The following is adapted from Posamentier and Salkind's Challenging Problems in Algebra, published by Dover Books.
(a) Prove that for any positive integer n, the final digit of n5 is the same as the final digit of n.
(b) Without using a computing device, find all positive integers whose 5-th power is a 7-digit number that ends in the digit 7.
Don't Be a Math Teacher
I have enjoyed teaching math for several decades and
currently teach K–12
math teachers about education and mathematics. So you'd think that I would
recommend that a mathematically-inclined person should give strong
consideration to becoming a math teacher. You'd be wrong.
If when I began my career the state of the mathematics
teaching profession, both at the school and university levels, was what it is
today, I don't think I would have gone into it. Math teaching now has the
following difficulties.
First, digital technologies have resulted in students with
shorter attention spans and more dependence on instant gratification. You would
be surprised how many high school students are unable to add or multiply two
two-digit numbers without a calculator, and in many cases unable to add or
multiply even one-digit numbers. This matters, in much the same way that it
would matter if students couldn't read words of more than one or two syllables,
and depended on print-to-voice readers to “read” literature. Also, most students faced with a problem
that requires more than five minutes of work, most will say that they can't do
it. A Korean student that I had in a Calculus course in an elite U. S. woman's
college earned an A+ told me privately that she felt she was not good at
mathematics but was best in the class because the rest of the girls were lazy.
I think we will only see more and more students who are unwilling to do the
hard work required to become good at math.
Second,
in the United States, virtually all K–12 students must pass tests in order to
advance in grade or to graduate. Teachers must teach to the test. While
teaching to the test is, in my opinion, a good thing for formative assessments,
teaching to the test for these high-stakes one-size-fits-all tests at best
limits the ability of the teacher to approach the curriculum in innovative ways
and at worst results in much time wasted with students taking practice tests
and learning test-taking tricks that have nothing to do with mathematics. And
of course, if the students do not do well on these tests, the teacher's job is
on the line.
Third,
computers are in the process of making math teaching obsolete. If a student can
learn a subject online, where they can have their instruction completely
personalized and can have their homework graded instantly, why should a school
system invest in expensive and inefficient human beings to do the teaching?
Most teachers will be replaced by low-paid “facilitators” of online courses who
do not need to even be trained in mathematics. Instead of there being thousands
of college instructors nationwide teaching Calculus I, a dozen or so master
teachers will make videos and consult with courseware designers to put together
online courses. Math teaching, as we know it, will exist only at the graduate
school level where economies of scale do not apply, or at pricey private
schools.
So my career
advice is only to become a math teacher if it is really a calling for you. Good
teaching will be harder than it has ever been, with fewer rewards and little in
the way of job security.
Fun with divergent series
A friend sent me a link to a Numberphile presentation in which Tony Padilla and Ed Copland show that the infinite series S = 1 + 2 + 3 + 4 + ... has (in some sense) the value - 1/12. It seems that this result is used in superstring theory in physics and goes all the way back to Euler in 1740. It has generated a lot of heat (and some light) on one of the math forums I follow, so I looked into it and have written it up a bit.
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