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.<p>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.
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!<p>"A minority of students then wend their way through geometry, trigonometry and, finally, calculus, which is considered the pinnacle of high-school-level math.<p>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."<p>...<p>"“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."<p><a href="http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/" rel="nofollow">http://www.theatlantic.com/education/archive/2014/03/5-year-...</a><p>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 <i>patterns and structures</i> is the foundation of not only math but also of language and reading.
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.<p>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.<p>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.
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.<p>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.
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.<p>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.<p>Anyone interested in LA an its applications should check out my upcoming book, the <i>No bullshit guide to linear algebra</i> availale on pre-order here <a href="https://gum.co/noBSLA" rel="nofollow">https://gum.co/noBSLA</a> (it's almost finished; just beefing up the problem sections).
> 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.<p>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.<p>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.<p>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.
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.
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.
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.<p>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.
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 <i>do</i> 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.<p>understanding what derivatives and integrals and trig functions <i>mean</i> should come before we are forced to execute their use, or learn proofs, or even really go beyond basic notation.
Personally, I have always wondered about the prominent position trigonometry and geometry get.<p>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.<p>The rationale is probably history.
>This is not true for students who take Advanced Placement calculus<p>I <i></i>hated<i></i> BC calc. Bad teacher. I learned the same material at the same time in AP Physics, and loved it.<p>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.
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.<p>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.<p>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.
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.<p>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.
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?<p>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.<p>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?<p>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?
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.<p>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.<p><a href="http://www.ansep.net/programmatic-outcomes/statistical-data" rel="nofollow">http://www.ansep.net/programmatic-outcomes/statistical-data</a><p>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.<p>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.<p>Full Disclosure: I am good friends with a few of the people running this program.
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.<p>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 <i>will</i> add to infinity. All too often, humans fail to see the cumulative effect of an integral.<p>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.<p>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.<p>tldr - None of this matters until your institution decides to truly educate its students.
I do recommend to read these articles:<p><a href="http://www.artofproblemsolving.com/articles/calculus-trap" rel="nofollow">http://www.artofproblemsolving.com/articles/calculus-trap</a><p><a href="http://www.artofproblemsolving.com/articles/discrete-math" rel="nofollow">http://www.artofproblemsolving.com/articles/discrete-math</a><p><a href="http://www.artofproblemsolving.com/articles/what-is-problem-solving" rel="nofollow">http://www.artofproblemsolving.com/articles/what-is-problem-...</a><p>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.<p>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.
We can teach intuitive integral calculus in kindergarden:<p>Have lots and lots of ping pong balls handy, and try to measure the volume of everyday things in ping pong balls.<p>Exact results are not a requirement and in fact they don't matter at all for this early age, only the intuitive visualization.<p>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.<p>All this is intuitive, children can understand it easily, and it will help them in the future.<p>And all this is before they learn about equations and algebra, it's even before fractional numbers.
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.<p>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!<p>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.
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.
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.<p>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.
Terrible teachers making difficult subjects impossible for students.<p>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.<p>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.
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.<p>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.<p>Either comment here, or hit me up - louis@connecteducation.com.au
I think a Physics course taught with "Just in Time" Calculus would be fruitful for motivating the Calculus. Calculus was invented to do Physics.
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.
"Instead, students who took high school calculus often find they have to retake college calculus, or even pre-calculus. Many flee to the humanities."<p>Well no sh!t. I find it funny that people who aced calc in high school get their ass handed to them in college.
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.
I accidentally took Combinatorics as a fill in and it has been surprisingly useful.<p>The need to count things well comes up on a regular basis, not to mention helps with learning poker.<p>Would recommend this over some caclulus for a lot of folks.
There should a high school class on numeracy, followed by statistics.<p>Wrote learning of algebra does not necessarily lead to numeracy.<p>Sweating over the students that are (mostly) ready for calculus and college is missing the forest.
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<p>1) We teach algebra, geometry, and their relationship wrong in many ways<p>2) We don't teach enough calculus at lower levels of mathematics, instead opting to prefer a "poor man's" algebra.<p>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.<p>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.<p>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.
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?<p>Before even starting on algebra