Do the math: too much calculus? (2012)
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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.
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.
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.
Though stats/prob are perhaps more 'useful' in non-academic life. Imparting the notion of what probability actually is and how Bayes' rule fits into it requires a very gifted teacher with a fundamental grasp of the real mathematics and on top of that the ability to communicate it and link it to previously learned material aka algebra/ pre-calc. You can make it visual, but it would seem to require defining the concepts of sets, sampling, distributions etc.
The issue imo is that teachers tend not to teach the intuitions of math, rather they focus on the mechanics of it. As such, most students turn into machines calculating things as opposed to viewing math as a way to reason.
Yes. I agree generally--often even at relatively advanced levels.
The stats course I took in engineering grad school was so heavy on the math that I'm not sure I appreciated what the math was in service of. By contrast when I took a less math-heavy stats course later I much better appreciated the underlying principles.
But yeah, it's very easy to turn any of these math-related topics into: memorize the formula, plug the numbers, come up with a result.
That's why calc is central. I wouldn't be surprised if it's the same in the US.
This means most people have no idea what the math they learn is used for, because they never reach a level where they have to use it.
So they leave school with the idea that math is annoying bullshit that they don't understand and can't use.
I think it would be better to stream math by ability much earlier. Make everyday math compulsory for everyone - interest rates and loans, basic stats, currency rates, areas/volumes, simple trig, maybe some slightly more advanced finance, perhaps touch on some calc ideas.
Only teach calc and scientific math to those with the talent and interest for it. Give everyone else a taste so they understand a little, but don't try and get them ready for a career they'll never go into.
Why most people don't understand the math they learned and what it is used for, I would argue has much more to do with how they are not taught to interpret what they see around them mathematically- granted that takes years of practice, but we do not care to teach this to children.
Now streamlining math is all well and good, but we may loose some of the greatest mathematical minds by doing so. Many famous mathematicians (Euler, Lagrange for example) were slated to be someone else- professionally that is. And it took other great mathematicians to notice them.
Right before any of the doors is opened we have no idea what to expect, so there are 3 equaly likely possibilities:
1. prize is behind A 2. prize is behind B 3. prize is behind C.
In case (1) you win by staying with your choice of A. In case (2) Monty Hall must open C (by the rules), so you'll win if you switch to B and in (3) the host will open B(by the rules) and you should switch to C.
In 2 cases out of 3 (2/3) you win by switching and in only one case out of 3 (1/3) you win by staying with your first choice. Since 2/3 > 1/3, you should switch.
This problem is easy if you know the defintions of sample space, event and the Equally Likely Probability Formula.
You are shown 100 doors 1 ... 100. Imagine you pick 1 door.
Of the other 99 doors Monty tells you which 98 of them don't have the prize.
He now asks if you want to switch to the 99th door that he didn't cancel out.
Intuitively that seems to make much more sense to me.
I think that commitment bias or what it's called is also an explanation for the misunderstanding in the first place. The argument against the increased chance, as I would use, works both ways and that is much more apparent the smaller the initial chance is. But I only intuitively used it to defend my position.
A sample space is the set of all possible outcomes of a random experiment. An event is a subset of a sample space.
Let S be the finite sample space with equally likely outcomes in it. Let E be an event. Then the probability of E is the number of outcomes in E over the total number of outcomes in S.
Now consider 52 cards in a deck. The sample space of outcomes here are the 52 cards. What is the event that the chosen card is a black face card? E = {J-club,Q-club,K-club,J-spade,Q-spade,K-spade}. What is the probability that the chosen card is a black face card? 6/52 by the formula above.
Monty Hall problem can be solved by using the above as a guide.
Sample space here : {switch, switch, stay-with-your-original-choice}. By the formula above, the probability of switching is 2/3 and the probability of staying put is 1/3.
If you're doubting the formula, I can provide you with a proof.
Two thirds of the time, you pick the wrong door on your first choice. So far, so obvious. Now, you picked a wrong door, Monty opens the other wrong door, and there are only three doors. Obviously the remaining door has the prize.
Every time you pick the wrong door, and then switch, you win. That means two thirds of time, if you switch, you win. That makes switching the right strategy.
From a computer science, more specifically programming, background I find linear algebra, logic, calculus and the combination thereof to be more insightful when handling abstract ideas.
I fear that if too much focus is put on statistics then we many end up with another generation of heavy statistical machine learning (and programs that consume high amounts of power to function).
At my high school (Texas public school), Calculus and Statistics were both optional AP courses.
You can usually choose whether or not to take whichever combination you want. Some people took neither. I took both.
If high schoolers had to take one advanced class, I think it should be discrete math. Keeps their algebra skills sharp, introduces them to proofs, and still introduces them to basic probability. Obviously the availability of this offerring makes this impractical at the moment.
the class really needs to be taught well for people to understand it. also. i think it should include probability too.
I think a good stats education requires heavy emphasis on doing real problems with what you learn. It's so easy to do and apply to the world around you. Unfortunately, most books that even touch on this have absurd text and problem sets that revolve around pulling jelly beans out of a bag or rolling a dice (I mean yeah, but super literal). Mostly the professors and teachers I knew that taught statistics at the lower levels always seemed bored and wanted to be elsewhere. A shame, really.
If we can't have better math instruction, let's at least have more relevant math instruction. Calculus is very useful in some disciplines, like physics, but startlingly useless elsewhere. Cool, yes. But useless.
More here: http://short-sharp.blogspot.ca/2015/11/grade-12-math-somethi...
Until calc is needed in the statistics. On which level do you even compare this, space or time complexity? I mean, are you saying, the complexity of the sum of the stochastic in use exceeds that of calc?
I think you gave an unfounded statement, without proof. i'm guessing you are biased to think statistics is harder, because you were less exposed to those classes and now you have remorse. If it was the other way around, you'd want more calc.
The way curricula are laid out, I'm getting the impression it is not so much about what is taught, school is rather a benchmark of abilities and a day care.
The problem with statistics education is that it is generally heavily watered down, particularly the classes at the university level for business, psychology, and other majors that need the knowledge but aren't math or engineering majors. People heavily misunderstand the usage, difficulty, and nuances of stats education if they are only exposed on this level. It's better than nothing, but leads to serious issues that remind me a bit of the novice arrogant programmer vs. the grizzled veteran smart senior programmer. Generally, most people I see with limited stats education come out of it thinking mean, median, normal distributions, and at best linear regression are the whole of stats.
Another problem that also related to the parent with regard to stats vs. calculus is that calculus is actually a pre-requisite for most useful statistics work. It is true there are plenty of topics in stats you can cover without knowing calculus, but you are at a severe disadvantage and won't really understand the "why" and just be regurgitating the memorized method. I feel a lot of statistics is a combination of the "why" and "how" because so much of it is applying math at a more discretionary and opinionated level, like is done in science. Really learning stats requires a lot of math, from advanced algebra to diff eq possibly to matrices to calculus. To do good work in stats, you need a lot of tools to draw from, otherwise you're stuck with tools that often don't work (common example is people who only are familiar with normal distributions and nothing else).
Yet another problem with stats is that you can really make up whatever you want when you conduct statistical applications such as an analysis. The rule is that as long as you provide justification and proof for what you are doing - i.e. how and why, it is acceptable along with some text to explain. This is why good statisticians do things like describe and reveal their sampling methods, explain what did and did not work, and why they selected various tools. Obviously most dubious work can be quickly disputed and identified by someone who knows what they are doing, but the reality is that so many people never follow-up, fact check, or even read anything but the conclusion.
Stats is in so many ways about showing the process and justifying your answers than the quantitative answers. This is similar to financial forecasts as well for example which also use a lot of applied math, and people just believe what they read even though the evidence to disprove the report is right there. The science comparison applies here too. Just like with research studies, a lot of people don't check, verify, and re-run the results so people just believe what they read or rubber stamp it. I recall I had a very mean professor that once gave a final exam question that actually could not be solved at all using the methods we learned because the types of analysis and models we were asked to try to apply were invalid for the data. The simplest version of this familiar to many people is trying to create a best fit linear equation to non-linear data - simply draw it on paper against a data plot and you'll see it makes no sense.
If there's one important thing I learned in stats, it's to be skeptical and prove things with evidence. Nearly every time I watch the news, read an article, etc., I go nuts because they are making huge conclusions without giving you a way to understand who, what, where, why, and how about the study. The presidential elections in the US are an obvious example I see a lot with things like "Survey Monkey" passed off as scientific data in some contexts. Even represented as a casual, unscientific process, such things can be dangerous because people still accept statistics at face-value usually.
In summary, yes, people should learn statistics but it requires a lot of hard work to develop the tools and analytical skills to do it right. Like calculus, it can require many courses which can equate to years of study. Like anything, the skills involved also tend to get better with time and increased knowledge. At least an extra course or two beyond what is taught now would go a long way, but people really need to emphasize more than stats as repeating some processes of functions, formulas, proofs, and such, and more as a way of thinking and set of tools. Stats should teach you to be very paranoid and skeptical, and verify what you see in the world around you as best you can before you believe it. Once you do that, like the parent says, you will see so much clearer what is true, false, dubious, could be better, etc. in the world around you.
I still feel like calculus is one of the more valuable types of math. It really does teach you a lot, but like anything, it requires a good teacher. I find it all ties together for me in the end, like when doing graphics programming and games, I use a lot of trig, geometry, algebra, and calculus even, and stats some (and even more for business-type stuff).
I had the opposite problem, which was that I learned stats as a bunch of proofs with practically no applications. I only use fairly basic stats today, but I often test my conclusions by running some sort of monte carlo analysis. I graph everything.
It sounds like a lot of what you're recommending could be described as teaching general quantitative scientific methodology.
This is the general problem with math education in general. Just equations. Solving equations is relatively easy. Coming up with them is where the real challenge and innovation is.
These days you can solve many practical statistical problems with Monte Carlo, but understanding why those curves look the way they do is still very useful.
Perhaps linking integrals/derivatives into stats could be done from a more discrete approach as opposed to a continuous one first.
For example: Starting with a representation of an integral as sum of areas of rectangles (something most students can do) and then saying what happens if the width of all the rectangles gets smaller and smaller. A la Riemann
This could then tie into the pdf quite reasonably?
Vs
Trigonometric substitute to solve analytically.
For example.
When we got to continuous distributions, we kind of hit a wall. Here's the puzzle. Suppose the data are recorded to two decimal places, and are reasonably bounded, for instance they are between 0 and 100. That's a discrete distribution. All of the data in the examples and problem sets were fixed-point.
I literally couldn't justify to someone who had never taken calculus, why a wholly different kind of distributions with its own collection of formulas, was necessary or useful. The teacher had never explained it either.
Instead, to deal with this 'a lot of really small things' we need a slightly different approach. Then draw a pdf, and talk about area under the curve. Perhaps relate this to the discrete case where the curve is just really blocky.
I took a fantastic one-off class as an undergraduate called, IIRC, "Statistics for Physicists". The class taught us Bayes' Law, what tests were, what significance was, and how one might go about deriving and using simple tests. Everyone came out of the class with a very clear understanding of what statistics meant, and no one who took that class would take a statement like "this drug had a significant effect on such-and-such" without an appropriate grain of salt (specifically, how big was the effect). The class didn't include a giant grab-bag of tests.
One of my friends took a regular introductory stats class the following quarter. At one point, he asked me if I could help him do a such-and-such test (Student's t, maybe, but I don't remember which one). I asked him what he was trying to do with the test, and he didn't know.
I think it would be enormously valuable to teach most students what statistics means, what it can do, and what it can't do. I'm not sure how valuable it would be to teach everyone how to evaluate a big pile of magic statistical functions.
Fortunately, I had a great professor in Calc III who more than made up for this.
I was a very math oriented student at a high school with a heavy bias towards liberal arts ... So when it came to picking senior year electives, I was delighted to enroll in AP statistics.
The classes turned out to be tutorials on the built-in TI-83 stats functions, with step-by-step instructions on how to answer the various types of questions we would encounter on the test. I forgot everything and gained virtually no statistical intuition.
Thankfully I had a similar experience to you ("Probability and Statistics for Engineers") with a fantastic professor who researched computer networks. His lectures were sprinkled with tangents tying the current topic back to his own field / research, which to me made the biggest impact. Contextualizing a taught subject markedly improves the course experience.
They should teach it from the perspective of what problems you are trying to solve, what your options are, and how those options work in practice.
The "in practice" part should be lots of simulations with something like a ChromeBook, so people could really see the ideas working. Create a (simulated) bucket with 70% red balls and 30% blue, then pull samples of various sizes and see how well the sample means match what you already know is the "truth" (70/30). You'll soon have everybody seeing that bigger samples aren't always better, but they tend to be, and they'll see how much better they get as a function of size. Make charts of the improvement. Talk about how good the estimate really NEEDS to be.
Then try some tests where the students DON'T know the "truth" about the population and have to use samples to estimate it. How far off do they think they are. How confident about various intervals, etc.
Then start explaining some of the math that lead to formulas that let you calculate these estimations, sample sizes, confidence intervals, etc.
Have estimation games: you get points for getting an estimate within some delta of the population mean, but you have to pay points to buy your samples. So, how big a sample does your team want to buy?
Then continue with other basic prob/stat ideas such as bayesian cancer tests (does this mean you have cancer or not?) and courtroom dramas (does this mean he's probably guilty or not?), commonly misunderstood situations (ex: Simpson's Paradox), how Gaussian estimators begin to fail when things aren't independent, and all sorts of other introductions to quantitative thinking about real life scenarios.
same way you can do linear geometry without calculus.
Most who teach stats at the undergrad or hs level have little choice but to defer a lot of things to a later course.
Or do you mean statistics as a bunch of applied methods, where you follow some recipes without understanding the math behind them? It's taught that way even at university level, but I can't see how this could be helpful in the long run.
"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-...
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.
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.
For example I taught my daughter factions this way.
Me: "These factions you see them? Now see these decimal points?"
Her: "Yeah I hate factions they are stupid!"
Me: "Factions are the only consistently real numbers below 1 and decimal points for the most part are made up numbers. 1/3 is 1/3 and never .3333."
Her: "I always thought it was the otehr way around."
I wish you would teach me calculus =)
This has nothing to do with calculus, though.
You can construct a complete ordered field from infinite decimals
FYI: infinite sums are calculus.
Scottish first degrees are 4 years rather than 3 in England - presumably to allow for this.
Makes no sense to me. All those things are related. There's a picture that shows you why Pythagoras' Theorem is true. And why the difference of two squares is never prime.
You'll have had a whole summer holiday between trig and algebra if they're separate things. Surely they are taught as a kind of whole?
Got a bit more separate at university, but not really.
Where I live, it was just called 'math'. The first time I encountered algebra was in university, and that started with groups (a set and an operation on pairs of items in the set)
As to the 'pre': I have a masters in math, and lots of what we did was in some sense pre.
For example, calculus on real variables showed you how to define differentiation and integration and how to rigorously prove lost of theorems (pro tip: if you can't make heads or tails of an exam question, at least write down "let epsilon > 0")
Next, we basically went through the same set of theorems but in multiple dimensions, but showed that those rigorous proofs had glaring holes, but hey, now you know better. Also: infinities are weird (pro tip: the area under a 2D curve can depend on the order in which you add up its parts).
Next, calculus on complex numbers extended that and showed that those rigorous proofs had glaring holes, but hey, now you know better (pro tip: every contour integral you compute at an exam has a value of plus of minus 2n pi, with n typically being plus or minus one)
For math majors, the next step was measure theory: that definition of integration we started with? Forget it. Lebesgue came up with something that's better. And by the way, those proofs we told you about have glaring holes (let's spend a few hours proving that cutting a rectangle across a diagonal gives you two parts that each have half the rectangle's area). A few of the functions that we proved cannot not be integrated can be, if you use this new definition of how integration works.
That seems a waste of time, but I don't think it is. Very few people can jump in at the deep end and survive; the first iterations are necessary to prepare one's mind for the next steps.
Given the above, I think you will have had various 'pre-' things, even though they weren't named so.
"pre-calculus" is whatever isn't covered in intro to algebra and geometry, but needed for differentiation and integration:
https://www.khanacademy.org/math/precalculus
* Trigonometric equations and identities
* Conic sections
* Vectors
* Matrices
* Imaginary and complex numbers
* Sequences, series and induction
* Probability and combinatorics <-- more of a side topic, usually.
After Geometry we had Algebra 2 (quadratics, etc.), then Advanced Math (trig, pre-calc). To my detriment I made up "pre-trig" to illustrate being thrust back into a cold curriculum after (what I felt were) warmer concepts in Geometry.
Then the solution is simple! We teach calculus to 3rd graders, then no one will be afraid of it!
But seriously, I agree that calculus is taught way to late. I wasn't introduce to calculus until college, and if I had been even by senior year of high school, I would probably be a mathematician right now.
For example, my 8th grader is being taught "Higher School Algebra" by a teacher who last year taught 5th grade and by all accounts has to leave the classroom often to get tips on the material from the 7th grade math teacher.
Boy, am I ever glad!
I am much more financially comfortable, enjoy the hell out of my job, and don't have to deal with the annoyances and politics of being a high school teacher. Would I be technically capable? Absolutely. Would I be good [enough] at it? Probably. Would I do it for a mid (or even high) five-figure sum, even with the good retirement package? Not on your life...
Unfortunately, the fragmented (ie, redlined) us public school system means that higher teacher play is impossible in lower income areas. So private schools will pay more for the calculus teachers, and the low income public schools will continue multi-classing pe teachers for the task. And in another twenty years we will continue to lament the increasing lack of social mobility in this country...
The lowest-income school districts around here pay higher salaries than the nicest districts by a solid 20-30%. Most teachers with options still won't work there because the environment is frustrating, unrewarding, dangerous, and toxic.
I agree it's nice to know the logic and background that is the foundation of how we got _here_, but sometimes the older details aren't really as important and take away time from students learning the modern, more useful techniques and concepts.
One of the big things about kids and math these days is that you have less and less of their attention. If they see it's getting too hard, most don't take the challenge and they just put less effort in.
I don't see how students can truly learn if they don't understand the formulas they're applying and their relevance to the real world.
Like you said, I didn't have to bother memorizing anything, because everything could pretty much be done with simple integration or differentiation.
Please drop me a line if this sounds interesting. Email in profile.
All calculus ever taught me was to apply these formulas to those problems (for some reason) and that dx/dy meant derivative of x with respect to derivative of y.
It boggles my mind that nobody simply bothered to tell us what a goddamned derivative was; what the word meant, and what real-life examples were. The "with respect to" was the most confusing and least useful concept. If we knew what the things were in the first place, the fact that one was "with respect to" the other would have been obvious. And "with respect to" could have been replaced with "relative to" in our minds, because we'd know what the things were.
Calculus seemed to be entirely repeating nonsensical mantras and applying impenetrable methods to problem sets and receiving an answer out the other end.
I'll also echo calls for more time on algebra. Surely the problem with algebra was that I wasn't a devoted enough student to repeating problem sets ad nauseum, but I just never got the 'knack' for it. I understood what I was supposed to be doing, but I was never satisfied with the answer "you'll start getting a feel for what to simplify to what after you do enough problems" as an explanation. It was frustrating to not feel like there was a concrete process to it all.
This made later Calculus and Statistics hard for me, because many exams would simply be 2 enormous problems ; the calculus I could get through but I couldn't simplify the problems quickly enough to be able to apply calculus or statistics at all.
I keep seeing how society needs better math education, whether it be statistics or calculus. Well no shit. Unfortunately those are extremely hard subjects to teach in high school. I'd argue statistics is harder to teach than calculus. In calculus you usually derive things by following the formulas. In statistics there isn't always a formula, logic, or path you can follow.
I took both AP Calc BC and AP Stats in HS and I'll tell you I learned a lot of calculus then and am learning a lot of statistics now (I'm 32).
The huge problem as I mentioned elsewhere is that stats is really applied math. You do learn a lot of regurgitation-style stuff - formulas, proofs, blah blah, but it's all useless without applying it. Moreover, stats draws from so many different areas in math and requires analytical skills that are closer to many of the things scientists emphasize in learning to conduct studies and experiments. This makes it almost unsuitable to teach the more valuable parts of stats at lower levels because the students can't possibly be prepared.
At best, I think you can teach some general things about stats and the mentality I described of being analytical and skeptical. Unfortunately, stats is just really hard for anyone to properly and comprehensively learn who isn't going to be able to invest a lot of time both learning pre-requisites and then all the different areas of stats. It's like learning to be a carpenter and only understanding how to work a hammer, but not a saw, measuring tape, or anything else.
I TA'ed physics at one of the top Ivy League universities in the US, and what I found amazing was that we didn't require calculus for our introductory mechanics course. You don't get into these universities without taking all the highest level courses in high-school, which means that almost everyone in the class must have taken at least one semester of calculus. But we still avoided the simple v = dx/dt calculations, the logic being that physics was too difficult to combine with the math they'd all learned years before.
My friends in economics told similar stories: "Don't worry, we don't have to use calculus to calculate marginal cost, we can use 'the midpoint method'", followed by 20 minutes of explaining an arcane nondeterministic procedure to approximate a derivative.
Math should be motivated by science and modeling, otherwise it's just an exercise in diddling numbers.
I like a lot of what you said, but I do disagree with that point. Immediate application of mathematical concepts may be gratifying, but making it the _only_ motivating factor behind maths can result in students who conflate the underlying machinery with its application [1].
Taking a pure maths course in undergrad with related subject matter prior to one of my Controls classes made the class substantially easier to understand. Learning the abstract concepts beforehand let me see the common applications more quickly than others.
I think the biggest difference is that when I learned some Calculus through Physics, it was a little more difficult to go from a concrete basis to a more general one. My understanding ended up being based on analogies to other concepts until I went back and covered the theory again.
[1] This is all my own opinion, I'm not an educator so the most I can do is pull from my own pedagogical experience :)
While not wrong, that sentence is devoid of meaning, with or without the synonyms swapped out.
I used to think this until I had kids and started teaching them math. When you know math you can abstract it away and really focus on the concepts. But you don't realize how many simple calculations are done when abstracting these concepts.
The problem is that elementary and middle school math teachers can't teach except out of a book. Your child's math teacher isn't teaching your child math, she is just the proctor for their math textbook, which isn't teaching them anything at all really.
This is not a dig on (all) teachers. These workbook-driven curriculum are a band-aid for bad teachers, and all they really do is harm good teachers and students.
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.
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.
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.
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.
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.
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_Ins...
edit:
Where are you getting "many" (rather than "some")? The page you link to discusses colleges in addition to universities, and it's well known that colleges exist primarily to educate students.
> Even if you subtract the research budget out first you will still arrive at a figure less than 50% for most universities.
Where is that numbers coming form?
I am having a hard time finding aggregate data online that supports (or contradicts) my claims on a national level beyond what I have already provided. Unfortunately the edit timer has already expired for my comments so I can't delete or alter them. If you can find reliable data that contradicts my claims reply to my root post and I'll be happy to upvote it.
> Universities are composed of colleges in USA.
This is pretty rare terminology in the US, being mostly found in Ivy league or other old schools. Most "colleges" in the US are independent institutions.
https://studyusa.com/en/a/107/what-is-the-difference-between...
I don't know anything about these people but there's two "good" reasons for this:
* Some colleges don't take certain AP credits e.g. the college I went to didn't take AP Bio credit, so I didn't take the exam for it when I took the class
* Some high schools give GPA boosts for AP classes. For example, an A in an AP class was worth a 5.0 vs a 4.0 in a regular class. Some students may think it's worth it even if they don't do that well
In general I agree with you because my AP Calc class wasn't rigorous or interesting at all. It was "here's how you do X, now do X 100 times for homework and take a test on it".
> A lot of the students in my AP Calculus class had no business being there
Maybe I'm cynical but this was the case at my college Calculus class as well. The entire back row was basically people cheating. There'll always be lazy people in freshman and sophomore courses.
Is this not usually the case? My high school used to make you fill out a short form and make the case for why you should be allowed in the AP class, and my senior year I had to go in and speak with a counselor who was concerned that I was taking too many. Sure, it was a pointless hurdle for me personally, but I can see how it would be a good idea for students who don't realize what they're getting into, or who are trying to take the course because their parents want them to.
That's too bad. Either his parents were paying the cost of the test (I think $86 per test when I was in school) or he was getting a paid-for test from some scholarship or another.
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 (it's almost finished; just beefing up the problem sections).
Take a lower triangle matrix multiply it by a vector, and "boom" you just explained causal convolution. Wash and repeat for several other topics.
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.
I don't think you can get rid of Calc in HS though. It's just too useful. I think room can be made for both subjects, though.
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.
I wished they were more consistent
1. Young children seem to have the biggest capability in learning languages.
2. There's the John Stuart Mill example -- homeschooled by his father in the classics and philosophy. Interesting biography (Wikipedia has a fair account).
3. There are children who are naturally skilled in art, storytelling, maths, sciences, music, etc. An opportunity to work to those strengths might be useful.
I've also noticed that a short period of instruction at the appropriate time often seems disproportionately effective.
Montessori schools are said to be better on this score, though I haven't looked into them.
Neal Stephenson's Cryptonomicon, The Baroque Cycle, and Anathem, for example, are all explorations of ideas. The first two of business, banking, and money, in contemporary and historical times, respectively. The latter covers a great deal of historical thought.
Better authors of topics often incorporate humour in their writing -- someting the past few generations of academic literature would do well to consider. OTOH, it's a cure for insomnia.
Is there any evidence that this is actually due to their capability? I would say it's far more likely it's their environment. IE, people are constantly talking to young kids. They simplify what they say to make themselves understood. And children have a huge incentive to learn to communicate - to get what they want from parents, or just understand the world around them. The same isn't true for a grown adult trying to learn a second language, in isolation, that they don't really need in their day to day lives.
Disclaimer: this is pure conjecture.
I'm also aware of a few other age/skill type relations.
Musical ability frequently manifests early -- by age 4-5 in numerous historical cases.
Talents requiring physical coordination -- ballet and gymnastics -- around age 9-10 by Russian tradition.
Maths and scientific curiosity perhaps early adolescence.
Complex logic and activities requiring a fully developed inhibitory function (avoiding reflex and rash actions) possibly early 20s.
I'm suspecting other forms of more synthetic (as in: assembling from parts) logic might not develop fully until the 30s or 40s. Other skills might similarly not develop until later.
Let me qualify the above a little. You might ask, well what are the purpose of high-school teachers or university lecturers? Once we are given these tools shouldn't we just be given the books to progress on our own? Hopefully, teachers are people who can take a complex subject and guide us towards an understanding of that subject. To me teachers are like guides through a jungle of knowledge. We need a guide so that we don't get lost. For me, a teacher is part guide and part "composer of problems". I consider teaching to be guided problem-solving. When I am teaching a course, I pose problems, whether it be in Math or Chemistry, and guide the students as they try to solve those problems.
And as for your comment about "wasted time", I am not sure what to think about this. The difficulty with learning is that we need repetition and reinforcement. Yes, I see a lot of pointless repetition, but we do need some repetition. I am presently struggling with improving my chess skills, as well as trying to learn to program an Arduino. Mastering these skills require a lot of repetition in order to cement that knowledge. Wish me luck.
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.
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.pd...
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.
SymPy is the best CAS I know of. You can use the methods like expand, factor, collect, and solve on any expression.
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.
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.
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.
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.
Also, geometry gives students something concrete to visualize, and so might be easier for some to understand than more abstract kinds of math.
Maybe trig is over-emphasized at the high school level, but it is pretty useful in a lot of real-world situations, and it's easy to apply without having to know a lot of other stuff.
I hope that's not universally true, and that proofs are still actively taught in some high schools.
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.
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.
What gets measured gets managed.
I remember trying to figure out how the Cha Cha Slide could be a decent Calculus problem. How does the song change the pattern?
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.
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.
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.
if you think about it, it's just a rephrasing of 'survival of the fittest'.
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.
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?
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
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.
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.
http://www.artofproblemsolving.com/articles/calculus-trap
http://www.artofproblemsolving.com/articles/discrete-math
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.
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.
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.
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.
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.
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
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 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.
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.
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.
I recently-ish helped my stepdaughter with an algebra assignment where she was asked to find the maximum of some polynomial. I totally blanked on how to do this with only the tools that she was supposed to have. I didn't know how to do it or explain it without calculus.
a(x - b)^2 + c
In lucky cases, fourth-order polynomials could be put in such a form. Odd-degree polynomials will not have a unique maximum.
Further, we should all remember to draw a graph as our first step. Symbolab, Desmos and Geogebra (my favourite) are all fantastic graphing software.
Also, for a quadratic if you know the zeroes (roots), it's half-way between them. So for the equation -(x-1)(x-5) = 0, the roots are x=1 and x=5, and the mid-point (maximum) is at x=3. Your graphing will confirm this.
Your stepdaughter might also have learnt that the mid-point of a quadratic ax^2 + bx + c = 0 is x=-b/2a. (This can be derived by setting the derivative equal to zero, but is generally just given to the students.) Her teacher may be expecting her to use that formula.
Alternatively the students might be required to determine and plot various data points using, say, Excel, and find the maximum this way.
The other point is that solutions often do not spring to mind immediately. That is why I ask students to send me their problems before our tutoring sessions. I often need time to think about them.
The fun of math is this exploration to find an answer. So try to get problems and give yourselves time (days) to explore them together.
To be honest I don't really care what kind of Mathematics they teach: I just want it to be taught with passion and rigor such that the students gain insight and a better understanding of how the world is put together; how to solve problems and express ideas.
Before even starting on algebra