What algebra problem did Serge Lang give to calculus students?

Joel Spolsky tells this story:

Serge Lang, a math professor at Yale, used to give his Calculus students a fairly simple algebra problem on the first day of classes, one which almost everyone could solve, but some of them solved it as quickly as they could write while others took a while, and Professor Lang claimed that all of the students who solved the problem as quickly as they could write would get an A in the Calculus course, and all the others wouldn’t. The speed with which they solved a simple algebra problem was as good a predictor of the final grade in Calculus as a whole semester of homework, tests, midterms, and a final.

What could this algebra problem have been? (I considered this question for matheducators.se, but I'm not sure they consider such historical questions on-topic.)

• I'm afraid it is off-topic here as well: almost any problem would do the job, and certainly he would give different problems at different times. Jan 14, 2019 at 18:03
• FYI, on many occasions in the Math Forum group math-teach, Wayne Bishop has stated that reducing algebraic rational functions to lowest terms correlates extremely well with success in algebra-based quant skills. For example: Our statewide Entry-Level Mathematics has shown that simplifying a single "rigged" to be simple but compound rational function (numerator and denominator each consisting of sums/differences of rational functions) correlate so well with future success in anything needing algebra that it could be a one item test. Jan 15, 2019 at 20:01
• @DaveLRenfro that link is unfortunately broken! Jul 12, 2021 at 5:19
• Lang, a Yale mathematician who taught calculus for years, was legendary for throwing chalk at recalcitrant, sleepy or uninformed students. Jun 12, 2023 at 11:54

In 1969 Lang wrote an article for the Columbia Daily Spectator, Don't Blame Us if You Flunk Math (Volume III, Number 4, December 8, 1969). The phrasing of the subline illustrates how much the times have changed:"A fifteen minute quiz raises questions what kind of people should be taught what kinds of math at Columbia".

He provides a basic introductory test for calculus courses and argues for its adequacy to decide who should or should not take them. The amazing predictive success is not mentioned, though. It seems it was used as a placement test for sections with three different levels of difficulty, but the prediction story is more fun, I guess.

"During the last few years there have been some complaints concerning the calculus courses. The situation was particularly tense last year when I was away, and the math department got a lot of flak. Since I am departmental representative this year I have tried to pin point some of the causes for the difficulties. We separate our first and second year calculus students into three groups, A, B, C, of which A is supposed to be the more computational and easiest, B more theoretical and harder, and C for the very interested and the talented students.

[...] Following complaints which Mr. Wyer regarded as comjng from unqualified students, the course was watered down and became less rewarding to the more responsible students. Mr. Wyer vouches for the opinions of at least ten others who agree with him. Mr. Wyer questions the insight of students criticizing math courses as being too "theoretical". As he says: "It is not an infrequent occurrence in the fields of physics and engineering for a student to do very well in his introductory mathematics courses and subsequently to run into difficulties".

After receiving Mr. Wyer's letter, I decided to give a short test to check Mr. Wyer's opinions. The test consisted of five problems, and was given to the 1A sections. One can draw some conclusions: a) The test is very easy and students unable to do reasonably well on such a test should not be taking a calculus course."

BASIC TEST

You have fifteen minutes to do the following problems:

1. Solve the system of equations: $$3x - 2y = 1;$$ $$4x + 7y = 15$$

2. Solve the system of equations: $$3x - y + 2z = 1;$$ $$x - y - z = 0;$$ $$2x - 2y + 3z = 3$$

3. Solve the following equations: a) $$2x^2 - 4x + 5 = 0;$$ b) $$3x^2 + 2x - 8 = 0$$

4. What is the sine of an angle of: a) 45 degrees b) 30 degrees c) 90 degrees

5. Show that: $$1/(x+y) - 1/(x-y) = -2y/(x^2 - y^2)$$

• Yes. I knew Serge L. a bit in the 1970s-80s, and, though at times his rhetoric was overblown, he was astute about college-and-beyond mathematics education, especially at mid-to-high-end... and, also, was a very ethical, moral person, so would not have jerked anyone around. I was not aware of this particular episode... but these are stunningly easy questions. It's not that being able to answer quickly is a sign of gifts, but that inability to answer is a very troubling indicator... (And several people have told me that Columbia students are often "entitled"... which might have irked Serge :) Jan 14, 2019 at 23:10
• If this is really the same test, then Spolsky's description of students who "solved the problem as quickly as they could write" seems goofy. It's not hard to scribble a solution to #1 or #3 quickly, but it is time-consuming to make sure you're not making a mistake along the way, and to check your answer at the end. If I was given this as a placement test, I would work slowly and carefully and spend all available time checking my results. Why would I rush through it and hand it in early?
– user466
Jan 15, 2019 at 4:34