Do the math: too much calculus? (2012)

https://www.washingtonpost.com/blogs/class-struggle/post/do-the-math-too-much-calculus/2012/08/16/9a35034e-e4fd-11e1-8f62-58260e3940a0_blog.html

by carlosgg • 10 years ago

104 212 10 years ago

washingtonpost.com

spike0213• 10 years 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• 10 years 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• 10 years 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• 10 years ago

imperfect incentives/objectives + strong competition = perverse outcomes

if you think about it, it's just a rephrasing of 'survival of the fittest'.

hprotagonist2• 10 years 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.

baldfat5• 10 years 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.

hprotagonist2• 10 years 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.

baldfat5• 10 years 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-...

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.

CalChris2• 10 years 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• 10 years ago

Circuit analysis turns differential equations into algebra, so you might have been using calculus more than you realize.

dboreham3• 10 years ago

That really makes no sense, unless the "EE" part was optional in your case.

baldfat5• 10 years ago

The beauty of calculus is that you actually understand the Real World.

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."

surrey-fringe• 10 years ago

Berkeley's a really good university.

wfo• 10 years ago

Berkeley lets you skip learning asymptotic runtime analysis of algorithms if you're doing EECS? Weird.

FYI: infinite sums are calculus.

rhizome2• 10 years ago

I would have done much better in math if the class after Geometry was something other than pre-trig.

jameshart3• 10 years ago

As an alien to the US education system, I am always completely confused by these 'pre-' course designations. What on earth is 'pre-trig' or 'pre-calc'? I mean, I got all the way up to undergraduate level in mathematics and I never needed to learn 'pre-' anything.

kj01a• 10 years ago

BUT people are always afraid of anything past 4th grade math!

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.

dboreham3• 10 years ago

In the US, problem #1 is that it is very hard to find people who understand Mathematics sufficiently well to teach it effectively, who are willing to work for the salary typically paid to a teacher.

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.

M_Grey• 10 years ago

We should teach early calculus along with entry-level SR/GTR instead of rehashed Newtonian crap. Unite math and physics for a while, and kids might actually see the point of it.

YZF• 10 years ago

Exactly. Physics becomes so much simpler with calculus. Once you realize e.g. the relationship between position, velocity and acceleration you don't need to memorize 20 different equations for all the different flavours of problems that are basically down to integrating or differentiating ...

yequalsx3• 10 years ago

I think that a student not well versed in how to perform calculations will not do well with the theory. Last night in the calculus class I am currently teaching a student could not correctly translate the fact that f is increasing where f' > 0. This on a problem where f' was linear. He just stared blankly at the inequality 2x-3>0. Anyone well versed in this sort of calculation is not going to fumble around at this problem. He thinks calculus is the problem when it's his knowledge of algebra that hinders him.

ivan_ah5• 10 years ago

I'd be very interested to hear your opinion about a book I've been working on. It's a 3-in-1 combo, math+mech+calc, I wanted to teach mechanics and calculus in an integrated manner and I realize, from 15 years of private tutoring, I need to review all the high school math material because many students don't know the basic.

Please drop me a line if this sounds interesting. Email in profile.

rconti4• 10 years ago

IME the #1 problem is the failure to teach anything WELL. I took AP Calculus in high school, and today, as a 35-year-old, I'm still continually surprising myself with the problems I realize must be related to Calculus-- but nobody bothered to tell me. Things like the 'ol speed vs acceleration vs jerk kind of problems.

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.

eanzenberg3• 10 years ago

Dude, the "with respect to" clause is a pretty deep idea in calculus that really matters in multi-variate but not so much in uni-variate.

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).

dguest• 10 years ago

When calculus is taught badly no one knows how to apply it even when the problems are staring them in the face.

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.

posterboy5• 10 years ago

This pattern of patterns and structures is the structure of structures

While not wrong, that sentence is devoid of meaning, with or without the synonyms swapped out.

kenjackson2• 10 years ago

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.

debacle• 10 years ago

BUT people are always afraid of anything past 4th grade math!

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.

Yhippa• 10 years 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• 10 years 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.

castratikron2• 10 years 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• 10 years ago

Isn't it kind of weird that a college won't let you take whatever math class you want?

gnarbarian5• 10 years 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.

castratikron2• 10 years ago

Exactly!

ebola1717• 10 years ago

At my university, they would usually let students sit in on any lecture classes they wanted. But students taking classes they aren't qualified for isn't the most efficient use of educational resources

sotojuan2• 10 years ago

another guy didn't even take the AP exam, and another guy just put his head down on the desk during the AP exam.

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.

dataharry• 10 years ago

How do you cheat at Calculus? Asking for a friend.

wutbrodo2• 10 years ago

There should be some vetting process for the students who want to take AP classes.

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.

sotojuan2• 10 years ago

Depends on the school district and size of the school. For example, Texas high schools are for the most part huge and can accommodate anyone who wants to take AP classes without question. We did check with our counselor but they were more than happy students took AP classes.

spike0213• 10 years ago

another guy just put his head down on the desk during the AP exam

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.

rconti4• 10 years ago

Most of my high school classes in the later years were AP, and there was no pressure to take AP tests we didn't see the value in. shrug

eanzenberg3• 10 years ago

Same here. I took both AP Stats and AP Calc BC in high school and I'd be more worried about stats being taught wrong to a bunch of teenagers than calculus.

zumu• 10 years ago

This sounds like a defense of AP classes and not calculus in particular. Mind you there are also other AP math courses, notably statistics.

gnarbarian5• 10 years 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

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• 10 years 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• 10 years 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.

dredmorbius4• 10 years ago

Source?

douche2• 10 years ago

What infuriates me, looking back on it, was simply how much of my time the educational system was allowed to waste. Thousands and thousands of lost hours, some of them filled with pointless busy-work, but many more just squandered on inanity. At least some of the teachers would let you finish the daily penance quickly, and sit quietly in the corner reading something interesting. Too few, though.

pirocks2• 10 years ago

Learning cursive comes to mind.

pdm553• 10 years ago

I look at this way: early education focuses on learning to read and write and do arithmetic. The basic purpose is to give you tools whereby they can continue to educate yourself throughout your life. So basically, our early education aims to put our future education into our own hands.

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.

AstralStorm• 10 years 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• 10 years ago

You can't possibly understand calculus without first understanding trigonometry and, to a lesser extent, geometry

mmmpop• 10 years 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• 10 years 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.

thesimpsons10223• 10 years 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• 10 years 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.

douche2• 10 years ago

I always found trig and geometry to be the most interesting and useful part of math. It maps pretty well onto real things that you might have to do. Maybe this is more relevant in the rural area I grew up in, but geometry and trig is really useful when you're trying to do carpentry, or figure out if you've got room enough to chop down a tree, pathfinding on a map, etc.

elihu2• 10 years ago

For a lot of people, geometry is their primary or perhaps only exposure to proofs. Not that everyone needs to understand proofs in their day-to-day lives, but it's a good mental discipline to at least be exposed to, and geometry proofs are fairly straightforward and relate to things that many people will have a good intuition to convince them that the thing proved by each correctly-constructed proof is really true.

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.

analog313• 10 years ago

From what I can tell, proofs are gone. I got my daughter all excited about geometry, by promising her that she could do proofs. The curriculum was almost exclusively focused on problem solving based on remembering formulas. She went ahead and did all of the proofs anyway.

ivan_ah5• 10 years 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 (it's almost finished; just beefing up the problem sections).

FT_intern• 10 years ago

Thinking in matrices and vectors is much more complicated and abstract than calculus.

microDude• 10 years 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• 10 years 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• 10 years ago

I'm guessing you did applied math in grad school? I think very few (pure) math majors had any issue with calc.

tnecniv2• 10 years ago

LA wouldn't be any harder to teach than Calculus. In fact, if you wanted to teach proof based math that isn't Geometry, it's probably the easiest subject area.

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.

dver23• 10 years 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.

euroclydon3• 10 years 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.

Oletros2• 10 years ago

AS an Spanish not knowing the US education system, at what age one kid goes to 5th grade?

euroclydon3• 10 years ago

My 5th grader is currently 10 years old.

colechristensen• 10 years ago

The median age in 1st grade is 6.

Oletros2• 10 years ago

Thanks to both of you

ivan_ah5• 10 years ago

You should also consider SymPy, available as a webapp 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.pd...

euroclydon3• 10 years 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.

pdm553• 10 years ago

Have you tried Geogebra?

hanoz• 10 years 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.

ThrustVectoring2• 10 years 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.

0xCMP2• 10 years ago

Good ones at least.

LgWoodenBadger• 10 years 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• 10 years 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.

jpfed2• 10 years ago

Finding the maximum of a relation using Calculus is much easier than strictly using algebra for this reason

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.

mturmon• 10 years ago

If it was a quadratic, you could (in effect) complete the square and thereby get it to a form like

  a(x - b)^2 + c

At that point, you get the answer by inspection. I.e., in this case, if a is positive, then the function is minimized when x = b, and the minimum value is c.

In lucky cases, fourth-order polynomials could be put in such a form. Odd-degree polynomials will not have a unique maximum.

pdm553• 10 years ago

Use a tool like Symbolab. It shows you the steps for different methods. Very useful.

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.

dboreham3• 10 years ago

I came to the US (where I have children in school) from Scotland (where I obtained my education). US K-12 Mathematics education drives me nuts. Peeve #1 is the idea that the subject should be divided into these seemingly separate subjects : Algebra, Geometry, Calculus and so on. These are not different subjects -- they're all deeply interconnected, and furthermore : observing the connected nature of seemingly disparate areas in Mathematics is one of the key insights to be had.

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.

anu7df• 10 years 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• 10 years 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.

strathmeyer• 10 years 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• 10 years ago

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

japhyr• 10 years 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_ah5• 10 years 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

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• 10 years ago

Rubik's cube can be used to teach group theory, see http://geometer.org/rubik/group.pdf or http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20an...

Koshkin• 10 years 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• 10 years 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• 10 years 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.

yequalsx3• 10 years 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.

rconti4• 10 years 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.

krschultz2• 10 years 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.

lordnacho3• 10 years 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.

ghaff2• 10 years 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.

posterboy5• 10 years ago

As if algebra or calc was common sense. I did learn about the Monty Hall Paradox but I'm still doubting it

spynxic• 10 years ago

I don't think it's possible to make the assertion that one branch of math is more relevant than another, given that the set of students have diverse interests.

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).

sn9• 10 years ago

Is it really a question of focus, though?

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.

asdf333• 10 years ago

One issue is statistics can be super boring if taught incorrectly. I am an engineer too, loved probability but found statistics to be 'shoot me in the head' boring.

the class really needs to be taught well for people to understand it. also. i think it should include probability too.

optionalparens4• 10 years ago

I agree, but I thought most lower level classes were often titled things like "Intro to Probability and Statistics." Further, it's been my experience meeting and being around a lot of people who thought they knew stats that they only learned some basics which included but not limited to some basics such as: factorials, normal distributions, coin flips, t-test, permutations, combinations, etc.

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.

johan_larson• 10 years ago

Yes, stats. Also logic.

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...

posterboy5• 10 years ago

Statistics is relevant to more fields than calc

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.

optionalparens4• 10 years ago

One of my degrees is in statistics, and the others are in computer engineering, finance, and computer science. As such I not only took a ton of math, but studied and worked in the contexts you describe where stats is relevant.

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.

kenjackson2• 10 years ago

Where I went to school there was an undergrad stats sequence for engineers, which was mathematically more intense than the general stats sequence. But it still didn't use much, if any, calculus. And I think in general I'm on board with most here that believe that for most people who aren't going to be engineers, a year of stats is more valuable than a year of calculus.

analog313• 10 years ago

For students who aren't going to learn calculus, I wonder if a better introduction to prob and stats (and perhaps math in general) would be through simulation and visualization rather than learning formulas by rote.

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.

pps43• 10 years ago

The problem is, it's hard to understand statistics without knowing at least basic calculus. As soon as you move from coin flips to probability density, you need derivatives and integrals.

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.

danielmorozoff4• 10 years ago

Though i think pdfs are useful, they may be a bit advanced.

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?

patrickthebold• 10 years ago

There's a difference between defining the integral,then taking about how it's estimated numerically and to use some software to get numbers.

Trigonometric substitute to solve analytically.

For example.

krschultz2• 10 years ago

That's a good point. I'm not an educator and it's been a while so I can't recall the sequence of stats 101, calc 1, more stats, more calc, etc. It all blurs together at some point.

analog313• 10 years ago

I had kind of a strange experience. While in college, I worked as a math tutor. Since I had taken one semester of stats, the tutoring center decided I was qualified to run a tutoring session for the non-calculus stats course. I was happy for the money, and had been reasonably successful with other tutoring assignments.

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.

amluto• 10 years ago

I would love to see statistics taught more universally. But there's statistics and statistics.

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.

apricot• 10 years ago

There are good stats classes and bad stats classes, just like there are good calculus classes and bad ones. When I was a student, we spend a full term learning integration techniques, with almost no class time devoted do learning what an integral actually is. The fundamental theorem of algebra was glossed over in a way that makes me think the lecturer didn't really understand it. Differential equations was a catalog of recipes, with almost no modeling and certainly no qualitative analysis. I got an A+ in the course but learned almost nothing from it.

Fortunately, I had a great professor in Calc III who more than made up for this.

orangutango• 10 years ago

Completely agree, and this hits close to home.

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.

SiVal• 10 years ago

This is an area where I think the usual proper math approach of abstract patterns and theorems and proofs is all wrong for the general public (the ones in a general public school).

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.

eli_gottlieb2• 10 years ago

The problem with trying to teach statistics, to anyone really, is that once you get past the very basics, you have to stop every time the lecturer/textbook invokes measure theory and figure out how to explain the concepts without just making everyone sit through two to three semesters of real analysis.

hilop2• 10 years ago

You just use discrete scenarios, not continuous. Same concepts, easier math.

same way you can do linear geometry without calculus.

thearn42• 10 years ago

Agreed, there is definitely a huge gap between doing, for example, a t-test and deriving the t-test (or giving any substantial justification for its form). You need a class in analysis, a class in measure-theoretic probability, then a mathematical statistics class to bring you back to the topic. It's a tough square to circle.

Most who teach stats at the undergrad or hs level have little choice but to defer a lot of things to a later course.

cageface• 10 years ago

Linear algebra is also a lot more useful in real world applications than calculus. As it stands calculus seems as much like some kind of college hazing ritual as it does an essential part of anybody's education.

JohnStrange• 10 years ago

But doesn't understanding statistics require a solid understanding of calculus anyway?

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.

CN7R• 10 years 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• 10 years 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• 10 years 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• 10 years 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• 10 years 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

danielmorozoff4• 10 years 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

tnecniv2• 10 years 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• 10 years 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• 10 years ago

I do recommend to read these articles:

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.

thearn42• 10 years 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• 10 years 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• 10 years 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.

Do the math: too much calculus? (2012)

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