th Mersenne prime, would have 10.5 million digits. There was absolutely no systematic basis for that guess, but I did suggest having an office pool for the number of digits, so what I lack in mathematical sophistication is made up for by my instinct for good nerd party games. On the other hand, Isabel at God Plays Dice predicted 14.5 million digits based on a number theoretic argument. Since I am merely a wannabe number theorist, I can’t compete with that sort of thing. But I can make up a mean Excel spreadsheet, so I figured I’d do a little data plotting and see what happened.

If you make a plot of the number of digits in , the nth Mersenne prime, going all the way back to antiquity, here’s what you get:

The horizontal axis is n and the vertical axis is the number of digits in .

Admit it — one look at this plot and you’re itching to add some trendlines. Here’s what you get when you add both an exponential trendline (perhaps the obvious choice given the shape) and a 6th-degree polynomial:

The exponential one has a higher value, but that’s perhaps misleading because of the really good fit for all those low-digit Mersenne primes that happened prior to around . We’ll take that issue up in a moment. But for now, let’s put those trendline equations to work. The exponential trendline would predict that would have a digit count of

which is obviously rather a lot more than either my prediction or Isabel’s; and if you put in into the 6th-degree polynomial, you get a digit count of 11819349, which is in the ballpark of both my rough estimate and Isabel’s estimate.

It doesn’t make much sense, though, to include all Mersenne primes, since Mersenne primes didn’t even cross the 100-digit mark until in 1952. A more accurate idea — if you can call this kind of reasoning accurate in the first place — would be to run the numbers starting at around and seeing what we get. I’ll save that for later, unless somebody wants to beat me to it.

Filed under: Crypto, Geekhood, Math , excel, Math, mersenne, mersenne prime, number theory

New Mersenne prime discovered?

GIMPS is reporting that on 23 August a new Mersenne prime was reported to their server. Verification began today and should take about two weeks to complete. No word on what the prime was, how many digits, etc.

The last Mersenne prime discovered was , back in 2006 (blogged about here) and weighed in at a whopping 9,808,358 digits. Any bets on how big this new one is, if it’s really a prime? I’m guessing 10.5 million digits. Sounds like a good occasion for a nerd office pool.

Update: Isabel at God Plays Dice likes 14.5 million digits instead, and she’s actually using math and stuff to make that estimate instead of just shooting totally in the dark like I am.

Filed under: Math , GIMPS, Math, mathematics, mersenne, number theory, prime

Wednesday morning links

• Walking Randomly has an interesting discovery about the Fibonacci sequence and linear algebra.
• The Productive Student offers up some advice on how to be a leader and conduct killer team sessions. It’s good stuff not only for students who are doing collaborative work but also for anybody who goes to meetings. Are there people who don’t have to go to meetings?
• InsideHigherEd reports on an interesting setup to attract Chinese students to study in the US — the 1+2+1 degree, which involves one year in China, two in the US, and then the final year back in China. (Unfortuately, as the article notes, you can’t Google “1+2+1″ because all you get is “4″.)
• Also at IHE and a lot of other places, Rice University is now using an open textbook for its elementary statistics course which is not only free but open for rearrangement and adaptation by any user. A shot across the bow of traditional textbook companies?
• Study Hacks offers advice to students on cutting out the single biggest source of stress (according to them) — the killer course load. There’s something to be said for having an unbalanced course schedule — I found grad school to be easier in some ways than college because I was only taking math courses — but I do remember the worst semester I ever had as an undergrad had me taking three senior-level math courses… plus German, orchestra, and concert choir, with a 20-hour work week at a donut shop to boot. Balance is important.
• Reasonable Deviations writes about a hack of the Boston subway system by three MIT students (for what appear to be purely academic purposes). Predictably, the subway authorities have sought legal action against the students. They ought instead to be thanking them, or hiring them outright, for pointing out a security flaw that eventually could have cost the company millions.
• Java is the most popular programming language in the world, but some are saying that using Java as the language of choice in intro programming courses (as is currently done, for example, in the standard AP Computer Science courses) is hurting students in the long run. To me, this article raises the question of just what computer programming is these days.
• A new Zogby poll is indicating that online university programs (meaning, it seems, online programs offered by existing brick-and-mortar universities as opposed to online universities) are rapidly gaining mainstream acceptance, despite the perception (possibly justified) that they offer less academic rigor than traditional university programming. Unfortunately you have to drill down into the Chronicle article mentioned at the above link to discover that the Zogby poll was administered online! So much for unbiased sampling. But at any rate, the trend seems to be limited mainly to older adults who are looking for college coursework, which makes sense. I think if you restricted the polling to a traditional college population — for example, high school seniors who are looking at colleges to attend — I don’t think you’d see nearly as much of a trend toward online programming.

Filed under: Education, Higher ed, Life in academia, Links, Math, Student culture, Teaching, Technology , higher education, Math, mathematics, Higher ed, fibonacci, online university, china, group work, subway hack, java