18 ms·

Do the math: too much calculus? (2012)

104 points by carlosgg 10y ago

spike021 10y ago
I think this is just another example of standardized testing (specifically AP course exams in this story) almost creating a niche-like education for students. Maybe that's an odd way to put it, but some students wind up spending hours and hours studying AP/SAT/ACT/etc.-specific materials (literally 500+ page textbooks for each type of standardized test). Sure this method of studying helps them achieve great scores on the standardized tests, but my impression as a recent student in academia (finishing college now) is that this creates an issue where students are almost "overly" dependent on more specialized areas. I don't know if it's an issue of not having time to learn "street smarts" (so to speak) because standardized studying takes up all their time, or if it's something else. But there's definitely some improvement that can be made.
- smallnamespace 10y ago
  A pattern that you see almost everywhere is: imperfect incentives/objectives + strong competition = perverse outcomes Over the last decades, we're increasingly seeing this in academia (publish or perish), the corporate world (boost the bottom-line at the expense of everything else), politics (win the election even if you have to burn your own party to the ground), and many other walks of life, as the world has gotten more competitive. To fix this, you can either fix the objective (which may be very difficult), or you can reduce the level of competition.
  - Retric 10y ago
    This also causes some odd results. If the MIT/Caltech/Harvard/etc student bodies are generally less intelligent than you might expect and often boring people. You see the same thing at most national level high school competitions. The surface details may seem very diverse, but underneath that they tend to have a lot in common.
  - baq 10y ago
    > imperfect incentives/objectives + strong competition = perverse outcomes if you think about it, it's just a rephrasing of 'survival of the fittest'.
hprotagonist 10y ago
>This is not true for students who take Advanced Placement calculus I hated BC calc. Bad teacher. I learned the same material at the same time in AP Physics, and loved it. I think a related issue to the author's point, though, is that students are taught in a way that implicitly suggests that calculus is the pinnacle of mathematics. Which is blatantly wrong.
- baldfat 10y ago
  I have had the pleasure of pushing many female students into taking calculus but my success rate was about 50%. Each one of my children's friends LOVED STEM and were thinking about going into engineering. All of them said that they were the only girl in the class and were intimidated and the other said it was REALLY boring. Calculus should be easily made fun just exploring the actual concepts in real life similar to a good physical science teacher.
  - hprotagonist 10y ago
    As it happens, my teacher was female. She was a firm believer in "teach to the test", "do these exercises and follow the magic rules of differentiation, shut up and just do it don't ask why it works". That's not my jam.
    - shostack 10y ago
      When teachers' performance is judged based on test scores, it makes sense where that sort of unfortunate mentality comes from. What gets measured gets managed.
    - baldfat 10y ago
      I think many "Math" people get the mechanics and like the functions and miss out on the application. I remember trying to figure out how the Cha Cha Slide could be a decent Calculus problem. How does the song change the pattern?
baldfat 10y ago
I think we teach Calculus wrong in a deep fundamental way. We should incorporate more high math concepts earlier into education BUT people are always afraid of anything past 4th grade math! "A minority of students then wend their way through geometry, trigonometry and, finally, calculus, which is considered the pinnacle of high-school-level math. But this progression actually “has nothing to do with how people think, how children grow and learn, or how mathematics is built,” says pioneering math educator and curriculum designer Maria Droujkova." ... "“Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than “little manipulations of numbers,” as she puts it. It’s akin to budding filmmakers learning first about costumes, lighting and other technical aspects, rather than about crafting meaningful stories." http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/ http://www.theatlantic.com/education/archive/2014/03/5-year-... Why is calculus the pinnacle when I actually feel that pre-calculus would be better students even make it into middle school? These concepts of patterns and structures is the foundation of not only math but also of language and reading.
- CalChris 10y ago
  Berkeley EECS grad. I didn't go the device physics route but I used calculus about twice in 4 years except in physics classes.
  - supahfly_remix 10y ago
    Circuit analysis turns differential equations into algebra, so you might have been using calculus more than you realize.
    - CalChris 10y ago
      Yeah, I might have been guilty of some hyperbole. I remember a nasty integral on the Linear Systems midterm. It probably would have taken an hour or so if I didn't make any mistakes; but you could transform it and solve it as a linear equation in 10 seconds. That was the point of the class, that there area two kinds of problems, linear problems and problems you can't solve. I'm not in the least suggesting that an engineer shouldn't go through three semesters of Calculus, Linear Algebra+DiffEq and a Discrete Math course.
Yhippa 10y ago
> Too many students experience a secondary-school calculus course that drills on the techniques and procedure that will enable them to successfully answer standard problems but are never challenged to encounter and understand the conceptual foundations of calculus,” he said. As I get older I've realized that I learn better when I understand the concepts about things. I then get excited to apply what I've learned into practice. I think that if I learned about "why calculus", what problems did it solved at the time it came around, and how we use it in practice I would have been able to grok it quicker and deeper. I didn't end up getting above a 3 on the AP calculus exam. I did however get a 5 on the US government exam. I had a teacher who did a great job throughout the course having us work out essays and testing arguments in class to help solidify our knowledge as opposed to memorizing facts.
- hood_syntax 10y ago
  Conceptual background is so important for understanding something. I was lucky to have a great teacher for BC calc, and I got an easy 5 on the exam. The concepts make sense, they are consistent, and they build on each other, so with the right progression of ideas students can get a lot farther than they would otherwise.
castratikron 10y ago
AP Calculus was the most useful course I took in high school. It was rigorous, and I scored high enough on the exam to get two free college classes and start freshman year at the sophomore level, which left room at the end of my college career for more specialized courses. There should be some vetting process for the students who want to take AP classes. A lot of the students in my AP Calculus class had no business being there. One guy got a 4 out of 50 on a test once, another guy didn't even take the AP exam, and another guy just put his head down on the desk during the AP exam. But this is really a problem for individual schools and has nothing to do with College Board.
- mhb 10y ago
  Isn't it kind of weird that a college won't let you take whatever math class you want?
  - gnarbarian 10y ago
    No. Tuition only covers a portion of a student's cost to a university. I think it's around 30% at mine. 1) If a class is way over your head the university is wasting money having you there. 2) Having stragglers in the class that are far behind slows down the whole class. 3) Every seat that is filled with a student who is not ready for the class is a seat that can't be filled with a student who could actually pass it.
    - jessriedel 10y ago
      > Tuition only covers a portion of a student's cost to a university. I think it's around 30% at mine. Where are you getting that from? Are you just dividing the tuition revenue by the university's budget? Because the university is there for a lot more than educating undergrads.
      - gnarbarian 10y ago
        Most universities aren't primarily focused on research.[1] They are there to educate students. Research does occur at every university, but looking at their research budget as a fraction of their whole budget will show what sort of institution it is. Even if you subtract the research budget out first you will still arrive at a figure less than 50% for most universities. [1] https://en.wikipedia.org/wiki/Carnegie_Classification_of_Institutions_of_Higher_Education https://en.wikipedia.org/wiki/Carnegie_Classification_of_Ins... edit: https://nces.ed.gov/fastfacts/display.asp?id=75 https://nces.ed.gov/fastfacts/display.asp?id=75
gnarbarian 10y ago
It sounds like the issue is there are too many high schools offering classes in AP calculus which don't do a decent job of actually teaching the concepts to the students. Having a good teacher for these classes makes all the difference. Additional testing at the college level would help assess a student's ability level but it does not solve the problem of time wasted when the course must be retaken. Another issue this brings up is it causes a larger gap to form between trigonometry and calculus 2. Where memorization of hundreds of trig identities builds on top of the identities learned in trig. If a student takes Trig -> pre-calc->AP calc -> Calc 1 -> Calc 2 They will have a much harder time in Calc 2. That is what happened to me. I had to take Calc 2 a couple times before I got it sorted out. http://www.ansep.net/programmatic-outcomes/statistical-data http://www.ansep.net/programmatic-outcomes/statistical-data I think the answer is to get more qualified teachers in the schools teaching the AP level classes. There is a program in Alaska called ANSEP (Alaska Native Science and Engineering Program) who have this figured out. Middle school students who enter the program come to the university during the summer time and take math classes from university professors. Many of the students are completely finished with their math track (for engineering programs) before they start university their freshmen year. These kids are all the way through partial differential equations at 17. The graduation rate for ANSEP students who enter engineering programs is around double the national average. There are many other important aspects to it. Including mandatory study sessions and living with other students who are also taking the same classes. Students are also given well paid internships in the industry during the summer time. When looked at as a whole, there is no doubt that they have figured out a far more effective means for getting students through STEM programs and jobs once they graduate. Full Disclosure: I am good friends with a few of the people running this program.
jimmydddd 10y ago
For lower level math, my kids seemed to revisit the same subjects over and over from k-9th grades. For example, mean, median and mode, year after year. The first few years it was just robotic problem solving. After a few years, they started to get a broader understanding. Maybe some people need to take several passes through the higher level stuff too.I guess the problem is that there is not enough time.
- cgag 10y ago
  I believe there's decent support for the idea that you could just teach them nothing until they reached the age of understanding, and theyd pick it up pretty much just as quick without the years of rote.
  - dredmorbius 10y ago
    Source?
    - abecedarius 10y ago
      A U.S. elementary school principal in the 20th C. got his school to try leaving out any math in the first three grades (iirc), and the kids reportedly came out fine. I can't remember his name, and I'd like to be reminded too.
      - asimeqi 10y ago
        It was L. P. Benezet, superintendent of schools in Manchester, New Hampshire, who experimented with teaching no math in the first five grades in the early 1930s. See for example https://www.psychologytoday.com/blog/freedom-learn/201003/when-less-is-more-the-case-teaching-less-math-in-school https://www.psychologytoday.com/blog/freedom-learn/201003/wh...
        abecedarius 10y ago
        Thanks! There's also his own report: http://www.inference.phy.cam.ac.uk/sanjoy/benezet/1.html http://www.inference.phy.cam.ac.uk/sanjoy/benezet/1.html
hanoz 10y ago
I remember once when when driving I was pleased when it occurred to me that I should be seeking to minimise the third derivative of distance with respect to time for the comfort of my passengers. That was the sum total of calculus's contribution to my adult life. Not sure it was worth it.
- ThrustVectoring 10y ago
  Curves in freeways are designed that way as well - they don't just start turning suddenly, but gradually transition into and out of the curvature.
  - 0xCMP 10y ago
    Good ones at least.
    - ThrustVectoring 10y ago
      Don't quote me on this since I don't remember why I believe it, but I think it's part of the legally mandated design requirements for the US interstate system.
      - 0xCMP 10y ago
        No I'm sure there are requirements, but sometimes there are also space or budget constraints (I assume) since some curves are definitely safer and easier than others for the same highway (I-95 for me). I wished they were more consistent
  - LgWoodenBadger 10y ago
    One of my calculus classes in college involved calculating the required spline for a curve of a theoretical portion of interstate in order to meet the interstate requirements/regulations. It was pretty interesting.
ThatGeoGuy 10y ago
I am not sure how much of this is specific to the American education system, as I am Canadian, and have not experienced it, but my own take after going through high school to my current graduate studies is that 1) We teach algebra, geometry, and their relationship wrong in many ways 2) We don't teach enough calculus at lower levels of mathematics, instead opting to prefer a "poor man's" algebra. I'm sure there's too much to write on for one HN comment, but a lot of 1) comes down to how we teach "solve for x" type problems which are more interested in re-ordering and simplifying equations than we are about giving people a fundamental understanding of what our relations and equations actually mean. Often I notice that many students couldn't graph or draw or even begin to reason about what they're trying to solve for. If you give someone a graph and ask them to find the maximum point of a line on a graph, they can very easily point it out. The next step is doing it numerically, but a lot of times the relationship between what we can determine visually and what we can determine computationally is lost. Which is sort of what prompts 2). You can't learn Calculus without learning limits, and Calculus places explicit meanings to relations that we can graph, and how those relations change according to specific variables or unknowns. Finding the maximum of a relation using Calculus is much easier than strictly using algebra for this reason, because we can make the relationships between a curve and it's derivative explicit. Most students cringe at the thought of Calculus because we place it on a pedestal and students just assume Calculus is the peak of mathematics, but we really need to introduce Calculus as more "normal" math earlier on, IMO. I'm talking basic tasks like derivatives finding the equation of a line tangent to a curve, or finding the limit as we approach a point on a curve, or even just (re)factoring equations so that we can plot them easier and / or take their derivative easier. In almost every example I can think of where I learned Calculus, it was easier than building intuition for Algebra because there is less robotic crunching and more reasoning about what specific problems mean. What does it mean to take the derivative? Why are volume and area and perimeter related? I know in many ways it seems like I'm just advocating for thinking spatially or visually about mathematical problems, but that's part of what (I think) makes Calculus more approachable. You can get a lot farther with a basic understanding of Algebra and a very small amount of Calculus than you can with just Algebra alone. Should our curriculum be entirely Calculus? No, that's ridiculous; however we should ease up on the systems-of-equations type problems and obtuse word problems that require students to produce or remember all manner of expressions and formulae, and instead focus on building that first intuition of how we can use Calculus as a tool for problems that are much, much harder without. I expect that it would give Calculus a more realistic reputation, and would probably put off less students who are already giving up on math.
anu7df 10y ago
I have no idea why calculus gets such exalted status in the annals of highschool nightmares. No I am not bragging, humble or otherwise. I found calculus to be simple neat and logical, and not so mind bending. Linear algebra on the other hand was far less trivial; not the mechanics, the aha moment bit. Geometry.. that for me was genuine mind bending pleasure pain.
maxerickson 10y ago
There should a high school class on numeracy, followed by statistics. Wrote learning of algebra does not necessarily lead to numeracy. Sweating over the students that are (mostly) ready for calculus and college is missing the forest.
AstralStorm 10y ago
Personally, I have always wondered about the prominent position trigonometry and geometry get. As it is typically taught, it is the most useless thing, the time spent there would be much better spent on algebra, calculus. From basic theorem of algebra flows trigonometry and a lot of number theory, while from calculus with some linear algebra flows geometry. The rationale is probably history.
- hossbeast 10y ago
  You can't possibly understand calculus without first understanding trigonometry and, to a lesser extent, geometry
- mmmpop 10y ago
  So I guess you've never framed up a house have you? Basic trig is so incredibly important to anyone that ever desires to build a thing. Don't you remember learning the unit circle in Calc 2??
- jandrese 10y ago
  IMHO, high school math misses the one course that would be most useful to the students. Statistics. While there are a LOT of statistical models that aren't especially useful in every day life, knowing the most common ones can be a big help. Of all the math classes I took in college the only one I use on a regular basis is statistics. Thinking back on high school I spent a whole lot of time on basically calculus prep and in the end I hardly ever take a derivative or integral of anything.
- thesimpsons1022 10y ago
  trig has seemed extremely important to me. specifically using sin and cos. helps a lot in physics as well as computer graphics and many other applications.
- gozur88 10y ago
  Trig is a pretty useful subject for the skilled trades. Framing a house, as mmmpop notes, but also things like plumbing and cement work. If you don't go on to college, or you don't major in engineering or math, calculus is pretty useless. I've never once used calculus outside work, but I have used trig.
ivan_ah 10y ago
I think linear algebra is a much more valuable and general-purpose math tool. I'm not sure how feasible it would be to teach linear algebra topics "properly" to high schoolers, but even a half-assed linear algebra would be more useful than calculus. The connections to geometry, general systems thinking, formal math methods, and countless applications of linear algebra are just the kind of thing students need to get them interested in learning more math. Anyone interested in LA an its applications should check out my upcoming book, the No bullshit guide to linear algebra availale on pre-order here https://gum.co/noBSLA https://gum.co/noBSLA (it's almost finished; just beefing up the problem sections).
- FT_intern 10y ago
  Thinking in matrices and vectors is much more complicated and abstract than calculus.
  - microDude 10y ago
    I disagree. Especially, if you combine it with a graphical tool like Matlab or the Python libs. Take a lower triangle matrix multiply it by a vector, and "boom" you just explained causal convolution. Wash and repeat for several other topics.
- lqdc13 10y ago
  They taught me about matrices and vectors and common matrix/vector operatoins in pre-calculus and even earlier in a shitty public school. Calculus needs to happen in high school so that people are more prepared to take it again in college. For me, calculus 2 was the hardest class in college. This is coming from someone who went to a math grad school. Some people need to study way more for calculus than for any other class in high school or college.
  - sidusknight 10y ago
    I'm guessing you did applied math in grad school? I think very few (pure) math majors had any issue with calc.
dver23 10y ago
Terrible teachers making difficult subjects impossible for students. It's the same experience for my kids that I recall from my time. Industry can pay more, and outside the rare person who has come to teaching to give back, the instructors are those who can't. Even when I was at Cal Poly, the only instructors in eng/sci who I recall being helpful were those from JPI that would come in and teach a class or two. One taught us three classes worth of material in course when he realized how much we had missed from our previous instructors.
euroclydon 10y ago
I have a 5th and a 6th grader. I'm going to buy them an educational copy of Maple soon, so they can have a beautiful platform to plot 2-D and 3-D graphs, solve equations, do derivatives and integrals, and factor algebraic expressions. I want them to do all this in high school, whether or not they take Calculus, so they can get an intuition for it. Then they can memorize the trig rules in college.
- Oletros 10y ago
  AS an Spanish not knowing the US education system, at what age one kid goes to 5th grade?
  - euroclydon 10y ago
    My 5th grader is currently 10 years old.
  - colechristensen 10y ago
    The median age in 1st grade is 6.
  - Oletros 10y ago
    Thanks to both of you
- ivan_ah 10y ago
  You should also consider SymPy, available as a webapp http://live.sympy.org/ http://live.sympy.org/ or in jupyter notebook form. Plotting function graphs is a bit harder (matplotlib) but overall I find SymPy to be much more logical and easy to grok than Maple's weird syntax. Here is a short tutorial on how to do basic math using SymPy: https://minireference.com/static/tutorials/sympy_tutorial.pdf https://minireference.com/static/tutorials/sympy_tutorial.pd...
  - euroclydon 10y ago
    I love Jupyter. I set up pydot in a notebook, and made high level functions to create and print trees and graphs. My 10 yr old picked it up quickly while we killed time at his brother's swim meet. But, I'd love for them to learn to use a CAS engine. I appreciated, in college, being able to quickly factor a polynomial, solve an equation, and find derivatives.
strathmeyer 10y ago
Why did we take two years of Calculus in high school when there weren't any jobs when we came out of college?
- Retra 10y ago
  Those things have no relationship, and the expectation of one is a fault in your understanding of the world.
japhyr 10y ago
I have a 5 year old, and I was talking to some acquaintances who also have a kindergartener. The kids are learning math, by playing with manipulatives. It's great; they're learning mathematical concepts, and they have no idea anyone in the world doesn't like "math". They don't know what "math" is, they are just starting to discover that the world is made up of interesting but identifiable shapes, and that numbers have some really interesting patterns as well. These parents, however, don't see anything they recognize as "math". They're afraid their kid is going to fall behind, so they're giving their kid worksheets at home focusing on two-digit addition and subtraction. Their kid doesn't like math already, because at home math is just writing things on a piece of paper with little meaning. This is sad to me, because there are so many interesting things you can do with your kids at home to help them develop their mathematical understanding. Our educational issues with math start young, and they come from adults. I'm fortunate to work in a small high school where I get to treat each kid individually. It's wonderful to meet kids where they're at, and help them move forward and start to enjoy math again.
- ivan_ah 10y ago
  I don't have kids, but if I had a 5 year old, I'd sure get them some of these: https://en.wikipedia.org/wiki/Cuisenaire_rods https://en.wikipedia.org/wiki/Cuisenaire_rods I've often thought about inventing math toys, e.g., group theory for toddler: S_3 = triangle toy, S_4 square toy, etc. I'm sure there are many advanced math topics that could be turned into toys. No symbols, no equations, just toys, but subtly you're getting kids familiar with important math structures.
  - kol 10y ago
    Rubik's cube can be used to teach group theory, see http://geometer.org/rubik/group.pdf http://geometer.org/rubik/group.pdf or http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20Rubik's%20Cube.pdf http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20an...
Koshkin 10y ago
The best thing about learning calculus is that you see how everything you have learned so far - algebra, geometry, trigonometry - everything comes together and becomes equally useful and important. This is, in fact, what mathematics is all about.
waynecochran 10y ago
I think a Physics course taught with "Just in Time" Calculus would be fruitful for motivating the Calculus. Calculus was invented to do Physics.
abakker 10y ago
Sample size of one: I think I understand calculus, the needs of calculus, and how it is profoundly meaning to understanding things around me. Can I do calculus? no. I couldn't do calculus because I really couldn't do algebra. I think that if I had been taught the concept and relevance of calculus earlier, and only had to worry about the execution of calculus later, I would have been much better off. understanding what derivatives and integrals and trig functions mean should come before we are forced to execute their use, or learn proofs, or even really go beyond basic notation.
- yequalsx 10y ago
  I'm skeptical that a person can really understand why x^(2/3) is not differentiable at x=0 but has an absolute minimum at x=0 without understanding the algebra. I think one can get an idea of the concepts without knowing algebra but I doubt one can understand calculus without knowing algebra. We always teach the meaning of a given mathematical concept before doing using them or doing calculations. It's just that true understanding comes from the doing the calculation part.
  - rconti 10y ago
    Not the parent, but I had the same problem. I understood that kind of algebra but could never figure out how to simplify things well, which hamstrung me in Calculus. Also, they never bothered to explain the purpose of any of the equations in calculus.
krschultz 10y ago
I graduated with a mechanical engineering degree, meaning I took 3 semesters worth of Calc and 3 more semesters worth of differential equations. So I'm not one to hate on math or calculus specifically. But I think high schools should focus more on statistics. Statistics is relevant to more fields than calc. It matters a lot more in business and in all the jobs where you use calc you will also have to use statistics. It's also useful for citizens to understand statistics better - and educating citizens is truly one of the purposes of public education.
- lordnacho 10y ago
  I second that. Stats is so much more applicable, and it is also the area where common sense will fail you the most. Simple things like birthday paradoxes, Gambler's fallacy, Monty Hall problem, and so on are all contrary to common sense yet not taught in schools. And you need stats all the time. Every time someone says crime has never been worse, you need to know how to decide if you believe it. When someone says flying is safer than driving, you need to know how to compare. A lot of stats thinking is also not formulas, but qualitative things like "Am I comparing apples to oranges". Calculus by contrast is pretty common sensical. Acceleration is the derivative of velocity, you find the local extremes by finding the zeros of the derivative, the integral of a net shaped field is the rim... that all makes sense with some pictures. I also haven't come across it much in finance, apart from derivatives work, where funnily enough you can't do with high school calculus, but you need stochastic calculus. Which contains some quite statistics related concepts.
  - ghaff 10y ago
    I suspect calculus is less commonsensical to most people than you think :-) That said, I've argued the parent's point for years. Basic stats/probability/economic biases are really fundamental stuff and don't require a lot of complicated math if you don't want them to. There are some basic calculus concepts that are useful. Area under a curve. Basic derivatives (for finding maxima/minima, etc.). But the truth is that I've rarely used calculus outside an academic setting and I use at least basic statistics/probability thinking all the time.
CN7R 10y ago
Schools should adopt more dynamic curriculum, instead of a one size fits all, for individual students. Not everyone will grow up to be an engineer, but that doesn't mean those who have an interest in advanced mathematics should be held back. Encourage students to pursue their academic passions, and don't limit certain subjects depending on grade level. A personal anecdote: I didn't start liking math until Calc AB, as subjects beforehand were presented as dry and were relatively easily so I didn't individually pursue mathematics outside of school. But by that time, I was already a junior and as a result, I often wonder what better math teachers and a non static curriculum would've done for my education.
noobiemcfoob 10y ago
I feared calculus, right up until circumstances forced me to not only take calc in high school but to take the hardest level of calc my school offered (AP BC Calculus, counted as 2 semesters of college calculus). I was greatly benefited by an amazing teacher. I've always felt that calculus is one of the most important branches of mathematics for the simple insight it should hammer home time and again: An infinite amount of infinitely small things can and will add to infinity. All too often, humans fail to see the cumulative effect of an integral. Every branch of math has these types of insights built into them. Calc the power of instantaneous change. Geometry shows the beauty of a proof - and if you scratch a little bit, the shattering epiphany that triangles are literally the same everywhere, always and forever. The reality is that many of these concepts are simply absorbed into our culture, and we fail to realize it until someone explains that it took centuries of work to formulate the concept of 0. When it comes to education, I believe the struggle has far more to do with testing standards than the material. It's easy to teach to process, much harder to teach insight and further harder to test. In reality, none of this matters until your institution decides and commits to truly educating its students. tldr - None of this matters until your institution decides to truly educate its students.
madengr 10y ago
"Instead, students who took high school calculus often find they have to retake college calculus, or even pre-calculus. Many flee to the humanities." Well no sh!t. I find it funny that people who aced calc in high school get their ass handed to them in college.
WhitneyLand 10y ago
I accidentally took Combinatorics as a fill in and it has been surprisingly useful. The need to count things well comes up on a regular basis, not to mention helps with learning poker. Would recommend this over some caclulus for a lot of folks.
LouisSayers 10y ago
We're building out an online solution over the next year to help prepare final year high school students for exams - with an initial focus primarily on Maths. Would love to hear from people with thoughts around Math tools they feel would be valuable - especially taking into account current resources such as KhanAcademy, and how we could potentially provide something complementary. Either comment here, or hit me up - louis@connecteducation.com.au
danielmorozoff 10y ago
Maybe logic should be the first mathematical discipline to be taught in middle school. The notions of and / or and the representation of language in mathematical form? Before even starting on algebra
tnecniv 10y ago
I have a specific problems with the HS math curriculum I experienced (as opposed to the general problems with the whole math curriculum I had before my junior year of college). Mainly, there was just so much wasted time it was unbelievable. I actually cannot remember a single thing that I learned in Algebra II because I either relearned it in Pre-Calc or never used it again. Similarly, we spent the whole first semester of Calc BC redoing all of Calc AB. What a waste! If I could, I would replace all that wasted time with Linear Algebra (with proofs) and Probability (throw in statistics if you want). Calculus should stay, but teachers should motivate it better (this goes back to the general problems I alluded to). Those two classes are immensely useful and teach a whole different modes of thought compared to the regular pre-college curriculum.
jiaweihli 10y ago
I grew up 3 years ahead of my classmates in math. I found everything up until calculus to be easy but elegant - algebra, geometry, probability, trigonometry. Much like programming, there were often multiple ways of the solving the same problem, and deriving a formula was feasible and logical. In contrast, I found calculus to be mind-numbingly boring. It required memorizing and reciting formulas upon formulas. It was the first time I felt that math was tedious, and pushed me away from majoring in math in college.
hal9000xp 10y ago
I do recommend to read these articles: http://www.artofproblemsolving.com/articles/calculus-trap http://www.artofproblemsolving.com/articles/calculus-trap http://www.artofproblemsolving.com/articles/discrete-math http://www.artofproblemsolving.com/articles/discrete-math http://www.artofproblemsolving.com/articles/what-is-problem-solving http://www.artofproblemsolving.com/articles/what-is-problem-... Short recap: Mathematics is not bunch of dull rules. Education system puts way too much emphasis on memorization of dull rules instead of problem solving and developing strong intuition. Calculus overrepresented and discrete math underrepresented in education system. As a person who currently is rediscovering math from scratch, I find these articles very insightful. I rediscover math from problem solving/intuitive point of view rather than beating my head against the wall of formal definitions.
thearn4 10y ago
As someone currently teaching college calculus on the side this semester & works in aerospace engineering, I agree. The two math subjects that I notice most people are deficient in post-college are statistics, and linear algebra. In my opinion, we teach way, way too much calculus.
angry_octet 10y ago
So let me get this straight: kids who didn't study enough math in middle school have difficulty doing harder math later on, so they rote learn it to get into college, then drop out of math. Okay? And the kids that studied math, e.g. because they enjoy it and/or have good teachers early on, take AP math and go on to do well in math in college. Is this a Captain Obvious moment? When will people learn that there is no shortcut and they actually have to do the work, and pay a living wage to teachers? It reminds me of the 'coding in schools' thing. Who exactly is going to take a massive pay cut to teach computing? And do we just give them a walled garden to code in?
Shorel 10y ago
We can teach intuitive integral calculus in kindergarden: Have lots and lots of ping pong balls handy, and try to measure the volume of everyday things in ping pong balls. Exact results are not a requirement and in fact they don't matter at all for this early age, only the intuitive visualization. We can teach the concepts of squares, cubes and prime numbers using the ping pong balls as well: Get for example nine balls and form a square shape. Squares are numbers that can form that shape. Similar for cubes. Factorization in two factors is arranging the balls in a rectangle. A prime number is any number that can't be arranged in a rectangle. All this is intuitive, children can understand it easily, and it will help them in the future. And all this is before they learn about equations and algebra, it's even before fractional numbers.

104 points by carlosgg 10y ago

I think this is just another example of standardized testing (specifically AP course exams in this story) almost creating a niche-like education for students. Maybe that's an odd way to put it, but some students wind up spending hours and hours studying AP/SAT/ACT/etc.-specific materials (literally 500+ page textbooks for each type of standardized test). Sure this method of studying helps them achieve great scores on the standardized tests, but my impression as a recent student in academia (finishing college now) is that this creates an issue where students are almost "overly" dependent on more specialized areas. I don't know if it's an issue of not having time to learn "street smarts" (so to speak) because standardized studying takes up all their time, or if it's something else. But there's definitely some improvement that can be made.

smallnamespace 10y ago
A pattern that you see almost everywhere is: imperfect incentives/objectives + strong competition = perverse outcomes Over the last decades, we're increasingly seeing this in academia (publish or perish), the corporate world (boost the bottom-line at the expense of everything else), politics (win the election even if you have to burn your own party to the ground), and many other walks of life, as the world has gotten more competitive. To fix this, you can either fix the objective (which may be very difficult), or you can reduce the level of competition.
- Retric 10y ago
  This also causes some odd results. If the MIT/Caltech/Harvard/etc student bodies are generally less intelligent than you might expect and often boring people. You see the same thing at most national level high school competitions. The surface details may seem very diverse, but underneath that they tend to have a lot in common.
- baq 10y ago
  > imperfect incentives/objectives + strong competition = perverse outcomes if you think about it, it's just a rephrasing of 'survival of the fittest'.

hprotagonist 10y ago

>This is not true for students who take Advanced Placement calculus I hated BC calc. Bad teacher. I learned the same material at the same time in AP Physics, and loved it. I think a related issue to the author's point, though, is that students are taught in a way that implicitly suggests that calculus is the pinnacle of mathematics. Which is blatantly wrong.

baldfat 10y ago
I have had the pleasure of pushing many female students into taking calculus but my success rate was about 50%. Each one of my children's friends LOVED STEM and were thinking about going into engineering. All of them said that they were the only girl in the class and were intimidated and the other said it was REALLY boring. Calculus should be easily made fun just exploring the actual concepts in real life similar to a good physical science teacher.
- hprotagonist 10y ago
  As it happens, my teacher was female. She was a firm believer in "teach to the test", "do these exercises and follow the magic rules of differentiation, shut up and just do it don't ask why it works". That's not my jam.
  - shostack 10y ago
    When teachers' performance is judged based on test scores, it makes sense where that sort of unfortunate mentality comes from. What gets measured gets managed.
  - baldfat 10y ago
    I think many "Math" people get the mechanics and like the functions and miss out on the application. I remember trying to figure out how the Cha Cha Slide could be a decent Calculus problem. How does the song change the pattern?

baldfat 10y ago

I think we teach Calculus wrong in a deep fundamental way. We should incorporate more high math concepts earlier into education BUT people are always afraid of anything past 4th grade math! "A minority of students then wend their way through geometry, trigonometry and, finally, calculus, which is considered the pinnacle of high-school-level math. But this progression actually “has nothing to do with how people think, how children grow and learn, or how mathematics is built,” says pioneering math educator and curriculum designer Maria Droujkova." ... "“Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than “little manipulations of numbers,” as she puts it. It’s akin to budding filmmakers learning first about costumes, lighting and other technical aspects, rather than about crafting meaningful stories." http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/ http://www.theatlantic.com/education/archive/2014/03/5-year-... Why is calculus the pinnacle when I actually feel that pre-calculus would be better students even make it into middle school? These concepts of patterns and structures is the foundation of not only math but also of language and reading.

CalChris 10y ago
Berkeley EECS grad. I didn't go the device physics route but I used calculus about twice in 4 years except in physics classes.
- supahfly_remix 10y ago
  Circuit analysis turns differential equations into algebra, so you might have been using calculus more than you realize.
  - CalChris 10y ago
    Yeah, I might have been guilty of some hyperbole. I remember a nasty integral on the Linear Systems midterm. It probably would have taken an hour or so if I didn't make any mistakes; but you could transform it and solve it as a linear equation in 10 seconds. That was the point of the class, that there area two kinds of problems, linear problems and problems you can't solve. I'm not in the least suggesting that an engineer shouldn't go through three semesters of Calculus, Linear Algebra+DiffEq and a Discrete Math course.

Yhippa 10y ago

> Too many students experience a secondary-school calculus course that drills on the techniques and procedure that will enable them to successfully answer standard problems but are never challenged to encounter and understand the conceptual foundations of calculus,” he said. As I get older I've realized that I learn better when I understand the concepts about things. I then get excited to apply what I've learned into practice. I think that if I learned about "why calculus", what problems did it solved at the time it came around, and how we use it in practice I would have been able to grok it quicker and deeper. I didn't end up getting above a 3 on the AP calculus exam. I did however get a 5 on the US government exam. I had a teacher who did a great job throughout the course having us work out essays and testing arguments in class to help solidify our knowledge as opposed to memorizing facts.

hood_syntax 10y ago
Conceptual background is so important for understanding something. I was lucky to have a great teacher for BC calc, and I got an easy 5 on the exam. The concepts make sense, they are consistent, and they build on each other, so with the right progression of ideas students can get a lot farther than they would otherwise.

castratikron 10y ago

AP Calculus was the most useful course I took in high school. It was rigorous, and I scored high enough on the exam to get two free college classes and start freshman year at the sophomore level, which left room at the end of my college career for more specialized courses. There should be some vetting process for the students who want to take AP classes. A lot of the students in my AP Calculus class had no business being there. One guy got a 4 out of 50 on a test once, another guy didn't even take the AP exam, and another guy just put his head down on the desk during the AP exam. But this is really a problem for individual schools and has nothing to do with College Board.

mhb 10y ago
Isn't it kind of weird that a college won't let you take whatever math class you want?
- gnarbarian 10y ago
  No. Tuition only covers a portion of a student's cost to a university. I think it's around 30% at mine. 1) If a class is way over your head the university is wasting money having you there. 2) Having stragglers in the class that are far behind slows down the whole class. 3) Every seat that is filled with a student who is not ready for the class is a seat that can't be filled with a student who could actually pass it.
  - jessriedel 10y ago
    > Tuition only covers a portion of a student's cost to a university. I think it's around 30% at mine. Where are you getting that from? Are you just dividing the tuition revenue by the university's budget? Because the university is there for a lot more than educating undergrads.
    - gnarbarian 10y ago
      Most universities aren't primarily focused on research.[1] They are there to educate students. Research does occur at every university, but looking at their research budget as a fraction of their whole budget will show what sort of institution it is. Even if you subtract the research budget out first you will still arrive at a figure less than 50% for most universities. [1] https://en.wikipedia.org/wiki/Carnegie_Classification_of_Institutions_of_Higher_Education https://en.wikipedia.org/wiki/Carnegie_Classification_of_Ins... edit: https://nces.ed.gov/fastfacts/display.asp?id=75 https://nces.ed.gov/fastfacts/display.asp?id=75

gnarbarian 10y ago

It sounds like the issue is there are too many high schools offering classes in AP calculus which don't do a decent job of actually teaching the concepts to the students. Having a good teacher for these classes makes all the difference. Additional testing at the college level would help assess a student's ability level but it does not solve the problem of time wasted when the course must be retaken. Another issue this brings up is it causes a larger gap to form between trigonometry and calculus 2. Where memorization of hundreds of trig identities builds on top of the identities learned in trig. If a student takes Trig -> pre-calc->AP calc -> Calc 1 -> Calc 2 They will have a much harder time in Calc 2. That is what happened to me. I had to take Calc 2 a couple times before I got it sorted out. http://www.ansep.net/programmatic-outcomes/statistical-data http://www.ansep.net/programmatic-outcomes/statistical-data I think the answer is to get more qualified teachers in the schools teaching the AP level classes. There is a program in Alaska called ANSEP (Alaska Native Science and Engineering Program) who have this figured out. Middle school students who enter the program come to the university during the summer time and take math classes from university professors. Many of the students are completely finished with their math track (for engineering programs) before they start university their freshmen year. These kids are all the way through partial differential equations at 17. The graduation rate for ANSEP students who enter engineering programs is around double the national average. There are many other important aspects to it. Including mandatory study sessions and living with other students who are also taking the same classes. Students are also given well paid internships in the industry during the summer time. When looked at as a whole, there is no doubt that they have figured out a far more effective means for getting students through STEM programs and jobs once they graduate. Full Disclosure: I am good friends with a few of the people running this program.

jimmydddd 10y ago

For lower level math, my kids seemed to revisit the same subjects over and over from k-9th grades. For example, mean, median and mode, year after year. The first few years it was just robotic problem solving. After a few years, they started to get a broader understanding. Maybe some people need to take several passes through the higher level stuff too.I guess the problem is that there is not enough time.

cgag 10y ago
I believe there's decent support for the idea that you could just teach them nothing until they reached the age of understanding, and theyd pick it up pretty much just as quick without the years of rote.
- dredmorbius 10y ago
  Source?
  - abecedarius 10y ago
    A U.S. elementary school principal in the 20th C. got his school to try leaving out any math in the first three grades (iirc), and the kids reportedly came out fine. I can't remember his name, and I'd like to be reminded too.
    - asimeqi 10y ago
      It was L. P. Benezet, superintendent of schools in Manchester, New Hampshire, who experimented with teaching no math in the first five grades in the early 1930s. See for example https://www.psychologytoday.com/blog/freedom-learn/201003/when-less-is-more-the-case-teaching-less-math-in-school https://www.psychologytoday.com/blog/freedom-learn/201003/wh...
      - abecedarius 10y ago
        Thanks! There's also his own report: http://www.inference.phy.cam.ac.uk/sanjoy/benezet/1.html http://www.inference.phy.cam.ac.uk/sanjoy/benezet/1.html

hanoz 10y ago

I remember once when when driving I was pleased when it occurred to me that I should be seeking to minimise the third derivative of distance with respect to time for the comfort of my passengers. That was the sum total of calculus's contribution to my adult life. Not sure it was worth it.

ThrustVectoring 10y ago
Curves in freeways are designed that way as well - they don't just start turning suddenly, but gradually transition into and out of the curvature.
- 0xCMP 10y ago
  Good ones at least.
  - ThrustVectoring 10y ago
    Don't quote me on this since I don't remember why I believe it, but I think it's part of the legally mandated design requirements for the US interstate system.
    - 0xCMP 10y ago
      No I'm sure there are requirements, but sometimes there are also space or budget constraints (I assume) since some curves are definitely safer and easier than others for the same highway (I-95 for me). I wished they were more consistent
- LgWoodenBadger 10y ago
  One of my calculus classes in college involved calculating the required spline for a curve of a theoretical portion of interstate in order to meet the interstate requirements/regulations. It was pretty interesting.

ThatGeoGuy 10y ago

I am not sure how much of this is specific to the American education system, as I am Canadian, and have not experienced it, but my own take after going through high school to my current graduate studies is that 1) We teach algebra, geometry, and their relationship wrong in many ways 2) We don't teach enough calculus at lower levels of mathematics, instead opting to prefer a "poor man's" algebra. I'm sure there's too much to write on for one HN comment, but a lot of 1) comes down to how we teach "solve for x" type problems which are more interested in re-ordering and simplifying equations than we are about giving people a fundamental understanding of what our relations and equations actually mean. Often I notice that many students couldn't graph or draw or even begin to reason about what they're trying to solve for. If you give someone a graph and ask them to find the maximum point of a line on a graph, they can very easily point it out. The next step is doing it numerically, but a lot of times the relationship between what we can determine visually and what we can determine computationally is lost. Which is sort of what prompts 2). You can't learn Calculus without learning limits, and Calculus places explicit meanings to relations that we can graph, and how those relations change according to specific variables or unknowns. Finding the maximum of a relation using Calculus is much easier than strictly using algebra for this reason, because we can make the relationships between a curve and it's derivative explicit. Most students cringe at the thought of Calculus because we place it on a pedestal and students just assume Calculus is the peak of mathematics, but we really need to introduce Calculus as more "normal" math earlier on, IMO. I'm talking basic tasks like derivatives finding the equation of a line tangent to a curve, or finding the limit as we approach a point on a curve, or even just (re)factoring equations so that we can plot them easier and / or take their derivative easier. In almost every example I can think of where I learned Calculus, it was easier than building intuition for Algebra because there is less robotic crunching and more reasoning about what specific problems mean. What does it mean to take the derivative? Why are volume and area and perimeter related? I know in many ways it seems like I'm just advocating for thinking spatially or visually about mathematical problems, but that's part of what (I think) makes Calculus more approachable. You can get a lot farther with a basic understanding of Algebra and a very small amount of Calculus than you can with just Algebra alone. Should our curriculum be entirely Calculus? No, that's ridiculous; however we should ease up on the systems-of-equations type problems and obtuse word problems that require students to produce or remember all manner of expressions and formulae, and instead focus on building that first intuition of how we can use Calculus as a tool for problems that are much, much harder without. I expect that it would give Calculus a more realistic reputation, and would probably put off less students who are already giving up on math.

anu7df 10y ago

I have no idea why calculus gets such exalted status in the annals of highschool nightmares. No I am not bragging, humble or otherwise. I found calculus to be simple neat and logical, and not so mind bending. Linear algebra on the other hand was far less trivial; not the mechanics, the aha moment bit. Geometry.. that for me was genuine mind bending pleasure pain.

maxerickson 10y ago

There should a high school class on numeracy, followed by statistics. Wrote learning of algebra does not necessarily lead to numeracy. Sweating over the students that are (mostly) ready for calculus and college is missing the forest.

AstralStorm 10y ago

Personally, I have always wondered about the prominent position trigonometry and geometry get. As it is typically taught, it is the most useless thing, the time spent there would be much better spent on algebra, calculus. From basic theorem of algebra flows trigonometry and a lot of number theory, while from calculus with some linear algebra flows geometry. The rationale is probably history.

hossbeast 10y ago
You can't possibly understand calculus without first understanding trigonometry and, to a lesser extent, geometry
mmmpop 10y ago
So I guess you've never framed up a house have you? Basic trig is so incredibly important to anyone that ever desires to build a thing. Don't you remember learning the unit circle in Calc 2??
jandrese 10y ago
IMHO, high school math misses the one course that would be most useful to the students. Statistics. While there are a LOT of statistical models that aren't especially useful in every day life, knowing the most common ones can be a big help. Of all the math classes I took in college the only one I use on a regular basis is statistics. Thinking back on high school I spent a whole lot of time on basically calculus prep and in the end I hardly ever take a derivative or integral of anything.
thesimpsons1022 10y ago
trig has seemed extremely important to me. specifically using sin and cos. helps a lot in physics as well as computer graphics and many other applications.
gozur88 10y ago
Trig is a pretty useful subject for the skilled trades. Framing a house, as mmmpop notes, but also things like plumbing and cement work. If you don't go on to college, or you don't major in engineering or math, calculus is pretty useless. I've never once used calculus outside work, but I have used trig.

ivan_ah 10y ago

I think linear algebra is a much more valuable and general-purpose math tool. I'm not sure how feasible it would be to teach linear algebra topics "properly" to high schoolers, but even a half-assed linear algebra would be more useful than calculus. The connections to geometry, general systems thinking, formal math methods, and countless applications of linear algebra are just the kind of thing students need to get them interested in learning more math. Anyone interested in LA an its applications should check out my upcoming book, the No bullshit guide to linear algebra availale on pre-order here https://gum.co/noBSLA https://gum.co/noBSLA (it's almost finished; just beefing up the problem sections).

FT_intern 10y ago
Thinking in matrices and vectors is much more complicated and abstract than calculus.
- microDude 10y ago
  I disagree. Especially, if you combine it with a graphical tool like Matlab or the Python libs. Take a lower triangle matrix multiply it by a vector, and "boom" you just explained causal convolution. Wash and repeat for several other topics.
lqdc13 10y ago
They taught me about matrices and vectors and common matrix/vector operatoins in pre-calculus and even earlier in a shitty public school. Calculus needs to happen in high school so that people are more prepared to take it again in college. For me, calculus 2 was the hardest class in college. This is coming from someone who went to a math grad school. Some people need to study way more for calculus than for any other class in high school or college.
- sidusknight 10y ago
  I'm guessing you did applied math in grad school? I think very few (pure) math majors had any issue with calc.

dver23 10y ago

Terrible teachers making difficult subjects impossible for students. It's the same experience for my kids that I recall from my time. Industry can pay more, and outside the rare person who has come to teaching to give back, the instructors are those who can't. Even when I was at Cal Poly, the only instructors in eng/sci who I recall being helpful were those from JPI that would come in and teach a class or two. One taught us three classes worth of material in course when he realized how much we had missed from our previous instructors.

euroclydon 10y ago

I have a 5th and a 6th grader. I'm going to buy them an educational copy of Maple soon, so they can have a beautiful platform to plot 2-D and 3-D graphs, solve equations, do derivatives and integrals, and factor algebraic expressions. I want them to do all this in high school, whether or not they take Calculus, so they can get an intuition for it. Then they can memorize the trig rules in college.

Oletros 10y ago
AS an Spanish not knowing the US education system, at what age one kid goes to 5th grade?
- euroclydon 10y ago
  My 5th grader is currently 10 years old.
- colechristensen 10y ago
  The median age in 1st grade is 6.
- Oletros 10y ago
  Thanks to both of you
ivan_ah 10y ago
You should also consider SymPy, available as a webapp http://live.sympy.org/ http://live.sympy.org/ or in jupyter notebook form. Plotting function graphs is a bit harder (matplotlib) but overall I find SymPy to be much more logical and easy to grok than Maple's weird syntax. Here is a short tutorial on how to do basic math using SymPy: https://minireference.com/static/tutorials/sympy_tutorial.pdf https://minireference.com/static/tutorials/sympy_tutorial.pd...
- euroclydon 10y ago
  I love Jupyter. I set up pydot in a notebook, and made high level functions to create and print trees and graphs. My 10 yr old picked it up quickly while we killed time at his brother's swim meet. But, I'd love for them to learn to use a CAS engine. I appreciated, in college, being able to quickly factor a polynomial, solve an equation, and find derivatives.

strathmeyer 10y ago

Why did we take two years of Calculus in high school when there weren't any jobs when we came out of college?

Retra 10y ago
Those things have no relationship, and the expectation of one is a fault in your understanding of the world.

japhyr 10y ago

I have a 5 year old, and I was talking to some acquaintances who also have a kindergartener. The kids are learning math, by playing with manipulatives. It's great; they're learning mathematical concepts, and they have no idea anyone in the world doesn't like "math". They don't know what "math" is, they are just starting to discover that the world is made up of interesting but identifiable shapes, and that numbers have some really interesting patterns as well. These parents, however, don't see anything they recognize as "math". They're afraid their kid is going to fall behind, so they're giving their kid worksheets at home focusing on two-digit addition and subtraction. Their kid doesn't like math already, because at home math is just writing things on a piece of paper with little meaning. This is sad to me, because there are so many interesting things you can do with your kids at home to help them develop their mathematical understanding. Our educational issues with math start young, and they come from adults. I'm fortunate to work in a small high school where I get to treat each kid individually. It's wonderful to meet kids where they're at, and help them move forward and start to enjoy math again.

ivan_ah 10y ago
I don't have kids, but if I had a 5 year old, I'd sure get them some of these: https://en.wikipedia.org/wiki/Cuisenaire_rods https://en.wikipedia.org/wiki/Cuisenaire_rods I've often thought about inventing math toys, e.g., group theory for toddler: S_3 = triangle toy, S_4 square toy, etc. I'm sure there are many advanced math topics that could be turned into toys. No symbols, no equations, just toys, but subtly you're getting kids familiar with important math structures.
- kol 10y ago
  Rubik's cube can be used to teach group theory, see http://geometer.org/rubik/group.pdf http://geometer.org/rubik/group.pdf or http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20Rubik's%20Cube.pdf http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20an...

Koshkin 10y ago

The best thing about learning calculus is that you see how everything you have learned so far - algebra, geometry, trigonometry - everything comes together and becomes equally useful and important. This is, in fact, what mathematics is all about.

waynecochran 10y ago

I think a Physics course taught with "Just in Time" Calculus would be fruitful for motivating the Calculus. Calculus was invented to do Physics.

abakker 10y ago

Sample size of one: I think I understand calculus, the needs of calculus, and how it is profoundly meaning to understanding things around me. Can I do calculus? no. I couldn't do calculus because I really couldn't do algebra. I think that if I had been taught the concept and relevance of calculus earlier, and only had to worry about the execution of calculus later, I would have been much better off. understanding what derivatives and integrals and trig functions mean should come before we are forced to execute their use, or learn proofs, or even really go beyond basic notation.

yequalsx 10y ago
I'm skeptical that a person can really understand why x^(2/3) is not differentiable at x=0 but has an absolute minimum at x=0 without understanding the algebra. I think one can get an idea of the concepts without knowing algebra but I doubt one can understand calculus without knowing algebra. We always teach the meaning of a given mathematical concept before doing using them or doing calculations. It's just that true understanding comes from the doing the calculation part.
- rconti 10y ago
  Not the parent, but I had the same problem. I understood that kind of algebra but could never figure out how to simplify things well, which hamstrung me in Calculus. Also, they never bothered to explain the purpose of any of the equations in calculus.

krschultz 10y ago

I graduated with a mechanical engineering degree, meaning I took 3 semesters worth of Calc and 3 more semesters worth of differential equations. So I'm not one to hate on math or calculus specifically. But I think high schools should focus more on statistics. Statistics is relevant to more fields than calc. It matters a lot more in business and in all the jobs where you use calc you will also have to use statistics. It's also useful for citizens to understand statistics better - and educating citizens is truly one of the purposes of public education.

lordnacho 10y ago
I second that. Stats is so much more applicable, and it is also the area where common sense will fail you the most. Simple things like birthday paradoxes, Gambler's fallacy, Monty Hall problem, and so on are all contrary to common sense yet not taught in schools. And you need stats all the time. Every time someone says crime has never been worse, you need to know how to decide if you believe it. When someone says flying is safer than driving, you need to know how to compare. A lot of stats thinking is also not formulas, but qualitative things like "Am I comparing apples to oranges". Calculus by contrast is pretty common sensical. Acceleration is the derivative of velocity, you find the local extremes by finding the zeros of the derivative, the integral of a net shaped field is the rim... that all makes sense with some pictures. I also haven't come across it much in finance, apart from derivatives work, where funnily enough you can't do with high school calculus, but you need stochastic calculus. Which contains some quite statistics related concepts.
- ghaff 10y ago
  I suspect calculus is less commonsensical to most people than you think :-) That said, I've argued the parent's point for years. Basic stats/probability/economic biases are really fundamental stuff and don't require a lot of complicated math if you don't want them to. There are some basic calculus concepts that are useful. Area under a curve. Basic derivatives (for finding maxima/minima, etc.). But the truth is that I've rarely used calculus outside an academic setting and I use at least basic statistics/probability thinking all the time.

CN7R 10y ago

Schools should adopt more dynamic curriculum, instead of a one size fits all, for individual students. Not everyone will grow up to be an engineer, but that doesn't mean those who have an interest in advanced mathematics should be held back. Encourage students to pursue their academic passions, and don't limit certain subjects depending on grade level. A personal anecdote: I didn't start liking math until Calc AB, as subjects beforehand were presented as dry and were relatively easily so I didn't individually pursue mathematics outside of school. But by that time, I was already a junior and as a result, I often wonder what better math teachers and a non static curriculum would've done for my education.

noobiemcfoob 10y ago

I feared calculus, right up until circumstances forced me to not only take calc in high school but to take the hardest level of calc my school offered (AP BC Calculus, counted as 2 semesters of college calculus). I was greatly benefited by an amazing teacher. I've always felt that calculus is one of the most important branches of mathematics for the simple insight it should hammer home time and again: An infinite amount of infinitely small things can and will add to infinity. All too often, humans fail to see the cumulative effect of an integral. Every branch of math has these types of insights built into them. Calc the power of instantaneous change. Geometry shows the beauty of a proof - and if you scratch a little bit, the shattering epiphany that triangles are literally the same everywhere, always and forever. The reality is that many of these concepts are simply absorbed into our culture, and we fail to realize it until someone explains that it took centuries of work to formulate the concept of 0. When it comes to education, I believe the struggle has far more to do with testing standards than the material. It's easy to teach to process, much harder to teach insight and further harder to test. In reality, none of this matters until your institution decides and commits to truly educating its students. tldr - None of this matters until your institution decides to truly educate its students.

madengr 10y ago

"Instead, students who took high school calculus often find they have to retake college calculus, or even pre-calculus. Many flee to the humanities." Well no sh!t. I find it funny that people who aced calc in high school get their ass handed to them in college.

WhitneyLand 10y ago

I accidentally took Combinatorics as a fill in and it has been surprisingly useful. The need to count things well comes up on a regular basis, not to mention helps with learning poker. Would recommend this over some caclulus for a lot of folks.

LouisSayers 10y ago

We're building out an online solution over the next year to help prepare final year high school students for exams - with an initial focus primarily on Maths. Would love to hear from people with thoughts around Math tools they feel would be valuable - especially taking into account current resources such as KhanAcademy, and how we could potentially provide something complementary. Either comment here, or hit me up - louis@connecteducation.com.au

danielmorozoff 10y ago

Maybe logic should be the first mathematical discipline to be taught in middle school. The notions of and / or and the representation of language in mathematical form? Before even starting on algebra

tnecniv 10y ago

I have a specific problems with the HS math curriculum I experienced (as opposed to the general problems with the whole math curriculum I had before my junior year of college). Mainly, there was just so much wasted time it was unbelievable. I actually cannot remember a single thing that I learned in Algebra II because I either relearned it in Pre-Calc or never used it again. Similarly, we spent the whole first semester of Calc BC redoing all of Calc AB. What a waste! If I could, I would replace all that wasted time with Linear Algebra (with proofs) and Probability (throw in statistics if you want). Calculus should stay, but teachers should motivate it better (this goes back to the general problems I alluded to). Those two classes are immensely useful and teach a whole different modes of thought compared to the regular pre-college curriculum.

jiaweihli 10y ago

I grew up 3 years ahead of my classmates in math. I found everything up until calculus to be easy but elegant - algebra, geometry, probability, trigonometry. Much like programming, there were often multiple ways of the solving the same problem, and deriving a formula was feasible and logical. In contrast, I found calculus to be mind-numbingly boring. It required memorizing and reciting formulas upon formulas. It was the first time I felt that math was tedious, and pushed me away from majoring in math in college.

hal9000xp 10y ago

I do recommend to read these articles: http://www.artofproblemsolving.com/articles/calculus-trap http://www.artofproblemsolving.com/articles/calculus-trap http://www.artofproblemsolving.com/articles/discrete-math http://www.artofproblemsolving.com/articles/discrete-math http://www.artofproblemsolving.com/articles/what-is-problem-solving http://www.artofproblemsolving.com/articles/what-is-problem-... Short recap: Mathematics is not bunch of dull rules. Education system puts way too much emphasis on memorization of dull rules instead of problem solving and developing strong intuition. Calculus overrepresented and discrete math underrepresented in education system. As a person who currently is rediscovering math from scratch, I find these articles very insightful. I rediscover math from problem solving/intuitive point of view rather than beating my head against the wall of formal definitions.

thearn4 10y ago

As someone currently teaching college calculus on the side this semester & works in aerospace engineering, I agree. The two math subjects that I notice most people are deficient in post-college are statistics, and linear algebra. In my opinion, we teach way, way too much calculus.

angry_octet 10y ago

So let me get this straight: kids who didn't study enough math in middle school have difficulty doing harder math later on, so they rote learn it to get into college, then drop out of math. Okay? And the kids that studied math, e.g. because they enjoy it and/or have good teachers early on, take AP math and go on to do well in math in college. Is this a Captain Obvious moment? When will people learn that there is no shortcut and they actually have to do the work, and pay a living wage to teachers? It reminds me of the 'coding in schools' thing. Who exactly is going to take a massive pay cut to teach computing? And do we just give them a walled garden to code in?

Shorel 10y ago

We can teach intuitive integral calculus in kindergarden: Have lots and lots of ping pong balls handy, and try to measure the volume of everyday things in ping pong balls. Exact results are not a requirement and in fact they don't matter at all for this early age, only the intuitive visualization. We can teach the concepts of squares, cubes and prime numbers using the ping pong balls as well: Get for example nine balls and form a square shape. Squares are numbers that can form that shape. Similar for cubes. Factorization in two factors is arranging the balls in a rectangle. A prime number is any number that can't be arranged in a rectangle. All this is intuitive, children can understand it easily, and it will help them in the future. And all this is before they learn about equations and algebra, it's even before fractional numbers.