What tests are currently required to graduate from high school/college in California? (Besides algebra.) Is geometry really not required? What about liberal arts besides mathematics?
How about in New York? France?
I’ll grant you that calculus has very little relevance (I won’t say “none”) to computer science. But algebra is certainly still relevant.
And geometry as it’s currently typically taught contains more insight into the nature of math than algebra as it’s currently typically taught, but I think that’s more an indictment of the way that algebra is currently taught than anything fundamental about either topic. For most students, geometry is the only subject where they’re expected to produce proofs, even though proving theorems is really the heart and soul of mathematics.
As someone who teaches in the California State University system, let me assure you that literally thousands of students are admitted into our universities every year without a decent grasp of how to write standard English.
I’m no advanced math expert. I’m a humanities person, and i teach history. But i was pretty good at math in high school, i appreciate its utility, and i agree with everything you’ve said here. I think a decent grasp of basic probability and statistics would be a much more useful skill for many college graduates. And it doesn’t need to be especially complex stuff. In terms of statistics, if people grasp some of the fundamental concepts about mean, media, mode, standard deviation, and things like confidence level, they are better equipped to understand a whole raft of important issues. Same with probability.
Regarding the LA Times problem, i have done no formal mathematics in over 20 years, and i haven’t done this sort of algebra equation since i was in high school 30 years ago. It took me a moment to fire up the dormant part of my brain that deals with this sort of thing, but i managed to simplify the equation in about a minute.
I may have been able to tackle this right after taking the class, but now I have no idea. Is something covered in first or second year Algebra? I had an extremely hard time passing high school Algebra, but had no trouble with the logic class I took to satisfy the quantitative requirement.
I’m not sure what you mean by “abstractions of Algebra” … Algebra is very deductive … step by logical step … the troubles understanding Algebra mostly come from a lack of desire to learn, and then the lack of effort to even try … getting away from Learning Disability, let’s focus on the 14-year-old who wants to be a marital counselor, someone who hates math, sees no reason to learn math and just can’t bring themselves to focus on learning math … and we’re saying they have to pass Algebra, even though it’s obvious they will never use these skills ever again … been there, done that … I suffered to get those credits in English, but I wanted to be able to be a junior in college … no university in the nation would let me without those credits …
“It’s not a great medium …” … I agree, but this says it better:
… although I think accounting is the heart and soul of mathematics … money is the only thing worth counting …
[raises hand] … I didn’t even know what a dictionary was for …
In partial answer to my own question, a basic New York Regents Diploma requires passing exams in Global History, U.S. History, English, Mathematics (one of Algebra/Geometry/Trigonometry), and 1 x Science.
France has a list of subjects depending on one’s specialization; in fact for Bac L (literature) applied mathematics is optional, whereas it is required for Economics/Social Sciences. The relevant exam seems to cover probability, statistics, calculus, analysis, algebra and geometry (and seems more serious compared to the California exam question we saw, or at least a bit closer to real-world problems).
Let’s see. I went to Junior High School, Middle School these days, and Algebra 1, was in 7th grade. Algebra 2, the next year. (High school kids, why I outta…) Totally aced two years of algebra. Then it got tough and I lost it. Geometry, forget it. Never even had to try trig or calculus. But I realized that algebra was so logical and understandable that it truly helped you figure out different styles of problems. And it turns out, one of the things I used to do, for a long time, was measure, among other things, every cap and lowercase letter, individually, every ascender and descender of each letter, of whatever actor or director’s name appeared in some advertising artwork, aggregate those numbers and prove legally that any individual names involved were within their contractural rights of percentage of visibility. Pure Algebra 1.
Algebra is an abstract concept in and of itself. Or can you tell me what x is in the OPs problem?
And as a math teacher I can safely say that your claim that “the troubles understanding Algebra mostly come from a lack of desire to learn, and then the lack of effort to even try” is not factual.
What you’re trying to convey with the rest of this paragraph I do not know.
Algebra isn’t usually considered abstract, thus my question about how you’re defining “abstract” here … x in the above statement is a number, just like any other number … I suppose we could nitpick and say it’s a symbol for a number, just like “5” is a symbol for five … but generally this is all concrete math, we don’t know the value of the number x, but we can certainly say there is a numerical value, as opposed to a solution set of partial differential equations, for example …
So, first thing I did when I saw the problem is I determined the unique matrix that represents the upper left term (x[sup]2[/sup] - 25) and quickly found it’s factors (x + 5)(x - 5) … THAT is an abstract concept, for every Algebraic expression there exists a unique matrix that represents that expression … something I learned in hushed whispers in some dark corner of my high school’s library …
I’m fine with you saying I’m wrong … but I sure would like to know what’s right … why do students who are competent in deductive reasoning have so much trouble learning Algebra? … Again, we are excluding students with Learning Disabilities …
In case anyone’s interested:
[spoiler]For any equation ax[sup]2[/sup] + bx + c = (kx + L)(mx + n) where abckLmn are constants, there exists a unique matrix:
| k L |
| m n |
Such that: a = km, c = Ln and b = kn + Lm … in the case of the OP’s upper left:
| 1 5 |
| 1 -5 |
[/spoiler]
Reprinted from Chrono’s Game-the-Test Methodology[sup]TM[/sup] … and without permission.
=====
Keeping the General Question in mind … the college degree means, in part, the individual is willing and able to learn anything an employer tells them to learn, whether the worker wants to or not … as demonstrated by having learned Algebra or Ethnic Studies perhaps against their will …
Algebra can be abstract:
• Every non-constant polynomial in x with complex-number coefficients has a complex root.
• Every group of order p[sup]a[/sup]q[sup]b[/sup], where p and q are prime numbers, is solvable.
Etc. (note these are no longer problems trivially solvable by a computer; you actually need to prove something)
Of course this is true if we use the normal and more general definition of abstract … but then this is also true for basic arithmetic … consider “5 + 3” … the graphemes “5” and “3” are symbols of things, we understand what these symbols mean so we can give the answer with another grapheme “8”, yet another symbol … all very abstract using the common definition of the word … think abstract art for example …
Algebra simply introduces the grapheme “x” to symbolize something we cannot define exactly … and the creative arts on how to manipulate such a symbol to obtain useful answers … but the “abstraction” of this symbology is hardly new to the student of Algebra …
There does exist a thing called Abstract Algebra … generally introduced to the college junior in the Linear Algebra class … although the high school Algebra students will have been given a taste of this stuff with Cramer’s Rule presented right after the section on solving for three variables in three equations …
Quoted for truth.
Me, graduated 1976 Stuyvesant High School, NYC, “science and math” specialized high school with perennial freaky world-class math teams and eventual Nobel winners, etc.
Learned to plug in Princess Sohcahtoa, got by fine. Gotta ace the tests, right? But to this day I lack any natural sense of the ways constrained lines operate, let alone in agebraic expressions. My senior-year Calculus courses were a sloppy inconsequential muddle.
So it’s an abstraction from specific numbers, which in cognitive sciences is an abstraction from specific and concrete dimensions and countable objects.
In the field of pedagogics and math instruction it’s been shown that it requires different cognitive skills to deal with the mathematics of, from the concrete to the abstract:
- Actually making a right angle triangle out of physical bits of wood utilizing pythagora’s theorem.
- Solving a math problem described as figuring out the necessary lengths to make a right angle triangle out of bits of wood.
- Solving a math problem described as figuring out the necessary lengths to make a specific right angle triangle.
- Solving a math problem described as finding the right x,y and z to fulfill the equation x^2 + y^2 = z^2
Another example: it’s a well known dead end in lower level algebra instruction to concretize the simplification of basic expressions such as 2a + b - a + 3b by saying a’s are apples and b’s are bananas and that’s why the expression can be simplified to a +4b. Algebraic expressions are abstractions, counting fruit is concrete, but 2a + b -a + 3*b isn’t an abstraction of counting fruit.
And the lesson here isn’t “algebra” isn’t abstract. It’s “abstract and concrete isn’t binary, it’s a continuum”. And just because this abstraction isn’t new to any student doesn’t mean all the students have the same, or even sufficient, understanding of what’s “really” going on, which I’ve painfully experienced with numerous students who have attained fairly good grades at lower levels, but have appallingly poor understanding of what they are doing.
That was very well said imo.
I think this is a huge problem with high school in general. So much information is presented in such a short range of time, there is no time to understand any of it. It’s not fair to the kids or the teachers.
I was speaking to a friend who is a 2nd grade teacher and a mother of three, aged from college to 13. She said the pressure put on kids is ridiculous from the time they are in kindergarten and we expect them to read. Then we wonder why the hell when they get to middle school and high school and there’s more pressure so many of them have anxiety issues.
It should be clear that poor, inadequate preparation can make even incontrovertibly “easy” questions, like the one in the OP (Chronos is right that it’s a stupid question to boot), difficult to answer. If it’s systematic my inclination in such a case would be first to blame the crappy high school educational system, not the students.
If we are debating what an otherwise intelligent student who finds him- or herself in that position and is unable to earn a diploma as a result should do, the article mentions remedial classes (surely at least one is taught by a good instructor like naita?) In New York the student could fulfill the mathematics requirement by testing in Geometry instead. There could also be something said for the French option of graduating with a humanities degree which requires no mathematics at all but a huge dose of Philosophy.
Flattery will get you nowh… oh, who am I kidding.
I worked at my college’s math tutoring center … so I’ve had “somewhat” the same experiences … trying to help an Algebra student who couldn’t add fractions … for me it was more cut-and-dry, “You’re in the wrong class, transfer to the remedial math class” …
The difference here is that high schools are trying to teach everybody Algebra, in college we’re only having to teach Algebra to people who need to learn it … and there’s the presumption that anyone has an “appallingly poor understanding” of arithmetic isn’t getting a high school diploma and thus not being allowed into a four year college in the first place … in California the student would go to junior college to learn and once completed they have to demonstrate that they are ready for college level math … if California drops this requirement, then the college degrees they issue don’t mean as much …
Let’s look at the other half of the brain … do we want college juniors who are unable to write an essay at the high school level? … college graduates in the workforce who can’t issue intelligible written instructions to their subordinates? …
Anyway … I claimed that problems learning Algebra is a lack of desire to learn it … you say that’s factually wrong … instead you say it’s because of a lack of understanding of arithmetic … but that’s just kicking the can down the road … why didn’t the student learn arithmetic if the wanted to learn it … at some point here we’re going to have start thinking about a Learning Disability … we’re talking about six years to pass Algebra, eight if the junior high school offers that class … I just don’t see this as overly burdensome requirement to become a college junior, anymore than the requirement to write an intelligible sentence …
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Your lack of understanding of modern cognitive thought theories, not to mention modern pedagogy theories is considerable, to judge from this post. I won’t bother to categorize them. But you are making a simple basic error when you say that the only reason someone might have problems learning Algebra is a lack of desire. That statement is laughably wrong, and unfortunately buys into a stereotype of what holds some people back in a given subject area.