What mathematics should any well-educated person know? It’s rather rare that people have a chance not just to think about this question, but do something about it. But it’s happening now.
There’s a new college called Yale-NUS College starting up this fall in Singapore, jointly run by Yale College and the National University of Singapore. The buildings aren’t finished yet: the above picture shows how a bit of it should look when they are. Faculty are busily setting up the courses and indeed the whole administrative structure of the university, and I’ve had the privilege of watching some of this and even helping out a bit.
It’s interesting because you usually meet an institution when it’s already formed—and you encounter and learn about only those aspects that matter to you. But in this case, the whole institution is being created, and every aspect discussed. And this is especially interesting because Yale-NUS College is designed to be a ‘liberal arts college for Asia for the 21st century’.
As far as I can tell, there are no liberal arts colleges in Asia. Creating a good one requires rethinking the generally Eurocentric attitudes toward history, philosophy, literature, classics and so on that are built into the traditional idea of the liberal arts. Plus, the whole idea of a liberal arts education needs to be rethought for the 21st century. What should a well-educated person know, and be able to do? Luckily, the faculty of Yale-NUS College are taking a fresh look at this question, and coming up with some new answers.
I’m really excited about the Quantitative Reasoning course that all students will take in the second semester of their first year. It will cover topics like this:
• innumeracy, use of numbers in the media.
• visualizing quantitative data.
• cognitive biases, operationalization.
• qualitative heuristics, cognitive biases, formal logic and mathematical proof.
• formal logic, mathematical proofs.
• probability, conditional probability (Bayes’ rule), gambling and odds.
• decision trees, expected utility, optimal decisions and prospect theory.
• sampling, uncertainty.
• quantifying uncertainty, hypothesis testing, p-values and their limitations.
• statistical power and significance levels, evaluating evidence.
• correlation and causation, regression analysis.
The idea is not to go into vast detail and not to bombard the students with sophisticated mathematical methods, but to help students:
• learn how to criticize and question claims in an informed way;
• learn to think clearly, to understand logical and intuitive reasoning, and to consider appropriate standards of proof in different contexts;
• develop a facility and comfort with a variety of representations of quantitative data, and practical experience in gathering data;
• understand the sources of bias and error in seemingly objective numerical data;
• become familiar with the basic concepts of probability and statistics, with particular emphasis on recognizing when these techniques provide reliable results and when they threaten to mislead us.
They’ll do some easy calculations using R, a programming language optimized for statistics.
Most exciting of all to me is how the course will be taught. There will be about 9 teachers. It will be ‘team-based learning’, where students are divided into (carefully chosen) groups of six. A typical class will start with a multiple choice question designed to test the students understanding of the material they’ve just studied. Then the team will discuss their answers, while professors walk around and help out; then they’ll take the quiz again; then one professor will talk about that topic.
This idea is called ‘peer instruction’. Some studies have shown this approach works better than the traditional lecture style. I’ve never seen it in action, though my friend Christopher Lee uses it in now in his bioinformatics class, and he says it’s great. You can read about its use in physics here:
• Eric Mazur, Physics Education.
I’ll be interested to see it in action starting in August, and later I hope to teach part-time at Yale-NUS College and see how it works for myself!
At the very least, it’s exciting to see people try new things.
Azimuth 
Great post! I am a physics professor at a small liberal arts college on the East Coast (of the US, which shouldn’t need specifying). We are constantly concerned with just this problem, i.e., what skills/concepts should our students engage in order to build competency in (basic) mathematical/quantitative reasoning. Our “student learning outcomes” (probably trade jargon, but I’m not completely sure) are intended to produce young people who can assess information and interpret the world around them in order to form their own opinions. Really, we are concerned with this question across all of the disciplines contained in our curriculum, but the issue is especially contentious in mathematics.
I have been amazed in recent months about the backlash against such ideas. Perhaps some of you were unlucky enough to hear this piece on NPR’s Diane Rehm Show:
http://thedianerehmshow.org/shows/2012-08-29/algebra-necessary/transcript
Listen at your own peril.
[Also, while googling for the link above, I found the following page:
http://homeschooling.penelopetrunk.com/2012/08/16/5-reasons-why-you-dont-need-to-teach-math/
Um, enjoy?]
Anyway, to get back to the point, our stumbling block has always been delivery mechanism. Do students who pass Calculus 1 (thus fulfilling the school’s math requirement) really engage the concepts of innumeracy, probability, uncertainty, visual display of quantitative information? I would answer “No, not at all, please don’t be silly.” A survey course such as the one described in this post would be delightful (definitely for instructors, and probably also for students), but dedicating nine teachers to a single class would require significant redistribution of teaching loads.
To my mind, though, if this is the technique that best serves students, the it is the technique that we must use. Thanks for posting, John; this is great food for thought.
Thanks for the comments! I should add that at Yale-NUS it’s not required of all students that they learn calculus—though I expect a lot of the high school students entering this college will have already studied some. The topics in the Quantitative Reasoning were deemed more essential. And I think that’s right.
I couldn’t agree more. And I should specify: our students are not required to take calculus, but they are required to complete some course deemed to have a quantitative reasoning component. Many of them, because they’ve studied calculus in high school, opt to complete this requirement by simply taking the next calculus course (but most simply take Calculus 1).
As the token homeschooler around here, I’d like to note for the record that Penelope Trunk enjoys taking contrary positions without actually knowing anything about what she’s talking about. People who follow her figure that out quickly enough. I’d be more worried about E.O. Wilson than Penelope Trunk.
“What should a well-educated person know, and be able to do?” 1) Deter government, 2) deter social activism, 3) deter Enviro-whinerism, 4) deter religion, 5) deter advertising, 6) chose a good combat rifle and its ammo; 6) read, write, math through trigonometry, reason; 7) create and produce, 8) reproduce, 9) be responsible, 10) have fun, and avoid being caught succeeding at it.
A society inundated with rules from religious and secular political classes, ignoring productive ends, will collapse. Flashy mediocrity holds a vast audience diligently converting glitter into a tyranny of immersive falsehoods. God save us from the congenitally inconsequential.
The Yale-NUS course seems to have a methods focus (or at least the description only talks about methods, although I’m sure they must plan to cover applications). Some liberal arts curricula have an applications focus, with the methods playing a supporting role. One example is the textbook For All Practical Purposes. It looks at politics (voting systems / Arrow’s theorem, apportionment / districting), law (fair division), management science (scheduling, optimization), and economics, as well as basic methods like game theory, statistics, etc. The COMAP organization behind the book is interesting too. They try to develop curricula for math methods and math modeling in an applied context from elementary school through college. We used some of this in a math modeling course I took in high school.
Yes, the Yale-NUS quantitative reasoning class is organized by methods, though they’ll all be illustrated with lots of examples—for example, there will be contests to find the worst examples of flawed quantitative reasoning in newspaper articles, and the students will have projects to take data and convert into good newspaper articles, and so on.
This talk “The top ten things that math probability says
about the real world” by David Aldous
Click to access talk.pdf
may be of interest. It includes a list of probability-related questions asked by ‘random’ people on the internet:
Query: chance of pregnancy on pill
Query: how to improve chance of getting pregnant
Query: chance of getting pregnant at age 41
Query: chance of getting pregnant while breastfeeding
Query: can you increase your chance of having a girl
Query: if twins run in my family what’s my chance of having them?
Query: does a father having diabetes mean his children have a 50% chance of getting diabetes
Query: chance of siblings both having autism
Query: chance of miscarriage after seeing good fetal movement heartbeat at 10 weeks
Query: chance of bleeding with placenta previa
Query: any chance of vaginal delivery if first birth was Caesarian
Query: probability of having an adverse reaction to amoxicillin
Query: does hypothyroid in women increase chance of liver cancer?
Query: does progesterone increase chance of breast cancer
Query: which treatment has the least chance of prostate cancer
recurring?
Query: what is the chance of relapse in a low risk acute lymphoblastic leukemia patient?
Query: chance of getting a brain tumor
Query: probability of flopping a set with pocket pair in poker
Query: does a ring of wealth a ffect the chance of the dragon pickaxe drop in runescape?
Query: chance of surviving severe head injury
Query: chance of snow in Austin Texas
Query: is there chance of flood in Saint Charles, Illinois today?
Query: calculate my chance of getting into the University of Washington
Query: what are the chances of becoming a golf professional?
Query: chance of closing airports in Mexico because of swine flu
Query: any chance of incentive packages for government employees to retire
Query: chance of children of divorce being divorced
Query: chance of food spoiling if left out over night
Query: what does it mean 50/50 chance of living 5 years?
Query: probability of life and evolution
Thanks! I forwarded that list to some of the people who will be teaching the course next spring. Even the questions that are “clearly too hard to answer” are worthwhile, because they illustrate how nonexperts are just as interested in these questions as ones that a mathematician could hope to give a useful answer to.
Speaking of probabilities in everyday life, this book sounds fun:
• Michael Blastland and David Spiegelhalter, The Norm Chronicles: Stories and Numbers About Danger, Profile, 2013.
From the Economist review:
For more information, click here:
David Spiegelhalter was recently interviewed on BBC R4, its a fun interview.
http://www.bbc.co.uk/iplayer/episode/b02x7h0z/The_Life_Scientific_David_Spiegelhalter/
*Great* class and topic list. Taught creatively, such a class could simultanously top many students’ most-fun and most-valuable lists.
Additional topics which strike me as valuable:
• Estimation aka back-of-envelope calcs aka Fermi problems
• Particularly useful problem-solving heuristics a la Polya
• At least one significant math abstraction plus a glimmer of how broadly powerful it can be (scads of possibilities here: e.g. the concept of optimizing an objective function over an n-dimensional space w/ a set of constraints – one could build to this in stages)
The Monty Hall problem (or variant) would be an entertaining and instructive example of how counterintuitive probability can be. To soften the blow, the instructor can close by relating how even Paul Erdos was taken in by it (after 1st fact-checking this scarcely-credible factoid).
The story of John von Neumann’s solving the fly-and-colliding-trains problem is both memorable and instructive.
A potentially indelible example: a) ask the students why the ‘average class size’ stat as cited by most universities is very misleading, b) ask them what Yale-NUS should do instead, c) (the tricky but cool part:-) do it.
Thanks for the suggestions!
I’m guessing your average class size puzzle is hinting at this: as an average student, the average size of the classes you take is larger than the average class size. This is a fun thing to ponder.
Similarly, if you have friends on Facebook, their average number of friends will typically be higher than the average number of friends of someone on Facebook.
Right John. The ratio between the two averages can be significant, and it’s arguably the first which is more relevant to students.
That said, perhaps the reason no one’s up in arms about this is that the largest classes a) tend to have associated smaller discussion sections, b) have smaller attendance relative to enrollment (though I’m not sure the latter would hold in Singapore:-)
I like your Facebook example as an updating of the (antiquated?-) family size example.
I concur with estimation being an important topic. A not-insignificant portion of incorrect answers I see students give on homework, quizzes, etc. could be caught before they submit their answers IF they could properly evaluate a simple question: “Does this make sense?” And that can often be gauged with a quick estimate.
For a simple practical example, I calculate my fuel economy every time I fill my gas tank. Even ignoring past values, if I come up with a number that is off by an order of magnitude, I can tell something went wrong if I estimate the fuel economy using front-end rounding. This was an absolutely essential skill when calculators were slide rules (they do not supply the order of magnitude), but even now it’s an important skill to be able to tell if the calculator answer makes sense.
I saw a good illustration of why the common erroneous answer to the Monty Hall problem is wrong. In this variant there are 1000 doors and 999 goats. All except the door you selected and one other door have been eliminated for you as definitely having goats behind them.
For estimation, “Street-Fighting Mathematics – The Art of Educated Guessing and Opportunistic Problem Solving” by Sanjoy Mahajan is excellent, although it does require some background.
Might be worth having a look at it! There’s even a free download of the pdf! (If I remember correctly, the author is currently working on a similar book targeted at a more general audience).
You’ve reminded me of another practical example, described by David Mermin, I think in his “Boojums All the Way through: Communicating Science in a Prosaic Age”-
The gist is that Mermin sanity-checks his grocery bill by doing a quick mental scan of his cart in the checkout line, and finds that his totally is usually within a dollar or so. Is Mermin some kind of savant? No, he just estimates each item to the nearest dollar (which he found pretty easy to do after a little practice); so the error per item averages $0.25, so the total error averages sqrt(n) times this, i.e. roughly a dollar for his typical purchase.
Interesting architecture!
Perhaps of interest for liberal arts dissertations, some work published years ago may already support an application of Category Theory to the liberal arts, specifically to the theory of literary interpretation. (Jon Barwise. The Situation in Logic. “On the circumstantial relation between meaning and content.”)
https://docs.google.com/file/d/0B9LMgeIAqlIENENELXoycjdFTWM/edit?usp=sharing
Some, more current work presents the idea of a general theory of interpretation. But it is not yet a theory based on category theory. The theory would comprise a theory of legal interpretation, a theory of psychoanalytic interpretation, as well as a theory of literary interpretation. (See for example P. C. Hogan. On Interpretation: Meaning and Inference in Law, Psychoanalysis, and Literature.)
http://books.google.com/books?id=E0uePv2wkC8C&lpg=PA6&ots=3uP1WeCAwN&dq=%22theory%20of%20literary%20interpretation%22&pg=PA6#v=onepage&q=%22theory%20of%20literary%20interpretation%22&f=falsellb
Imagine as a result: new career-relevant techniques, perhaps for future lawyers or other professionals, from the fields of Rhetorical Analysis or Rhetorical Criticism. More concretely, imagine using Category Theory to interpret stories, as above, with diagrams that help specify radically better decision-support-systems based on “case-based” reasoning. In this application, “case-based” would mean “story-based”– where the story in the case-based reasoning system is the story discussed in the theory of literary interpretation. A test would be that the decision support tools prove useful.
Because I have been trying to read what I can of John’s blogs, I am starting to think of the kind of story occurring in Socratic “dialectic,” (which is discussed in these fields) to actually be a game where hierarchical self-rewriting Petri nets are manipulated, where places either hold types or situations, where players move by putting either types or situations in appropriate Petri net places, and where the game ends when either (a) the listener to the story quits listening before the end of the story then breaks off and leaves, in which case both lose– especially in the case with Socrates– or (b) the listener stays to hear the end of the story, in which case both the storyteller and the listener win. Winning Socrates’ game of dialectic means enough information has been communicated to transform all the important possibilities into impossibilities.
By comparison, in the rhetoric described by Socrates there is not a two-person game, as in his dialectic. Instead there is a group listening to the person telling the story. The rhetorical story is like an animation, not an interaction, because it is replayed identically each time unless there are unwelcome hecklers. Success would be measured in a different way as well. For example, if half or more of the crowd stays to hear the whole story, then the storyteller wins. But if more than half the crowd leave before the story is finished, the storyteller loses.
So I’m starting to see the act of reading a story– for example a story about Sherlock Holmes– itself to be a process of “inquiry” and describable in language now being used on the Azimuth blog.
PS, Barwise used the word “inquiry” in his second-to-last paper titled “Information and Impossibilities”– in which paper he quotes Sherlock Holmes: “How many times have I said to you that when you have eliminated the impossible, whatever remains, no matter how improbable, must be the truth?” Apparently Holmes was an “informationalist”– a person who takes the inverse relationship between information and possibility as a tenet. “Whenever there is an increase in available information, there is a corresponding decrease in possibilities, and vice versa.”
John wrote “… we don’t know what ‘p’ and ‘q’ mean in the stochastic context…”
I wonder about these inverse relationships between probabilities and available information– could they be candidates for conjugate variables in some context? Channel theory would be involved. Channel theory is about distributed systems where each system has parts with the power to carry information about other parts in the same system– which can happen because they are all parts of the same system.
Each part has its own unique pair of possibilities and information available. That might be like the i-th pair of conjugate variables in the Hamiltonian equation.
A human being could be this type of system. There is something constantly invariant in a human being’s life. (Wherever you go, there you are– It’s you!) But the invariant involved is simply not the real number for energy, as in a physical Hamiltonian. I don’t know what this function giving the real number H(p,q) in dH/dt = 0 would mean for this kind of system. I’m just conjecturing that it must exist for this idea to work.
At the same time there is constant change. The breath moves between two different points of two different types. But if this movement were to stop forever, any feelings of “Wherever you go, there you are (dH/dt = 0)” cease to exist. Clearly, the process of breathing must be be actively moving from state to state before a human being can feel anything.
Then there’s the solar system. Stop the planets from moving in their orbits around the sun (by somehow sucking away enough energy so they stop dead in their paths), then the planets immediately fall into the sun and our solar system ceases to exist. (dH/dt = 0 — NOT)
In these two scenarios, admittedly not standard, dynamic balancing acts by the all the parts of the system create and hold the system invariant at dH/dt =0 (whatever H means). But when the performance of these balancing acts stops, the system is changed maybe to the point of destruction. (dH/dt = 0 — NOT)
From this I wonder about the large number limit for any distributed system having for its conjugate variables the (p) possibilities and (q) information available for each part of it. The perhaps unusual idea is that each part of the system must perform its own balancing act with its own conjugate variables so as a result, the total system maintains existence in a non-degraded state. At the other extreme, destruction of the system would be guaranteed when each part stops performing its balancing act between possibilities and available information, and violates their inverse relationship. The system is created when all the parts of the system begin their balancing act and destroyed when all the parts of the system fail to perform their balancing act.
Far away from the large number limit, evidence that (a) possibilities for each part of the system exist, and (b) for each part, a history exists of its states and, for each of the states, the information that was available– taken together mean that the balancing processes were at work up to the last move and that the system was not degrading, in fact every part has possibilities, and therefore by all evidence the system is maintaining existence at some dH/dt = 0 (whatever H means). At the other extreme, if the possibilities for each part cease to exist, and simultaneously each part also proves to have been “live” on its last move (as evident from the above history), then under these conditions a self-destruct transition in the model gets fired to represent destruction of the system. (dH/dt = 0 — NOT)
From an evolutionary point of view we can compare this to a different kind of system. Say that dH/dt = 0 means a system balancing on a tightrope. For example, two bicyclists support a pole on their shoulders on which another bicyclist rides, etc. There are gears and axes involved tying everything together. When these kinds of systems degrade, the entire system has to be taken off-line to be maintained.
On the other hand, we can consider a system based on a number of tightrope walkers, each balancing on one’s own tightrope. Somebody getting off-balance on a different tightrope does not physically interfere with any other part of the system, each part of which has its own tightrope. Rather than gears connected to axes and each other, this is more like the juggler in the Ed Sullivan show who gets a zillion plates spinning by means of supporting, off-axis sticks. If any part degrades, it can be replaced on-the-fly while the total system keeps running in a degraded state. The new part begins its balancing act and the system can be restored to dH/dt = 0.
The latter system is probably much easier to create and maintain, while the former is probably much easier to destroy.
Would evolution agree?