Down with Algebra II!

Miscellaneous and Personal Stuff I Must Share
61

Take me.

Started in 1967 to be an Engineer (Aero).
Got through Calculus with D’s by using extensive cheat-sheets.
This was a continuation of K-12: never understood math.

I once worked on a Survey crew. All of a sudden, Trig made tons of sense.

I was able to scrape together enough Math credits to score an AS in Computer Science.

In 30 years, I used math higher than arithmetic exactly ONCE - I needed to code A = B + (C**2).
Notice the extra parens - I knew better than to expect a coder to remember Algebraic sequence.

I still have a slide rule - which I never did learn to use.

I don’t see the problem with using terms more intuitive. Is Std Dev somwhow a lesser concept?

62

Stat majors can tell you how appallingly ignorant you are about the principles of std dev, but it’s mainly an arithmetic trick to get rid of negatives cancelling out with positives, and making your usual average too small to be useful.

As a geologist, we had a little calculus during geophysics. In financial analysis, lots of algebra and stats. However, my knowledge of stats doesn’t go past mean, median, mode, variance, standard deviation, regression, and anova. Everything like VAR, time series, and all those tools for technical analysis are done by Bloomberg.

63

I’m an electrical engineer, and one of my master’s degrees is in applied math. Before I became a manager, I used all variety of math - calculus, analytical geometry, trig, stats. I also teach math and engineering part time.

So, I’m not hostile to math - I realize that it’s absolutely essential for some careers.

That said: IMO we have a confused system for teaching many tough subjects (especially math). Fifty or more years ago, the system was set up with a ‘sink or swim’ mentality. If the majority flunked, the system’s attitude was, hey, you weren’t cut out for higher education. (It was also true that not as many people were attending college (or even high school) so the ones who were were likely better prepared, motivated and academically capable. But that’s another story).

More recently, we’ve realized that it’s more efficient to improve teaching methods so that more people ‘get it’. This is partly due to the rise of college students seeing themselves as customers who are paying increasingly sky-high tuition, and therefore demanding their money’s worth. (This has also led to lowering academic standards, and students’ constantly haggling for better grades. But that’s another another story).

We should acknowledge that different career tracks need different sets of math knowledge, and especially different levels of detail in that knowledge. I realize this is already being done somewhat (e.g., business calculus versus calculus for math majors). It should go further.

One example:

Everybody who should be in college is capable of understanding basic analytical geometry (the concept of the x-y plane and the equation corresponding to simple line graphs [y = mx + b]), and will benefit substantially from it. Require this subject of everybody majoring in business, nursing, etc. But leave the detailed study of the behavior of 4th-degree equations to the math/science/engineering majors. Replace the more advanced parts with real-world applications. Create major-specific, level-appropriate math courses (Math for Business Majors, Math for Nursing Majors. Hell, Math for Music Majors!) And don’t worry about people who can’t pass these classes - they need to find another major.

Similarly, the basic concepts of calculus are not that hard to understand, and will benefit all educated people. Everyone can be taught how to use the area under a curve and the relationship of rates of change. Should we require people to solve non-linear differential equations, evaluate triple integrals, and master integration by parts? Only if they’re majoring in a subject that requires it (math/physics/some engineering).

Draw a distinction between exposing students to a subject that might be enlightening, and requiring them to master it on pain of dropping out.

*It’s more important that people understand the basic concept - and be able to apply it as needed - than sweating them through more advanced study of things they’ll never use, and will forget as soon as they leave the class. *

I understand that education is about more than job training. But if smart, capable people are unable to graduate in fields that they could have succeeded in because they couldn’t demonstrate skills that aren’t necessary, they’re getting ripped off.

We make people take phys ed. We don’t require them to run a five-minute mile to graduate.

64

To me it smacks of “what I am interested in is General Culture and everybody should know it; the courses I had to take that I wasn’t interested in are a waste of time for everybody.”

(I’ve had more than a conversation along those lines with for example Treasury Inspectors who claimed the manuals on “how to fill your taxes” were perfectly clear… perfectly clear for someone with degrees in Taxation Law, that is, I’m just an engineer. What they are experts in and actually spent years in college about is “basic knowledge”, what I spent years in college about is “yucky”)

65

You are starting way to high.

There is the +50% of the population that is not even college bound. To divide them up:

You have mainstreamed special needs kids on IEPs who don’t have the capacity beyond basic math. They’ll graduate through a series of exemptions.

You have the kids who won’t graduate. Maybe they have discipline problems. Maybe they are sick of school. Maybe they get no support at home. Or maybe it becomes obvious to them that they won’t get through Algebra or English and they say screw it.

You have the kids who will graduate, but don’t intend on continuing school at all. Maybe they are joining the military. Or their life’s ambition is to be a stay at home parent. Or they have a friend who makes good money laying tile and will get them a job. Or they live in a town where there is still a factory they can work at.

You have the kids who won’t go to a four year school, but will go to a trade school. They want to be a stylist or a plumber or a medical tech or a mechanic.

These kids won’t ever apply to college to not get into their first choice school.

So what is a high school diploma. Does it mean you are college ready? Why - 50% of students aren’t heading there? Does it mean you’ve done your time? That seems like you’d graduate kids you shouldn’t graduate. It should mean that you did your time and have basic life skills to tackle whatever is next - for you. Whether that is the Navy or chasing toddlers or Harvard.

66

This actually got me thinking. For a lot of math courses/ideas, there’s a more or less concrete example or field where it’s actually applied. And for a lot of students, having that applied example is VERY helpful in understanding it. I’m one.

So for me, geometry, trigonometry, and even calculus weren’t that hard to understand, as you can relate them back to something you can see on a page, or some kind of physics problems, or even just things like surveying. Even basic arithmetic has currency as something you can relate it to (“If you have five $5 bills, how many dollars do you have in total? 5+5+5+5+5 = 25, or 5x5 = 25”)

Algebra is kind of unique among the lower mathematics in being nearly totally abstract. The quadratic formula or FOIL for multiplying polynomials is abstract. I’m sure a teacher could gin up problem examples to display the concepts, but the concepts themselves are abstract, unlike say… a derivative, which in physics terms is the acceleration at a point in time along a curve describing speed vs. time. (I think I got that right… it’s been 20 years since I took calculus or physics)

67

You get word problems in Algebra and some concrete examples, but 15 and 16 year old kids who aren’t into school still don’t see it applying to their lives outside of work. Few fifteen year old kids believe they’ll end up as surveyors. And even if they do, they don’t need to understand the abstraction when the computer is doing the calculations now - they just need to know where to look for the answer.

And word problems, which is where you tend to get concrete examples, further confuse a lot of the kids who don’t think well in abstractions anyway. They have to turn the abstraction that is language into the abstraction that is math. They may now understand why its being taught to them, but getting those answers can be very frustrating to a kid who can’t do those translations. It adds a layer of mental gymnastics to a kid who is already confused - or maybe can follow the steps to get the right answer, but doesn’t really understand (or care to understand) the theory behind it.

I have an Alg II textbook around here somewhere if you want me to dig up the types of applied math problems it presents.

68

I feel I should say something here but I am not sure what. Let me start out by saying that Algebra II is entirely different from what it was when I took it 62 years ago, judging by the description that Shagnasty posted. I took a lot of courses in HS that I have never “used”. Have I made any use of the 8 Shakespeare plays I read, one each semester for four years? I don’t even know what it means to use such a thing. Do I use the verb paradigm we were tested on, to conjugate the verb “to be” in the past perfect progressive (“I shall [sic] have been being”)? Of course not. Do I “use” the history courses I took? Of course not. On the other hand, they made me what I am today, for better or worse. Whatever I learned in my four years of HS math, the 13 math courses I took in college, the math courses (I don’t recall how many) I took in grad school, also contributed to what I am today. So did the literature courses (which I mostly didn’t enjoy), the English composition courses (ditto), the two anthropology courses I took (which I really enjoyed, but could not get a mark above C in, no matter what I did), or the chemistry and physics courses.

So the entire discussion is besides the point. If you don’t know any math, you are not educated.

That said, it would be nice if the average citizen knew enough science, math, yes and stats, to be able to read a graph, understand a statistical claim and understand enough of Bayes’s theorem to understand that someone with a positive test for a rare disease probably doesn’t have it. That’s probability, not statistics, by the way.

69

There are in fact plenty of concrete applications for quadratic equations and multiplying binomials (people were doing this stuff long before there were surly high school students, after all.)

I’m a firm believer in including applied concepts in math education.

But showing that a concept has practical applications is not persuasive to the “why do I need this?” argument. And while applications and conceptual understanding are beneficial pedagogically, at the end of the day all the applications and conceptual work don’t mean squat if you can’t write it down and solve the equation.

Certainly there’s a lot of room for debate about how much math education in high school makes sense, and how deep it should be. I’ll also note that the peculiar American obsession with single-tracking nearly every student and assuming everyone wants to or even should attend an academic postsecondary program is unhelpful.

But I’ll repeat my premise that I posted earlier, in a more digestible form:

Education isn’t only about what you need.

70

I guess what I was trying to say is that if you have a graph of speed vs. time, each part of it means something concrete, or at least real-world. The speed is the Y-axis value at a certain given X axis value for time, and the first derivative at that point is the acceleration. The area under the curve is the distance traveled. Those are all things that can be conceptualized and understood.

Even trigonometry can be related back easily to things like things like surveying (I also worked as part of a survey crew in college)

But if someone hands you (3x+4)(4x-3) and tells you to multiply them, it’s abstract. There’s not an easy real-world example you can use to relate this to a student who’s not thinking math.

And I agree that learning some amount of math is essential to a good education, but I’m not convinced that getting much past basic algebra, geometry and trigonometry is necessary for anyone not getting into STEM, and even then, it’s not universally important. I mean, I have a computer science degree, and something like 18 math hours. When I took the GMAT, I had to go look up a huge amount of algebra that I hadn’t seen in 5 years, and haven’t seen since. I haven’t solved a series of equations since the last college math course (linear algebra) that required it. I do use the set math I learned in my database courses all the time, if somewhat informally. I also use some basic first-year algebra / pre-algebra stuff pretty frequently as well.

71

Note that the main point of taking a lot of Math courses isn’t so much to learn every little formula and such, but how to think in a logical way, analyze things, break problems down, etc.

In addition, taking the next class in a sequence helps significantly to reinforce the knowledge of the core class. I.e., you take Algebra II so you’ll remember Algebra I better.

Also, the statement “I never used X so why was I forced to learn it.” all too often comes from people who probably could have used X quite a lot but didn’t realize that a situation could be helped by it and so they didn’t. It’s quite a fallacious argument.

The Slate article is so … illogical … that the author could have clearly benefited from taking a lot more Math.

72

I’ve been a software developer for nearly a dozen years. I had to take a ton of math classes for the tech degrees. I’ve never used any of it for work. If I needed it at this point I’d have to relearn it, but I severely doubt I’ll ever have to. (Decades later I’m still pissed about having to take Calculus, because I knew I’d never use it.)

The most advanced math I’ve done lately was figure out the circumference of a circle, and that was for a DIY project at home.

73

Three statisticians go hunting. Out in the woods, a deer runs by them. One fires his gun ahead of the deer, and the other shoots behind it.

The third one shouts, “We got it!”

74

I don’t think math is necessarily the best way to do this though; I learned a lot of how to really do that as a result of programming, not from mathematics courses.

Various forms of puzzle solving and games can do it just as well, and without all the abstract mathematical garbage that turns people off, confuses people, or otherwise hinders the learning process.

And I can honestly say that there was no freaking point in making me take Calculus II or III; I have never had to do integration by parts or calculate the arc length of a catenary curve at all since college. It didn’t even come in handy in linear algebra or any of my CS courses. What would have come in handy would have been a course in how to write proofs; that seems to have been something math professors thought we’d learn by osmosis instead of pedagogy, and that CS profs assumed we’d already know how to do from our math coursework.

75

Through an outreach program at my workplace, I used to give presentations at a local middle school demonstrating applications of math.

One example used line graphs to pick the best cell phone plan. This was in the days when plans had different rates for different amounts of time (e.g., first 200 minutes free, 5 cents for every minute over 200 vs. no free minutes and 2.5 cents per minute) so it’s no longer practical.

Another example used linear equations to determine how much of a head start a mile runner would need to consider challenging a faster runner.

Sports are actually a very effective way to teach math to kids. I first became fluent at doing mental arithmetic to calculate my batting average in Little League (I’m 3 for 10. What’s my average if I get a hit? An out? A walk?). And there was the scene in “Big” where Tom Hanks explains ratios to his boss’s kid by asking him, “Michael Jordan scored 11 points in the first quarter. If he keeps that pace, how many points will he score in the game?”

76

Ya know - I’d really love to be able to calculate the area under a parabolic curve (at least as long as both X and Y are positive).
But: making it through two semester of CALC (which were specifically designed as “flunk out” course, BTW) I still have no idea of how to go about it.
OTOH, I once dated a woman who would amuse herself by running through quadratic equations she’s dream up.

The only time I ever heard of advanced math being used in software design was the formulations required for the first Airline Reservation systems. I suppose the military uses some heavy number-crunching for weapon design and strategic games.
But, from a practical viewpoint, the solution was not to develop armies of mathematicians - it was to have a few hundred of them design computers and systems to allow any idiot to plug in numbers and get results.

I suspect that math is still being used as “flunk out” (drown the bunnies) courses. This is not good for mathematics. I don’t know enough to speak for Academia.

In HS, we quite blatantly had “Levels” for the classes - one was for the “going to College” group, one for “at least a HS Diploma”, and one for “might yet pull out”.
I thought it a good idea at the time. Is it now forbidden?

And: while flunking the Calc, I spoke to a Counselor - who assured me I “needed” the Math to be able to handle the Physics (which was an easy B).
Maybe we should look at the basic assumptions about exactly how “critical” math really is?

Or, has it been the flunk-out subject for so long that we can’t discuss its actual necessity?

77

In my day we learned to write proofs in 9th grade geometry. Math isn’t just the formula. It’s the systematic and rational thinking to come to conclusions based on axiomatic statements and logic.

How do people make financial decisions without being able to calculate exponential functions? How do people answer the Monty Hall goat problem without basic math?

78

About a year ago, a computer science grad student asked me for help with probability calculations in a class on the design of search engines. The calculations (which determined the average time to find the right website) were used to choose between different candidate search engines before doing the coding.

In the wake of Sputnik, the U.S. had the misguided notion that the only way we could beat the Rooskies was to turn out armies of math geniuses. Equally misguided: that mathematicians could perform any job below that station, because they were, like, really smart. “Flunk-out” classes (we called them “weeders”) were part of the outdated “sink or swim” concept.

With all due respect, I think a lot of good math teachers need more experience applying the subject, in order to teach it meaningfully.

79

No it isn’t and they still have them. But since I went to high school the graduation standards have gotten a lot tougher. When I was in high school you needed two credits of math - any math would do - practical math (balancing a checkbook) was the course taken by a lot of people who barely got through high school. That came after basic math. You didn’t even need an Algebra course to graduate.

Now kids need to pass (or get an exemption from) Algebra II to graduate from high school. Which basically means that in order to do anything other than ask “would you like fries with that” for the rest of your life, you need to be able to solve quadratic equations and log functions and polynomial equations. I like my hairdresser, she’s a great woman and does color really well, but there is no way she needs even conceptual knowledge of polynomials.

And it isn’t just math. English requirements have gotten harder. My non-college bound Junior got a C on a paper, because it “wasn’t college level writing” - why should it be, he isn’t in college and doesn’t intend to go! He’s a Junior in high school and he isn’t even in the college prep English class! His World History class meant he had to know who Spinoza and Ibn Battuta were.

Now, I want that sort of rigor and options for my college bound daughter.

Weeding young adults out of the Engineering program at a University is fine - they can go major in Education and teach math or switch to business and become accountants. Weeding kids out of high school seems like a bad idea.

80

Huge chunks of software engineering are closed off if you can’t do math.

Anything that remotely resembles a simulation needs heavy math. This includes real simulations (weather, structural analysis, etc.) as well as pretty much everything in a game (which involves physics and rendering).

The new fad is deep learning. Guess what–most machine learning algorithms (including the stuff needed for training neural nets) requires some basic knowledge of calculus (specifically, you need to compute a derivative to know where to move).

Optimization requires, at the least, basic algebra. It’s not that hard to recognize that Ax[sup]3[/sup] + Bx[sup]2[/sup] + Cx + D can be rewritten as ((Ax + B)x + C)x + D, but it is nevertheless algebra. Recognizing these kinds of things requires a broad background.

Of course many people make a career out of plugging together libraries that other people wrote. There’s nothing wrong with this but it’s not a particularly valuable skillset, and breaks down as soon as there’s some problem with the library.