17 September 2011

Mathematics screening test for university students

I am writing this in response to my friend Joe who tends to believe that things in US education are bad. In fact, for anyone who thinks that things are bad in the particular place he or she happens to work.

This is a screening test I gave to second year university students (school of mathematical sciences of a UK  university I worked at earlier). The rationale behind a screening test is to alert the students that they should not take a further course without having basic skills acquired in earlier courses and that, if they have not learnt earlier material, they should repeat the courses before proceeding further.

The front page of the linked document contains statistics of students' responses. The remaining pages contain the questions I asked, together with sample responses. I would say that out of 60 students who took the test, there was probably one who could, perhaps, qualify as a university student. The remaining ones had no clue.

Although the document is self-explanatory, here are some of the questions, along with the most funny answers:

Q:  Given two polynomials $p(x) = \sum_{k=0}^n a_k x^k$ and $q(x) = \sum_{k=0}^m b_k x^k$, express the coefficient of the term $x^k$ of the product $r(x) = p(x)q(x)$ in terms of the coefficients $(a_k)$ and $(b_k)$.
A: $a_{k^{1/2}} b_{k^{1/2}}$.

Q: Define the concept of the derivative of a function $f : R \rightarrow R$ at a point $x$.
A: This is the distance of the point $x$ from the origin on a plain  [sic].

Q: Explain what we mean by the integral $\int_0^1 f(x) dx$ of a function $f : [0; 1]  \rightarrow R$. (The answer "area under the curve" is not acceptable.)
A: By integrating this function, we are being asked to calculate an area, and by providing definate [sic] integrals, the question asks us to provide a specific area.

Q: In how many ways can you put 5 indistinguishable balls in 7 distinctly numbered boxes and why?
A: 21/5.

Q: Expand $(a + b)^5$, where $a, b$ are real numbers.
A: $(a + b)^5 = \binom{a}{0} + \binom{a}{1} a^4 b + \binom{a}{2} \frac{a^3 b^2}{2!} + \binom{a}{3} \frac{a^2 b^3}{3!} + \binom{a}{4} \frac{ab^4}{4!} + \binom{a}{5} \frac{b^5}{5!}$.

Q: Compute the (indefinite) integral $\int dx/\sqrt{x}$.
A: $-2u+C$.

The huge problem in education, around the world, is that the meaning of the verb "to learn" is frequently disassociated from the verb "to understand". This is convenient for students. It is also convenient for many teachers who do not want to bother to understand and explain. It is convenient for politicians. It is convenient for administrators. In short, it is convenient for everyone. Except that the result is the production of generations of students who get a degree in, say, mathematics, but know very little mathematics. What is worse, is that they think they know. It is more dangerous to have people who believe they know rather than people who know they do not know (and, therefore, may try to learn whenever necessary). Someone who is convinced of his/her skills will do nothing to improve them.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant