HS Math teachers; students: How is Algebra I today harder than of yore?

[asterion]

One more perspective. Chemistry-major college senior here. I started getting the real basics of algebra as early as 6th or 7th grade (you know, 5+x=12). I had a various slew of math in middle school, but I think most of it was algebra. I had more algebra as a freshman in a class I was thoroughly bored in (too easy and slow), then moved to the honors class for geometry/algebra II. I lasted a semester as it was too advanced after that bad freshman year. The school really needed a mid-level math class. Anyway, I was then bored out of my mind again that second semester. I’m trying to remember what I took junior year–probably more algebra II. Senior year was precalc, which had basically no calculus in it, a whole bunch of jocks, and a teacher who, while not a bad guy, was too prone to prefer talking basketball to math. Got a dose of both single and multivariable calculus here in college.

Anyway, I’ve learned over the years that I learn math better in context. I’m not talking about one of those “somewhat-related-to-something-in-business-or-science-in-the-problems-section-of-the-math-book” problems, but in a full-blown chemistry or physics course. Logs, for example, as somebody else said, aren’t that useful for lots of math calculations when you’ve got a calculator instead of a slide rule. I kinda know how to read a log table, have no idea how to use a slipstick, and would rather punch it into my calculator than look up the numbers anyway. But I know logs well, not because of math, but because of chemistry and physics. Actually, I first learned to use logs while doing radioactive decay calculations back in high school chemistry.

I took calc II here at school after I took a semester of physics, and we used matrices in physics far before I saw them in calc. So by then I had matrix operations down for the level the calc II required, though not at the level something like linear algebra would’ve taught me. I’ve got a far better handle on calculus in general now thanks to physics and physical chemistry than I ever got out of calc I.

As far calculators and graphing calculators, I didn’t have a graphing calculator of my own (a TI-86) until I was a junior in high school. I still use the same calculator now, 6 years later (it’s complete overkill for my accounting class.) I used a scientific calculator for several years before that, and the school’s collection of old T1-82s were only given out occasionally for specific uses in graphing applications. I don’t think I knew anyone, other than the teachers, who had a graphing calculator, at least not in middle school. To me, graphing calculators are a mixed bag, like I guess any tool. They make tedious things like graphing easy (I can draw a graph by hand, but I’d rather just let the calculator do it) and you can do a lot with them if you learn the advanced functions. However, too many people just trust what the calculator tells them instead of thinking about whether or not it’s a reasonable answer. I swear, sometimes I think that if the calculator said that 2+2=11, people wouldn’t think to question it (or at least check the base.) Calculators are wonderful tools, especially things like the TI-89 that can do things like solve multivariable calculus, but you have to know how to use the calculator and have the ability to do it without the calculator so that you know if the answer is reasonable.

[zoid]

@Spectre of Pithecanthropus

Can you tell us what the problem was?

This is not the exact wording but it was along the lines of:

You have one bottle of scotch (A) which is 20% alcohol and another bottle (B) which is 40% alcohol. What ratio would yield a mixture which is 30% alcohol?

[GargoyleWB]

@Valgard

I used to love one particular problem which came down to finding the area of a chunk of a triangle. A right triangle with known width and area. Get the height, use your really basic geometry (area of similar triangles) and zip zip zip you’re done in two minutes. You barely need a pencil and paper.

I was astounded at the number of bright sophomores who would come to my office for help. Their approach to the problem was to find the coordinates of the endpoints of the hypotenuse (argh!), which many would get wrong (sheesh), then derive the equation of the line (gnaaaa) followed by integrating that (nononono) to find the area.

All carried out to something like 12 significant digits on a $200 calculator.

You can certainly DO it that way but WHY would you want to? .

This is my huge criticism with math education today. All to often, blind adherence to procedure is what is taught and evaluated, instead of logical thinking or creative solutions to a problem. I would be delighted at any student that took the calculus approach (even if they didn’t get the nitpicks of the mechanics correct) and had the insight to use calculus as a flexible tool to approach a problem.

Nothing burns me up more than a teacher criticizing a student for not doing something the “textbook” way. Instead, you really should have helped the students refine the logical path that they were working on to solve the triangle problem.

An example of my own: I’m currently pursuing a second degree in Electrical Engineering (first degree was in Physics). I routinely am questioned or criticized (and sometimes accused of being flat wrong) when I solve a problem using physical laws as a basis as is my habit from physics, instead of the rote “cookbook” approach often used by EE textbooks. Nothing throws of the teaching assistants grading my EE homework more than starting off with F=MA

[Mathochist]

@zoid

This is not the exact wording but it was along the lines of:

You have one bottle of scotch (A) which is 20% alcohol and another bottle (B) which is 40% alcohol. What ratio would yield a mixture which is 30% alcohol?

Follow-up question: why would you want to dilute the stronger liquor?

[Valgard]

@GargoyleWB

This is my huge criticism with math education today. All to often, blind adherence to procedure is what is taught and evaluated, instead of logical thinking or creative solutions to a problem. I would be delighted at any student that took the calculus approach (even if they didn’t get the nitpicks of the mechanics correct) and had the insight to use calculus as a flexible tool to approach a problem.

Nothing burns me up more than a teacher criticizing a student for not doing something the “textbook” way. Instead, you really should have helped the students refine the logical path that they were working on to solve the triangle problem.

Don’t get me wrong, it’s not that I told people that they had to solve it any particular way, it’s just that so many people were so eager to apply the biggest and baddest tools in their toolbox that they forgot a couple of things:

1. In the simplest terms, what are you trying to find?
2. What’s the simplest way to find that?

In this case they were dealing with probability density functions. That’s just a curve f(x) with a couple of characteristics, including that the integral equals exactly 1.

Students were given a picture of the curve (so they know it’s a triangle) and they had to find the probability of the random variable x being between two values. What I was getting at is that a lot of students just leaped right to a general formula from the book (“The probability will be equal to the integral of f(x)dx between the two values ‘a’ and ‘b’…”) and charged headlong from there without stopping to think for a second. Integrating under a curve is just finding the area. In this case the curve is an angled line and everyone knows how to find the area under a triangle in three seconds flat, in their head. If somebody really wants to use calculus to arrive at area = 1/2(b*h) they are welcome to do so, but they’re simply making a lot of extra work for themselves.

To quote Feynman again, it’s like they don’t know what they know.

Addressing your other point I wasn’t upset at people trying to solve the problem “outside of the textbook way” but rather I was disappointed at blind adherence to the textbook way, specifically at taking the most general form given in the textbook and then going about it in about the most complicated way possible, rather than stepping back and looking at the big picture.

When people would come to my office hours and start to show me their pages (literally) of complex calculations I’d make them put it all away and turn off their calculator. Then I’d get a little Socratic on 'em:

-“What are you trying to find in this question?” (probability)
-“How do you find that?” (well you integrate under the curve to get the area)
-“OK, what shape is that curve?” (it’s a triangle)
-“And how do you find the area of a triangle?” (that’s all I had to do? d’oh!)

[Valgard]

…and yes GargoyleWB I gave full credit to the people who did it the “complicated” way and got it right. It’s just that a large number of them made many, many math mistakes doing it that way and didn’t even recognize the fact. They’re 19 year old engineering students, I trust them to know basic math and it’s perfectly OK to save the Big Guns (calculus and expensive calculators) for Big Problems.

In civil engineering this was something that many, if not all, of my professors stressed - have an idea of what the answer is before you even start, by which they mean a ballpark answer. Blind adherence to A Method has caused a lot of problems, including loss of money and loss of lives, because people made mistakes in a complicated calculation and weren’t looking at the big picture to see if their answer made any sense, they just ran it through, got a number, plugged that into the next step, and so on. If I was designing a skyscraper and there was some garbage in my finite element analysis that resulted in a computer model spitting out main columns 2 inches on a side, I really need to notice that and know that it can’t possibly be right, rather than just taking “2 inches” and going on with it. Sorry, that’s a little off track but the point is not to make things harder than they need to be.