There is one problem with this approach and that is if you do not understand the calculation then it is very, very hard to find errors. If you understand the calculations you have a much better chance of having a ballpark idea of what the answer should be even if you don’t do the full calculation.
Not a calculus example or particularly hard, but kinda funny. I went with my Dad to assist in buying a car. I was there so that he could ‘talk’ about it with someone. In other words, I was there as an excuse to get the salesman to go away so he could think.
Anyway, he decided to buy the car at an agreed upon price plus taxes. The sales guy starts adding up stuff and sez something like “Ok, the total is going to be $42,650*.” I looked at my Dad kinda cross eyed because I had added it up in my head and had a different price though I only estimated. So did my Dad. My Dad said, “No, that isn’t the price. The price is $41,451.27.”
The sales guy did a double take and then said “Well, the calculator sez 42,650”. To which my Dad replied “Well, my PhD in math says you did the calculation wrong. Add it up again.”
So the sales guy adds it up a second time and, lo and behold, came up with a different and lower price that matched ours.
The point of this little story being that, if you don’t understand the numbers and how they were arrived at, you won’t know when you are being taken for a ride.
Advanced Algebra does a poor job of doing that IMO. Like countless students before me, I asked my Algebra II teacher why we needed to learn the more arcane points at all. He replied like they always do that it was to teach us how to think logically. If that is the case, why not just teach Logic directly instead? Logic courses already exist at the college level and you could create a multi-disciplinary and interesting one for high school students rather easily.
I believe Stats does a much better job of teaching logical thinking and teaching how math in general can help people understand the world better. In my experience, people that aren’t especially mathematically inclined can memorize the concepts necessary to pass Algebra II test by test but they don’t usually truly learn anything from it long-term. Statistics is not like that. You may forget some of the formulas but most people retain many of the larger concepts which is the main goal.
There is nothing holy or even special about advanced Algebra even within math and science education. It was simply picked as a core subject by committee at some point and managed to hang on through tradition and inertia. It can and should be replaced as a near universal requirement.
To get the answer to that, all you have to do is look at a couple of Algebra II high school textbooks. They are very standardized in terms of content across the U.S. and that is part of the problem. Textbook publishers indirectly control the curriculum and they have to remain very conservative to sell their version because the administrators they are trying to sell them to across the country already know what an Algebra II class consists of with very little variation. The publishers can compete on physical quality, a little on style but very little on core content. Combine those expectations with the rise of standardized testing and there is not much room for deviation from the norm whether you are in California or Maine.
That said, I took it myself and couldn’t tell you what it was about at all so that was time really well spent. It was something about moving letters and some numbers around in increasingly more complicated patterns for mysterious reasons. We never knew what those letters and numbers actually represented or even could represent and the teacher got offended if you asked that question. From what high school students tell me today, it is exactly the same in 2016 Massachusetts as it was in 1991 Louisiana.
I am not bad at math and I did quite well in all of the undergraduate and graduate Statistics classes I took but advanced high school Algebra still escapes me because it is taught in a way that is a really bad fit for some learning styles.
Here is an example of typical curriculum. Many of those concepts could be useful if they were taught in a way that balanced theory and application but the classes tend to be so abstract that students don’t even realize what anything they are learning could be used for.
I don’t disagree with that. It’s a problem with math education in general (especially HS level), that they tend to focus on rules and specific ways of doing things instead of abstract principles. It doesn’t have to be that way.
I think it’s actually gotten worse in that online grading tools are common, and are frequently less lenient than a human in how they accept answers. If it expects x+1 and you’re graded wrong because you entered 1+x, then it has done you a double disservice–not only has it given you the wrong score, but it has effectively taught you that the two expressions are different. It would be no surprise at all for someone to walk away from this with an utterly wrongheaded view of math.
At any rate, I’m not trying to defend algebra over stats. Stats might well be the better choice at that level. What I dislike is that math (and to a lesser extent, science) has to defend its existence on practical grounds, while more humanities-like subjects get a pass. The whole notion of “we should teach people how to do their taxes, not solve the quadratic equation” is as stupid as “we should teach people how to write resume cover letters, not analyze the symbolism in poetry” (not trying to say that you’re claiming this, just that the attitude does exist).
That’s really the problem. High school graduation requirements are very close to four year college entrance requirements - at least here in Minnesota. Most colleges require at least two years of a foreign language, and Minnesota doesn’t. And colleges will want to see four years of Science, Minnesota wants 3 for graduation. Colleges will want you to pass Algebra II (and maybe Trig and Pre-Calc), Minnesota wants you to pass Algebra II.
When I graduated from high school 35 years ago, you could graduate by taking Basic Math. That was frankly all some of my high school peers could handle. You could graduate having written no more than a single three page essay - no research paper - and having completed a single novel. And frankly, that’s all you need if you are going to go to trade school and become a manicurist or welder or a dental hygienist or work childcare or construction (most people I know who work construction picked up a lot more math on the job than they ever bothered to take in high school).
Now I have a trade school bound kid who has had to get through The Tempest and Algebra II in order to graduate. They’ll make him take an applied Algebra + Trig in trade school for what he wants to do. He’s been able to do it - he struggles, but he can do it - but I can’t really tell him why he is reading The Tempest or learning log functions - other than “get through it to graduate.” “Just in case you want to go to college” really isn’t a good reason.
I absolutely hated Algebra II, even more than I hated school in general. I just squeaked by with a D-, and that was with tutoring every day after school. I should have never taken it, since I ot a C- in Algebra I and got worse as the class went on. As an undergraduate the class I took for the one math credit I needed was more of a logic class, and I had no problem with it. In grad school I took a statistics class, which made me very skeptical every time someone says statistics prove anything when it comes to education. I a A-B student with minimal effort in everything but math. I had no trouble with the type of math I use in day to day life.
Our oldest son took Calculus B/C as a junior and got a 5 on the AP exam. He’s taken Multivariate, Linear Algebra, and political/sports statisics as a senior. All that, and he’s planning on majoring in film next year. The youngest is taking Algebra II as a sophomore and has made A’s both quarters. Obviously they didn’t get my math gene; they’ve succeeded in math despite seeing school as a necessary nuisance to be endured.
Frankly teach critical thinking. Teach a course for non-college bound kids that is a year long and has a little applied stats and a little logic and a little argumentation and a little source evaluation.
I learned how to think deconstructing literature. Feminist deconstruction, Freudian, Marxist - using this lens, pull this apart and communicate it in an eight to ten page paper. It was really useful for me, I’ve gotten a ton of mileage out of those skills. But its not going to be valuable to everyone. I’m betting that a lot of the people advocating for learning to think abstractly by doing advanced math would be driven nuts by having to do a feminist deconstruction of The Great Gatsby - yet its still abstract thinking.
Moreover, the more I deal with my own kids, the more I’m convinced that you can’t really teach people to think abstractly if they don’t have any innate talent for it - any more than I can be taught how to sing. Yeah, with practice and instruction, I could probably carry a tune without scaring the cat - but I’m never going to be good at it, my voice isn’t great and my ear isn’t either. (I have no talent at drawing either, but I can do a Freudian deconstruction).
Nowadays in Georgia, 18 of your 20 units are taken up by requirements of 4 math (including algebra II or its equivalent), 4 science, 4 English, and 3 social studies. When I last taught, a minimum of 26 units were required for graduation, and the choices beyond the required 18 were pretty well defined by program rules called pathways. It’s been a pile-on paradigm since the late 80s. Requirements for graduation just keep getting harder. I truly believe not all students need or should be subjected to classes which will make school seem like nothing more than an impossible task, and the algebra II requirement is one of those classes for many students.
If you’re wondering how 26 units fit into four years, we used a 4 x 4 block schedule. Four classes of 1.5 hours a day for one semester count as 4 units. This make 8 units a year, as compared to the six units possible a year when I went to school.
What immediately comes to mind is how populations get divided. First, there are those who didn’t pass the admissions test to the college of their choice.* Next those who can’t pass their freshman algebra and calculus, and therefore have to change degree programs.* And then those who somehow run out of steam during the upper years and have to drop out.* Then those who fail to graduate* and those who fail the board.*
Failing any of those hurdles can suck big time for some, while others just laugh themselves out of it. The only real failures are those who refused to try. Sounds cheesy but certain old school rules still apply.
NB: I made it through all the asterixes and still find myself fenced off from a lot of green pastures. I don’t think so much of things that don’t require algebra. It’s the things that do and catch me with my mind on food or sex that get my goat. It’s called being a professional.
Or you can realize that high school (all school for that matter) should be more about effective teaching rather than checking off little boxes. I did get into my first choice of undergraduate and graduate schools and I am classified as a senior consulting engineer (not just software but heavy manufacturing machinery as well) in one of the most prestigious companies in the world. I have fond memories of even my hardest college classes but I literally still have nightmares about Algebra II because I almost failed the second semester of my senior year in high school despite already having a full scholarship to a prestigious college already secured. That would have stopped me from graduating and derailed my life.
Granted, I wasn’t trying all that hard by that point but something is wrong when so many people that are eventually successful in the STEM fields in real life complain about those types of classes at the high school level.
The GRE (test to apply to graduate school after undergraduate completion) doesn’t even test for Algebra II concepts at all because they are considered irrelevant. That should tell you something.
Then can always propose an alternative means. Otherwise, I can’t see how you can be analytic without going through the rigors of college algebra and calculus.
I did propose alternate means. Statistics is just as good as both of those and is vastly more useful. Computer programming in a lower level language is another way. When your program won’t compile or run on the first try which almost always happens, you have to pick through it in excruciating detail. The next step is when it doesn’t give expected results or bugs pop up. You have to learn to think like a computer and step through every line and think through every possible scenario to figure out what is going wrong.
If you want to learn set theory, database programming usually using SQL is an excellent way to do that and you get instant feedback. Those types of skills happen to also be highly marketable skills as well but they are also grounded in math and logic.
The mistake that educational system makes most commonly is assuming that you have to learn the theories in an abstract sense before you can apply them. In fact, the opposite approach works much better for most people. Give them an assignment that requires certain theory and skills to complete and let them learn the knowledge to complete it themselves.
Our high school requires 3 years of math and their lowest is algebra 1. I too wonder about the kids who just dont get math.
I’m better off with taking a course in accounting, business math, and consumer math which we all need. Another would be a tech math covering issues like ohm’s law and math involving things like water pressure.
Engineer here and I’ve had to use calculus, diff eq, algebra, trig, and statistics professionally. Also have had to deal with non linear systems of equations.
Design work can require a wide range of math.
Truthfully the math isn’t much harder than conjugating verbs.
