High-School Algebra: Our Executive-Type Middle-Schoolers

I have been teaching college mathematics for many years, and quit recently.  (Well, I called it retirement, but actually it was quitting.)  But I was getting a little bored, and decided to put up a video on YouTube to show an alternative method for doing something students learn in Calculus 2.

Well, I just finished it, and it involved quite a bit of algebra.  And I was wondering just how much detailed explanation would need to accompany the slides, to make the calculations comprehensible to a typical viewer.  For instance, I had to compare the results we would have gotten, with an authoritative result from WolframAlpha, a program we used at our school, until it became a little too demanding for our majors, and some of our faculty!  (Our school is still ahead of the curve, but depending on how much incoming faculty at schools like ours like to trade interesting content for easier, more entertaining content, we might end up behind the curve.  Read on.)  The two answers are the same, except for a little algebra.  Here is a screen capture of the last slide:

Anticipating the question:  "What has 192 got to do with anything?", I added the little note at the bottom.

Now, I expect my readers to be divided pretty much in half between (a) those who think "It is perfectly clear that the answers were the same," and (b) those who think, "Well, that may be the case, but an explanation is in order, after all, these are just kids; heck I could not have figured that out, and I'm pretty good at math!"

Part of the problem is that many sorts of employers, both businesses and the government (or as we say in economics circles: the Private Sector and the Public Sector) want their prospective hires to know calculus.  Among other things, this means that Calculus teachers have to deal with students now who suspect that they will never actually use Calculus in their chosen fields, and their suspicions are probably right.  Over the last several years, things have let up a little, but there was a time when even Pre-Med students needed to get a good grade in Calculus to get into Medical School.  Why is this?  I don't know, but I can guess.  And my guess would give you pause as to the motives of the Medical Education Industry.

Coming back to those algebraic formulas above, it boils down to whether or not two formulas which look roughly like

are equal.  This material is actually learned (in most schools, at least a decade ago) in Grade 8, in better school districts.  Obviously, there's nothing to be done if you happened to live in a school district in which most parents prefer that their kids get better grades than a rigorous mathematics training.  This is one of the major pairs of opposing forces that keep battling inside most parent's heads.  The kid is hamstrung unless its grades are good, but the kid will struggle to even understand its math classes unless it knows its algebra really well.  Looking up these algebra facts just can't be done in real time; they have to be at the tips of the child's fingers.

In addition to school training, there is the problem of discipline.  Fractions are an obnoxious kind of mathematical thing; most kids prefer to use decimals.  Fractions are exact, while decimals are usually approximations, and for most purposes, including Chemistry, Biology, Physics, Economics, Political Science, Business, Engineering and Statistics, approximations are good enough.  (They have to be good approximations.)  So why do math teachers keep plugging these fractions?

The fractions are more logically useful.  Before the decimal approximation can be applied, the formula is developed using fractions; the relationships between various quantities are given using fractions.  Once the number you're interested in is narrowed down in terms of fractions, it can be approximated using decimals.  If anyone takes a nap while the fractions are being thrown around, and wakes up just when the calculator is needed, he or she will know a number, which may not be any use for the next problem, and will not know how it is arrived at.

This little post about fractions and elementary algebra describes only the tip of the iceberg.  Better minds than mine must address the problem that Indian and Chinese and Russian and Brazilian kids learn algebra a lot faster than American kids.  Their lives are tough enough that algebra is hardly something to complain about.  In contrast, for American kids, algebra is the worst kind of torture they have to face, so that for many men and women, algebra is the poster-boy for the unpleasant subjects they had to deal with in grade school, and which thankfully they did not need to suffer with in Adult Life.  If we keep up this level of intolerance to mathematics, kids might end up refusing to subtract, even if they reluctantly agree to occasionally add.  Think that's funny?  If Medical Schools keep up the mathematics requirements for admission, their numbers will fall, and we will find most of our medical professionals coming from overseas.  Which is not entirely a bad thing, I have to add.  On the other hand, if med schools relax the math requirements for admission, many citizens will harbor the (entirely unfounded) suspicion that medical professionals of the years after 2018 are not quite up to the standards we're accustomed to.

What the Monkey should have said is that There is no progress without some drudgery.

Arch

About that string quartet . . .

I finished it.  There are just three movements.  (Most string quartets have four movements, I read, but I'm not cruel.)

• The first movement is here; I have not made very many changes since I put up a link on this blog a few weeks ago.
• The second movement is here (second movement); lots of changes were made, but if you're checking this out for the first time, you won't notice any of them.  There is a sort of refrain between various segments, which I am sort of planning to take out.  Just one of them (the third) is nice, and I might keep that...  I did orchestrate it for a small orchestra, and it sounds lovely, which is probably a big indictment on the quartet movement.
• The third movement is here (third movement), in a sort of rondo form.  It has a speed (tempo) indication of Presto, which is very fast.  I don't think most professional string quartets can play it that fast, so it's going to be essentially allegro.

Added later: here's a video of the whole thing.

Just in case you were wondering: I did set the last movement going at M-M ♩=200, which is fast.  Here it is, if you're interested.  Try and relax.  By the way, it sounds as if the music is getting faster and faster, but I'm fairly sure it keeps a steady pace, because, after all, the music is played mechanically by a program.  (There is a complicated way to verify whether the music is speeding up, but I haven't tried that approach yet.)

Solving Problems: Georg Polya and Beyond

Okay, this is going to be just a shell for a post.

In the nineteen forties, a Cambridge scholar called George Polya gave his research students a checklist of strategies that could help them get started on solving a problem (usually a research program), or get past some sticky obstacle.  Since Polya, lots of people have extended and elaborated this checklist, and little kids in elementary school are taught some of these strategies (which they probably proceed to forget right away!), and I felt I should read up on the latest thinking on it.  But it is such a useful list of strategies that getting it written down ASAP, even if it is a little premature, is probably going to be useful.  OK, straight onto the list.

01.  Understand the problem.  State the problem clearly.  This is less useful for those of us with simple problems that are easily stated, but often doing this clarifies a misunderstanding that has been an obstacle.

02.  Define your terms.  (This is a little technical, but examining the words used in stating the problem is often surprisingly helpful.)

03.  Study similar problems that actually have been successfully solved.

04.  Simplify your problem.  If a simplified version of your problem can be solved, the original problem might have a solution obtained by slightly tweaking the simpler solution.

05.  Break your problem into parts, and solve them separately.

06.  Draw a picture, or make a graph, or build a chart.  (Often problems that have to do with dealing with numbers sometimes suggest a solution when you study the numbers visually.)

07.  Make a list, or a table.  (Online sales points offer tables for comparing different models of electronic equipment.)

08.  Look for a pattern.  This makes most sense when trying to find a property of a number.

09.  Work backwards.  This strategy is in the context of trying to find out why something is the way it is (which comes up in various sorts of research).  Often trying to understand a chain starting from one end of it leads to a major obstacle.  Starting at the other end often does the trick.

A formal description of George Polya's methods is found in Wikipedia.