Category Archives: education

Singapore Math

CJ’s school is using it — this already comes as a surprise, since I have heard of it only in the context of “I hate my school’s discovery-based math curriculum and I’m supplementing with Singapore Math” and didn’t know it was actually used in schools.

The second surprise, given that context, is that Singapore Math is not brutally drill-oriented; in fact, I’d call it fairly balanced.  No flashcards, no pages with 100 problems all essentially the same.  Instead, when students are asked what 3×5 is, there’s a 3 x 5 rectangular array of stars drawn next to the problem so that the kid can see what the expression 3 x 5 actually means, not just what it evaluates to.  Similarly, kids are asked to color in 86 boxes in a 10×10 grid (the point being to learn that you can do this by filling in 8 whole columns and then 6 more.)  And with all this, there’s enough practice with basic arithmetic operations to build fluency and speed.  (Educational equity klaxon:  but of course I’ve already done a lot to teach CJ to be fluent and fast in arithmetic operations, because that’s what parents with cultural capital do.  (Subparenthetical: note that thanks to Peli Grietzer I now know the difference between cultural capital and social capital!))

Sometimes people criticize a curriculum for teaching students to do computations mindlessly, but I’m OK with that — mindlessness is a skill you need in life.  The right goal is to be able to do a computation either mindlessly or mindfully, as the situation demands.

Side question:  would Singapore Math have the cachet it does if it weren’t named for a country that’s both authoritarian and East Asian?  What if it were:

  • non-authoritarian, East Asian:  Korea Math?
  • non-East Asian, authoritarian:  Moscow Math?
  • non-East Asian, non-authoritarian:  Israel Math?

 

 

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Ten-second tragedy

Two undergrads sitting against the wall this morning outside Ingraham Deli.

First undergrad:  “So, what are you going to do with your life?”

Second undergrad:  “Good question!  (very long pause)  I think I’m applying to journalism school?”

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Historical textbook collection

I’m working in the math department library today and have gotten distracted by a collection of historical math textbooks that’s just gone on the shelves.

From College Mathematics:  A First Course (1940), by W. W. Elliott and E. Roy C. Miles:

The authors believe that college students who take only one year of mathematics should acquire a knowledge of the essentials of several of the traditional subjects.  From teaching experience, however, they are convinced that a better understanding is gained if these subjects are presented in the traditional order.  Students who take only one year of college mathematics are usually primarily interested in the natural sciences or in business administration.

The book covers algebra, trigonometry, Cartesian geometry, and calculus.  The definition of the derivative as a limit is given, but the epsilon-delta definition of limit is not.  Startling to think that science majors came to college never having taken algebra or analytic geometry.

Further back in time we get Milne’s Progressive Arithmetic, from 1906.  This copy was used by Maggie Rappel, of Reedsville, WI, and is dated January 15th, 1908.  Someone — Maggie or a later owner — wrote in the flyleaf, “Look on page 133.”

On the top of p .133 is written

Auh!  Shut up your gab you big lobster, you c?

You got me, Maggie!

I can’t tell what grades this book is intended for, but certainly a wide range; it starts with addition of single digits and ends with reduction of fractions to lowest terms.  What’s interesting is that the book doesn’t really fit our stereotype that math instruction in olden times was pure drill with no attention paid to conceptual instruction and explanation.  Here’s a problem from early in the book:

How many ones are 3 ones and 4 ones?  Write the sum of the ones under the ones.  How many tens are 6 tens and 2 tens?  Write the sum of the tens under the tens.  How do you read 8 tens and 7 ones?  What, then, is the sum of 24 and 63?  Tell what you did to find the sum.

From the introduction:

Yet the book is not merely a book of exercises.  Each new concept is carefully presented by questions designed to bring to the understanding of the pupil the ideas he should grasp, and then his knowledge is applied.  The formal statement of principles and definitions is, however, reserved for a later stage of the pupil’s progress.

Would these sentiments be so out of place in a contemporary “discovery” curriculum?

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Paul Ryan, science, money, television

From ScienceInsider’s summary of Paul Ryan’s approach to the federal spending on science:

“Instead of using its resources to fight life-threatening diseases like HIV/AIDS and cancer, the CDC has instead spent money on needless luxury items and nongovernment functions,” Ryan said in introducing his amendment to a spending bill. CDC had spent “over $1.7 million on a ‘Hollywood liaison’ to advise TV shows like ‘E.R.’ and ‘House’ on medical information included in their programming, clearly an expense that should have been covered by the successful for-profit television shows, not by our hard-earned tax dollars. … In a time when we are facing increasing risk of bioterrorism and disease, these are hardly the best use of taxpayer dollars.”

“E.R” and “House” are surely seen by vastly more Americans than all federal science education programming put together.  Doesn’t $1.7m  sound pretty cheap for ensuring that the medical information coming through those giant megaphones is correct?  In Ryan’s world, what’s the mechanism under which TV producers would spend their own money doing this?  Their own goodwill?  Or will scientifically sloppy doctor shows inevitably be rejected by the wise aggregate consumer, so that the market does the job for free?

I also think it’s not fair to ask that every US-funded program be the best use of taxpayer dollars.  I mean, do we really want a federal government that consists entirely of my NSF grant?

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David Mumford says we should replace plane geometry with programming and I’m not sure he’s wrong

MAA Mathfest is in Madison this week — lots of interesting talks about mathematics, lots of interesting talks about mathematics education.  Yesterday morning, David Mumford gave a plenary lecture in which he decried the lack of engagement between pure and applied mathematics — lots of nodding heads — and then suggested that two-column proofs in plane geometry should be replaced with basic programming — lots of muttering and concerned looks.

But there’s something to what he’s saying.  The task of writing a good two-column proof has a lot in common with the task of writing a correct program.  Both ask you to construct a sequence that accomplishes a complicated goal from a relatively small set of simple elements.  Both have the useful feature that there’s not a unique “correct answer” — there are different proofs of the same proposition, and different programs delivering the same output.  Both quite difficult for novices and both are difficult to teach.  Both have the “weakest link” feature:  one wrong step means the whole thing is wrong.

Most importantly:  both provide the training in formalization of mental process that we mathematicians mostly consider a non-negotiable part of general education.

But teaching coding instead of two-column proofs has certain advantages.  I am not, in general, of the opinion that everything in school has to lead to employable skills.  But I can’t deny that “can’t write five lines of code” closes more doors to a kid than “can’t write or identify a correct proof.”  People say that really understanding what it means to prove a theorem helps you assess people’s deductive reasoning in domains outside mathematics, and I think that’s true; but really understanding what it means to write a five-line program helps you understand and construct deterministic processes in domains outside a terminal window, and that’s surely just as important!

Computer programs are easier to check, for the teacher and more importantly the student — you can tell whether the program is correct by running it, which means that the student can iterate the try-check-fail-try-again process many times without the need for intervention.

And then there’s this:  a computer program does something.  When you ask a kid to prove that a right triangle is similar to the triangle cut off by an altitude to the hypotenuse, she may well say “but that’s obvious, I can just see that it’s true.”  And she’s not exactly wrong!  “I know you know this, but you don’t really know this, despite the fact that it’s completely clear” is a hard sell, it devalues the geometric intuition we should be working to encourage.

 

 

 

 

 

 

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Nobody likes us

When Ted Turner told his father he was majoring in classics, he got an angry letter in response.  The senior Turner had little patience for anything theoretical, academic, or abstract:

There is no question but this type of useless information will distinguish you, set you apart from the doers of the world. If I leave you enough money, you can retire to an ivory tower, and contemplate for the rest of your days the influence that the hieroglyphics of prehistoric man had upon the writings of William Faulkner. Incidentally, he was a contemporary of mine in Mississippi. We speak the same language—whores, sluts, strong words, and strong deeds.

It isn’t really important what I think. It’s important what you wish to do with your life. I just wish I could feel that the influence of those oddball professors and the ivory towers were developing you into the kind of a man we can both be proud of. I am quite sure that we both will be pleased and delighted when I introduce you to some friend of mine and say, “This is my son. He speaks Greek.”

I had dinner during the Christmas holidays with an efficiency expert, an economic adviser to the nation of India, on the Board of Directors of Regents at Harvard University, who owns some 80,000 acres of valuable timber land down here, among his other assets. His son and his family were visiting him. He introduced me to his son, and then apologetically said, “He is a theoretical mathematician. I don’t even know what he is talking about. He lives in a different world.” After a little while I got to talking to his son, and the only thing he would talk to me about was about his work. I didn’t know what he was talking about either so I left early.

If you are going to stay on at Brown, and be a professor of Classics, the courses you have adopted will suit you for a lifetime association with Gale Noyes. Perhaps he will even teach you to make jelly. In my opinion, it won’t do much to help you learn to get along with people in this world. I think you are rapidly becoming a jackass, and the sooner you get out of that filthy atmosphere, the better it will suit me.

And here’s Ben Franklin, talking about a member of his philosophical society, the Junto, who just couldn’t manage to fit in:

THOMAS GODFREY, a self-taught mathematician, great in his way, and afterward inventor of what is now called Hadley’s Quadrant. But he knew little out of his way, and was not a pleasing companion; as, like most great mathematicians I have met with, he expected universal precision in everything said, or was forever denying or distinguishing upon trifles, to the disturbance of all conversation. He soon left us.

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Should Harvard offer a “good enough, but no room” certificate?

There are people who think that the information conveyed by a Harvard diploma is almost entirely made up of the fact of admission to Harvard; that is, that Harvard graduates on average have no more skills than students who got into Harvard but chose to go somewhere else.

I’m not one of those people.  But it got me thinking — the fact of admission certainly conveys some information.  And there are unquestionably lots of students who the admission office feels are academically strong enough to attend Harvard, but who don’t make it into the entering class.

What would happen if the admissions office offered exactly this certification?  A signed piece of paper saying, “At age 17, student X had credentials which would have made academic success at Harvard very likely, had there been room.”  Would that be a valuable piece of paper for a 22-year-old to have?  Would it be in Harvard’s interest to offer a certain number of certificates of that kind?

Related question:  can a student who gets into Harvard, but goes to a lower-ranked school (say, for financial or family reasons) put on their CV that they were admitted to Harvard, but declined?  Something about that strikes me as strange.  But why?  Isn’t it useful information for a potential employer?

(Note:  obviously the above applies with any elite university in place of “Harvard.”)

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Reader survey: Have you advised — could you advise? a Ph.D. student online? Or have you been so advised?

Jason Starr asks a great  question in the comments to the previous post:  if you are a Ph.D. advisor, to what extent do you think you could advise a graduate student who you rarely or never physically met?  If you’re a graduate student, to what extent do you think you could thrive if you rarely or never saw your advisor in person?

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Online education and creative writing workshops

More about on-line education.  One hesitation people have, of course, is that it’s easier to dephysicalize some forms of education than others; and that if higher education gets redefined as something that happens online, the parts of higher education that don’t survive that transition get redefined as “not part of higher education.”

But what about creative writing workshops?  Right now, these sit somewhat uncomfortably inside English departments in universities.  What are you paying for when you pay tuition to attend a fiction workshop?  (I was lucky enough to go to a program with funding, but I think most MFAs don’t work this way.)  I think you’re paying to  have a known novelist read and think carefully about what you’re writing, and you’re paying to create some official sense that This Is The Year I Write My Novel.  (This last part might be the most important.  Of course, you could write your novel any time!  But having paid a great deal of money with the intent of doing a thing focuses the mind on the task extremely well.  Freud always said this was why he charged so much; he didn’t need the money, but the patients needed to spend it.)

What happens if a novelist decides to offer a writing workshop via Google Hangout, to 12 people, charging them much less than university tuition but enough to meet his expenses?  Like, say, $3K a person?  Does that work?  Or, since most novelists probably don’t care to run their own small business, what happens if a startup company collectes well-known but poorly paid novelists and runs the marketing/payment processing side of things, in exchange for a cut?

It’s not clear this is interestingly different from existing distance MFAs like Warren Wilson.  Certainly I don’t think you can scale up the offering of “serious and admired writer X read my work closely” to hundreds of thousands of people, which I suppose is a reason it might continue being possible to charge serious money for the service.

An online workshop wouldn’t reproduce what I got out of my MFA program at Johns Hopkins, but I was a special case.  I was on break between college and graduate school, I was pretty sure I was going to be a mathematician my whole life, and I really needed to be something else for a year.  The people I saw every day that year were writers, the professors whose opinions I valued were writers, the people I drank beer with and argued with and dated were writers.  And by the end of the year I was able to call myself a writer without feeling like I was half-kidding; not because I’d written a draft of my novel but because I’d lived in Writerworld for a year.

 

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Several attacks on the previous post

As promised, a few attacks.  I’m sure by tomorrow I’ll have thought of several more.  Oh, and also, I meant to link to this Crooked Timber thread about Coursera, with a richly combative comment thread.

  • I, along with lots of other people who succeed in traditional schools, love text and process it really fast.  Other people like other media.  Streaming video isn’t the same thing as talking to another person, but it’s plainly closer than text, and talking to another person is the way we’re built to take in information.  If streaming video weren’t a useful means of educational transmission for a substantial fraction of people, Khan Academy wouldn’t be popular.
  • Some people would say that we could get by with many fewer scientists that we have now, without compromising the amount of meaningful science that gets done.  That seems too simple to me, but I just want to record that it’s a belief held by many, and on that account maybe a small NSF-funded garden of science is sufficient to our needs.
  • Online credentials, whether from Udacity or future-ETS, could in principle lead to a massive gain in global equality.  Nothing is stopping 300 people from China and Brazil from being among the 500 people Google hires.  I was going to say the same thing about inequality within the US but here I have to stop myself; my sense is that massive availability of online resources has not e.g. made it just as good to be a 14-year-old math star in Nebraska as it is to be a 14-year-old math star in suburban Boston.  Reader comments on this point welcome, since I know there are lots of former 14-year-old math stars out there.
  • More on within-US equality; it’s easy to see gains flowing to kids whose parents are rich enough to buy them a prep course or just buy them the time to spend a year at home studying.  On the other hand, this seems no less rich-kid-friendly than the current system, in which kids whose parents can afford college graduate debt-free, and the rest, who still have little choice but to attend if they want professional jobs, spend decades of their working life chipping away at a massive debt.
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