While visiting my old high school’s website, I clicked on one of the math teachers’ links, and was surprised to read that first-year Algebra is much harder than what today’s teachers and parents went through.
Naturally, this made me curious. How is it harder? What has been added to the course content, and what, if anything has been omitted?
I didn’t do well in HS algebra, but managed to teach myself enough of it later on so as not to be completely clueless. It’s been a very long time, but the topics I remember covering were:
Factoring
Elementary manipulation of members and terms in solving equations
Linear equations
Quadratic equations
Application of quadratic formula, but not its derivation
Application of binomial theorem, but not its derivation
Rudimentary graphing.
There might have been others; as I say it’s been a long time and I didn’t do very well. But I’m curious about what today’s students are being taught. Has additional content been brought down from what used to be taught in the second year?
(Note, FWIW this is a California school, so replies from Californians are particularly desired, though others are welcome too.)
I’m an engineer, not a high-school math teacher, but I taught and tutored first-year college math, and I currently tutor middle school kids through an outreach program at my job.
Looking at the material covered by the middle schoolers I tutor, I think that students today are introduced to topics earlier than I was (I’m 45). So maybe that’s why it’s harder? Just my two cents.
Are you referring to run-of-the-mill students, or the ones who were particularly good at math? I’m around your age; even in our time, the better math students got to algebra much earlier than the rest of us…in middle school as you say. So maybe I should rephrase my question: Do first-year algebra courses today, regardless of whether they are being given in middle school or high school, cover substantially more material than they used to?
Well if Algebra 1 is still fundamentally Algebra 1, AND the teaching and representation of the fundamental ideas behind Algebra 1 have not degraded, but have in fact approved as teaching and math have both progressed to make things easier to get a grasp on, then it should be easier today.
When I was in high school, my first three years of math were elementary algebra, geometry and algebra/ trig. My children are now taking math A . which is a 3 semester course. As far as I can tell, the content is arranged differently than it was when I was in school - my daughter’s first year of Math A covered some topics that I didn’t get to until my second or third year of math. They need graphing calculators, which didn’t even exist when I was in high school, so I would imagine they are learning more complex graphing than I did… My daughter is not in an honors math course. (not sure about my son)
Topics do seem to be introduced somewhat earlier. I don’t remember encountering the idea of x standing for an unknown quantity until eighth grade. My children learned to solve equations starting in fifth grade. Their word problems were more difficult than I remember encountering in grade school, and they were taught to set up equations to solve them, something I wasn’t taught until high school. There were no honors classes in their grade school - it was a very small school, with ony one class per grade.
California mandates that Algebra I be taught in eighth grade - at least a year earlier than it used to be. Trouble is, the legislators who thought this was a really cool thing aren’t really acquainted with brain development, mathematics, adolescents, or education. Whether or not a student has the mental development to manipulate abstract concepts is pretty hit or miss at the age of 13. Maybe half of them actually have brains that have matured enough to grasp the concepts. The other half end up flunking their first time and taking it again their freshman year. Way to motivate the kids, guys.
As an ex-college prof, at the “output” end it is clear that the typical incoming college frosh is far, far less knowledgeable about Algebra than the frosh of years gone by. It was a quite noticable effect over the years. People had no idea at all what the definition of a logarithm* was, let alone anything of substance.
My opinion, is that the appearance of teaching Algebra has gone up, but the reality is that the standards have pretty much fallen off the face of the Earth.
America has been dumbing down its ElHi educational system for over 3 decades now.
*Logarithms are a sore point with me. My kids attended what is supposedly one of the best high schools in the state, but one of their Math teachers was completely unable to explain why the answer to a given question was so. She could recite from the answer key, but not explain it all. And it was an exceedingly easy problem. I also did summer programs for Math and Science teachers from urban schools, they were even scarier.
I think the calculators have a huge impact on algebra. They allow students to do a wider range of more complex calculations in less time, while having no real understanding of what they’re doing.
Or such is my experience.
At the middle school I attended, 8th graders took either algebra or pre-algebra. I skipped pre-algebra, so I’m not entirely sure what it involved, but I know there was at least some math and abstract manipulation, and no one seemed to have much trouble with it.
Can I ask over what period of time you noticed these changes? I’m asking because I’ve noticed some other differences between my high school and my children’s. One is that not nearly every student in my high school took algebra. If you weren’t going to be able to handle Regent’s algebra, you took business math or consumer math or accounting as your math class. Now, in my son’s high school, everyone takes algebra . Of course, not everyone is taking Regent’s algebra- those students who in my day would have taken business math are now taking non-Regents algebra, a concept that didn’t when I was going to high school. Same thing with biology. Another difference is that most of those business math students in my high school weren’t expecting to go to college. The few who did were expecting to go to a community college and get an associate’s degree in something practical at most. Most of them expected to go to trade schools or go to work immediately after high school. Now it seems as if every kid who manages to graduate from high school, even if it’s by the skin of the teeth is expected to attend a four year college. It would be taken as an insult if someone were to suggest that Junior might be better off going to a trade school and learning to repair cars rather than going to a four year colege.
I wonder how much of the perceived change in standards is really a change in students. Perhaps there would be less of a perceived difference if you were able to separate out those students who would have been college bound 20 or 30 years ago from those who would have gone to a vocational program back then.
My recollection of public school math is that, except in geometry, there was little emphasis on proving theorems. And that was too bad; without that, algebra seemed like a bag of unrelated tricks. Years later, I started studying an old college level algebra book of my father’s, got to mathematical induction and sequences, and thought, “this stuff is really interesting; if only my HS math courses could have been taught this way!”.
Not long after that, that book really helped me in studying for the GRE, and I raised my math score by about 150 points.
I was never a prof. but I was a TA (teaching assistant) at my university in some basic engineering courses. I used to give a simple algebra exercise as the first quiz of the semester. I would say about one third of the incoming freshman could solve it. It was a pretty good indicator of who would actually be obtaining an engineering degree. Needless to say, if you can’t do basic algebra, you’re in deep trouble in an engineering program.
I remember learning that x stood for something and how to solve for x in the 10th grade. We continued with quadratic equations, binomial equations and so on in 6th and up. I took algebra in 9th grade, geometry in 10th, trig in 11th and college credit algebra in 12th (we had a choice between algebra and calculus).
Applied Math degree holder here with a son who is currently in Algebra 2 as a junior. I’m a product of California public and parochial schools and my son is a product of parochial school up to 8th grade and then public high school from 9th grade onward.
My first discovery about the current “difficulty” of Algebra, Geometry, Algebra 2, and Trig/Pre-Calc is that it came down to the TEACHERS. The teachers I had back in the late 70s-early 80s actually knew how to prove theorems, not just recite them. The teachers that my son has/had don’t have enough background and time to thoroughly explain theorems, and the examples used are usually too simplistic at times. The current books used also seem to have more errors in them and the teacher will not point them out either, which confuses the students even more. I spend time with my son on some problems (especially word problems), and I have to get through his defeatist attitude to show him that it’s not as hard as everyone thinks it is. Sometimes he becomes enlightened, other times he just complains how tired he is and wants to go to sleep…at 9 pm!
I actually enjoyed my Geometry class that was taught by a crusty old WWII vet by the name of Mr. Critchlow…He was a pilot who always politely said “Please talk to towards my right ear, because my left ear spent 4 years next to an engine!” He took great interest of teaching the subject to us, and used some of his piloting knowledge to apply some of the theorems, giving us some appreciation towards the subject. I aced virtually all of his quizzes and always got A’s on his tests. I think there are no more teachers of this breed in our school system, and that’s one of the reasons why our kids of today don’t have much interest/knowledge in math.
Another thing that irks me about current testing in math…the tests are prepared by the school board/admin and not by the teachers. Guess what happens when a teacher doesn’t prepare/teach the entire subject matter that will be covered on the test that admin prepared? Yep, kids that finish half the test and getting crappy grades. Too much disconnect here, nowadays. :mad:
I’ve done tutoring for high school math. I think the use of graphing calculators has hurt. The students are being exposed to more statistics and graphing. However, I don’t believe many students understand the concepts behind them.
Just to point out that logarithms are not the best indicator of whether or not things have been dumbed down: as you’ll undoubtedly know (but should be pointed out for the non-mathmos) logarithms are a fundamental concept for operations involving a slide rule, so they had to be taught to students when the only means of performing tough calculations was a slide rule.
They can quite reasonably be moved up to a higher level of the course when students have calculators to perform difficult calculations. A better landmark would be techniques such as long division of polynomials, where your graphing calculator doesn’t help much. They fit very naturally into Calc I, IMHO.
It’s not just high school. As a grad student I was the TA for a great big required “engineering under uncertainty” class (probability and statistics for civil engineers). This was right about when those megalith HP programmable graphing super calculators were hitting the market (early 90s).
I used to love one particular problem which came down to finding the area of a chunk of a triangle. A right triangle with known width and area. Get the height, use your really basic geometry (area of similar triangles) and zip zip zip you’re done in two minutes. You barely need a pencil and paper.
I was astounded at the number of bright sophomores who would come to my office for help. Their approach to the problem was to find the coordinates of the endpoints of the hypotenuse (argh!), which many would get wrong (sheesh), then derive the equation of the line (gnaaaa) followed by integrating that (nononono) to find the area.
All carried out to something like 12 significant digits on a $200 calculator.
You can certainly DO it that way but WHY would you want to? The problem as I see it is that people are being given an enormous toolbox of mathematical methods and incredible technology (those calculators) but they don’t understand how a lot of it works and they try so hard to use that stuff that they forget to just look for the simplest approach. In “Surely You’re Joking Mr. Feynman” Richard Feynman talks about some similar experiences when he was a college student (pre-WW2) so this isn’t really a recent phenomenon. I still enjoy simplifying things and rounding and making assumptions and working out complicated problems on the back on an envelope, so to speak.
I graduated from high school in 1969, and my youngest daughter is a senior in high school this year. We’ve both taken the hardest math classes. I think it is tougher today. They did matrix multiplication last year, which I don’t remember doing until college. They did polynomial division in a much different way than we did. We did interpolation, which they of course don’t have to do with calculators. The book has a lot of “just graph this to see what you get” problems which I agree are stupid, but her class didn’t get assigned all that many.
I’ve also noticed that the homework problems are not designed to be come out as fairly simple numbers, because with calculators it doesn’t matter. I find that it is harder to tell if you’ve gotten off course.
This isn’t algebra 1, really, which she also took in junior high, and I took in 7th grade. Last year she had pre-Calc, and there was a lot more on polar coordinates and geometric transformations which I didn’t get until AP calculus. Since she has the Voyager family calculus deficiency gene, she’s taking AP Statistics this year, which is much more useful for what she wants to major in.
As for whether they learn it - that is dubious. Teachers aren’t as good as they used to be (lack of pay, no doubt.) My other daughter got a 4 on the AP Calculus test, but chose not to place it, which she was grateful for, since the first quarter at U of Chicago went past her AP Class.
I’ve only browsed through this thread, since I’m not a teacher, and probably can’t provide the necessary perspective. But here’s my 2 cents. I’m a junior in a California high school, just starting Pre-Calculus, having taken Pre-Algebra, Algebra 1-2, Geometry, and Algebra 3-4 Honors (With Trig) in that order.
The idea that algebra is “just a bag of tricks” seems to be the general feeling of most kids, and the fact that we all have shiny new TI-83s and TI-84s probably doesn’t help. I remember one day last year someone was looking through some flash cards, trying to memorize something, when she ran over to me saying, “Is this right?!” I think the card said
log[sub]b[/sub]x - log[sub]b[/sub]y = [sup]x[/sup]/[sub]y[/sub]
I said something along the lines of "Well, yes it’s right, but it would be a lot easier to derive it. Then you’d only have about 1 flashcard instead of 10 or so. And she said something like, “No! I don’t know how to derive anything! I don’t have time to derive anything! I have a test next period!” :rolleyes: :rolleyes: :rolleyes:
I was particularly surprised when we were told that the TI-83 can only do logs in base 10 and base e, but you could use this formula to find a log in an arbitrary base:
log[sub]a[/sub]b = (log[sub]x[/sub]b)/(log[sub]x[/sub]a)
where x is any number except the ones that don’t work, like 0 and 1.
I figured they would show us how to derive it if they could, but it was probably much too complicated. Just like we all know that the area of a circle is pi®[sup]2[/sup] but we won’t know why until much later. Turns out I could do it without help, using only what I knew from class, in about 1 minute. But I think everyone else in class just saw it as magic. Magic that they had to memorize. Makes me sad, it does.
:mad: