The Math Scam
June 26, 2012
By Stan Brin
Some of us are good at math, some of us struggle merely to get through it.
Whether we’re good at it or bad, few of us will ever again use anything we learned in calculus or trigonometry class ever again, not even once. After graduation, few will even be able to recognize such general terms as sine and cosine, much less be able to explain what they mean.
For those who want to become engineers, scientists or economists, math is the foundation of their careers. It’s vital, not to be questioned.
For the rest of us — and I include technicians and medical workers* among the rest of us — math is, more often than not, a painful and soul-breaking ritual that we are forced to endure if we hope to have a decent life.
The official line is that lots and lots of math is supposed to prepare us for work. It’s supposed to teach us to think logically. It’s also supposed to help America compete against Asian Tiger economies that are eating our national lunch.
These assumptions may be mistaken. For many, if not most students, math education, at least as taught in this country, is little more than a cruel and expensive obstacle course designed to force large numbers of them to fail.
Even wore, this torture machine produces generations of Americans who graduate utterly unprepared to tackle real-world studies.
Consider the following sample problem that all students bound for higher education are expected to understand:
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
This requirement, taken word for word from page 60 of the California Common Core standards, is among the norms used by 45 states and the District of Columbia to determine what every student should know. There are many, many, more examples, all equally opaque.
Obviously, somebody in 45 states and D.C. really thinks that all of us common folk really, really, need to know how to “traverse counterclockwise around the unit circle” or we won’t be able to think logically, as if mathematicians are known for their logical thinking. (Ted Kaczynski, Paul Erdos, Lord Bertrand Russell, John Nash, and Sir Isaac Newton come to mind — all of them brilliant, all of them mentally handicapped in various ways.)
During my career, I’ve written thousands of articles on subjects as varied as boxing and physics, I’ve designed software products that won two Editor’s Choice awards, but I’ve never had occasion to “traverse counterclockwise around the unit circle,” nor even to traverse it clockwise. Not once.
Nor did I ever have to understand “how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers.” In fact, I don’t even know what a unit circle or a coordinate plane is, or why anyone would want to traverse one. I’ve looked it up, and I still don’t know, other than the unit circle has something to do with a radius of a circle being equal to “one.” One what, no one says, at least not in English.
Of course, I’m sure that some people actually need to know all about the unit circle. I would include among them such truly and honest-to-gosh smart people as scientists, engineers, economists and artillery officers.
My brother, for example, needs to understand the unit circle. So does his wife. They’re both astrophysicists. They study the paths of comets and asteroids, and how to send space probes to meet them, what is known colloquially as “rocket science.”
Brainy folk like my brother — his face is familiar to the millions who like “history of the planets” shows — may account for 1 percent of the entire population. To that 1 percent, let’s add people whose work requires them to talk to scientists, engineers, economists and artillery officers, and we may have another 1 percent of the population. Let’s add another two percent for people who marry scientists, engineers, economists and artillery officers and those who know how to talk to them. Let’s also add another percentage point for math hobbyists who are actually interested in traversing the unit circle for its own sake – and we have a total of 5 percent, one out of 20.
And that’s being extremely generous.
The rest of us, 19 out of 20, are force fed higher math for up to four years. All college-bound students are required to pass three years of it. Vast numbers drop out in frustration, others manage to barely get by– and swear that they will never enter a classroom again.
And a day after our last finals, all of us who passed immediately forget absolutely everything. Meanwhile, very, very, few of us are taught how to use math to solve real-world problems, such as how to calculate the amount of wood needed to build a house, or how much concrete is required to pave a patio.
The average homeowner doesn’t have much use for the unit circle, but knowing how to buy just the right amount of materials, how to have it delivered on time and how much it will cost down the line, would save a lot of time and money.
But that’s not as important.
Blame History
There are those who believe that degrees are pointless scraps of paper. I disagree, but Peter Thiel and others have a point: We force young people to suffer obscure and useless subjects as a ritual — because it’s the way things are done, and the way things have always been done.
These obstacle courses — and that’s what they are, obstacles disguised as courses — exist because our grandparents and great-grandparents endured them, and if they learned to traverse the unit circle, well, by jiminy, today’s young whippersnappers had better learn to do it as well. We may no longer be expected to learn Latin and Classical Greek, thank Almighty Zeus, but the struggle with theoretical math still holds a mystical, untouchable holiness among well-meaning educators.
And yet very few young people study computer programming in high school, and those who do, don’t learn enough to obtain an entry level position. Think about it, 35 years into the PC age, and most kids put on their blue caps and gowns without ever learning Boolean logic, conditional loops, variables and arrays, terms that should actually mean something to the average, reasonably educated person.
Why? Because the starched-collared, monocle-wearing worthies who invented secondary education curricula for the unwashed masses, back in the olden days of bustles, shirtwaists, handlebar mustaches and buggy whips, didn’t think that practical subjects were as important as the skills they mastered at their exclusive private schools — such as translating Ovid or Caesar’s Gallic Wars.
Nevertheless, by 1900 times had changed. According to Professor Mark Ellis of Cal State Fullerton, a specialist in the history of math education, “The turn of the 20th century was the first era of large cities with diverse populations. Child labor was banned. Kids had time to go to school.”
The result was a vast increase in demand for secondary education, but the idea that students born of farmers or immigrant shopkeepers should study bookkeeping instead of Pericles’ Funeral Oration was difficult for academics from privileged backgrounds to understand.
Still, courses such as “shop” and “home economics” managed to infiltrate the system. Boys used to learn how to saw lumber, and girls learned how to cook. Perhaps, these days, boys and girls should study both subjects, or at least learn how to operate a microwave. Instead, they’re cramming math, yet falling even further behind international standards.
Latin and Classical Greek are now out of fashion, praise Jove and all the others. A few might want to study these languages so that they can name new species of slugs and jellyfish. (These days, dinosaurs are mostly given Chinese names.)
But California law still requires three years of higher math, including calculus from anyone who wants to go to the University of California or Cal State to study marketing, public administration or even history.
This doesn’t mean that that there are any monocled, pointy-bearded men and pince-nez wearing women out there eager to paddle the daylights out of nineteen out of twenty students with wooden rulers for failing to traverse the unit circle. Far from it. Their modern incarnations, such as Gerardo Loera, executive director of curriculum and instruction of the Los Angeles Unified School District, mean well. They just don’t get it.
“It should be as embarrassing to say ‘I can’t do math’ as it is to say ‘I can’t read,’” Loera says, which would make sense in a perfect world. “I still believe that math skills, such as critical thinking and problem solving, will transfer to other areas and are important even for liberal arts. Even if students don’t take any more math.”
Yet Loera, a fine and decent man who proved remarkably open and generous with his time, couldn’t cite any facts or figures to back up that belief. I asked him if calculus and trigonometry are useful in, say, journalism.
He sighed, and admitted that “I can’t cite any research that higher math helps journalists.”
Precisely my point.
The Asian Solution
One reason why Asian countries seem to be eating our lunch appears to be an understanding of a basic fact of the human brain: Only so much stuff can be forced inside.
So they teach math from staple-bound booklets less than a hundred pages long. Only the most important topics are covered, but students are given time to actually understand them. Contrast those books to the dangerously heavy bricks California students are forced to lug home every day, and skim through because there’s no time to really understand what’s in them.
Everyone involved in teaching mathematics admits that the situation is ridiculous and self-defeating. According to a famous paper by Prof. William H. Schmidt of Michigan State University, the math curriculum in the United States is “a mile wide and an inch deep.”
Everyone involved in math education that I’ve talked to, including Cathy Seeley, a past president of the National Council of Teachers of Mathematics, agree with Schmidt: High schools are trying to cram too much into kids’ heads.
Courses cover too many topics in too little time, the teachers have to move on before the kids have time to absorb anything, the lessons are so abstract that they mean absolutely nothing, and in the end, the students will forget absolutely everything.
Some people claim that the problem isn’t topics — Singapore students, we are told, study more math topics than American students and do better on standardized tests.
But Singapore is an island city state. It has a small, rigorously conformist and highly disciplined population that accepts a single-party dictatorship without complaint. Singapore also famously produces university graduates who haven’t the slightest idea where babies come from. Even worse, chewing gum is illegal there.
It is also one of the countries whose students are expected to brutally cram for admissions tests. Once admitted, Asian students find that university studies are less rigorous than they are in North America, hence the vast numbers that come here for post-secondary education.
There are those who still believe that narrowing math standards in high school — and adding flexibility to the system — will cause California and the rest of the country to go to hell in the proverbial hand-basket. These should remember that American high schoolers have been doing poorly, by international standards, for decades, yet our universities are still the envy of the world.
An American Solution
I would never say that higher math is only for nerds, or that it is unnecessary for those interested in fields that build on its foundations, nor would I ever say that students shouldn’t know what trigonometry is, and why Newton invented calculus.
But instead of frying their brains trying to traverse the unit circle counterclockwise, perhaps students should be given a year of natural history. Instead of trying to solve useless, abstract puzzles, they should try to plot the orbit of a Mars probe, or how much energy would be required to send an asteroid hurtling to Earth to wipe out the dinosaurs. Or how scientists were able to use math to analyze regular mutations in DNA, proving that we are all descended from a single woman who lived some 200,000 years ago.
Or how King George III used calculus and astronomy to test the first practical longitudinal chronometer. (Yes, King George was the villain of the American revolution, but ancestors of most Americans arrived on this continent in reasonable safety, and at a much lower cost, because that very odd King was able to prove that the longitudinal chronometer actually worked — and convince others that it did.)
Students would find these examples more interesting than anything in the Core Curriculum. They might not be able to traverse that unit circle counterclockwise when they were finished, but they would know a few more things that they might remember past prom night.
———————————-
* Note: Doctors and nurses don’t need higher math. Perhaps some doctor might like to throw a quarrelsome hypochondriac from a window and calculate the time it takes him to land — that would be higher math. But in the real world, they mainly need to know the metric system and be able to keep its infernal decimal points in the right place. They have too much to learn about the infinite frailties of human anatomy to be bothered with traversing the unit circle. Try asking your surgeon a trig question, and you might as well be speaking Latin or Classical Greek, but he or she is still required to learn higher math as a way of demonstrating an intelligence sufficient to remove an appendix.
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Great column, Stan.
The problem is: most kids cant even just add and subtract anymore, let alone do higher mathematics. I love it when I go shopping and the kid behind the counter cant get……..1 @ @2.00…..omg, they have 3 of them! What do I do!!! Our system is pitiful. And getting into college should not be a cake walk. Give me a break. I disagree with you. Higher education should be hard, it should be a challenge and it should make one more aware of how to handle complex “stuff” in life. It should not be a breeze. This is why our school system sucks. You dont expect anything from any of them.
Calif needs a dumbed work force to get wages down to compete internationally. Public schools are geared up…let them work their magic.
As a landlord I try to show collage grads tenants how to pro-rate their rents to the 1st of the month. Many have taken higher math. But the stupid looks on their faces as I try to explain 10th grade math to them is astounding.
High school should focus on 4 kinds of math (as should collages)
General math: 2+2=2 squared
Cooking math: IE, pints to teaspoons to liters to gallons.
Construction math: True story. I was working at Home Depot when one of my workmates came to me to ask what those lines on the tape measure are for? Trying not to laugh in his face I slowly explained half inch, quarter inch, ect.
When you get a bid for either yards of concrete or yards of carpet on your house you should know the simple math it takes to figure it out so you won’t get scammed.
Financial math: Pro-rate your rent, amortize your home loan, figure out your credit card statements, understand the rate of bond returns on you pension plans, ect.
Finally: Everybody who graduates from high school, male or female, should know how their toilet works: where the shut off valve is, how to change out the flapper valve, and what the float valve is. I’ve had dudes with engineering degrees screaming at me on the phone how to stop the turds from over-flowing onto the bathroom floor.
I tell them to not have their girlfriends flush their tampons down the toilet.
Hondo…..
P.S.
That dude a Home Depot is now the head of the lumber dept.
Hondo……
Someone has to do it…honest work should never be demeaned….ever…..
Even that of cubicle worker posters…
Yeah well that “dude” at Home Depot better know what the heck he is doing. Last time I had one of those idiots figure out how much rock I needed, he was way short. I told him he was and had to show him how to calculate the area. And the college student shouldn’t get it wrong at all.
Instead of “dumbing” down students, I would think the system should be smartening them up. Otherwise why even bother.
I never suggested that math be “dumbed down,” merely that students should learn the sort of “math” that would actually be of use to them. We may not think of accounting or programming as “math” but the logical thinking is the same — and they are much more likely to be put to use than traversing the unit circle.
On the other hand, maybe if girls admired guys who did well in math, perhaps… no, that would be asking too much.
I’m an engineer, know exactly what a unit circle is, and use trig functions and radians all the time. Maybe terminology has changed in the years since school, but I have no idea what is meant by, “enables the extension of trigonometric functions to all real numbers”.
That said, I’m not sure what the long-term implications would be if we only taught students practical math. It implies that only those whose career paths take them into professions that need higher math would learn higher math concepts. Thing is, how many students know back in 7th grade what their career path is going to be? If you don’t decide to go into engineering or astrophysics until high school or later, you’d have years of catching up to do, so maybe you’d pick a different career rather than have to invest that time (and cost) now. We might turn people away from math-intensive technical professions by having inadvertently erected higher barriers to entry.
Say hi to your brother David, astrophysicist and well-known sci-fi author. I’m a fan.
This article is all too true.
How many young people do any of us know who can handle the very simple math of personal finances? Do any high school math courses cover balancing a check book?
It MAY seem like a trivial example, but remember these folks end up voting for politicians who want to spend unlimited amounts of other peoples money because they have no grasp of their own personal financial situations beyond “Gimme more!”.
I’m with Hondo. How many college grads with economics degrees can calculate their monthly mortgage payment?
There is a PhD in economics from U.C. Davis who has a prominent website that is followed around the world. I had to correct him once on one of his posts because he couldn’t figure out amortization of a loan. He ended up removing his posting from his website.
Same thing with how much imported water do farmers use in California. Peter Gleick of Fake Gate fame at the Pacific Water Institue and academics in U.C. system say it is 80 percent. The California Dept. of Water Resources says it is about 40 percent. Gleick’s higher percent is based on a dry year with farmers not shifting to groundwater supplies. The Dept. of Water Resources percentage is based on an AVERAGE year of rainfall with no shifting to groundwater. The actual number based on the higher base of the total amount of RAINFALL in an average year is about 8 percent to 20 percent. The higher number is used for political purposes to shake down farmers for their contractual water allocations so water can be redistributed to others for political favors for tourism, lodging, and real estate development in return for political contributions. Numbers are highly politicized in California.
Yikes, figuring out how to answer that question only takes looking up a few words…answering it takes a few seconds, far less than the time you spent writing this article.
Absolutely, only people that want to be called educated should learn this stuff, for all others there’s trade school. Unfortunately, the “everyone must go to college” scam, combined with the “everyone can get a loan no matter how uninterested they are in learning” scam has, indeed, put us in the situations of many thousands of students taking subjects they need know nothing about.
Funny, I don’t see rants online about feminist ideology (also forced upon students), multiculturalism (also forced) and a host of other ridiculous subjects. It’s probably because the latter are stupid and any fool can figure it out (hate white males, and you’re set). At least math will still be true in a 1,000 years or more, giving it a distinct edge on most other college subjects.
The ironic thing is, the unit circle is not higher math – it’s basic math.
It helps to give a very clear understanding of the way trig functions work and their approximate values if they fall between its most common angles (marked on the diagram in the article).
Trig functions are extremely useful in calculating heights or other lengths when it is impossible to use a tape measure – e.g. when the height is taller than you can use a ladder for. They are also useful for doing rough calculations as a double check on laser measurers etc.
Measuring heights etc is of course basic for everyday tasks such as home / office DIY repairs and renovations and trig functions are often used by amateurs in this way.
So while I fully agree with the thrust of the article, you couldn’t have picked a worse example to try to illustrate your very valid point.
Stan, your issue does not appear to be with math but with the method of teaching. Math, in its own right, is an incredibly valuable mental tool. Math applied, accounting, finance, construction, etc. as discussed, is even more valuable. So is the ability to think with discipline, logically and rationally. I have a solid foundation in math that I apply to all types of things from aviation to fly my own plane to finance to pay for it.
In my opinion, if anything there should be a greater focus on math and it should be taught much more quickly. I do not see any reason most students cannot be finished with Differential Equations by 13-14 years old.
There is a reason the Robinson Cirriculum or Khan Academy stress math so strongly; because math is such a foundational and critical mental tool. Math is what it is. Therefore, if people find it difficult to labor through then it is because they do not think correctly or with enough discipline.
What an article.
First off…Your whine…
Why do I need to know Math?
Do you have children? Who teaches them? Your brother? Why not take some PERSONAL RESPONSIBILITY and TEACH THEM YOURSELF?
Oh…Because you FAILED TO LEARN MATH. If you learn Math…YOU DO NOT FORGET IT.
You posed a question that I will be happy to answer. You asked have I ever used Calculus to determine the amount of materials for a patio? Curiously enough…I HAVE. The patio slab extended outward to a curved pool deck. I used the principle I learned in Integration to determine the volume of concrete necessary to pour the slab. I was paid well for my services rendered.
BTW…I am not a professional scientist.
As for Medical Doctors knowing Math? I give them one question from my Third Semester Calculus class. If they cannot answer it…THEY ARE FIRED.
What is the Error Equation in N Variables? It is a Partial Differential Equation. The Human Body is a system of Functional Parts. If one part of the system is errant then the ENTIRE SYSTEM IS ERRANT. Systems affect one another.
I believe in a Holistic approach to the SCIENCE OF MEDICINE. Now Medical Science is QUANTIFIABLE. If it was not why do the Doctors place numbers on phenomenoa as Blood Counts…Electrolytes…enzymes…and other MEASURABLE QUANTITIES? Is it just Hocus Pocus to impress the patient? That would be insane.
No…Math is a necessity for Medical Science. For someone who is demonstrably mathematically challenged you could not even begin to surmise these underlying AXIOMS…er…UNDISPUTED TRUTHS.
That is the reason why you cannot think unless you understand Math. Your thinking is merely shallow and superficial…lacking in any depth. You are not capable of thinking in that way. It is sad as you will miss out on much of the underlying beauty.
In my Diff. Eqs class…We flew to Mars as the class project.
Tall Tom
I Cor 13
I disagree with you. I am a retired chemical engineer. Obviously I used my higher math education in my profession and obviously not everyone needs to learn or is capable of learning higher mathematics but… that said everyone will benefit from the advanced courses in algebra, trig, calculus and differential equations. I still use the logic of my math daily in my retirement. I do not do problems but I use the logic to solve everyday problems. Someone who has not had or comprehended these courses will never know how much they help in everyday life. My friend is a retired middle school math teacher. We met after she retired and we have had long conversations on education. She was astonished when I told her i never have nor will memorize individual facts but that I learn concepts and then apply them to whatever. Last year as we walked down a road with a large chemical plant on it we looked at a huge pile of stuff. She pondered what toxic waste it might be. I started in — lets this plant makes ya da ya da here and the reaction for making this stuff in a plant on the ocean would be ya da ya da and the by products would be ya da ya da so I suspect this is a mountain of ya da ya da. She looked at me and said you do that all the time — you start at the beginning and then answer my question after rambling on and on. Sometimes it go on for quite a while. She said I do not know how you learned to do that or why and we did not teach that way when I taught but it works and I think I taught math wrong all those years. American education does not teach the average person to think. Math teaches you to think. Period.
I have a PhD in aeronautical engineering, which means that I have studied every kind of applied mathematics under the sun and have solved problems that have never been solved before and I, too had no idea what the question was asking. The solution came to me in a dream at 3 am, meaning that the subconscious mind is remembering things long forgotten.
The key to this problem is that this is a geometry problem, not a trig problem. When I took geometry in the 80s in Ohio, we physically drew things using ruler, compass, triangle, etc., and that is what we have to do here. You also have to know the definition of the words radian, sine, and cosine, which every geometry student should have been taught.
radian = angle = length along the circumference of a unit circle
sine = opposite / hypotenuse of a right triangle
cosine = adjacent / hypotenuse of a right triangle
Suppose we want to know sin(2) and cos(2).
Take a piece of paper and draw a horizontal line across it. This will be the x axis.
Open a compass to, say 3 inches. 3 inches will be the radius of the circle
Put the pointy part of the compass in the middle of the x axis and draw a circle, the hole in the paper made by the compass is the origin.
We want sin(2), so cut a piece of string 6 inches long (2 times the radius)
Place the string along the circumference of the circle, one end of the string where the right side of the circle crosses the x axis the other end counter-clockwise along the circumference of the circle. Mark the other end on the circle with a pencil. This is 2 radians.
Draw a line from the origin to the circumference where you marked it. This is the hypotenuse of the right triangle.
Draw a vertical line from the point on the circumference to the x axis. This is the opposite side of the right triangle.
The x-axis part of the triangle is the adjacent side of the right triangle.
Use a ruler to measure the sides of the triangle. Calculate the sine and cosine using the formulas above. If the point on the circumference is above the x axis, the sine is positive. If it is below the x axis, the sine is negative. If the point is on the left side of the circle, the cosine is negative. If it is on the right side, the cosine is positive. For our case, sin(2) is positive and cos(2) is negative.
This math problem assesses a student’s knowledge of the fundamental relationship between geometry and trigonometry. It is entirely appropriate to have it on a standarized test. I didn’t have to look any of this up; it was all there buried in the subconscious. We’ve just been out of school too long to remember it in the conscious mind. Knowing the solution, how would you word the question so that it is less confusing?
Saw re-post of this on Lew Rockwell’s site. Please excuse if I run long. IN general, I appreciate the post, but I truly think the problem with deciding how much something is “needed” is that there is a chronic trend downward. I won’t go linking proofs simply because that will take longer to source articles and stats than typing, but I stand by my assertions. Several years ago it was demonstrated that there was a functionally catastrophic drop in literacy levels amongst university grads in the previous decade(and as they were lumped in with prior uni grads in compiling stats, the drop amongst newer graduates in literacy levels must have been tremendous to affect the average so much). SATs get renormed, and despite the greater access to educational resources available since the web came on, ipod university, MIT Opencourseware, KhanAcademy, etc., there seems to be a downward pressure and trend on actual educational attainment. Don’t get me started on the comparative poor performance and SAT to IQ conversion stats for “education majors” (presumably those who will teach) versus almost any other major. In short, there is a chronic weakening of educational standards and preparation of our young, lengthening of the process, etc. despite the fact that the US, and its workforce and economy face the twin pressures of increased foreign competition (and their increasing development and productivity of their workforces via education and capital improvement)and the siphoning effect of inefficient, expensive, burdensome government which sees greater and greater expense and intrusion into the economy, private lives, and fiscal health of the nation. Surely weakening educational standards, in math in particular is not the way to go.
Mathematics IS the language of science. One never knows when one may need to call on the higher faculties of mathematics, and one never knows what mathematics may become useful. Upon a time, boolean algebra, matrix mathematics, and the like was esoteric, the province of academia and professional mathematicians. If anything, I despise the lack of it in my primary and secondary educations, which, by the standards of society were excellent (and at least at the primary level, fell well below any meaningful challenge, and at the secondary level, it was only a challenge because I also was working an average of 35 or more hours per week to PAY for private Jesuit university prep high school).
In short, I think the real problem is that despite the massive infusions of information technologies, which should (and in the web do) reduce the time and costs and personnel requirements in schools since many jobs previously done by teachers can be automated and self supervised with instant feedback and correction, plus tele-teaching to import talent, the way IT is USED (in the main) has been simply to do what was previously done….just better. What is necessary, and if I could ever find someone with whom to partner, I have a business plan in basis sketched out to attain this, is to take existing technologies now broadly available, and utterly change the way things are done at the secondary and especially university level, including math. The Blitzkrieg was NOT the Wehrmacht taking new tech that no one else had and surprising anyone. Everything the Germans had, so did the Allies. The difference was, rather than using radios, tanks, aircraft, etc. to do old jobs better, they were combined in novel ways to do things not previously considered. THAT was the revolution in military affairs. It required NOT new technologies, but new ways of using them. Information technology needs the same approach, and THEN the access to university will fall, and the costs too, with profits driving up.
With math in particular, I agree with a lot of the criticisms, BUT what I find is that in almost all of primary and secondary education, math is taught badly. It is divorced from application and history. In home schooling my kids I have found that by discussing (and reading) about mathematics uses, engineering in history, pythagoras, Archimedes, etc., the history of zero, Pi, and so on, that interest in math is stimulated and maintained.
In general, when home schooling my kids, I find they can do twice as much work in half the time without serious stress. Their school day is half of what is typical in school, and they are learning well ahead. I don’t delay pushing advanced CONCEPTS and I combine subjects. To teach my son his times tables I also taught him to multiply and divide fractions. Using the number line to teach counting (then adding and subtracting), I also threw in understanding Cartesian coordinates. While in some cases, he still has the normal struggles with the memorized skills (getting 11 x 11 wrong when asked to spit it out on demand), ALL of the math he does know can be conceptually applied in a number of areas. At seven he is doing the normal, plus his times tables, AND ALGEBRA with add/subtract/multiply/divide components, negative integers, and so on.
I find that mathematics is just badly taught, it is either rote memorization but little imagination, or lots of shiny imagination (new math) but falling down on developing arithmatic and skills, when it should be both.
My wife returned to university after age 42, having previously left school at age 16 (in UK) to work for her father, than start her own businesses at age 18. She always worked her for herself, in various businesses she set up, made profitable, ran, sold, then took time off before doing it again. If it weren’t for a very bad marriage to a rotten guy who bled her and squandered the cash she made, I imagine she would have been very well off as she was making good money starting all over again at the time I met her. Life has had its ups and downs, and she has just finished a double degree in physiology and pharmacology, and now been awarded a scholarship to do a masters in radiopharmacy/radiomed. She put in extreme work, but what caused her the greatest problem on her course…the MATH. WHY? Because it is not taught well in general, deductive skills are taught, but inductive ones neglected, and it generally does not go far enough.
Once upon a time, a degree was NOT just a certificate that ensured access to the job market, which it increasingly has become. Education itself was not perceived soley as a utilitarian good (economics perspective), but was an INTRINSIC good, seen as good for its own sake. A degree was a certificate that stated that you had achieved a level of knowledge across a range of subjects considered necessary for any educated person, PLUS whatever your specialty might have been. Now (especially here in the UK), it is a specialization and from funding to focus, has become a very utilitarian pursuit.
I achieved my professional status in engineering post military, despite lacking ANY engineering training from any formal external source, but having more than enough credits for degrees in history and international business management (but lacked the consolidation of transcripts, AP, CLEP, DANTES, ACE military guides), and with “residency requirements” I did not feel like disregarding the credits in my portfolio and taking two years with X or Y univeristy to have THEIR degree, so economically, the value of my “almost degrees” is nothing. Engineering wise, DARPA invited me to present my ideas and systems concepts in conference years ago (which I did), and companies like GE, Tmobile, etc. hired me to teach their engineers courses in relevant seminar subjects and do years of design work as an employee too. I developed a great informal education, but I promise you, for my purposes and for a belated interest in the hard sciences (quantum physics, electric magnetism, etc.), I truly wish I had been afforded the opportunity to take more mathematics, and had kept it up post high school. It is always harder to get back into, and lack of mathematics prohibits progress in a range of engineering disciplines, chemistry, and physics. Knowing Calculus, and trig, are essential.
Educations are broad because people should be generalists, NOT specialists, and if we are going to throw people into a school system for an average of 13 years times at least six to eight hours/day and NOT have them to the level of calculus by the time they are 18 is a criminal waste of time and effort. Jaime Escalante proved beyond any doubt that you can take someone of any background with at least average or better motivation and teach them advanced mathematics to an incredible degree…if it is done right.
Home schoolers, on average, are four years ahead of their peers educationally by high school, which means that all things being equal, by the time a state or private school kid is graduating high school, a home school kid CAN be graduating with a four year degree, or equivalent knowledge. Having raised kids and step kids that run the gamut from (now) 24 to age 3, from graduates with masters in engineering (the 24 year old) and in uni, to home schooled. My kids and step kids have been in public schools, private schools, and home schooled. Some have zero motivation relative to their ability(the ones who spent their early years in government schools before I came on the scene), some have lots. I find that the longer someone has spent in a formal school, the LESS their general ability and willingness to study independently and really achieve relative to their abilities. Then again, for anyone who read John Taylor Gatto’s book “The Underground History of American Education” this should not be a surprise.
Ultimately, schools teach math poorly, and do in 12 years what is easily achievable in 8 if there were not this push to lump all abilities, motivations, and groups together in lockstep progress by age rather than achievement. Because we now take 12 years to teach 8 years worth of material (something I first learned was the problem about 14 years ago from a variety of excellent sources, books, articles, etc.), the issue with teaching advanced math becomes a problem.
No, not everyone needs trig, but in not having it, they do limit what they can achieve if they decide, later in life they want to go into a pursuit that does. If we were like the Swiss, where only 22% of the school population goes on to high school (and they are already tracked for university), and the rest apprenticed (including white collar professions like Banking, Insurance, etc.), starting their work experience by or before age 14 in conjunction with their remaining studies, it would be defensible to NOT push trig.
But as long as we incarcerate student inmates from age 5 (or sooner) to age 18 (or almost), eight months or so a year, five days a week, six to 8 hours a day, then to NOT get their mathematical education to that level is an absolute criminal waste. I think the whole system needs to be reformed, first by defederalizing, and decentralizing everything, and allowing localities total local control. For every mistake their will be ten successes, and the system will become self correcting. Allow vouchers with NO control in curriculum or standards beyond ensuring the money is actually being spent on education and not cashed by a mafia front. I can think of a dozen reforms that would help, from total privatization to charters, etc. What is certain is the current system is broadly failing, taking too long, keeping kids from work, responsibility, and a large degree of freedom too long while often delivering a mediocre, constrained education. I also think we place too much emphasis on degrees, and degrees that are liberal arts vs a specialized subject degree could and perhaps should be differentiated. The IT industry makes great use of certifications, and there is no residency requirement to achieve a series of high level technical certifications, unlike most universities, so a high level of independent subject matter learning CAN be credentialed. Universities would and should emulate this, but that strikes at the heart of ensuring you sit on a seat for two or more years (two years residency credits at least no matter how many credits you transfer in is typical)paying fees, taking their courses (and in many state university systems, they add things like rarified “cultural diversity courses” that are actually left wing in orientation, as I found in the University of Wisconsin System, since a raft of courses in various histories and cultures meant nothing, but African dance or Gay Studies, Women’s Studies, Latino Writers, African American studies were the ONLY acceptable categories to satisfy it).
I guess the whole system, when we think about it, could do with a revamp. In reality, a lot of people, companies, government agencies, etc. are invested in the current system, and really, the best way through is to get out (have people offer market solutions, which I hope someday to do, and to home school). Personally, with the variety of sources out there today, Robinson Curriculum, Khan Academy, BrainPop.com and co.uk, etc., there is a lot people CAN to to study math during school, unlike in “my day.” Schools and teachers can take advantage of this, but on the whole, the system cannot deal with a lot of kids who, in one or more (or alL) subjects, are two or four years advanced. The system is bad, it needs to be modular, and scalable, and though this is possible not against the entrenched interests.
Textbooks is one “simple” area. There is a powerful lobby selling texts, texts which are cheaply made, riddled with mistakes, politically vetted, increasingly dumbed down, and very expensive. Opentext book sourcing, PDF texts, electronic readers could make the whole issue cheap and moot. Getting a new addition would be a one minute download. Tinyurl it, etc. Every kid could buy, be issued, work off, lease, borrow, access to a simple electronic reader or tablet computer, with USB access, cheaper than it costs to get textbooks. Best textbooks could easily be achieved, including free ones. Huge money could be saved in that budget item. I recall at least a decade ago a load of Nobel Laureates offered to redesign the texts and science curriculae for California, and were turned down flat. Jaime Escalante was basically forced out by jealousies, and in the end, the system fights reform. I think rather than worrying about whether or not we get too much math for our needs, we ought to wonder how to ensure we get more, and all of what we do need to complement that which is perhaps a luxury to know, but a necessary luxury to avoid wasting the time spent confined in an institution supposedly concerned with learning. I have a dream, and in it, we won’t be judged by the color of our skin, but the radians in our circles instead….
(Tall) Tom, Tom, Tom…
I think you are needlessly harsh in your response, but I do so very much admire the cut of your jib, especially in regards to doctors, medicine, holistic systems, etc.
I have lived a horror story of medical issues, for over a decade now, for want of the smallest intelligence in the way medicine is administered, and the ONLY progress I have had from what could and was diagnosed fatal has been what I have personally learned, personally done, using nutrition and substances like Hydrogen Peroxide (a real miracle worker), thinking of the body as a system not a series of linked specialties. As the Dept of Education in the USA has done little more than preside over the collective collapse of the national intellect, doctors have done nothing for me (beyond palliative pain relief…sometimes…at a cost to my body and wallet) except preside over the deterioration of my health. From HMOs with “managed care” (thanks to government meddling in healthcare) to NHS (socialized medicine in the UK), I have nothing positive to say in terms of results that has come from a doctor in more than ten years.
BUT, on the plus side, I am learning a lot about my body, my wife took degrees in physiology and pharmacology (and soon radio med too…) to help support my efforts and second guess doctors, and I have opened my mind from being a systems engineer (a natural outgrowth of my faculty with tactics and strategy, applications in systems thought being what they are) to learning about the emerging science of systems and networks, which I may push and see if I can make a PhD in several years hence.
You sound like a guy with some interesting things to say/share. I can be emailed via s dot e dot murphy AT ancientengineers dot COM if you want to swap info sources. I think, from a lot I have studied about Tesla and other prophets of science is that many of their greatest achievements never happened because they were seeing the world, and its aspects, holistically, as inter-related systems. This is true of electricity, and I was just learning about Muller’s Global Scaling Theory, which sees reality (I think…) as a series of standing pressure waves, and large masses as convergent nodal points, with gravity not being the effect of warped space time, but both warped space time and gravity being the result of these convergent points. Some of the applications Tesla was witnessed as doing, and his further postulations into the “true wireless” and “longitudinal electricity” to my understanding seem to converge with this, and that explains a variety of phenomenon they experimented with. While no expert (yet) into this or the conventional theories either (yet), the fact that nature seems best operating in systems, AND according to mathematical models (Ian Stewart wrote books on it, “Life’s Other Secret, the new mathematics of the living world” and “Mathematics of Life” plus Marcus du Satoy contributed mightily to a show called “The Code” dealing with mathematical patterns in living systems), even biology has MUCH to benefit from newer uses of higher mathematics to a degree never before thought possible.
The reality, to me, is that a multi-disciplinary approach always benefits. Just as, in pop music, there has been an explosion of songs “featuring” another top artist (Like Tinnie Tempah feat. Eric Turner), I think the approach I saw at DARPA (integrating teams of disparate subect matter experts combining expertise into systems work) is something that can be modelled into degrees. Rather than a degree in “Biology” why not a degree in “systems biology” and one or more minors taking the place of some graduate work? I have found, in my life, that the broad reading I do has had disproportionate benefits on my ideas and work, and if I were in a position to take advantage of it now, I could push these ideas further.
But hey, absent that, I wouldn’t mind making another acquaintance, friend, and exchanging ideas, concepts, and article or book recommendations so as to mutually broaden minds…..
Hey tall tom, a couple of points.
— Homeschooling, something I’ve done, isn’t possible for everyone.
— People do forget things. As my mom once told my brother, “I’ve forgotten more things than you’ve even learned yet!” Even important things. E.G., I used to know without hesitation what the value and tolerance of a resistor was by looking at the colored stripes, but now that I haven’t used my electronics training in a while, I’d have to take a refresher course. The real issue is whether you still use the skills or training, and how a given individual’s brain happens to retain information. Some things “stick” for me that just shouldn’t, since I never use them, like phrases from my H.S. German class I took over 30 years ago.
— Did you memorize the calculus formulas? All of them? Why not? Unless you use it every day, you had to look up the right formula. And if you were one of those who struggled with calculus, would you trust yourself to get the answer right? Or would you ask for help? People are of different talents, personalities, and priorities for a very good reason; the world needs all kinds of them. Unless you are prepared to also mix the concrete, quarry the stones, make the truck that brings the stuff to your house, etc., you’re using a double standard just because you happen to have ONE of the skills involved in getting a concrete patio.
— Your statement about doctors and systems reveals another long-lost subject: logic. Everybody should take at least a basic course. Case in point: you claim that since the body is a system, and since systems must have all their parts, then any missing part renders the body nonfunctional. Needless to say, “logic fail”. Not only can a body live without a surprising number of subsystems, no math is necessary for every single aspect of the practice of medicine.
— signing off with 1 Cor. 13 after all the snark and SHOUTING kinda gives the wrong message
Hey Rick:
I might mention that I, at least, prior to reading your comment, did complain about the forced multi-culturalism. When I realized that I had to take such a course or I would be denied my degree at UW Milwaukee, I quit. In so called multiculturalism did ANY of the following (possibly appropriate classes)count:
History of Post Soviet Russia
History of Nazi Germany
Premodern Japan
Modern Japan
US History 1
US History 2
US Economic History
History of Third World Dictatorships
Political Reform in the USSR
History of China
Political Reform in China
US Military History
History of Modern France
European History
The Vietnam Conflict
History of WW1
History of WW2
Intercultural Communications
No. Not one counted for satisfying the cultural diversity requirement (actual term used). Growing up white in an all minority (95% black, 4% hispanic) neighborhood without another white family around for several blocks did not count, nor being in the Army, nor Special Forces, where a significant amount of my training and experience had to do with training foreigners from a variety of countries, living with, working closely with them (Phillipinos, Africans, Arabs, etc.). No, no, no, I had to submit to (that being the correct word) gay studies, latino studies, african american studies, or womens studies approved courses. It did not help me in long run, but I resented having to be forced to pay in time and money for a course in what amounts to political victimology by what are now often politically privileged groups. I HAD plenty of multicultural experience, training, and education, in and out of the univeristy, but the narrow, politically motivated focus in many state university systems over rides all other considerations (and many of those departments would not exist without mandatory class requirements shovelling victims into them).
Society, Universities, and individuals cannot reasonably be hurt in the long run by the “forcing” of mathematical material on them. I told my eldest step son, and surprisingly, he listened, to do degrees in engineering and learn his science, history, etc., on his own. In that way, he would be forced to learn NOTHING that was not literally, provably true. There is more to learn, but at least he would not be paying in time and money to learn someone else’s lies, propaganda, or misunderstandings!
I disagree for the reasons for the educational system cramming math and science pointlessly into the brains of students. I do not think it is because of history but because of political correctness and central planning.
It is politically correct to teach math other than some useful subject like philosophy because math doesn’t offend as many people because it does not challenge their beliefs.
It is central planning because some clown at the UN has given a bunch of kids from radically different cultures some math and science test that the US kids do worse than some kids do in Asia. Being central planners they do the easy thing which is to add more math and remove other stuff in the curriculum. Why, because it is much easier than examining the motivations kids have to learn and understand simple algebra.
My experience in High School math classes were horrible. I remember asking my Algebra teacher for some assistance to which he replied “You should know this”. Well, if I DID I wouldn’t have been standing there damn it all! Clammed up and for the rest of the school year watched as my performance went over a cliff. I figured that if asking for clarification was too much bother for the bastard then what was the point. My severe shyness was no help and he, like so many within “education”, really couldn’t give a damn.
Public school is a meat grinder that tosses everyone in and out come the proverbially educated sausages. What I’ve learned over all the years since is that failure lay in how the system made the fatal mistake in creating more subjects to study, calling that “progress”, instead of parring down the classes to what is fundamentally necessary and taking the required time in doing those well. This fatal conceit in being a mile wide and an inch thick carries over into the workplace where we’re asked time and again how well we “multitask”. Ye gods! So even the “Jack of all trades and master of none” has been devalued to the point where doing simple math and cash register transactions in ones head, as I do very well, have store employees staring at me blank faced.
Everyone has their particular gifts, and cramming unnecessary mathematics into a panalopy of politicized coursework does nobody, except perhaps those pimping their wares, any good. I do agree that classes in logic would be beneficial but I also believe people should be able to balance their accounts and cook a decent meal for themselves. Trig never did anything to help me with that. I also agree with Hondo’s mathematical recommendations as those are the things everyone needs to know in order to simply function in this world.
Did anyone challenge Loera to show any DiffEqu or Linear Algebra skills on the spot? If he can’t do it he is a typical government hypocrite.
Multiculturalism is a great stain upon college education. I had the misfortune of attenting a Diversity talk by the head of the department of African-American studies, with over 20 years of experience. It wasn’t simply the things he said that was frightening, but that in a room full of Ph.D.’s, I was the only one willing to take him to task repeatedly:
“France and England didn’t have any wars in North America because when they got here, they united to keep the black man down.” At this point I interrupted him to mention how George Washington’s military career began in the British military, fighting the French, but he was impervious.“This was how they solved the problems of war, and managed to not bring the European wars to the New World.”
“The US Education system is the best in the world,” he proudly claimed. He backed this up by mentioning the 36,000 institutions we have. Naturally, I asked him if by “best” he meant “largest”, but nope, he honestly thought the US system is superior to the rest of the world.
He quoted someone as saying “Diversity without equity is meaningless”…and from this he concluded from this that the most important thing the US needs is “diversity at all levels”.
“Memphis, in North Africa, is the oldest city in the world,”he said…and my interrupting him again to mention that Sumeria predates Memphis by 1,000 years did no good. He followed up with “When Alexander the Great captured Memphis, he found the Great Library there, and took the knowledge back to Greece, to form the foundations of Western Civilization”.
Again, I called him on this, making certain that he really was claiming Alexander the Great captured Alexandria (a city Alexander the Great founded, and named after himself, being something of a megalomaniac). Yes, that’s what he meant, and he doubled down by saying Alexandria used to be called Memphis.
After he gave his “lesson” on history, he asked us “How many of you were familiar with this chronology?” I had to admit, I wasn’t, but I’m also not intimately familiar with other fictional timelines, like the Star Wars or Lord of the Rings histories, either.
He then finished up with “Hitler only persecuted the Jews because there were no black people in Germany”. By the end of his speech I was covering my mouth with both hands, and yes, I was sitting in the front row. Only a few of us bothered to speak up…I can’t help but be disappointed that he was even taken seriously in the slightest.
There is much, much, hatred of mathematics, but I honestly feel knew how much absolute garbage is in a college education nowadays, they’d direct that hatred to the garbage, instead of math.
I don’t think math much past addition, subtraction, multiplication, division, including fractions, percents, decimals, a little geometry, (so you know how to figure out how to make your building square), enough algebra so you understand simple formulas like for the area of a square and cube (so you can order concrete), a little trig so you can measure the height of something without having to actually apply a tape is about all that is necessary for 95% of people. I do have an engineering degree. I have never worked in the field–sheetmetal bending and forming. I was able to manipulate the numbers to come up with the answer in calculus class, but never thought I truly understood it. I went to Navy Nuclear Power School. They started out at 2+2=4, and proceeded through differential Calculus in 3 months–and I actually understood it then. I have not used it in the succeeding 50 years. I think I’d be hard pressed to make it work, now. Actually, most of the normal world use of trig is just algebra. Algebra is pretty much adding, subtracting, Multiplying, Dividing, but the operations are combined, and you have to figure out what numbers to substitute for the letters.
I think the biggest problem with much of math instruction is that the teachers wants to impress the student with how much they know.
I just love people who openly trumpet their ignorance (in this case, of mathematics) as if it is a virtue. Stan, did you go to public schools?
Paul Simon “When I Look Back On All The Crap I Learned in High School-It’s a Wonder I Can Think at All.” In an editorial I wrote and sold to Yahoo for publication I noted that when 60% of Massachusetts college graduates failed a high school curriculum test as the last step to credentialing-it proves most of what you learn is useless and quickly forgotten (an astute viewpoint made in this editorial too):
http://voices.yahoo.com/government-credentialing-gone-wild-unnecessary-degree-11486786.html?cat=9
Math-the most OVER RATED subject in the history of the world-99% of it USELESS in everyday life-unless you’re an engineer or rocket scientist.
Anyone who says everyone needs the garbage that is cited in this article is a fool.
I just love people who openly trumpet their ignorance (in this case, that math is needed by everyone) as if it is a virtue. geoih , did you go to public schools? Do yo have ANY common sense?
I have a degree in electrical engineering. I had a lot of math in college and have used very little of it in my career. I know a few engineers who work for NASA or in very specialized fields who do use calculus and other forms of higher math often in their work. But those people are a small minority.
Even though I don’t use much of what I learned in school directly, I often realize that I am able to understand the details and boundaries of a problem I am attempting to solve because my knowledge has both depth and breadth.
The last phrase in your article tells us why things are the way they are, “but he or she is still required to learn higher math as a way of demonstrating an intelligence sufficient to remove an appendix”. That is true for my field. People who hire engineers from an accredited school can feel safe in the knowledge that they are hiring someone who is intelligent enough to learn difficult subjects and put forth the work it took to pass those courses. Is that a good way to judge the qualities that make a good engineer? Maybe not, but any alternative I can think of poses other problems or is just practically unworkable.
The trigonometry example is just basic school stuff. In the sense that being familiar with mathematical objects like the unit circle and trigonometric functions are necessary in practically all scientific fields, certainly in mathematics, physics, chemistry and engineering, and you will encounter these things in your studies even if you go for degree in computer science, biology or medicine, which are build on top of the basic sciences.
So your comments make me think about other school subjects. Like, I am a professional mathematician now, as it happens. Yet in school I had to learn about christianity and religion, which I do not seem to need now, and about history and ancient arts, which I do not have much use for either. I took Latin, probably you would say there is not much need for that either in science, but actually it helps quite a lot to have a good grasp of the grammar.
The point is that seeing a multitude of different subjects already during school years, it gave me the choice about pursuing one that I had a good feeling about. Like, I might have been inspired by the classes on religion to become a cleric. The fact that it didn’t happen does not immediately suggest to me that all education on religious issues are suspect in some way.
Anyway, you argue from ignorance, and you appear to have a clear anti-scientific agenda, so I am sure you will dismiss my thinking off-hand.
Our public education is stupid! I was a student who graduated high school and college. I was misled! I got a B in Precalculus. However in college they don’t admit that grade to get to Calculus! You either need to retake the class or take a math exit exam! Retaking the class did nothing! It’s just the same stuff my high school teacher taught! And when I took the exit exam, it’s obvious there were some problems that were impossible to do even with what my teacher had taught. I had to go and read a Precalculus textbook front to cover, annotate, derive trig functions and other stuff, and even a supplementary workbook! Fact is either our high school math system and the college math system is stupid! I get how colleges require students to pass a rigorous, difficult exam to get into Calculus. However, high schools should not fool students to believing they passed “Precalculus.” They should get students to read through the textbooks, annotate, figure why that trig function is derived that way, not be spoon fed a lecture and do the minimum to pass the class.
Fact is our education system is broken. We need to fix it but they won’t do anything because all they care is keeping their job, get the kids pass a class according to “their standard” and not think about the kids that pursue a college education. They only pretend to do so.