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I feel like perhaps you don't know enough people from the entire range of human abilities to understand why requiring calculus might be going too far.
It should certainly be an option, and it should be a requirement for certain career paths, but making it a high school graduation requirement would just unnecessarily result in more people dropping out of school.
I’m certified in special education and spent two hours of my day today teaching an adult how to do subtraction. I’ve worked with kids with Down syndrome. I entirely believe that it would be possible for 95% of students, if given the appropriate support, to learn how to take a simple derivative and have some vague understanding of what they did. It just takes visuals, good use of real world examples and metaphor, and patience.
I have family working in Special Education, most of them with kids under 12, some through early adulthood. All your points are correct. But from what I know of US Education, most schools - or schools in certain states - will not receive appropriate support and the students will ultimately be hurt for it. Think of the implementation of Common Core in the mid 2010s.
Students with proper support and encouragement can accomplish amazing feats, but most students don't have the resources to do that on their own (or with limited support and instruction.)
Looking at the state of the US right now, calculus wouldn't be where I'd devote my energy.
Let me ask you this, do you know how to budget?
We over provision for higher level arithmetic but don’t teach fundamental arithmetic for living successfully in our society.
Budgeting and more probabilities/statistics are where I think it should be.
Both of those directly relate to improving your life.
And fucking Excel. Better yet teach budgeting and spreadsheet courses in one.
If people had stats, budgeting and excel it would be an incredible improvement.
Budgeting also only gets you so far in our dystopian age when you need 2 full time jobs to pay rent.
When I got to college, I had to take two math course, which I dreaded. Because I was a music major, one of the math classes had to be Acoustics. For the other, I was terrible at Algebra, and didn't want that dragging me down, so I chose Statistics, since I was interested in politics, and would learn about polls.
I actually liked the class a lot, and to this day I track political polls closely. But I'm not a person who just accepts raw numbers. I want to know the sample size, the margin of error, etc. I know when a candidate is cherry picking his data, or leaning on a partisan poll, etc. It's been very helpful through my life.
BTW, it was standard procedure for every music major to procrastinate on the Acoustics class until their senior year, and we got a cool math professor who was also a pretty decent amateur trumpet player. He didn't want to be the guy to destroy our graduation prospects in our senior year by flunking us all, so he made the class interesting and challenging but not really difficult.
I learned a LOT in that class, and later I ended up working in sales for an audiophile classical record company, and my knowledge of sound and acoustics from that class allowed me to weasel myself into an additional part-time job helping out at recording sessions, some of which went on to win Grammys.
So Statistics and Acoustics were the math that worked for me, and I posted elsewhere that Business Math is something that I have also used a LOT, but picked it all up mostly on my own. NOT ONCE, have I ever said "I wished I paid more attention in Algebra." Those two quarters of high school Algebra might have been the two most painful quarters of my educational career.
The emphasis on advanced math at the high school level is detrimental to many people. It instills a sense of failure and stupidity early on, reinforced by parents and teachers, and often develops a sense of hatred toward those who are good at it. People who struggle with advanced math would be far better served by teaching them Business Math. First week lesson: put up a pay stub, and start figuring out all the percentages of all the withholding on that paycheck. Every kid in that class will be riveted on the screen, even the thugs, who will want to know who FICA is, and why is he taking all their money?
Budgeting and filing taxes, please!
*And understanding credit card debt
"The most powerful force in the Universe is compound interest."
My final year of high school (not in the US) had a finance class that had recently been split off from one part of the "current events" class into it's own thing. We were taught how to budget and handle interest, loans, taxes, savings, ect...
Also a bunch of BS about how big corpos are great and awesome because the teacher made money on the stock market.
I do think it should be a standard class everywhere though, it's ridiculous to not teach that stuff.
I just posted a similar take, but used a lot more words. Yours was much more succinct.
I would include statistics. So much everyday information is presented using statistics, often in ways that are misleading or deceptive. A bit better understanding would make people harder to trick.
In terms of utility for the average person, statistics >>>>> calculus.
I work in an engineering field, and can count on one hand the number of times I've had to do an integral in the last year. But I run into glorified statistics problems virtually every day both in personal and professional situations.
Having to constantly remind people of error bars, statistical significance, and the difference between correlation and causation, it would have been nice if those things were hammered home more thoroughly in school.
In my fourth semester im Uni I could choose whether to take numerical analysis or probability theory.
Most students took numerical analysis, even if the exam typically had a 80% failure rate. (Yes, one of five successed)
It was a completely different with probability theory (Wahrscheinlichkeitsrechnung). Oh, having chosen it due to these reasons now I know why: The prof loved teaching and was really good at explaining.
Ultimately this shows, people have no idea about probabilities.
Edit: fixed the nunerical typo. No it was not about catholic nuns.
I don't think the question is what level math to end on, but rather how math is taught. I teach psych statistics at University and the average student does the math parts mostly fine (it's just algebra) but their critical thinking and application of the math is usually what is sorely lacking regardless of their ending math course. And in the real world where we do everything with computers, the application is 99% what matters.
I've had people in middle age who dropped out in 6th grade in Mexico do better than fresh-from-US-high school calculus experienced students, and that's not even taking into account this more recent COVID-survivors generation that feels like they skipped a year of education. It's very... grim.
Yep, critical thinking enhances all other intellectual pursuits. It is so easy to fail at the critical thinking stage and go down a blind hole pursuing something absolutely nonsensical because you didn't check your basic assumptions.
I would want kids to learn about the Monty Hall problem, do a little Bayesian analysis, etc. I think they could learn through trying to smuggle some lies into a paper and then peer reviewing each others papers and finding the flaws. Kids are way more creative than they are given credit for and they would find ways of sneaking things through we wouldn't ever consider. Making it adversarial would prepare them for interacting with the huxters and frauds that make up a huge amount of modern life.
Here, stochastics and statistics are the key student filters in psychology.
. . . the fundamental ideas about rates of change seem like they’re something that everyone human deserves to be exposed to.
People understand the idea of instantaneous speed intuitively. The trouble is giving it a rigorous mathematical foundation, and that's what calculus does. Take away the rigor, and you can teach the basic ideas to anyone with some exposure to algebra. 6th grade, maybe earlier. It's not particularly remarkable or even that useful for most people.
When you go into a college major that requires calculus, they tend to make you take it all over again no matter if you took it in high school or not.
Probability and statistics are far more important. We run into them constantly in daily life, and most people do not have a firm grounding in them.
Some other countries build up math skills a little differently. For instance, in Portugal, they teach a little bit of Algebra, a little bit of Geometry, and a little bit of Calculus every year.
In the U.S. the students focus on Algebra, one year, then Geometry the next, then Algebra again, and finally Calculus (if they did well in the previous math courses).
So, if a student transferred for their senior year of High School from the U.S. to Portugal, they would have a different experience compared to their peers. They would find all of the Algebra and Geometry sections very easy and be able to help tutor the other students, but then they would struggle with the Calculus portions and need help from the others.
I'm not sure how common this is among other european countries. I would be curious to know how math courses are taught in other countries.
As a Norwegian, focusing on one kind of math per year sounds absolutely bizarre. We did a bit of everything every year in the 90s at least, and I doubt it's changed. How do you not forget everything if you learn it one year just to not touch it again for years?
In college each group of subjects have a separate class, but doing that in high school sounds nuts.
Honestly that sounds much better
I don't think rates of change or approaching a limit are things that an average person would find useful. I do think that some sort of statistics should be a requirement though, especially applied statistics.
No, and while I took calc in high school, I did fantastically bad at it.
When my brother had to do some word problems for his business classes, they were talking about coming up with splitting supply chains between products I realized some uses for it.
I think there are better ways to show it's application than "if you are filling a pool and have two hoses, one that fills at x gallons and another that fills at y. How long would it take to fill with both hoses?"
For me, if they talked about using it for drag racing and comparing the time accelerating to top speed and time at top speed to complete a quarter mile the fastest, I might have cared.
It's certainly possible to make it easier to understand and relatable, but I'm just saying that as far as useful things to know for all students, I think calculus is at the bottom of the list. On the other hand, nearly every single person will encounter some sort of statistics in their daily lives, and it is important to know how to interpret them.
I agree. Stats, z-scores, and significance would be way more useful. If only to offset how easy it is to lie with statistics.
I would follow the guide laid out by Lockhart’s Lament. Basically, teach math as an art.
That dream aside, I wouldn’t mind aiming at statistics as a target, instead of calc… specifically to lessen the impact of people who lie using statistics, and also demonstrate that not ALL statistics are lies.
I'm on page 3 and already sold.
I think statistics is far more important for people to know than calculus.
Nah, I already know the odds. Each time I lose, the chance of winning the next time goes up! Never fails!
I mean, who needs both their kidneys?
I see anything higher than the algebras as STEM focused, and certainly calculus is in that category. I do like the problem solving that comes with such studies.. but I’d argue there are more important civics focused courses that should come first. Time is limited after all.
Graduating high schoolers are newly minted adult members of society and grade school should focus on ensuring they are ready for just that responsibility. I don’t think forcing calculus fits that model.
If you can’t solve differential equations by the 4th grade, are you even learning?
The people that tell you that you will never need it are the ones too stupid to understand it.
Math is a universal language. It is the most important thing to know. Even more than the local spoken language.
I agree... Simple way of putting it is that it just makes you smarter. The same way that solving puzzles as a kid (well, at any age) makes you smarter.
Maths is really just a series of puzzles. I think people mainly despise it at school when they haven't engaged enough with puzzles as a youngster.
I disagree with calculus being mandatory. Most students still won't need it and it will increase dropout rates.
But a pre-calculus course with calculus as an optional offering would sure be beneficial. Most highschoolers get their ass kicked by college calculus courses because the logic jump from even moderately complex algebra to differentials and integrals is fairly high. Problems become significantly more abstract with more ways to solve things rather than rigid solution paths. A good precalc class gets them strong on the trig identities and more complex algebra rules that they'll need moving on.
Why, in the name of all that is good and holy, should we require someone whose dream it is to be a carpenter, to take calculus to graduate high school? In what universe will that requirement be doing any good in their life? What will it serve other than a potential completely arbitrary barrier to simply graduating from high school? And a carpenter is actually far more mathematically inclined than most career paths people pursue.
Yes, learning calculus can be a revelation in mathematical beauty. But the same is true for a thousand potential fields of study. In terms of practical use to most people, they would all be equally frivolous. A case could be made that a thousand fields of study are something that people simply must be exposed to. I'm more in favor of letting people choose their own path. We shouldn't be piling on arbitrary barriers on to a diploma that is only meant to signify basic competence.
How do you even walk if you don't know calculus?
In order to change the degree so that it allows studying in many universities abroad (such as Germany), this would be needed:
It's based on my subjective impression of weaknesses in the few Americans studying in Germany that I know.
stochastics & probability
statistics.
If everyone understood statistics and probability, no one would gamble.
Sorry, but I can't see the justification for it. I'm on board with everyone else who's suggesting statistics, though.
Algebra 1, geometry 1, statistics 1
I feel like calculus should probably start to be introduced in like maybe late-elementary-age? Certainly before high-school-age. I don't think everyone needs a dedicated 1-year course on it, but some of the concepts are certainly useful and understandable at that age. Regardless of whether its from compulsory education or some alternative education process.
Sorry for the rant. I long story short, I agree with you.
The quadratic formula.
When we learned to use it in algebra, it was just rote memorization that made little sense. We knew there was a proof for it, but we were told it was beyond our level and to just wait. When we finally touched on it again in Calculus, it was little more than a footnote. Since we had developed better tools for finding roots already, we did little more than note its existence and solve the problems more generally. I don't think we got around to the real proof of the quadratic formula until later with Linear Algebra. Most people aren't going to get that far. Most people don't have any need to. The quadratic formula is a bit of a chicken and egg problem. You need upper level math skills to prove it, but we learn it early in order to practice algebraic skills to get to that level.
I just wish that we'd have been taught some of those calculus fundamentals and ideas earlier. It would have been like a light at the end of the tunnel. Maybe we wouldn't be ready to rigorously work through limits and integrals before all that algebra practice, but even a child can understand acceleration and its relationship to changes in velocity. We have so many documentaries about special relativity, general relativity, and quantum mechanics. Almost no one watching these documentaries can do that math, but we don't worry about that. Our society could benefit from everyone having more general knowledge about the very broad strokes of calculus, differential equations, statistics, and combinatorics long before we worry about teaching the mechanics of those maths to them. Not everyone needs to know HOW to do them, but everyone can be taught to appreciate WHAT they do and WHY they are important and a part of every facet of our lives.
even a child can understand acceleration and its relationship to changes in velocity
I remember trying to figure out if a specific infinite sequence converges or diverges because I was playing Sonic. I didn't have any algebra nor calculators that could handle precise calculations nor the terminology I refer to it as now, so I was just trying to guess by doing calculations by hand to see if it looked like it was plateauing to something or if it looks like it was gonna keep growing.
Not sure if its actually useful, but it was something I cared about regardless. Children should play with math more.
Depends on your level and what you want to do afterwards. Then again, I'm in a country that offers different levels of highschool.
Hmm. I think algebra 1, 2, intro stats, and geometry for core curriculum. Anything beyond like calculus(I took) as elective or college credit. It’s been years but I think I took stats over trig.
Personal finance should be taught but not at the expense of other maths.
Yeah, I feel like what you're asking for would have to be a more holistic thing, like the concept of rates of change could be covered in physics quite naturally alongside momentum/impulse.
I have a math minor so have taken through calc 3, abstract algebra, and quite a bit of matrix math. Like 90% of concepts from those classes wouldn't really help people in the world around them, not just "they won't use integrals daily" but rather they won't be taught in a way where they'll connect differentials to what they see in the world or news.
Now, Finland, they have the right idea. They're teaching classes on spotting disinformation and how to find trustable answers. That seems way more important to me and likely to have a higher impact.
What do you propose we cut in favor of calc?
edit: core class, because calc is already an elective
