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Wednesday, August 27. 2014
That's an assertion by AVI, but I don't know whether he refers to high school or college students. Presumably every college-bound kid would have taken Alg ll in high school, if not AB Calc (most do that too, it seems). He also says:
I'm not sure what I think about this. How much math is enough to make a person functional and numerically-literate, and how much to be considered well-educated? I think all of these areas are excellent training for rigorous and critical thinking. It's basically a logical language, and seems best approached that way.
I have heard experts say that around 5-10% of high school grads are truly eager and ready for rigorous higher ed. The rest are just postponing adulthood.
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Okay, call me incredibly dumb. I have been so confused ever since my kids entered high school. When I was in high school in the 80s, this was the math you took:
1) Alg. I
3) Alg II
Some kids had taken Alg. I in junior high, so they would be in calculus by their senior year.
Anyway, would someone please explain to me what these classes are now? I never hear about Trig anymore. HELP!
Trig was rolled into Calc and Geometry when I was in school 10 ish years ago. Trig as a standalone topic isn't really interesting without having Calc and imaginary numbers to really make it go.
Thank you for the explanation. It is hard for me to know where my kids are in the scheme of things and if they will be prepared for the SATs based on the math track they are on now. Guess I'll find out once the oldest takes her SAT next year!
Trigonometry, taught using geometric methods, is where students learned to be very facile with algebra. Namely, the derivation of trigonometric identities involved both gaining understanding of the trigonometric functions and the algebraic manipulations required to establish the identities.
When that course is accompanied and/or followed by Analytic Geometry, complex numbers may be very naturally introduced using the plane mapping the real axis and the imaginary axis. Beautiful things like Euler's formula become accessible for students just beginning to get serious about math.
Trig as a standalone topic isn't really interesting without having Calc and imaginary numbers to really make it go.
My senior year math course consisted of trig the first semester and imaginary numbers etc. the second semester.
I found trig interesting enough before we even got to imaginary numbers. Trig was taught to me with an emphasis on the geometric/unit circle approach along with proving various trig identities. See WalkingHorse's comment.
I have never once used Calculus in real life. Never really understood why the hell I was taking it. Felt like mental torture that was supposed to make me a better person somehow.
I have used statistics, finance, financial accounting, cost accounting and even some basic economics out in the business world.
I think teaching every High School kid basic statistics and personal finance before graduation would be much more useful than calculus.
I tend to agree with NJSoldier
I believe the overwhelming majority would be in very good shape if they could just add, subtract, multiply, divide and understand fractions.
Sadly, a significant percentage of the population can't even do that.
I was going to post the exact same thing. There is no use trying to teach Algebra II if the student cannot add, subtract, multiply or divide. Can anyone in the Federal Government do any arithmetic? Most of the time in school these days is spent doing 'group projects' to learn teamwork.
I enjoyed Calculus, it was a fun game. But it would have been much more useful, to put it mildly, to have taken a good statistics/probability course along with a good finance course (Home Ec. anyone?)
Stats is increasingly important in a society that believes itself to be founded upon "settled science." In order to see through scams, primarily.
Well, as a PhD engineer, I guess my experience is a bit different. Math is the language of the physical universe, as far as I'm concerned.
I consider calculus to be an essential building block of understanding the physical world. I don't see how anyone can speak intelligently about things like climate science and climate modeling, without at least one course in partial differential equations and nonlinear equations, let alone calculus or basic differential equations. And a some background in computational numerical methods, which is how all this math is actually dealt with inside a computer, when there is no closed form solution. Otherwise, we have an ignorant population dependent upon the opinions of politicians and those with access to the microphone about important issues... and no good means of evaluating what they are hearing.
Financial math is a basic block anyone coming out of high school should have. In self-defense if no other reason.
I agree with you - but I agree more with the sense of the above comments stressing financial math. Get the kids out of HS with the basics drilled in, and applied to basic finances / interest / budgeting and schools will serve more kids better in their adult lives than if you leave out the financial stuff and make 'em take Calculus.**
As a young lawyer taking all work I could get, over and over I saw otherwise intelligent young people getting themselves into terrible binds b/c they couldn't wrangle financial math in much depth.
But I get what you're saying. A decent foundation in Calculus might at least reinforce the kid's BS detectors.
**Plus, it's the way match is taught. I took Algebra I & II for B grades, but nobody showed me what it was really good for. I could sort of the number & letters & get the right answers, but mostly I felt like I was chanting the right spells and the Math Gods smiled on me at test-time.
I taught math courses for 25+ years and agree with everything you stated. I believe that most "advanced" classes were used by universities as a filter.
I am not certain about the 10% figure, but it sounds about right to me.
I have used Calculus in my job, and occasionally at home. It's not frequent, but it happens.
I've used statistics MANY times, and I teach it at work, because it's important.
Probability is a tougher one. I think it's useful for everyone, though most people don't understand it very well. If I told someone that 90% of the work is done by 10% of the people, and 10% of the people in the office have post-graduate degrees, there's a strong chance most people (including some who understand probability) will say that the post-graduate degree workers do 90% of the work.
Part of what I teach at the office is not confusing percentage POINT changes from percentage changes (going from 90% to 100% is an 11% increase, though most people say it's a 10% increase), because those minor variations wreck havoc in reports and will often confuse upper management if not handled properly (most often, it's accounting and finance that go nuts when they try to make sense of the reports).
I also teach the new employees to be wary of rounding errors when working with statistics. 10% out of 1,000 units can be as few as 95 and as many as 104 when you round out the decimals. That's a .9% swing, which doesn't mean much, but again it can add up.
This is simple stuff that younger people don't pay much attention to, or even know, today.
Wish it were all so easy. Schools don't "teach" algebra and calculus. They have classes where a teacher reviews homework and assigns more homework each day. Schools don't "teach" much of anything anymore. If the child is very bright or inclined to study they learn. If the parents are willing and have the time they can "teach" the subject to the student or at least make sure the student completes assignments and learns something. To require students to take classes which they clearly won't put in the effort required to learn the subject is probably the single greatest cause of our high dropout rate. But the educators are stuck on insuraing there is a job for them and THAT is why everyone has to take classes where many will not learn and none will be taught.
The best thing I learned in school was the "present value of $1". That's how to calculate mortgage payments (coming in and going out) and the present value of an income stream. I had a hard time really understanding the ramifications even though the math is easy enough. It would be a good basis for Econ 101 or a personal finance course. I used it the first day on the job and still you it several times a week.
It defies reason why these people would think Algebra II disposable. One has to have that basic skill to do probability, statistics, calculus, and virtually anything mathematical. But I am old enough to have seen a different math curriculum in high school:
Algebra I, II
Elementary Analysis (essentially precalculus)
I am highly prejudiced on this topic, partly becaause I run across far too many innumerate people who are helpless with kitchen math, much less probability and statistics. When I was really up on my game, I didn't memorize formulae, but remembered how to derive them from first principles should I forget one. Doing that requires a lot more than just knowing how to "turn the crank"; one has to have a road map in one's head that relates mathematical concepts which regularly appear in different forms.
Mega dittos on that.
On the economics front, I found calculus concepts endlessly valuable when I was running my small business -- fresh out of school.
Those who don't grok calculus are not interacting with the real world at a nuts and bolts level... They're using abstractions -- perhaps dating into antiquity.
If your need is people to people, then, yeah, calculus is of no utility.
Only 10% of students should be taking Algebra II and Calculus.
Algebra II's main focus is functions. You cannot deal with Physics and Chemistry without an elementary grasp of functions. Some of my high school classmates who wanted to take Chemistry a year early were told that unless they had already taken Algebra II or were planning to take Algebra Ii concurrently with Chemistry, they wouldn't be permitted to take Chemistry. There was a reason for that.
Would AVI say that only 10% of students should take Physics and/or Chemistry?
A further point is that what students plan to do when they are in high school is not necessarily what they will end up doing. A broad background in high school will give them more options later.
When my sister was in high school, she wanted to become an artist. An artist would not have need of Algebra II. After dropping out of college and working for 5 years, she decided that she wanted to become an engineer. She would have had a much more difficult time getting admitted to an engineering school if she hadn't already taken Algebra II and Trig in high school. She got her engineering degree and worked as an engineer for 25 years.
I agree, what is the deal with limiting kids in high school. It's as ignorant as separating out vo-tech at that level. Really, a kid is suppose to decide to become a plumber at 14?
The purpose of school is to create potential. You build as much foundation as possible then you aren't limited as to how high you can build later. If you lay a poor foundation, few go back to catch up later.
Now, the whole math perfection or die practice is ignorant. Not to mention the continuation of mixed group progression. Just do the Khan Academy-flipped classroom with the kids moving along at their pace.
Most problems with kids and math come from being left behind when they hit a rough patch due group lecture instruction. Once left behind, kids become lost and lose all incentive. Same goes for those who do get a concept then must idle till the class average catches up so the teacher moves on.
I'm a mathematician by training and an Professional Engineer by trade. My dissertation was on Riemannian manifolds and their application to real space problem solving.
That and $5 will buy me a foot long at Subway.
Not everybody can use calculus. Not everybody needs to learn it as a "logical discipline" like TB posits. A much more practical approach is to teach trough geometry and maybe Algebra II if the student is interested and leave it at that. That is a well rounded education in mathematics for the average student and citizen. That much mathematics gives the student confidence because most problems can be solved at the pre-Algebra II level.
We're not falling behind in the complex mathematical regimes that calculus provides. We're falling behind in the basic everyday, figure it out with a pen and pencil brand of math and that's what we need to concentrate on.
I also thought that AVI was a bit off the mark with this. I can see where Calculus doesn't need to be a requirement but as Gringo and WalkingHorse point out, you need a grounding in basic algebra to work out problems in statistics and probability. I hadn't thought of Chemistry but that makes sense, and it's likely a useful skill in other science courses as well.
My main objection to his prescription was the emphasis on 'teaching mathematical thinking'. From my perspective (I see it's shared by a number of commentors) we appear to be doing a poor job of teaching people basic mathematical operations and a lot of the reason for that is we try to teach 'mathematical thinking' from the get-go instead of getting primary school students drilled in how to use mathematical functions.
I would call "basic algebra" the old "Algebra I." I think more than 10% of students should take that. 40%? 80%? I don't know.
Algebra II is more abstract. You need it if you are going on in math or science, or to any doctorate in which you will need statistics. I think that is a much smaller number.
One of my commenters thought personal accounting, or double-entry bookkeeping, was also an important math to teach, for a variety of practical reasons. I want people to move automatically to a picture of a bell curve in their head ins some situations, an exponential increase in others, and an arithmetic increase in others still. I want the automaticness that comes from repetition.
When I was in high school you took geometry, trigonometry, algebra and finally intro to calculus and analytical geometry. Don't know how that would translate into coures taught now. I am an electrical engineer and in your freshman year of engineering school you had to take and pass calculus. If you didn't, you would not become an engineer. Your senior year you had to take the mandatory course in electromagnetics. That required you to know complex variables, vector analysis and partial differential equations. This actually was useful because I used to work in antenna design.
We used to joke about Riemann manifolds for hot rods.
I had been a crappy math student, but when we got into Calc I loved it, got the feel for it, enjoyed it a great deal even though I have had far more use for understanding stats than calc.
Algebra 2 is required for trig, and everyone needs basic trig. in Everyday life trig is the most useful bit of math.
Good grief! Probability and statistics are two of the most difficult mathematical fields out there. Even world famous statisticians can't agree on what probability is or what constitutes a proper statistical test. What in the name of all that's Holy does p mean, or is r better than r^2? Go read the tirades by various Bayesians and frequentists if you doubt me.
What he obviously means is plain arithmetic, and maybe percentages.
My belief is that unless you are a working engineer, scientist or analyist , the concepts behind geometry, trig and calculus are more important than being able to solve work problems. If the subjects were taught that way to the 90 % who won't be working in STEM, there would be more interest and a better "take away".
As a working electrincs engineer, I often used algebra, geometry and trig but not so much calculus. However, an understanding of calculus was absolutely essential to the work, even if I didn't directly use it much. I found numerical analysis a "bigger hammer" to smash problems into submission than calculus. But it's probably not a suitable high school subject.
My personal take is that teaching in detail of algebra, a solid overview of geometry, trig and calculus, coupled with finance and basic statistics would be the best approach for the 90 % who won't be studying STEM subjects.
Of course, these days schools don't have time to teach much of anything other than socialism, "diversity", class envy and protesting.
100% agreement. Since my BSEE in 1961, I have never used calculus and applied differential equations exactly once, when I proved #12 AWG would be just fine for a buried loop antenna that could survive a 1 MT nuclear strike 100 meters away. Advanced math certainly was a filter back then.
But I was a design engineer, whose gadgets actually had to work, as opposed to the simulations that represent workproduct nowadays.
I am in favor of high school students being exposed to probability, statistics, and personal finance. As statistics will be bandied about in current events, it is a good idea to give future citizens the tools with which to analyze the various statistical claims that will appear in discussions of current events.
Perhaps these could be worked into the second semester of fourth year high school math, with trig done the first semester. Or vice versa.
These subjects would be useful for students who are not planning to take four years of high school math. Perhaps a year long course for them.
Could or should anybody gradaute from HS - or college without stats, calc 1?
I sure hope not.
I believe everyone should be able to calculate the interest on a home or car loan -- and the present value of the whole set of payments -- before he is considered an adult. Otherwise he will likely go broke and become a burden on the taxpayers, whether or not he ever gets to vote on tax or spending bills (perish the thought).
I should probably abstain from this discussion, but as a recently retired high school teacher, I felt I should at least speak for those youngsters who I had the privilege of teaching. I taught Econ/Am Gov to high school seniors. Some were indeed enrolled in AP Calculus, most had problems with basic algebra, not to mention fractions and decimals. I tried to help them as best I could, and tried to show them the logic behind the problems. I understand that you want the best for the youngsters and the country, but not all youngsters have the benefit of college-educated, graduate engineer parents. Many of my students were the first in their family to graduate from high school. Many of our parents did not have pleasant memories of school, or positive views on the value of education. Many of my students had a parent(s) in jail. I lost students to meth, other drugs or abandonment by parents. Some kids finished high school throughout the kindness of friends helping to sustain them. Many lacked a work ethic because they never saw that behavior modeled by parents.
So, my brothers, although you mean well, please understand that most youngsters in this country won't be discovering the joys of integrals and derivatives, but will do a pretty good job of repairing our machines, manufacturing our goods, raising our crops, caring for the sick, building the buildings and fighting our wars. Many will go to technical college, some to university. For what it's worth, I'm proud of most all of them, and I try to tell them that when I see them.
Oh, one more thing: we were discouraged from assigning homework. Sorry, that was policy. I can tell you some other policies that will have you shaking your heads, but that is for another time. Don't blame the kids for the policies, they do as they are told.
Please feel free to discuss among yourselves.
Add me to the engineer types who are pro calculus. I am by education an industrial engineer and by profession a systems and network administrator. Do I sit down and integrate trigonometric functions on a regular basis? No. But ask if I deal with quantities that change over time, and the answer is an emphatic yes.
There are two fundamental problems in elementary calculus: finding the slope of a line and finding the area under a curve. These seem to be pretty esoteric untill you recast them as finding the instantaneous rate of change of a quantity, or calculating the accumulation of a quantity over time. The great insight of calculus is that these are inverse operations.
I find it interseting that people are talking about how calculus is unnecessary, but students should be taught things like time value of money, compound interest, prob and stats, and marginal analysis. Without the knowlege of these basic concepts of calculus, these concepts are pretty much unknowable. They are directly derived from the kind of rate of change and accumulation over time (or probability space) that are taught in elementary calculus courses. Add in such concepts as an inflection point, point of diminishing return, exponential growth, or reaching an asymptote, and you'll see calculus principles used in the discussion of all but the simplest issues.
Otherwise they become, as others have characterized, the magical plugging of numbers and incantations that generate "the right answer".