As you might expect, several columnists/bloggers have been responding to the “Is Algebra Necessary?” article. In addition to the one ultrafilter already linked to, here are a few others I liked:
When Andrew Hacker asks “Is Algebra Necessary?”, why doesn’t he just ask “Is High School Necessary?”
Why Algebra Matters (and Why Andrew Hacker is Off-Target)
Er, yes, I’m sure that having a greater number of individuals without even a basic understanding of statistics could be better for society. I’m not terribly good at sarcasm, but, yeesh.
That’s only true if you’re going to redefine “high school algebra” to mean something like “word problems” (or even more arbitrarily, systems of linear equations). In point of fact, and contrary to your assertion, that subject is all about arithmetic — merely arithmetic with unknowns. Your example is hardly different in kind from the one I gave, just the tiniest bit less trivial. If you can solve for the number of cats that left the couch then you can learn to solve for the number of typists needed to finish 100 pages in 4 hours.
I have met literally retarded persons who could do that much.
Where did I advocate rote memorization? Where did I advocate anything except the point of view that people are generally more intelligent than you give them credit for?
I think geometry and trig could be profitably replaced by statistics or computer science for many students.
Algebra is essential though. If you don’t think so, that probably means you just “get it” without needing an entire class on it. It happens; teacher quality itself matters much less for top students in high school, who are going to get it anyway.
Your example I categorized as arithmetic because it’s discussed before 5th grade arithmetic (USA).
While my “algebra” example can be categorized as “harder” arithmetic or “disguised arithmetic”, it is the type of question kids don’t see until Algebra I. The question sets up a “relationship” and the student solves it from there. Arithmetic does not stress relationships between quantities. Algebra does. He can either use system of equations or adhoc common sense reasoning.
I’m not saying that. I’m saying we’re assigning more credit to algebra for uninterested students without proof of its benefits.
Some of the response articles are mischaracterizing it as “society giving up on algebra.”
First off, millions of students would still take the full 2 years of Algebra I & II.
The difference is that some students substitute something else that’s academically challenging. Could be linguistics. Or legal documents analysis. I’m sure our collective imagination could come up with quality alternatives besides “art” and “physical education”.
The discussion about the merits of algebra for non-motivated students assumes knowledge we do not have. It would require an experiment that would test 3 groups:
group 1: 8th graders right before they start their first day of Algebra 1
group 2: adults that have never taken algebra but have equivalent IQ to group 3
group 3: the 86% of teachers that failed that math portion of teaching certification exam (as a representative of those that have taken Algebra II and forgotten it)
We then create various test questions that do not look like algebra questions but test abstract reasoning.
To be clear, we are not testing the folks who already understand algebra, retained most of it, and appreciate it. (These include most posters in this thread.) Such a group would demolish the 3 mentioned above. I’m not sure how to design good questions that are unfamiliar to all groups and test abstract reasoning. Analyzing and categorizing pictures of knots? Using LEGOs and have the test candidates look at a picture and estimate minimum # needed to duplicate it?
If such an experiment has not been done, how can anyone say that “algebra helps with abstract thinking later in life even if it’s not used” ?
There is no proof.
Even Daniel Willingham had to use a weasle word “possibility”:
“In other words, Hacker overlooks the possibility that the mathematics learned in school, even if seldom applied directly, makes students better able to learn new quantitative skills.”
Consider:
"The study of 1000 Latin words for human anatomy is part of a medical doctors’ training.
Income charts show doctors earn six figure salaries.
Therefore, we have concrete evidence that children should study 1000 Latin human anatomy word because it will lead to six-figure salaries.
Do a search and replace: (Latin/algebra), (doctors/engineers)
"The study of algebra is part of a engineers’ training.
Income charts show they earn salaries above the median.
Therefore, we have concrete evidence that children should study algebra because it will lead to salaries above the median.
If the first premise (Latin) is flawed, why not the second one (algebra)?
Again, we’re not talking about quantitative-skill workers that already understand algebra as thinking tools in their career.
We are talking about the unmotivated students that muck through algebra without understanding, forget it, never use it again. Are we using their time effectively? (Doesn’t the real answer point back to the social experiment we haven’t done yet?)
I propose that we can only say the following statement:
*“We have no definitive proof that the people who struggle through algebra, pass the tests, and then forget it are better off in the long run. However in absence of actual evidence, we make them study it as a precautionary measure.” *
At least that statement is more defensible. It’s more accurate of what we actually know about the benefits of algebra for unmotivated students.
I don’t know that I agree with you. I think we just need a better way to teach math.
Math - at least up to calc II - is just a puzzle, and I don’t know any one who doesn’t like puzzles.
I think that if kids were taught math as recess, they would be done with algebra by third grade.
Please note that I am not slamming math teachers. Just what I remember of learning math in school, as opposed to learning with brightly color magnets on the refrigerator and Games Magazine.
Society has tried for more than 100 years of organized education and spending millions for new ideas including “New Math” reform, etc.
Nobody up to this point has figured out the magic formula for instilling the understanding of higher math beyond arithmetic to the majority of the population.
It’s 1000 times easier to just say, “we just need better math teachers” and yet no has definitive proof that these mythical math teachers for disinterested students are the ultimate answer.
A big part of the disconnect is that some people don’t learn well when faced with hopelessly abstract problems that just involve math symbols. I am an excellent problem solver in general and apparently know algebra pretty well based on all the sample problems given in this thread. I am an expert at anything Excel related and can always make a living consulting in it if I needed to but I can do things much harder than that.
The things is, I didn’t learn any of those things from Algebra class (you will have to take my word for it; I didn’t participate at all). I already knew the answers to the problems in Algebra I just by glancing at them or solving them by my own methods that I just figured out as they came so I just wrote the answer down without any steps shown.
It was Algebra II where the problems got too difficult to do that so I just stopped. You may think that if I just learned the methodology that the teachers wanted from the beginning, everything would have been fine but I disagree. I just wanted to know why they were teaching this apparently useless subject without any context. If it was given to me in the form of a word problem, I would have been all over that and I know other smart people feel the same way. There is a real problem with math education in that its application to real world is ignored with scoffs and a simple dismissal of learning the one true way.
Instead of presenting students with arcane symbols that have no context or any future apparent use, can’t we, in this day of Mythbusters and other popular media that makes science and math sexy do much better to present cool and solvable problems to students in the ways that they will care about and remember?
No doubt there exist such people. I can assure you, though, that my teaching experience has been littered with word problems, which have near-universally been the students’ least favorite, most imposing activity; they will quite vocally moan about their distaste for word problems as a category, how hard they are and why can’t they just have something more straightforward?, and so on.
This is probably not intrinsic; it’s probably a result of the need to transfer conceptual understanding back and forth between their mathematical tools and the (ever-so-slightly) non-formulaic context of the word problem clashing too greatly with the expectations set by the overall nature of their math training (which is by and large structured around memorizing algorithms or heuristics handed down from on high, and then chugging away, even in those same classes offering the (usually still quite contrived) word problems). So, simply littering the ground with word problems is no panacea… but then, I don’t imagine you felt it would be.
It’s just an observation I wanted to offer, in support of no particular point: In the world as it stands, students hate word problems.
[Formulaic, contrived word problems are just a superficial re-wording everyone sees through (exponential functions are always discussed in terms of bacteria growth, and this never does anything for anyone). But realistically detailed and messy word problems wouldn’t necessarily be exciting or motivating or accessible for many students either. Then again, that’s perhaps what others would want. The only thing I know to say is that everyone learns differently.]
True. Different (smart people and otherwise) have different learning styles. My mother makes living as a best selling author and speaker on that topic. She has done a lot for science education in the world by it making less abstract and simply more fun for people to want to know about. Most students do when it is presented in the right way like a combination of experiments, visual displays, and emphasis on what knowing such things have gained people already.
I have never seen such a movement in math. New math is a joke. Why not take these problems that people will eventually need Algebra for and use them directly? You can always work backwards to teach the basic tools of the language while students are seeing what it can do for them. I never got that and I don’t believe millions of other students did either. Algebra and all other higher level math is taught in a way that appeals to mathematicians and that is only a small minority of people. Why not teach it in a way that will immediately appeal to the future engineers or even homemakers as well?
Maybe part of our problem as a society, is that we think learning to graph parabolas is the main problem solving mechanism to figure out how to solve life’s problems.
Nobody here has given any conclusive evidence for mandating the levels of algebra we do. We do it cause it has always been done that way. As professionals, we tend to go along with the party line. We go with traditional thinking.
That’s the thing, though. The status quo doesn’t have anything to prove. It’s those who propose to change it that have to provide evidence, or else the status quo remains. That’s basic logic, and it doesn’t seem that math helps people learn it.
In fact, I don’t see how basic logic has anything to do with math at all, really. At least, the book I found being thrown away in college–which greatly improved my critical thinking skills to the point that I wished I’d learned it in grade school–did not have any math (at least as traditionally defined) in it. You had a bunch of statements, and you explained why they were or weren’t valid based on fallacies.
Also, because it’s been brought up, I never learned arithmetic in the rote way, and I wound up getting the highest grade in calculus in high school*. The thing is, I started with math in preschool/kindergarten, with very concrete visuals to help me learn the concepts. Because we started early, we had time to move more slowly. Plus, it was at a private school where we learned at our own pace, so there was no chance of not understanding something before you moved on to the next stage.
By the time you got through the math curriculum (which was all arithmetic), you knew all the stuff everyone else learned via rote, but with the added advantage of understanding it. For exact details, go look up the Montessori method and how it works with math.
Now, previously, that same level of teaching may not have been practical for public schools. But we have computers now. And I know we can use computers to teach math–they did that with us in fifth and sixth grade, along with, get this, a program that let you move at your own pace. If we actually used that for math teaching, making math into a game, I bet quite a few more people would learn it effectively.
*I think what helped me the most was that I’d learned how to simplify things on my own. I could take the complicated and make it simple. Plus, I had so little memorization in my head that memorizing those little formulas for derivatives and integrals wasn’t quite as hard–it wasn’t just yet another thing to memorize. Though, in truth, it’s the only part of the math I’ve learned that I no longer know–memorization doesn’t work long term. It wouldn’t surprise me if the reason some people fall behind is that they forget what they’ve memorized and thus get stuck, as they can’t go back to the basic concepts and memorize the information again.)
That’s not basic logic. That’s conservatism. Basic Logic is: Generally people don’t need to know how to graph a parabola. Because they don’t need to know how to do it, we won’t force them to do it. (The validity of the that statement is at the core of this discussion.)
What is being argued is “Sure people don’t need to know how to graph a parabola, but the learning to follow a step-by-step problem-solving process is what is gained.”
The problem is there is no scientific evidence for this being introduced in this thread.
And sure the status quo will remain. Like I said in my last comment, basically professionals are like all other humans. They value acceptance and want to appear knowledgeable… so they generally go with the standard party line.
That is bit of a strawman,
Who claimed " learning to graph parabolas is the main problem solving mechanism to figure out how to solve life’s problems"
How does this prove that making people learn math is bad and should be stopped?
Are you claiming that learning about things like equivalence or variables is useless?
Nobody here is disputing the need to learn math nor basic algebra. That can’t be derived at all from my statement. I am a little confused by your assumption.
Is the thread title not “Is Algebra Necessary?” and is that not the premis of the OP’s linked op-ed?
That isn’t really the way I interpreted even though that is the literal title. Some of us are arguing for alternatives to two full years of Algebra as it is currently presented. Basic algebra would still be taught but maybe in a different way although that depends on your definition of ‘basic’.
Part of the discussion is where basic math and numeracy end. (I’m still here, sort of.) It’s a lot easier to say basic algebra, whatever that is, is necessary. But completing the square?
Yes, and that’s the exact same title as the NY Times article. The title is nice and short 3 words for maximum provocative effect. The title is the teaser to encourage people to read the entire article for it’s actual content.
In the article he does say:
Algebraic algorithms underpin animated movies, investment strategies and airline ticket prices. And we need people to understand how those things work and to advance our frontiers.
…
Mathematics, both pure and applied, is integral to our civilization, whether the realm is aesthetic or electronic.
No. The author states that algebra is absolutely necessary — but not universally for everyone. The “algebra at all costs” traditional viewpoint has to be critically examined in light of that fact that the supposed benefits for the people who don’t use it has never been rigorously and scientifically proven.
The education community that prides itself on the rigors of geometry proofs and statistical thinking should welcome the critical assessment of the math curriculum for its advertised benefits bestowed on the uninterested students that will forget it. The force-feeding of algebra comes at the expense of other worthy classes that could (should?) be substituted and provide higher knowledge retention.
Why is it that Math always seems to be attacked under the ‘you don’t need to know that specific thing’ rule?
If we apply that to other subjects in school, what is there left to teach besides basic reading and writing and core arithmetic? Why expose students to Shakespeare? Why make high school kids dissect frogs or study biology at the level they do? Why should high school kids waste their time learning the basics of organic and inorganic chemistry, when the vast majority of them will never look at a periodic table again, or draw a chemical diagram? Why do we bother teaching astronomy when most high school graduates will never need to know how a solar system is formed? Why do we teach ancient history? Most people don’t need to know who the first ruler of the Roman Empire was.
The answer to all of these questions is that by the time you graduate high school you are supposed to be a well educated citizen with basic exposure to a broad range of knowledge, so that you can choose many different paths in life without educational roadblocks, so you can be a good citizen, so you can follow the news, solve problems on your own, and basically have the background knowledge that we expect people in a modern industrial economy to have.
Will you ever again need to graph a parabola? Maybe not. But maybe you will. And even if you don’t, having done it helps cement the notion of a parabola in your head, so if you read a news article about a bullet hitting someone after following a parabolic trajectory, you’ll have a clue what they’re talking about.
Or perhaps you want to get a pilot’s license. Lots of trigonometry in that. Or you want to learn how to navigate a boat or how to Scuba dive. All of these subjects require a basic understanding of a broad range of technical subjects. You may not have thought you’d ever need to worry about gas laws again, but if you fill a SCUBA tank too fast, it will overheat and you’ll have a chance of understanding why.
I could go on. We live in a technological age. You can’t navigate your way through it by being a half-educated dullard relying on computers and calculators to guide the way. Math is the LAST thing that should be dumbed down in the 21st century. It’s one of the foundational subjects that everything else builds on. If anything, we should be teaching more of it and dumping some of the newer, touchy-feely subjects that have wormed their way into the system.
To be more specific, I think kids need the concepts of variables and the very beginnings of basic Algebra. These concepts are usually taught in some type of pre-algrebra course (or used to be.) Thats enough for most. In other words, the Algebraic concepts we use in every day life. I threw graphing parabolas (which is actually one of the simpler Algrebraic things) out there as an example of the ridiculousness of mandatory algebra. Give ALL kids the some pre-Algebraic ideas, but certainly don’t hold back kids who are going in that direction. If kids need Algebra after High School for a career choice, there are many wonderful Community Colleges out there that can give it to them.