Casting Out Nines

I was looking at the web sites of a few colleges the other day which use a “Great Books” curriculum. This is an approach to a core curriculum in which students work their way through a listing of the great books from the past, across a variety of disciplines. Here’s an example from Thomas Aquinas College, a highly-regarded Catholic liberal arts college in Santa Paula, California. St. John’s College is probably the best-known example; I remember getting a mailer from them when I was a senior in high school, and I was fascinated by the idea of attending a Great Books university at the time.  There are also a few public universities which offer a great books curriculum as an option within the larger curricular structure of the university, for example as part of an honors program. 

Apparently Mortimer Adler is credited with coining the concept of the Great Books, and he gives three criteria for a book to be a Great Book (taken from the Wikipedia article): 

• the book has contemporary significance; that is, it has relevance to the problems and issues of our times;
• the book is inexhaustible; it can be read again and again with benefit;
• the book is relevant to a large number of the great ideas and great issues that have occupied the minds of thinking individuals for the last 25 centuries.

Just a few days after getting this request for introductions to mathematics for someone with an advanced degree in a humanities field, I received another email with a similar request. I’ll just quote from it:

Do you think you might be able to suggest some books or websites, for someone who had no math aptitude, to learn math? It is a great personal sadness to me that I was never able to master the subject. The older I become the more I wish I could understand it!

I was able to do basic arithmetic as a child but became lost beginning with algebra. If there was one thing that stands out in my memory it was staring at a word problem mystified as how to solve it, or staring at an algebra problem not knowing what sort of formula one should apply to solve it. In short, I suppose my greatest liability was the inability to see patterns.  I still believe that math could have been fun and challenging with the right teacher rather than overwhelming. Any suggestions on your part would be greatly appreciated!

This person has just finished graduate school in the arts, and the person’s last contact with math was “a sad dalliance” with a statistics course, which the person failed twice.

I get the sense from this email and the other one that there are actually a lot of people out there who are closet math lovers, or math-curious at least, who would like the chance to become literate in mathematics now that they’ve made it through school. That’s a far cry from the usual anti-math sentiment we math people get all the time from most people we meet.

The comments on the other post were so good, it makes sense to open the comments here for suggestions as well. Fire when ready!

I got an email this afternoon from a reader who is interested in learning mathematics — as an adult, post-college. The reader has an advanced degree in a humanities discipline and never studied mathematics, but recently he’s become interested in learning and is looking for a place to start.

I recommended The Mathematical Experience by Davis and Hirsch, How to Solve It by Polya, and any good college-level textbook in geometry (like Greenberg, or for a humanities person perhaps Henderson). I felt like these three books give an ample and accessible start at — respectively — the big picture and history of the discipline, the methodology of mathematicians, and a first step into actual mathematical content.

But what I thought this was an interesting question, and I wonder if the other readers out there would have similar suggestions for books, articles, movies or documentaries… anything that would be of use to an educated adult learner with little math background but a lot of genuine interest. Leave your suggestions in the comments.

Back in September 2006, I wrote about a new and innovative approach that Georgia Tech was taking towards its computer science curriculum. It appears that this approach, plus an improved job market for computing professionals, is helping turn around a fairly gloomy period for the field:

The Georgia Institute of Technology has revised its computer science curriculum to move away from a traditional hardware-software approach to much more emphasis on the creative process and the roles computer science majors go on to assume in their careers.

Giselle Martin, who directs student recruitment for the College of Computing at Georgia Tech, said that undergraduate applications are up 15 percent this year — in part due to new approaches to explaining the field. One key audience is parents, Martin said. Many remember the horror stories of the job market a few years back and Georgia Tech believes that it can break through that out-of-date mindset most directly with actual employers. So in April, when the college holds a series of events for accepted applicants, there is a panel for parents featuring employers who recruit at Georgia Tech talking about the jobs available and how much demand there is for new graduates.

And there’s this from Virginia Tech:

A new course focuses on problem solving, and several courses are being shifted to focus more on “how to think like a computer scientist,” he said. “We are thinking about how we portray ourselves and what we do,” [Cal] Ribbens [associate department head in computer science] said. “We do not want to be seen as just offering a bunch of programming classes.”

Indeed. There’s a lot of talk going around our campus and at the ICTCM about offering intro courses that focus on problem-solving and the methodology of the discipline, rather than just one little (but deep) slice of content. That certainly seems to make the front door of a major easier to get into. Right now, at least in math, it seems like many students who might do well in a math-related major are either turned off to the subject, or even shut out of it, because their first introduction to math is a technical calculus course, which is almost nothing like what the discipline of mathematics is actually about.

[h/t Inside Higher Ed]

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.

Check out the unofficial course description for Math 55 at Harvard’s math department web site (all emphases added):

This is probably the most difficult undergraduate math class in the country; a variety of advanced topics in mathematics are covered, and problem sets ask students to prove many fundamental theorems of analysis and linear algebra. Class meets three hours per week, plus one hour of section, and problem sets can take anywhere from 24 to 60 hours to complete. This class is usually small and taught by a well-established and prominent member of the faculty whose teaching ability can vary from year to year. A thorough knowledge of multivariable calculus and linear algebra is almost absolutely required, and any other prior knowledge can only help. Students who benefit the most from this class have taken substantial amounts of advanced mathematics and are fairly fluent in the writing of proofs. Due to the necessity of working in groups and the extensive amount of time spent working together, students usually meet some of their best friends in this class. The difficulty of this class varies with the professor, but the class often contains former members of the International Math Olympiad teams, and in any event, it is designed for people with some years of university level mathematical experience. In order to challenge all students in the class, the professor can opt to make the class very, very difficult.

The most difficult undergraduate math class in the country,  taught by faculty whose teaching ability is not necessarily guaranteed, designed for freshmen but requiring several years of university level math experience? I detect a distinct amount of satisfaction from whomever wrote this.

The funny thing is, according to the article I referenced earlier, dozens of Harvard freshmen sign up for the course each year — some with no intention whatsoever of staying on in the course but just wanting to say they’d enrolled in the course and watch the real math people at work.

If you’ve been submitting mathematics articles to refereed journals only to have them sent back to you every time, there’s hope. You can try submitting them to the new journal Rejecta Mathematica, which will consist only of papers which have been rejected from peer-reviewed journals. From their web site:

At Rejecta Mathematica, we believe that many previously rejected papers can nonetheless have a very real value to the academic community. This value may take many forms:

• “mapping the blind alleys of science”: papers containing negative results can warn others against futile directions; 
• “reinventing the wheel”: papers accidentally rederiving a known result may contain new insight or ideas; 
• “squaring the circle”: papers discovered to contain a serious technical flaw may nevertheless contain information or ideas of interest;
• “applications of cold fusion”: papers based on a controversial premise may contain ideas applicable in more traditional settings;
  
• “misunderstood genius”: other papers may simply have no natural home among existing journals.

Rejecta articles also allow the authors to speak out in defense of their rejected articles and include an open letter from the authors describing any known flaws in the paper.

And yes, although there’s no formal peer review process to get a paper into Rejecta, you can still have a paper submission rejected.

[ht Math-Blog]

We’ve got just 4-5 weeks left in the semester and until the textbook-free Modern Algebra course will draw to a close. It’s been a very interesting semester doing the course this way, with no textbook and a primarily student-driven class structure. In many ways it’s been your basic Moore Method math course, but with some minor alterations and usage of technology that Prof. Moore probably never envisioned.

As I mentioned in this lengthy post on the design of the course, students are doing a lot of the work in our class meetings. We have course notes, and students work to complete “course note tasks” outside of class and then present them in class for dissection and discussion. The tasks are either answering questions posed in the notes (2 points), working out exercises which can be either short proofs or illustrative computations (4 points), or proving theorems (8 points). We have a system for choosing who presents what at the board — I won’t get into the details here, but I can do so if somebody asks for it in the comments.

So the class meetings consist almost entirely of students presenting work at the board, where their responsibility is to make their work clear, correct, complete, and coherent — and ruggedized against the questions that I inevitably throw at them.

I was thinking yesterday that this method of doing class has really done a lot of good for the students in the class, in several key ways.

• Students ultimately rely upon the soundness of their own work. The students can work with others or with print or electronic resources — although with no textbook, they have to learn how to find those resources and tell the good ones from the bad ones, which is a great skill by itself. But it boils down to presenting that work, on your own and with nobody there to bail you out, in front of your professor and peers. I think this is a good antidote to the occasional over-reliance on cooperative learning that we (in education as a whole, and in my department) have. Group work is all well and good, but to be a complete learner you have to be able to rely on your wits and your skills and not just prop yourself up on the strength of peers.
• Students prepare for class in advance, several days in advance, every night. To do reasonably well on course note tasks, students need to plan on successfully completing 15-20 course note tasks throughout the semester, which comes out to about 1-2 per week. Combine that with the fact there are 8 students in the class all trying to do this, and it’s easy to see that working ahead is really essential. You want to get so far out in front of the class that you have no competition for a particular range of problems. Very often in college, there is no sense that you have to get ready for class the next day — unless there’s an assignment due — and we profs reinforce this by running classes that do not penalize the lack of preparation. (It’s not enough to reward the presence of preparation.) The course design here, though, rewards the students who have read and practiced ahead and learned on their own.
• Students become skeptical and tough-minded about their own work. It’s quite common in traditional math courses for students completing an assignment to simply barf up something on a piece of paper, hand it in, and see how many points it gets. When you are presenting work before a class, that route leads only to embarrassment. When most of the class time is spent doing these presentations, students learn something I didn’t learn until graduate school — that if you are going to hand something in or present something with your name attached to it, you had better make very sure that it works. I’ve noticed the students anticipating not only the fact that I will be asking them penetrating questions about what they are presenting, but also what those questions are. At that point they are learning to think like mathematicians.
• Students pay (more) attention to detail, especially terminology and the sensibility of a proof. It’s easy to write a proof or a solution to a problem that has no coherence or sense to it at all — but that incoherence and senselessness vanishes the moment you do something as simple as reading the solution aloud. Which is what these folks are doing every day. Example: A colleague told me a story of a student who was asked whether or not two groups G and G’ were isomorphic. The student answered, “G is isomorphic, but G’ isn’t.”
• Students base their confidence on the math itself, not on an external authority. Students aren’t allowed to ask me “Is this right?” or “Am I on the right track?” To clarify, they can ask me those questions, but I will only greet them with more questions — mainly, “What justifies this step?” or “How do you know this?” It’s not about me or what I like or what makes me happy with regards to their work — it’s about whether each step of the proof follows logically from the one before it, and whether that logical connection is clearly validated. Students know pretty well now when they have got something right and when they don’t, and if they don’t have it right they have a better sense of what’s missing or incorrect and what they need to do to fix it.

A lot of these effects I’m describing are just embodiments of what it takes to be successful in math after calculus in the first place.