Is Algebra Necessary?

[Ruminator]

@Sam_Stone

Then what exactly is the value of a high school diploma,

I think society has cognitive dissonance when it comes to the real value of a high school diploma. (And college degrees too for that matter.)

We force high school students to take courses, pass tests, get their official diplomas but is that really proof of anything? Who really trusts it? Do colleges trust it? Let’s see… reputable colleges use SAT/ACT for admissions and also require entrance exams. If the “high school graduate” doesn’t get a high enough score, he’s required to enroll in remedial classes. He must pay for these “college” classes that do not count towards a degree. High school transcripts are part of the review process of course but they are not the final word.

Can the threshold for “high school graduation” really be raised so the diploma has unquestioned respect and integrity? I don’t think so. Can you really force teenagers to constantly repeat grade levels to the sneers and mockery of their peers? Can you really have a 20-year old that’s a repeat slow learner in the same high school as 17-year old girls? (Statutory rape anyone?) Parents would be up in arms. So as society, we water down thresholds to “pass them.” – a 2.0 GPA for high school graduation. A 2.0? Really? Why not 3.0 or 3.2? I think we know why but can’t admit it to ourselves.

However, “reality” eventually catches up with that watered-down high school diploma. College entrance exams being an example of that reality.

College degrees also suffer the same problems. They create their own social pressures that contradict the integrity of academic achievement. Reality also catches up with college graduates too: employer interviews.

Here’s a popular forum where math/finance college grads (or soon-to-be grads) discuss interview questions. Quant Interviews | QuantNet

Obviously, the college degrees in themselves can’t be trusted. The employers constantly invent new creative questions that are not in the college curriculum to separate the graduates that really understand the mathematics from the pretenders. Could college professors raise their standards to the same levels as those quant interviews? No they can’t. The math department heads would get an avalanche of complaints that 90% of students are getting failing grades! (The 10% that do pass are the same guys that pass the gauntlet of quant interviews. :))

Perhaps for non-mathematics degrees, many employers will take the college degree at face value. Newspaper editor: “Oh, B.A. in English and Journalism? Ok, when can you start?”

The high school diplomas and college degrees are just first line filters. They do not convey unquestioned competency. Isn’t this an unspoken sham that we don’t like to admit?

A few years ago, there was an SMDB GQ question from a new math teacher about a systems of equations. (I vaguely remember it was about solving them or difference between notation of f(x) = ax +b instead of y = ax +b.) Obviously, to get her position as a math teacher, she had to major or minor in mathematics. Obviously, she “passed” whatever math exams were put in front of her. A few Straight Dopers mocked her for the nature of her simple questions. Why? What was that mathematics degree supposed to really represent? Did her university schooling do a “bad” job? I don’t think so. It did exactly what the institution was good at.

It would also carry more weight if there weren’t at least 25 other countries doing a better job of teaching math in high school. Canada has much higher math scores than does the U.S.,

Those standardized tests used to compare countries are what I affectionaly call “chimpanzee mathematics.” I suppose the USA could tweak things so that these comparative scores are higher but it really doesn’t get to the heart of the op’s (NYTimes) premise: algebra is not necessary.

@ultrafilter

There’s a very nice response to Hacker’s article by Dan Willingham, who’s one of the foremost experts on cognition and education. His short answer is that yes, algebra is necesary, and he has some pretty good arguments as to why.

He has the same flaws in reasoning that mixes up correlation with causation.

He also has zero proof that compulsory teaching of algebra increases the percentage of population with abstraction skills. The % could be exactly the same whether algebra was forced or not.

He also does not consider the possibility of students trading out algebra for some other academic subject they would prefer could lead to better outcomes. He doesn’t even put that up as an alternative scenario to shoot down. So having Jane study a semester of poetry instead of Algebra II could be for her and better for society. He has zero proof that it wouldn’t work out that way.

@fumster

I bet that 75% of the population would get it wrong if asked what the price of a $1 product would be if you raised it 50% one day and then cut it 50% the next.

If it makes you feel better, only 64% of the population would get it wrong if asked what the price of a $1 product would be if you raised it 60% one day and then cut it 60% the next.

[Jragon]

@Capt.Ridley_s_Shooting_Party

I have no idea why such a large emphasis on calculus is made in CS education. Perhaps in analogy to the education that mechanical and electronic engineers receive, to try to make it appear that CS or SE is a legitimate form of engineering, or an institutional bias amongst CS academics towards subfields like computer graphics and machine learning which make use of calculus, but at least from my viewpoint the overwhelming majority of research that I read is based on discrete mathematics and there’s whole subfields of the subject which are essentially being shafted by the insane focus on derivatives and integration in undergraduate years.

Yeah, but at any level worth talking about that uses calc, the calculation part is academic. For instance, if you’re writing a machine learning program that uses gradient descent, it really doesn’t matter whether or not you got a B in calculus because a very, very small portion of a calculus class is conceptual. It’s really not that hard to understand gradient descent if you understand the concept of a derivative, regardless of whether or not you find it difficult to calculate the derivative or integral of a given function.

The exception might be graphics or scientific/mathematical computing optimization, because in that case knowing off the top of your head not only the correct way to calculate things, but the way that requires the least total computing cost is very, very important.

[ultrafilter]

@Ruminator

He has the same flaws in reasoning that mixes up correlation with causation.

Do you actually know what this means, or are you just throwing it out because it sounds smart?

[Ruminator]

@ultrafilter

Do you actually know what this means, or are you just throwing it out because it sounds smart?

Which sentence(s) in his article proves the causation that force-feeding algebra predicts better outcomes for a country rather than a educational system that treats it as an elective?

Not these:

*Economic data directly contradict that suggestion. Economists have shown that cognitive skills–especially math and science–are robust predictors of individual income, of a country’s economic growth, and of the distribution of income within a country (e.g. Hanushek & Kimko, 2000; Hanushek & Woessmann, 2008).

Why would cognitive skills (as measured by international benchmark tests) be a predictor of economic growth? Economic productivity does not spring solely from the creativity of engineers and inventors. The well-educated worker is more likely to (1) see the potential for applying an innovation in a new context; (2) understand the explanation for applying an innovation that someone else has spotted.
*

[Mekhazzio]

@Sam_Stone

And if you study computing science in university, are you planning to only go into ‘line of business’ programming? Do you really think we should be graduating people from CS who are wholly incompetent in many fields of computing science?

Oh, we’re already well along that path. I just recently picked up a CS degree, and I wouldn’t trust 3/4ths of my peers to be able to write and compile “Hello, world!” without a template. “Group” work was agonizing.

[suranyi]

@Mekhazzio

Oh, we’re already well along that path. I just recently picked up a CS degree, and I wouldn’t trust 3/4ths of my peers to be able to write and compile “Hello, world!” without a template. “Group” work was agonizing.

Oddly, in a previous job my boss said she had better experience with physics and math majors who learned to program on the side, than CS majors. They were better problem solvers, she said.

[drewtwo99]

High Schools are already essentially glorified kindergarten. Algebra is not really required, because any student who can’t pass it on their own, who complains to administration enough with angry enough parents, will graduate. Gauranteed. I’ve seen it happened. I’ve had it forced on me when I was a teacher (physics is a required subject now in Texas, students had to get the credit to graduate. Immense, insurmountable pressure to pass them no matter what if it was the only subject they were failing). It was the long running joke as a teacher in the faculty room. We would all wait till graduation day to see if someone waved the magic wand and got a student to pass and walk at graduation, even though they didn’t earn it.

Students simply do not fail out of high school because they can’t pass just one class. Anyone who thinks otherwise is living in a fantasy world.

[Capt.Ridley_s_Shooting_Party]

@suranyi

Oddly, in a previous job my boss said she had better experience with physics and math majors who learned to program on the side, than CS majors. They were better problem solvers, she said.

Hardly surprising given the deep connection between programming and proof.

[KarlGauss]

Studying and doing “math”, regardless whether algebra, geometry, statistics, etc, requires mental discipline. So, not only does it hone your analytical skills, it teaches you, nay, requires you, to avoid fuzzy thinking. And, most of us have to work at it. For many kids, actually having to work at something comes as quite a shock:

“I suck at math” often just means “I’m not prepared to actually persevere and work on it”.

More importantly, studying math serves as a wonderful counterbalance to the subjectivity and arbitrariness which pervade so many other areas. You don’t get a shiny trophy just for participating. It shows you that there’s at least one area where not every idea is created equal, and where no one gives a hoot about your opinion. No matter what your mother told you, you’re not special. You can’t bullshit your way through an algebra proof.

Or, as Feynman put it, “Reality must take precedence over public relations, for Mother Nature cannot be fooled”. Getting people to appreciate that fact is why math should continue to be taught.

[Jragon]

Because so many people do proofs in high school algebra? Which I’d argue is part of the problem, if I had my way I’d halve or third the amount of rote that first graders do (you still need some, so you can’t eliminate it) in favor of making them answer questions like “why does 2+3=3+2?” And continue the same trend in every math subject from then on, but alas.

ETA: Though this may ultimately favor discrete math over other maths, at least for one semester, because in my experience discrete math has proofs that are easier for people to understand intuitively.

[Paranoid_Randroid]

@KarlGauss

“I suck at math” often just means "I’m not prepared to actually persevere and work on it”.

Hah. In high school I failed Algebra I; then, after barely passing it, I failed Algebra II. A number of years later I graduated summa with a baccalaureate degree in mathematics — and how? I was forced into a business calc class and I wanted dearly to understand it. Of course I’m a sample of size one, but it’s difficult for me to empathize with those who claim that understanding basic mathematics is utterly insurmountable for persons of otherwise normal intelligence.

No matter what your mother told you, you’re not special. You can’t bullshit your way through an algebra proof.

Hm. Maybe not, but on at least one take-home exam I bullshit my way to generous partial credit. “You mean I can’t assume such-and-such result?” Wish I could find that …

[Thudlow_Boink]

@Paranoid_Randroid

Hah. In high school I failed Algebra I; then, after barely passing it, I failed Algebra II. A number of years later I graduated summa with a baccalaureate degree in mathematics — and how? I was forced into a business calc class and I wanted dearly to understand it.

I, for one, would be interested in hearing the long version of this story, especially if it throws any light on how to turn the sort of person who fails Algebra I into the sort of person who majors in mathematics.

[The_Lurker_Above]

@Jragon

Which I’d argue is part of the problem, if I had my way I’d halve or third the amount of rote that first graders do (you still need some, so you can’t eliminate it) in favor of making them answer questions like “why does 2+3=3+2?” And continue the same trend in every math subject from then on, but alas.

As a former math tutor I really don’t support cutting down on the rote math drill in elementary school; though I do support your second suggestion. Some of my best math teachers really tried to get students to understand the ‘why’.

Drilling the basics of arithmetic (addition/subtraction, multiplication/division, fractions and I’d include exponents/logarithms) until they are second nature is key to success at algebra and above. You don’t want to know how many times someone would come to me for help and we’d go over their (failed) test to find they were actually pretty clear on the new concept, but their poor grounding in the basics were causing what should have been a B/B+ to drop to a fail.

If people don’t learn the basics with there classmates it’s almost impossible to get them to catch up later. “You’ve got the idea on solving systems of equations. But if you want to pass the course I suggest you do several hours of times-table drills.” does not go over well with grade 11 students.

[suranyi]

@KarlGauss

No matter what your mother told you, you’re not special. You can’t bullshit your way through an algebra proof.

My mother told me I was special, because I got an A+ in algebra.

[KarlGauss]

@suranyi

My mother told me I was special, because I got an A+ in algebra.

Me too; but I had just turned three.

[Indistinguishable]

@Jragon

Because so many people do proofs in high school algebra? Which I’d argue is part of the problem, if I had my way I’d halve or third the amount of rote that first graders do (you still need some, so you can’t eliminate it) in favor of making them answer questions like “why does 2+3=3+2?” And continue the same trend in every math subject from then on, but alas.

Yes, I’d agree with this.

But I think one of the main reasons we can’t do this now for higher math courses is because those students who lack interest just won’t engage with it at all. (As is their prerogative). So instead we settle for having them memorize algorithms and heuristics and mindlessly chug through, all without, all too often, understanding a whit of what they’re doing, just so they can be said to be doing something.

Actually, quite possibly, if we did something like this, we’d discover more people were interested in math than realized it. Everyone should be exposed to this style of math education at some point, so they can see what real math is (a goal not nearly accomplished by the current curricular standards). But I think it’s a brute fact that some people still won’t care about it, and that’s alright.

@The_Lurker_Above

Drilling the basics of arithmetic (addition/subtraction, multiplication/division, fractions and I’d include exponents/logarithms) until they are second nature is key to success at algebra and above.

What do you mean by “rote drill” and “drilling the basics of arithmetic”, apart from the conceptual understanding? I usually see these terms used to reference doing, say, 50 problems in a sitting of digit-string calculation by hand, in blind accordance with some memorized algorithm. Do you make your students calculate logarithms by hand?

[Paranoid_Randroid]

@Thudlow_Boink

I, for one, would be interested in hearing the long version of this story, especially if it throws any light on how to turn the sort of person who fails Algebra I into the sort of person who majors in mathematics.

I wish I had some narrative replete with insight, but from the perspective of hindsight everything just seems to melt together into a muddy and uninterpretable sequence of events. My failures in high school weren’t due to a special animosity toward math, as I failed most classes (except, peculiarly, French, in which I excelled); I ended up graduating from an adult learn-at-your-own-pace school.

No longer a total screwup, my first semester in college I earned a B in introductory algebra. While having difficulty with a business calc class — and petulantly refusing my girlfriend’s pleas that I find a tutor — I began obsessing over it; I taught myself the algebra and trigonometry I’d missed from a book; and at some point it must’ve clicked.

I suppose to learn anything you have to want it.

[rat_avatar]

The funny part is that remembering your algebra is what makes calculus hard

The idea that someone could program or do statistics without a basis in elementary algebra is quite absurd, you have to know how to solve for these problems to do any of this.

for i in 1 2 3 4 5
do
x=x+$i
done

That is pure and basic elementary algebra.
in “basic html” how do you calculate image width or deal with page scaling? how do you solve for those problems if all you know is how to add, subtract and multiply?

[The_Lurker_Above]

@Indistinguishable

What do you mean by “rote drill” and “drilling the basics of arithmetic”, apart from the conceptual understanding? I usually see these terms used to reference doing, say, 50 problems in a sitting of digit-string calculation by hand, in blind accordance with some memorized algorithm. Do you make your students calculate logarithms by hand?

Calculate logarithms by hand? No, that’d be nuts to do more than once or twice as part of getting a feel for what a logarithm is.

But when you break down an algebra test not only are you being tested on the new concepts you’ve learned but also on how well you can do basic elementary school arithmetic. It’s as if the test asked ‘what strategy would you use to solve this problem?’ and then gave a half page of elementary school level arithmetic quiz. You only get full marks if you got both parts perfect.

I’ts all well and good to have a conceptual understanding of arithmetic; as I said in my previous post my best teachers made a real effort to instill and understanding of the ‘why’ behind the algorithms. But to have a chance to succeed at algebra and higher a student needs to be able to do the boring, mechanical, chung-n-plug arithmetic quickly and near perfectly.

I’ve seen way to many students think that they’re no good at math due to poor marks. But when I sit down with them one on one I often found that it’s not that they have trouble conceptually, it’s that they’re trying to to build a house on a shaky foundation.

[rat_avatar]

@Mekhazzio

Oh, we’re already well along that path. I just recently picked up a CS degree, and I wouldn’t trust 3/4ths of my peers to be able to write and compile “Hello, world!” without a template. “Group” work was agonizing.

I will agree, CS majors use to come out of college with a deep understanding how computers functioned, now many schools graduate people who only know a few specific products and high level languages.

After a couple of decades in the industry I will say these graduates better hope for a management track in their career because they are never going to keep up unless they learn the basic concepts on their own.

This is why people who know the basics, like algebra can move from Unix to Windows to micro-controllers. If you understand how thinks work under the covers you can adapt.

If you don’t understand each new technology is another “god box” that you have to learn how to “dance with”

This is not a new problem but it is a pity that they have diminished the quality of what a “CS” degree curriculum should include.