Calculus vs Statistics

[Septimus Prime]

This thread was inspired by the latest, Levitt-hosted (Levitt episodes are the best) Freakonomics about revamping the math curriculum.

One of the points that stuck me was that, apparently in the UK, statistics play makes up a significant portion of high school math. Meanwhile, in American high school, while I did actually take both, we had it arranged so that calculus was in the main line of math classes, and stats was an elective.

I took a few more classes in both lines, but hell if I can still do a lot of the calculations now.

But if you could only choose one, which is now important? I'm leaning toward stats, since its concepts seem to play more of role in understanding the world.

 

[Masoyama]

Calculus all day.

I literally use it every day, can't even begin to think how the world would run if people started dropping its importance.

 

[bobnowhere]

If I had to choose one, it would be stats. Calculus can be "cheated" in the background but if you need to present any sort of data and don't have an understanding of stats it really shows. Stats also crosses far more disciplines than calculus. In an ideal world it would be both and linear algebra.

 

[Big Boss]

Honestly, personal finance should be taught way before that shit.

But to stay on topic. Calculus.

 

[Deleted member 12790]

Calculus is basically the fundamental building blocks on all sorts of mathematics that govern the universe. To give a comparison, if addition, subtraction, etc. are learning the alphabet, and algebra is learning how to use those characters to create words, then Calculus is learning how to write essays, form arguments, structure a paragraph, etc.

i say this as someone who has spent nearly his entire life mystified and, yes, even scared of calculus. As someone who picked it up later in life -- and actually learned probability and statistics in middle school and highschool -- I wish it had been flipped. Calculus is everywhere and we just don't realize it because people are taught it so poorly that they don't even recognize it.

My introduction to calculus was trying to bite off more than I could chew in college and taking Differential and integrated calculus as a single course, taught to a class of 500, with a professor with a thick chinese accent, and a TA with a thick russian accent. I got rocked by that class. I wish I had learned it first in a relaxed, calm, slow environment.

 

[Acorn]

Is calculus just basic arithmetic? British and I've never heard of it outside of TV shows so was never sure what it was.

 

[CHC]

Statistics by a long shot. Calculus gave me a deeper appreciation for mathematics and the order of the universe in general, but statistics knowledge is basic and invaluable.

An example: It's kind of amazing that if something with a 1% chance of happening happens, soooo many people still say that the original chance of it happening was "wrong." And things like people not knowing the difference between a median and a mean, or what a standard deviation is, the list goes on. I really don't mean to sound condescending or anything but it's just that stats intersects with almost every field in some way, and it's very useful to have a basic grasp of what you're looking at when you're presented with numerical data.

I feel like learning stats is like learning a social science and calculus is like learning philosophy. It's less practical (for most people) but it opens up deeper gulfs of understanding and illuminates otherwise hidden structures.

 

[Deleted member 12790]

Is calculus just basic arithmetic? British and I've never heard of it outside of TV shows so was never sure what it was.

Click to shrink...

calculus is the study of change

it's the type of mathematics that isn't concerned with (solely) discreet numbers, it's the mathematics of curves, instances in time and space, motion, etc. it's mathematics that can't be understood in single instances, but need to be seen over a series to discern the pattern.

Classic calculus example: find the area of a curve. Real life curves are non-discreet figures, boolean logic is governed by discreet values, and calculus is the mathematical attempt to reconcile this incongruity. You break the curve up into small discrete chunks, and thus "approach" a real answer. The smaller the chunk, the more "correct" the answer.

Calculus is where you start to deal with those logical paradoxes you might have once thought up, like "in order to walk across the room, I need to walk half way across it first. In order to walk half way across the room, I need to walk a quarter of the way first. In order to walk a quarter of the way across the room, I need to walk an eighth of the way first. Repeated infinitely, it is thus impossible to walk across the room."

It's pretty much the opposite of basic arithmetic, although use use basic arithmetic to do calculus.

watch this:



Calculus gets really, really cool when you start applying it to advanced concepts. anything that operates on a spectrum is where calculus shines. That goes for things like light and physics -- you'll find a ton of calculus if you go into hardcore graphics studies for example, or things like motion detection, or brain interfacing, any sort of time you need to understand the relationship (read: ratio) between one phenomenon and another phenomenon, you'll likely be using calculus in your study. Any time you hear words like "per" or "rate" or you deal with things "over time," you're dealing with calculus.

A super easy example everyone intuitively knows but doesn't even realize it's calculus: you're driving a car, and someone is slowing down in front of you. you don't directly control your cars position in space through time, rather you only control it's accleration and speed. By understanding the relationship between speed, accleration, time, and distance, you can calculate in your brain how to work your foot to slow your car down over time to avoid ramming into the car in front of you. You can work out each component of that problem to better, more minutely understand what is going on, and doing that is called calculus.

 

[Ashdroid]

I chose to take stats in high school instead of calc (1 semester of stats and 1 of trig instead of 2 semesters of calc), and I kind of regret it, even though they were more interesting to me. It meant I had to play catch up in university. Calc was basic required knowledge, and stats and trig didn't really matter at all.

A basic understanding of stats is definitely more useful in day-to-day life outside of schooling and STEM fields, though.

 

[mbpm]

An example: It's kind of amazing that if something with a 1% chance of happening happens, soooo many people still say that the original chance of it happening was "wrong."

Click to shrink...

You mean to say that when a tails result in coinflip is estimated to be 50% and it lands tails, it should not, in fact, have been estimated to be 100%?

/s

 

[ChrisR]

I say calculus, but I hated stats. Sitting in Calc 1 and having all of physics start to make sense with derivatives kinda blew my mind.

 

[Acorn]

calculus is the study of change

it's the type of mathematics that isn't concerned with discreet numbers, it's the mathematics of curves, instances in time and space, motion, etc. it's mathematics that can't be understood in single instances, but need to be seen over a series to discern the pattern.

Classic calculus example: find the area of a curve. Real life curves are non-discreet figures, boolean logic is governed by discreet values, and calculus is the mathematical attempt to reconcile this incongruity. You break the curve up into small discrete chunks, and thus "approach" a real answer. The smaller the chunk, the more "correct" the answer.

Calculus is where you start to deal with those logical paradoxes you might have once thought up, like "in order to walk across the room, I need to walk half way across it first. In order to walk half way across the room, I need to walk a quarter of the way first. In order to walk a quarter of the way across the room, I need to walk an eighth of the way first. Repeated infinitely, it is thus impossible to walk across the room."

It's pretty much the opposite of basic arithmetic, although use use basic arithmetic to do calculus.

Click to shrink...

Ah, ok I know what you're talking about now. Algebra etc

 

[GamerJM]

If you end up doing some kind of engineering calculus is more important, but I feel like for the average person stats comes up more often in everyday context.

 

[MajesticSoup]

I'd say calculus only cause it kicks your ass in college and a lot of students are unprepared.
First year stats is like a course you take to fluff up your gpa.

 

[turbobrick]

Things must have changed in the last 15 years, or I just went to a terrible high school. Only the really smart kids took calculus in high school, everyone else only went up to algebra 2, trigonometry, or maybe pre-calculus(which barely touches on calc subjects, but mostly just prepares you for it).
I personally didn't take calculus until my second semester of college.

I feel like for high school, a basic understanding of stats would be more helpful for the average person.

 

[jviggy43]

For average person its certainly statistics.

 

[Terra Firma]

Stats has a much wider application than calculus.

 

[Septimus Prime]

Keep in mind I'm talking about primary school, though, so it would still be pretty surface level courses (like basic, single variable calculus or stats interpretation where the calculations are already done). What's more important up understand superficially?

 

[ash32121]

I don't know which is more important but Stochastic is kicking my ass right now

 

[Deleted member 12790]

Ah, ok I know what you're talking about now. Algebra etc

Click to shrink...

not really, algebra is used to solve calculus. Algebra is more like the "how," calculus is more like the "why" so to speak. Calculus is more like a set of principles and tools and theories you use, to take situations that present with no hard figures, to turn them into a series of algebra problems, that when we solve for them all, gives us a picture into the larger problem we solve.

Just like how algebra uses addition, subtraction, multiplication, etc to solve it's problems, calculus uses algebra to solve its problems.

 

[Acorn]

When I was in primary school we never did anything further than addition, subtraction, multiplication and division. Unless I'm forgetting.

 

[Chikor]

If you want to understand physics on any serious (or even semi-serious) level, you need calculus, but outside that, there are very few practical application for calculus that the average person will use in their life.
Statistics is way more useful.

I still think it's a good thing to teach calculus, because school should not be exclusivity about teaching things that have obvious practical application. But if we talk about usefulness for the average person, it's not close.

 

[GK86]

I'm taking stats right now...fuck stats.

 

[Tamerlane]

You need a solid understanding of calculus before you can learn statistics

 

[Ionic]

I feel like an understanding of calculus would help people interpret information and the world even more than a study of statistics would. Basically all quantities in the world are dynamic, yet you really have to train people to understand the concept of the rate. I don't even think you need to necessarily go into Calc 2 integral and series stuff to greatly increase the population's analytical abilities, just knowing what a derivative is and how change relates to the amount of something over time would mean a lot.

On the other hand, basic literacy in understanding statistical plots would be very helpful too.

 

[Deleted member 12790]

Keep in mind I'm talking about primary school, though, so it would still be pretty surface level courses (like basic, single variable calculus or stats interpretation where the calculations are already done). What's more important up understand superficially?

Click to shrink...

I went to a magnet school funded by NASA, not a regular middle or elementary school. They did a random study where a group of students was admitted into a course intended to teach them calculus by the 5th grade. I wasn't a part of that group, but a bunch of my friend were. They are all amazing engineers currently. Like some of the smartest math people I know, and the purpose of the study wasn't to pick students who were good at math to begin with.

I feel like learning calculus as early as possible is incredibly important, because it seems like it's one of those things that becomes much more natural, and useful, the earlier you learn it. Myself, I struggled with calculus so mightily. But in my mind, I always wonder if I would be amazing at it had I been selected for that study.

 

[Deleted member 12790]

You need a solid understanding of calculus before you can learn statistics

Click to shrink...

This is a good point, too. Knowing calculus makes you understand statistics way better. The opposite isn't nearly as true.

 

[Brashnir]

Voted statistics, because the complete lack of understanding regarding probability is among the biggest ills infecting modern society.

 

[Septy]

Calculus is important for the study of physics. So calculus

 

[Yourfawthaaa]

Took and passed statistics few months ago.

It was confusing at first but i locked in and something just clicked.

 

[Kieli]

Stats is more important and more applicable for the daily person. I am also convinced it's harder at the freshman and sophomore level than calculus. I've taken up to vector calculus and I've never had my mind blown the way I did with probability and statistical/machine learning.

 

[Acorn]

not really, algebra is used to solve calculus. Algebra is more like the "how," calculus is more like the "why" so to speak. Calculus is more like a set of principles and tools and theories you use, to take situations that present with no hard figures, to turn them into a series of algebra problems, that when we solve for them all, gives us a picture into the larger problem we solve.

Just like how algebra uses addition, subtraction, multiplication, etc to solve it's problems, calculus uses algebra to solve its problems.

Click to shrink...

Ah ok, thanks.

Looking it up alot of what is in American calculus is just in physics here.

 

[Deleted member 12790]

I feel like an understanding of calculus would help people interpret information and the world even more than a study of statistics would. Basically all quantities in the world are dynamic, yet you really have to train people to understand the concept of the rate. I don't even think you need to necessarily go into Calc 2 integral and series stuff to greatly increase the population's analytical abilities, just knowing what a derivative is and how change relates to the amount of something over time would mean a lot.

On the other hand, basic literacy in understanding statistical plots would be very helpful too.

Click to shrink...

This is a great article about teaching math as art: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

I have this other math book I read on intro to calculus and what not, and one of the opening chapters goes into the history of negative numbers, and how for the common person until the 1800's, negative numbers was considered a very, very advanced math topic, things that only people in very technical fields would know. The book uses this as an example of how over time, through changes in education, what once seemed like a useless, and extremely hard type of knowledge has become basic. The point it is trying to make in the opening is how approaching calculus in a different manner, over generations, might yield impossible to predict gains.

 

[Deleted member 12790]

Ah ok, thanks.

Looking it up alot of what is in American calculus is just in physics here.

Click to shrink...

Well, physics are applied calculus. Physics are a specific application of calculus, usually involving time and constants like gravity. You can use that same type of math in much more abstract ways, you don't have to consider your samples (a discreet instance of the spectrum you're analyzing) over time. They can be over dimensions, over anything. Physics are just one of the most easily understood applications of calculus because we can readily see it and study it.

Every physics course I've ever taken has had a calculus per-requirement. You can get much more specific about all the math you are talking about, like calculus is usually (but not always) concerned with linear algebra.

 

[Ionic]

Ah ok, thanks.

Looking it up alot of what is in American calculus is just in physics here.

Click to shrink...

Calculus is in our physics too (in fact calculus was created largely to answer specific questions in physics the contemporary math at the time couldn't solve), but it's important to note that calculus is not necessarily physics. It's a type of math(s) that happens to be used to explain a lot of what shows up in physics from the relationship between kinetmatic equations to Maxwell's laws and more. Calculus can exist entirely independent of physics. After all, dy/dx has no physical meaning until you ascribe it one.

This is a great article about teaching math as art: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

I have this other math book I read on intro to calculus and what not, and one of the opening chapters goes into the history of negative numbers, and how for the common person until the 1800's, negative numbers was considered a very, very advanced math topic, things that only people in very technical fields would know. The book uses this as an example of how over time, through changes in education, what once seemed like a useless, and extremely hard type of knowledge has become basic. The point it is trying to make in the opening is how approaching calculus in a different manner, over generations, might yield impossible to predict gains.

Click to shrink...

I'll give this a read soon. But it really hits at the point of how powerful mathematical concepts are to learn. Like how the words we know allow us to describe things better, the math we know allows us to quantify and analyze things better. We spend a lot of time bolstering our qualitative abilities, but we could also go much further in expanding our quantitative abilities. Both allow us to interpret the world much better. Your example with the negative numbers is great. How many descriptive abilities would we lose if we couldn't internalize negative numbers? How would we describe our debt?! Well, perhaps there are better examples out there...

 

[Deleted member 10629]

Calculus, but I hate both.

Every physics course I've ever taken has had a calculus per-requirement. You can get much more specific about all the math you are talking about, like calculus is usually (but not always) concerned with linear algebra.

Click to shrink...

My Physics professor of college refused to teach any physics before he made sure we understood calculus.

 

[SRG01]

Calculus is basically the fundamental building blocks on all sorts of mathematics that govern the universe. To give a comparison, if addition, subtraction, etc. are learning the alphabet, and algebra is learning how to use those characters to create words, then Calculus is learning how to write essays, form arguments, structure a paragraph, etc.

i say this as someone who has spent nearly his entire life mystified and, yes, even scared of calculus. As someone who picked it up later in life -- and actually learned probability and statistics in middle school and highschool -- I wish it had been flipped. Calculus is everywhere and we just don't realize it because people are taught it so poorly that they don't even recognize it.

My introduction to calculus was trying to bite off more than I could chew in college and taking Differential and integrated calculus as a single course, taught to a class of 500, with a professor with a thick chinese accent, and a TA with a thick russian accent. I got rocked by that class. I wish I had learned it first in a relaxed, calm, slow environment.

Click to shrink...

I agree 100% with this. A lot of people say that statistics is more relevant, but Calculus underpins a lot of readily observable physical processes out there.

Hell, understanding rates is fundamental on a day to day basis.

 

[Dr. Nothing Loud]

calculus is the study of change

it's the type of mathematics that isn't concerned with (solely) discreet numbers, it's the mathematics of curves, instances in time and space, motion, etc. it's mathematics that can't be understood in single instances, but need to be seen over a series to discern the pattern.

Classic calculus example: find the area of a curve. Real life curves are non-discreet figures, boolean logic is governed by discreet values, and calculus is the mathematical attempt to reconcile this incongruity. You break the curve up into small discrete chunks, and thus "approach" a real answer. The smaller the chunk, the more "correct" the answer.

Calculus is where you start to deal with those logical paradoxes you might have once thought up, like "in order to walk across the room, I need to walk half way across it first. In order to walk half way across the room, I need to walk a quarter of the way first. In order to walk a quarter of the way across the room, I need to walk an eighth of the way first. Repeated infinitely, it is thus impossible to walk across the room."

It's pretty much the opposite of basic arithmetic, although use use basic arithmetic to do calculus.

watch this:



Calculus gets really, really cool when you start applying it to advanced concepts. anything that operates on a spectrum is where calculus shines. That goes for things like light and physics -- you'll find a ton of calculus if you go into hardcore graphics studies for example, or things like motion detection, or brain interfacing, any sort of time you need to understand the relationship (read: ratio) between one phenomenon and another phenomenon, you'll likely be using calculus in your study. Any time you hear words like "per" or "rate" or you deal with things "over time," you're dealing with calculus.

A super easy example everyone intuitively knows but doesn't even realize it's calculus: you're driving a car, and someone is slowing down in front of you. you don't directly control your cars position in space through time, rather you only control it's accleration and speed. By understanding the relationship between speed, accleration, time, and distance, you can calculate in your brain how to work your foot to slow your car down over time to avoid ramming into the car in front of you. You can work out each component of that problem to better, more minutely understand what is going on, and doing that is called calculus.

Click to shrink...

I'm an engineer and I agree with all this, except I still come to the conclusion that statistics and probability are more practical and useful skills in every day life.

More people need to know statistics, maybe then you wouldn't have so many moronic responses to science, vaccines, climate change, etc where people think anecdotes rule the perception of reality.

I don't need the voting populace to understand that velocity is the integral of acceleration. I need more people to understand that meta analyses and p values govern what's real, and that toxicity of a substance is the effective dose at which it presents health problems in a given percentage of the population.

 

[backpropaganda]

Linear Algebra

 

[Deleted member 1963]

It's statistics and it's not even close.

Understanding calculus is needed for STEM careers. Understanding statistics is needed for just about every career.

 

[Interested Observer]

It's statistics and it's not even close.

Understanding calculus is needed for STEM careers. Understanding statistics is needed for just about every career.

Click to shrink...

This was my immediate thought as well. It 100% is stats. And it isn't just for every career, it's for life in general.

 

[Jubbe]

Engineer here, who uses calculus and does not use statistics.

I would say statistics. Most people will look at calculus as a completely foreign thing that they will never use once they are out of school, whereas the concepts of statistics are easily applied to every day life.

 

[Acorn]

Well, physics are applied calculus. Physics are a specific application of calculus, usually involving time and constants like gravity. You can use that same type of math in much more abstract ways, you don't have to consider your samples (a discreet instance of the spectrum you're analyzing) over time. They can be over dimensions, over anything. Physics are just one of the most easily understood applications of calculus because we can readily see it and study it.

Every physics course I've ever taken has had a calculus per-requirement. You can get much more specific about all the math you are talking about, like calculus is usually (but not always) concerned with linear algebra.

Click to shrink...

Calculus is in our physics too (in fact calculus was created largely to answer specific questions in physics the contemporary math at the time couldn't solve), but it's important to note that calculus is not necessarily physics. It's a type of math(s) that happens to be used to explain a lot of what shows up in physics from the relationship between kinetmatic equations to Maxwell's laws and more. Calculus can exist entirely independent of physics. After all, dy/dx has no physical meaning until you ascribe it one.

Click to shrink...

I see, thanks. I was never sure what it referred to because we didn't have a specific class called calculus.

 

[sauce]

Uh, well you need to know calculus to do statistics (e.g. expectation of a random variable with a continuous distribution), so I pick calculus.

If you're talking about the layman, things like knowing how to spot biases in sample data is pretty important. It won't help when people fudge their numbers if you don't know how to repeat their analysis though.

 

[Pau]

Can I be terrible and say both? (Because I do love both.)

But really it depends.

If you aren't really going to be doing anything involving mathematics, an understanding of statistics will probably be more useful. Everyone should understand basic statistical concepts.

I think it'd be pretty cool if everyone also understood basic calculus, or at least knew enough to know what calculus is useful for, but given the state of math education in the United States, that's an uphill battle.

If you are going deeper into particular fields, including statistics, you just need calculus. You can't really take anything beyond a introductory statistics course or light data analytics course without a decent calculus background.

At the end of the day, I think most people have the capacity to learn both at an introductory level.

 

[Zushin]

Statistics and quantitative literacy in general to promote critical thinking and logic would solve a lot of issues.

 

[opticalmace]

I'd say stats. Also, here in Canada (or at least where I am) primary school means K-7. Though from the OP I imagine we're talking about high school level.

 

[Deleted member 12790]

I'll give this a read soon. But it really hits at the point of how powerful mathematical concepts are to learn. Like how the words we know allow us to describe things better, the math we know allows us to quantify and analyze things better. We spend a lot of time bolstering our qualitative abilities, but we could also go much further in expanding our quantitative abilities. Both allow us to interpret the world much better. Your example with the negative numbers is great. How many descriptive abilities would we lose if we couldn't internalize negative numbers? How would we describe our debt?! Well, perhaps there are better examples out there...

Click to shrink...

What's funny that I've found is that, at the top of most (I guess STEM) fields, the math converges into calculus. I honestly feel like knowing calculus is like knowing a secret language of the universe. I have a handful of friends who are late in life math nerds like myself, and each of us has a different window into calculus that governs our appreciation. I approach calculus from the perspective of light, had really dive in once I started working in VR. One of my very first VR projects involved medicine and BCI (brain control interfaces), which involved Fourier Transforms and linear algebra, and suddenly I found myself having to get up to speed very, very quickly. But i have another friend, works as an architect, and his window into calculus is understanding the vector forces in hurricanes and such, to make sure the buildings he designs are structurally sound. I have another friend and she is big into AI, convolutional neural networks. Funny thing about AI: it's really just a fancy word for "probability and statistics." But, as someone mentioned above, calculus is necessary to work with the type of probability that we classify as AI. I have another friend, he's an enormous hippy, went to india and lived with a guru for 2 years and learned sitar. Believe it or not, he also has an appreciation for calculus, in a spiritual sense. He was explaining to me about this belief he has about life being vibrations in some cosmic fabric and how he can use calculus to gain introspection into how life works. Some deep shit that went over my head.

Calculus is mad useful. It's just that we don't really know how to teach it well yet. Doesn't help that, in the US at least, we treat math like RPG levels: addition is level 1, algebra is level 2, trig is level 3, integrated calculus is level 4, differential calculus is level 5, boolean algebra is... uh....

Math shouldn't work like that. That's like saying you learn green after blue after red, that you move on from water colors to charcoal. There are definitely some fundamentals that need to be laid down, but treating math like a linear course is what makes people quit at it, because they think "well I just can't jump 5 steps to this math I actually need without going through these other 4 steps I don't first."

an aside, but that common core stuff? That, if they stick to it and teachers know how to teach it better, will yield a much more mathematically sound generation of people. If we approach math as an art, not a science, we will ultimately come to a point where people will be lead to calculus, physics, statistics, probability, etc on their own, rather than thinking of them as requirements to merely move on from and forget.

 

[Deleted member 197]

The kind of statistics/probability you can actually understand without knowing calculus is something you can teach people like in a week, i think for high school calculus is just way more important.

 

[backpropaganda]

What's funny that I've found is that, at the top of most (I guess STEM) fields, the math converges into calculus. I honestly feel like knowing calculus is like knowing a secret language of the universe. I have a handful of friends who are late in life math nerds like myself, and each of us has a different window into calculus that governs our appreciation. I approach calculus from the perspective of light, had really dive in once I started working in VR. One of my very first VR projects involved medicine and BCI (brain control interfaces), which involved Fourier Transforms and linear algebra, and suddenly I found myself having to get up to speed very, very quickly. But i have another friend, works as an architect, and his window into calculus is understanding the vector forces in hurricanes and such, to make sure the buildings he designs are structurally sound. I have another friend and she is big into AI, convolutional neural networks. Funny thing about AI: it's really just a fancy word for "probability and statistics." But, as someone mentioned above, calculus is necessary to work with the type of probability that we classify as AI. I have another friend, he's an enormous hippy, went to india and lived with a guru for 2 years and learned sitar. Believe it or not, he also has an appreciation for calculus, in a spiritual sense. He was explaining to me about this belief he has about life being vibrations in some cosmic fabric and how he can use calculus to gain introspection into how life works. Some deep shit that went over my head.

Calculus is mad useful. It's just that we don't really know how to teach it well yet. Doesn't help that, in the US at least, we treat math like RPG levels: addition is level 1, algebra is level 2, trig is level 3, integrated calculus is level 4, differential calculus is level 5, boolean algebra is... uh....

Math shouldn't work like that. That's like saying you learn green after blue after red, that you move on from water colors to charcoal. There are definitely some fundamentals that need to be laid down, but treating math like a linear course is what makes people quit at it, because they think "well I just can't jump 5 steps to this math I actually need without going through these other 4 steps I don't first."

an aside, but that common core stuff? That, if they stick to it and teachers know how to teach it better, will yield a much more mathematically sound generation of people.

Click to shrink...

Seems like a very narrow engineering perspective of maths