Is Algebra Necessary?

[Lil' Lavery]

Originally Posted by Michael Hill

Because not once has anyone likened anything in a briefing to the Habsburg rein of Austria. Also, I used that in the sense that it has no real-life use outside of high school for most people.

I find the whole "look at this aspect of the course that I haven't used" argument very disingenuous. Most people do (or, in reality, should) use the knowledge learned in history classes in their political behavior. Even if certain pieces of information never come to fruitition, others (and more importantly themes and historical trends) most certainly will.

Originally Posted by Michael Hill

I realize that isn't the full premise of the article (though a generous portion of the article is precisely that, complaining that math is too hard). However, his whole argument is based on the fact that students aren't doing well in Math. If students were able to grasp algebra easily, this article would have never been written in the first place. The author is a prime example of someone who advocates the "dumbing down" of America just like No Child Left Behind. Algebra is something that many middle schoolers (including myself) took. It's not too much to ask of someone in high school to be able to take it either. High school today is no longer about educating students, but giving them a degree to get them out of the door....a McDiploma.

A large portion of the article is indeed spent on the fact that students aren't passing algebra. That portion of the article served to demonstrate that a problem exists. Unless you're advocating that high failure rates are the desired outcome, I don't see how you can take offense to that. The solution to that problem, however, was not to "dumb down" education or give out "McDiplomas," but rather to adapt our educational system. The article is not advocating easier classes, simply different ones. If anything, it's suggesting the opposite of "McDiplomas," as the author is calling for more specialized math courses that apply more directly to different career fields.

[Ian Curtis]

Originally Posted by ManicMechanic

Motivating students to do the boring work helps, and connecting math to previously discovered concrete examples (rather than to what they can imagine/discover) seems to work well with my lower level students. For example, whenever students ask "Why do we have to know about singularities?", the Tacoma-Narrows Bridge, coupled with a discussion of resonance caused by undefined frequencies gets their attention.

In that case, you should definitely talk to them about flutter. 747 Wind Tunnel Flutter Test

It's been really interesting reading everyone's responses. I went to a magnet school, and for us Calculus AB was a graduation requirement. I thought that was pretty awesome, even my friends who said they were "bad at math" still got mostly 4s and above on the AP.

Originally Posted by JVN

It is my hope that natural curiosity would drive people to seek out this understanding. Algebra is of course (imho) a big part of this understanding.

Spot on! I think Linear Algebra was probably the coolest class I took even if I liked it the least. I always get excited when something I thought was unrelated pops up in a new place, and Linear Algebra has this tendency to show up just about everywhere. But that sort of natural curiosity means that I would be okay with really any class so long as it wasn't pointless.*

*I can't think of a pointless class I took with a good teacher...

[Taylor]

Student: This has no application. I don't get it. How can I use this? I don't understand it. It has no meaning to life.

Teacher: Well, this is how it's used every day. And here. And here.

Student: But those are story problems. I don't like story problems.

... and so the cycle continues.

[Ankit S.]

One thing that I don't think was brought up earlier is what happens if Algebra is no longer made mandatory in school. Many of my peers are lazy and would cut corners in any way possible. If given the option to take a moderate-workload Algebra class or to take the relatively homework free class Art 1, many would choose Art 1.

As a result, they would not even try to take Algebra. And who knows? Maybe some of them really will not benefit from Algebra and skipping it was a great choice, but for others who do no know what they want to study later on, the skipped Algebra course also narrows their options further along the road(either not do something related to math, or confront Algebra years later).

In summary: many people are lazy, if given the option to skip Algebra for an easier course they probably will. This skipping will narrow their study options later on.

[Tom Ore]

One of my favorite quotes that applies to some of the discussion here:

"Culture is what's left over after you've forgotten the details of what you've been taught."

EDIT: Another quote applies also: "The purpose of computation is insight, not numbers."

[Siri]

Originally Posted by Lil' Lavery

I find the whole "look at this aspect of the course that I haven't used" argument very disingenuous. Most people do (or, in reality, should) use the knowledge learned in history classes in their political behavior. Even if certain pieces of information never come to fruitition, others (and more importantly themes and historical trends) most certainly will.

Isn't that exactly the argument for the importance of algebra? I really don't see the difference you're pointing at.

[Lil' Lavery]

Originally Posted by Siri

Isn't that exactly the argument for the importance of algebra? I really don't see the difference you're pointing at.

"I never used the quadratic equation" vs. "I never used algebra."

[Siri]

Originally Posted by Lil' Lavery

"I never used the quadratic equation" vs. "I never used algebra."

huh. ok. I see it more as "I never use American literature or algebra" versus "I never employ the themes and trends I learned in American literature or algebra" first statement true for both, second statement false for both. As far as actual information once told to me by a teacher that I then called upon and executed on in my daily life, algebra beats all my history and literature courses [distinguished from writing and foriegn language] combined by an underwhelming 2 to 0 (but it wasn't polynomial). As far as trends, understandings and thinking skills I learned in them being useful day-to-day, they're about tied approaching infinity. Do you not find yourself using the trends and themes from algebra in your daily life? Maybe I just think weirdly. Never mind then.

[ManicMechanic]

Originally Posted by Lil' Lavery

But what I meant by "discrete skill" was that algebra as a tool isn't particularly useful to a large portion of the population (though ManicMechanic makes a strong counterpoint to this). You don't need algebra to balance your checkbook or figure out your monthly finances. Most people are probably never going to balance an equation outside of an academic setting.

Most people DON'T use algebra, but most people COULD, and if they did, it would make their lives better. I teach lower-level math (prealgebra through statistics), and I tell my students, "Not every formula or technique in this course will be used by everyone, but every formula could be used by someone, and you never know when the person who needs that formula could be you." I've always been able to think of a plausible useful real-life scenario for every formula I teach.

For example, one of the nastiest formulas from statistics is for the standard deviation of the difference of difference of means for 2 populations -- a 6-layered complex fraction. When would you ever use that? Well, I came across an article stating that some cholesterol drugs may be far less effective in women than in men, and that it wasn't certain whether more tests were needed for the women (but not the men). Two populations: men and women. Difference of means: cholesterol level before and after medication. Difference of difference of means: men's improvement is different from women's improvement. How to evaluate whether more studies needed to verify hypothesis: evaluate standard deviation.

One of my neatest classroom experiences was a student who, after hearing multiple tie-ins to formulas said, "I bet you could never find an application for the problem 1/(1+(1/(1+(1/1+1)))), our homework problem." As I thought about it, this formula is tied to the Fibonnaci sequence, whose real-life application has to do with the fractions that ensue: 1/2, 2/3, 3/5, 5/8. As you traverse these fractional increments around a circle and draw rays to the edge of the circle, this mimics a bird's eye view of a corn plant that shoots out leaves at these fractional increments, for maximum sun exposure. I suggested that this pattern could be used to optimize water coverage for certain patterns in the design of a sprinkler system. The student came the next day with a box of apricots -- his family is in the farming business -- they grow corn and design sprinkler systems, and this was something they could use.

It's our job as "math literates" to see these connections and help the people around us to use these connections to improve their lives.

[MechEng83]

I found this NPR story this morning quite applicable to the conversation at hand.

http://www.npr.org/2012/07/31/157637...-bed-with-math

One of the points they make is that society accepts when an educated person says "I'm not very good at math" but would be appalled if and educated person said "I'm not very good at reading"

[artdutra04]

The problem with algebra is that a large plurality of people today seriously don't know how much they can actually use it to improve their lives. Ignoring the time-value of money*, simple algebra can be used in many real life problems.

For example, do I buy a more expensive house closer to my job that lower commuting costs, or do I buy a cheaper house that is farther away but has a significantly longer and more expensive commute? At the simplest level, these are two linear y=mx+b equations that may or may not intersect.

All too often people fail to grasp the concept that things in life have both initial and ongoing costs/benefits (again, y=mx+b), and only balk at or support various things based upon their initial cost and ignore the annual costs/benefits. It's even worse when this mentality is used in the voting booth for public/government proposals/projects. "$X is too much money for a new highway interchange/light rail line/rebuild the schools/support robotics teams/etc" and ignore the long term costs/benefits ("it will collectively save drivers $50m per year and have a cost/benefit ratio well into the positive").

One of the greatest examples of this is the Clean Air Act; while this law annually costs the federal government about $50 billion to run/enforce, the benefits to the people and the economy from having cleaner air end up totaling over $1.3 trillion every year. In other words, the law has an insanely huge return on investment (about 26:1!!) that not even Berkshire Hathaway can match.

At the same time, most people also fail to grasp p=e^rt, and then they get screwed over on credit card debt, home mortgages, 401(k) and investment accounts, and anything else that involves interest. Many see the value of things rise but fail to account for inflation; for example if something is worth $100 now but will be worth $500 decades in the future, you lost value if the inflation-adjusted price ended up being $600.

Or if people had even the most basic understanding of statistics they would likely be more hesitant to gamble. Or at the very least, they would accept the cost of occasionally gambling as the price of a few hours entertainment that may have a small chance of ending up with more money than they started.

This list can go on and on and on.


* Which is never a good idea if you want accurate projections, but even simple analysis that has an interest rate of 0% is better than nothing.

[Nemo]

Originally Posted by JVN

Building along the same line that Karthik started: I recommend everyone read "A Mathematician's Lament" by Paul Lockhart (sometimes just referred to as Lockhart's Lament).
http://www.maa.org/devlin/LockhartsLament.pdf

Wow - thanks for posting the link to that paper. This paper has triggered a rant.

Trying to come up with a good way to teach math or science can be extremely frustrating, because in some ways we have both hands tied behind our backs. We are given a curriculum and told to teach it. Never mind that the curriculum misses the point and kills curiosity and creativity in favor of teaching certain procedures because they can be tested easily on exams, and because that's what will lead students to the next procedures in high school or college or whatever. It's frustrating to exist within a system of standards/curriculum/exams that constrain what we're able to do. I'm very interested in trying completely new ways of doing things, but I have a list of topics that have been deemed important, and I'm required to teach those things*. They're on the tests that I'm required to give. Some of them fall under the category of teaching notation without actually letting the students tackle a problem (see the linked article), and it's tough to salvage that type of curricular material with clever tweaks in the way you teach it. A new approach is required, and it is necessary to let go of some of the things that we deem essential. I think that's what the author of article in the OP's post is getting at. If you're not actually teaching people algebra, then you're not actually gaining anything by forcing them to take it. (Trust me, I personally enjoy and value algebra.)

*That said, I prioritize trying to teach well over following rules or doing exactly as I am told.

What I find really tiring is that teachers are constantly presented with requirements and standards and demands to measure student progress in new ways, but the experts always stop short of writing (or funding the writing of) a good curriculum for us to use as a resource. No, I don't want or need a step by step guide, but a list of great problems and investigations to use would be a nice start. You'd think this would be out there somewhere, but I certainly haven't found it. The idea seems to be that teachers are supposed to read the standards (which are full of concepts I completely agree with), and then concoct our own curricula to satisfy all of the standards. I think I could go on a pretty long rant about why this is impossible, but here's a medium length one: creating a curriculum is a very large and intricate puzzle; to do it right, you'd have to interweave all of the subjects in ways that reinforce each other and provide context for the others; you'd have to make sure each year builds on previous years; you'd have to make sure it's challenging for fast learners and accessible enough for your struggling learners to still make progress; you need to interweave a lot of "soft" (and more crucial) concepts with the so-called "content" learning, including inquiry/argument/critque (including doing them in social studies AND science AND language, etc). So... what's the master list of interesting and challenging problems that I should include in a chemistry class that will not only pique student's curiosity and creativity, not only get them the procedural skills that will allow them to survive a test from an outside entity, but also make up for the various deficiencies and bad attitudes they've collected from their previous education? I'm not saying it can't be done, but what I AM saying is that this isn't the type of problem that an enterprising teacher is going to successfully tackle over the summer as a summer project that I decided to try for shiggles. It's a large puzzle with a lot of constraints and moving parts, and I don't even have a great list of really interesting problems and investigations to start with as raw material.

If I have a point, it's this: the curriculum in math (and science) is way wrong, and we need to start over and write something that does a better job. The emphasis on specific content (such as stating electron configurations, or writing arcane geometry proofs) should be scaled back, and the emphasis on general thinking skills (like critiquing an argument or suggesting possible explanations for a surprising result) should be scaled up. Teachers shouldn't be expected to wave a magic wand and create this curriculum in our spare time... smart people, like the people who wrote the National Science Education Standards, should help in this effort instead of copping out and saying that districts are supposed to do it. If a brilliant curriculum is created, it should be a freely available resource, not a set of rigid requirements - let teachers use better stuff if they are clever enough to come up with it. And... more effort should be placed on this type of thing than is currently placed on creating and administering ever more standardized exams that will, predictably and inevitably, prove that our current system is still, in spite of doing the same thing repeatedly over many decades, not working as well as we would like.

[Tetraman]

Remove algebra - truly an american answer to an american problem.

After remembering how awful I was at math in high school, thanks to this opinion piece, I remember the times when math was easy for me - and those where the days where we were taught not "how to solve the problem", but "how problems are solved". By this, I mean instead of the teacher putting an equation on the board and explaining how to solve it, the teacher explained how the symbols were utilized. It was like, solving a sudoku or Rush Hour puzzle. We were taught how the parts work, how the parts are solved - it was a series of systems that could be pulled from a shelf of knowledge and then worked with to get an answer. Instead of learning how to solve a problem, I learned how to solve a series of individual puzzles and their connections to a larger puzzle.

Maybe this is why I like designing board games so much - I love to work with systems as puzzles.

[Katie_UPS]

I wasn't going to post, but I wanted to highlight something.

Originally Posted by KrazyCarl92

I will preface this by acknowledging that I am heavily biased on this topic, but I see this bias as justified.
This would be akin to if you asked someone from a robotics team why they were losing 90% of their matches and they said to you "because of our ball collector". Is it then the right move to take away your ball collector entirely? Will that make you win? Certainly not, and likewise removing algebra will not better educate our society!

I'm surprised no one else has responded to your robot simile* because its kind of awesome. Because you don't get rid of your intake system, you just make it better. So we don't remove math education, we just make it better.

GUYS, THIS MAKES SO MUCH SENSE.**

How do we make math education better? At a practical level, not just conceptually? Most people are saying very broad things that are easy to say, but hard to apply.

Personally, the best teachers I had taught relationships in concepts to make sense of things (like the relationship between the y-value of f'(x) and the slope of f(x); that blew my mind in high school, physics formulas and derivatives made so much sense)(calculus is why I love math). What are concrete ways for teachers to teach?

*akin means "similar to" according to google. Similes use phrases in the range of "is like a". Just in case anyone cared. I needed to justify it for myself.
**Sometimes I just want to talk like we're all facebook friends, you know?

[DampRobot]

That article was certainly illuminating. Although I never thought of it that way, the author certainly has a point. The fact that are students are falling behind compared to their international peers is hardly surprising considering how they waste their time studying needlessly abstract concepts. When will they ever need to solve for x or use a quadratic equation? We should abandon algebra in order to concentrate on real, concrete gains such as on test scores.

In fact, the view portrayed in the article can and should be applied to many other subjects. Most students will never play the clarinet or tuba outside school, and learning these useless skills takes up valuable time in the classroom. While they could be working to achieve real and positive gains on their SAT scores, they are spending time chasing the rather silly goal of learning to make music and express oneself.

On further consideration of the article, I realized that it could be applied to science as well. A car salesman or an advertising executive will not need to know what chromosomes are or the properties of gallium. They would benefit far more from working towards higher test scores, which would allow them to be admitted to the all important top college. Going to Princeton or Yale will take them much farther in their careers than attaining any meaningless "scientific" knowledge.

In fact, why pursue equally abstract and silly "educations" in the liberal arts or the humanities? Except for the all important Five Paragraph Essay, the mainstay of the SAT and several APs, learning literary analysis or many historical facts will not create more productive, higher testing citizens. These subjects should probably be eliminated as well.

I concluded that nearly all subjects should be eliminated from high schools in favor of concentrating on the holy trinity of tests: the SAT, the ACT, and the AP. Only these will allow the high school student to attain the true purpose of high school: the Ivy League School. Although some weak, liberal intellectuals may whine about the value of an "education" or "creativity," most schools are in fact already slowly coming around to this pragmatic view.