“The real reason why the US is falling behind in math” is a math professor’s take on “Why Johnny can’t do math.”
This is consistent with my experience as a tutor in the Cambridge Public Schools. High school students would be given about 50 identical problems on one page, each one starting with the equation of a circle with a goal of finding the center and radius. There was some sort of trick that one could use and they were better at it than I, but they struggled sometimes with arithmetic on the coefficients. I would ask them to back up and graph one of these equations so that I could understand it and they would say that they didn’t want to, didn’t know how to, etc., even though they all had graphing calculators. The goal of the school was to get the students to the point where they could crush the MCAS and some other standardized tests but the problems had nothing to do with what I would call “math” (and I was an undergrad math major before I found that I had to agree with Barbie that “math is hard [and frustrating]”).
Math has been taught by equations and repetative problems for years in traditional classes (that is how I learned it in the early 70s). My son who is graduating high school (in Massachusetts) took the IMP track in his school.
How does IMP differ from traditional high school mathematics courses?
Conceptual Understanding
The IMP curriculum challenges students to actively explore open-ended situations, in a way that closely resembles the inquiry method used by mathematicians and scientists in their work. While the traditional curriculum emphasizes rote learning of isolated mathematical skills, IMP calls on students to experiment with examples, look for and articulate patterns, and make, test, and prove conjectures.
Updated Mathematics
IMP integrates algebra, geometry, and trigonometry with the additional topics recommended by the national reports, using calculator and computer technology to enhance student understanding.
He aced the MCAS and was able get a good understanding of statistics (different then the standard algebra, geometry, trig and pre-calc curriculum). He is taking traditional calc now and doing well and finding his physics class pretty easy (physics is a lot about math in the real world and his IMP experience included a lot of real world problems).
He doesn’t see himself as a math major or scientist; he is interested in government and english (and maybe econ) however he is getting a great grounding in Math (as did his older sister who also did IMP).
From the MathIMP.org website.
Problem-Based Units
The IMP curriculum is problem-based, consisting of five- to eight-week units bound into a single techbook. The units are each organized around a central problem or theme. Motivated by this central focus, students solve a variety of smaller problems, both routine and non-routine, that develop the underlying skills and concepts needed to solve the central problem in that unit.
In the late 80’s I worked as a chemical tech in a brand new CRT factory. We were told that the people were selected for these jobs based on better than average math skill, although this bar was fairly low, just basic algebra with a calculator and the formulas provided.
While the factory was still under construction we were asked to figure the approximate capacities of 30 or so tanks ranging from 20 to 1000 liters–either rectangular or cylindrical. We had various tools tools available, but I was the only one out of about 15 to use a tape measure and calculator instead of some variation of “count how many 10 liter buckets it takes to fill the tank”. Quite a few people remembered that they had learned my method in school, but hadn’t thought to use it now.
The problem with mathematics education in the US, in my opinion, has to do with several things:
1) Ed schools are terrible at training teachers to teach content. They are mostly concerned with form (lesson plans, drawing a straight line, etc). My experience has been Ed schools have tried to water down curricula.
2) The content, as it is now constituted, makes very little sense. For example, I recently looked at my daughter’s algebra II curriculum and realized it is chock full of topics which are not appropriate for 11th grade. Algebra II/precalculus was originally supposed to train students for first year calculus in college. The first year college calculus curriculum has not changed all that much in the last fifty years. However, the 11th grade mathematics curriculum has changed dramatically. Trigonometry is no longer taught properly and the students are instead learning Gaussian elimination and bivariate statistics—all at a very superficial level. There is a huge gap between what the students are taught in high school at what they are expected to know when they enter college. This accounts for a good deal of the problems I am seeing: the transition from HS to College is anything but seamless.
3) At one time, K-12 teaching was a more honored profession which used to attract first-rate people. It is hard to make that statement today. Compare the situation with Finland, in which teaching is an elite profession.
This site
http://quadrivium.info/MathInt/MathIntentions.html
is an interesting take on why the math curriculum is so nonintuitive. It begins with equations that define curves, whereas historically and in fact the curves come first.
Historically, our system of algebraic notation is relatively new, and while it makes it easy to manipulate and solve equations, it is also a bar to understanding of the underlying concepts. It’s easy to come up with the “right” answer without having any idea whatsoever of what the equation does or what the answer means. You know that x = 5 but what does that signify?
My late father-in-law, who was an engineer, was a big believer in word problems. He felt that unless you could take the word problem and convert it to notation and use it to solve the problem (or even solve the problem without notation – graphically or whatever) then you didn’t really understand the subject matter.
Of course, he was from the age before computers so he was used to setting up even very complex real world problems in a way that could be solved manually (with a slide rule). Remember that the most advanced technology of WWII (B-29 bombers, nuclear weapons, V-2 rockets, Me262 jet fighters, the computer (ENIAC) itself). were all developed and built completely without the aid of computers or even electronic calculators – it was all slide rules and adding machines.
So the same thing goes for the graphing calculator – if you are fluent enough to set up a graph yourself, then you can graduate to the graphing calculator (but you probably don’t need it), but if you are just punching in x= , y = into the calculator and watching the pretty little graph appear without really understanding slope, intercept, etc. then the calculator is a bar to understanding instead of an aid.
It seems to me that mathematics education has more fads than dietary science. Every ten years, someone points out that the current method is wrong, dogmatic, and either fails to appeal to children’s natural intuitions, or fails to implant sufficient discipline. Possibly both.
Since most people wish that other people were better at mathematics, and since the subjects are invariably minors who can’t vote or even articulate an opinion about it, very few people argue with this.
Are the people who decide educational rules and guidelines proven experts in their respective fields?
Izzie is making me laugh. X is introduced at ages 7-8 in both China and East Europe – the places that rate very high on math. In US it comes often at 12-13 in middle school as a complete shock. I think it was 4th grade when my kids were given problems requiring systems of linear equations but instead were taught “guess and check” method. If you just start using any letters early like A for apples, p – pears or say J for John and M for Mary, it becomes extremely easy and intuitive and nobody is scared by x,y,z. It’s just another letter. Also here we don’t teach physics until 11 grade. If you use math to solve relevant albeit simplified physics problems it becomes very natural. All math studied in school has been created centuries ago to solve real world problems.