Help Needed: High school & college education

[LegionOnomaMoi]

Some time ago I started a thread asking about the utility of requiring students to take pre-calculus and/or calculus (as they are currently taught, anyway). Many members supported doing so, and gave as reasons mostly that mathematical literacy is important and that even if elementary calculus can't really be used in the "real world" it teaches critical thinking.

I disagree. This is not to say that I think mathematical literacy unimportant (nothing could be farther from the truth) but that I think the topics covered don't really teach mathematics. I'm interested in writing up preliminary portions of a textbook or textbooks designed to introduce the foundations of mathematics for those with only a very basic grasp of math (for those in the US, middle school mathematics) and that is not only useful practically immediately but also interesting even to those who generally dislike mathematics.

However, I have to contend with those who favor the traditional route, in which most of pre-college mathematics is designed to prepare one for calculus.

Recently, I've been doing problem sets for a mathematics doctoral examination. These, interestingly enough, include problems from elementary calculus. I know we have members here who have a background (some an extensive background) in mathematics. I have scanned and cropped a few problems an "elementary calculus section" of a problem set. I am interested in what those who have taken elementary calculus think about these problems (i.e., how difficult they are, what the answers are, how much they resemble whatever calculus each member was exposed to, etc.). I am also still interested about opinions on high school and college mathematics curricula, and I invite members to post their views here or in this thread if they would be so kind

And so, without further ado, some "elementary calculus" -

Spoiler: some additional problems

 

[Wirey]

Is this an attempt to get us to do your homework?

 

[Laika]

ahhhh.... maths!

Based on what you're saying I think you're a teacher so on a slightly more serious note it is very difficult to tell what exactly the maths is communicating. Maths, like Music, is a form of communication of knowledge. I think it was Issac Newton who invented Calculus (probably in order to understand the motion of the planets and stars in mechanical terms, etc.) It is hard to know what real world problems it is going to be applied to because it is abstracted away from the context which it is going to be used in (e.g. Science, engineering, etc.). Learning by route can be an obstacle to depth of understanding and feel a bit intimidating.

I'm probably being a typical maths student [sorry], but I hope this is more thoughtfully put. beyond knowing "it means something", my maths background is no more than (UK) secondary school level. I confess I have forgotten an awful lot of the maths I was taught but only because I haven't used it very much since then.

 

[LegionOnomaMoi]

Is this an attempt to get us to do your homework?

No I have the answers, they're not particularly difficult. I can post them if you are curious, but as the answers are one of the things I was interested in, I would prefer not to post all the answers at one. Pick any 3 you wish and I'll provide the answers to those now, and the rest later.

 

[LegionOnomaMoi]

Based on what you're saying I think you're a teacher so on a slightly more serious note it is very difficult to tell what exactly the maths is communicating.

That is one thing that I am interested in. These are problems designed for students studying for a doctoral examination for a PhD in mathematics, but the questions I chose are deliberately from elementary calculus. That is, there is nothing in the symbols or language that a student who has passed AP calculus or a first semester of university calculus wouldn't recognize. Unfortunately for my purposes, many members haven't and many who have doubtless don't recall much of what they learned. I had hoped, though, that a few who use mathematics in their work or who do remember might recognize the terms and symbols used enough to at least comment on the difference between the nature of the questions and what they recall or know of "elementary calculus".

That said, as I am really interested in mathematics education more generally and in particular that it should be reformed drastically (it is mainly geared towards preparing students for a calculus course they might not take and if they do, will be useless anyway). So I am interested in any and all comments to do with mathematics education, and would like to thank you for your comments!

I think it was Issac Newton who invented Calculus (probably in order to understand the motion of the planets and stars in mechanical terms, etc.)

It was independently "invented" (or "discovered"; the difference gets into a philosophical debate of no relevancy here so we'll stick with invented) by Newton and Leibniz. However, British mathematics suffered drastically after Newton because Leibniz' notation was superior to Newton's. Like you said, being clear what is being communicated is vital, and Leibniz' notation did a better job, so German, French, and Italian mathematicians progressed much more than their British counterparts.

Learning by route can be an obstacle to depth of understanding and feel a bit intimidating.

This is my belief and much of my motivation. High school (and the European equivalents to high school) teach math as a bunch of rules. At a certain age, of course, this necessary (you can't explain the logic of algebra or the mathematical structure of algebra, probability, statistics, etc., to a 10 year old). But once students turn 14 or 15 they are beginning to be able to learn the concepts that make up the foundations of mathematics. Instead, they are taught how to solve standard questions using route application of rules mostly in preparation for a calculus course that, if they take it, will not teach them calculus so much as it will allow them to solve calculus route calculus problems using mostly pre-calculus skills (which turns out to hinder them enormously if they go on to take courses in linear algebra, complex analysis, abstract algebra, etc.).

 

[Shadow Wolf]

I have a brain for words. I can do math, but I'm very slow at it, I have to mess it up a bunch of times, and if it's not taught to me in a certain way I will not get it. Personally I think Calculus is abit high to be a requirement for non-math related majors. It would be like requiring a biology major to write a thesis about Nietzsche. Without doubt it's good to know, but the time could be better spend learning lighter and less time consuming philosophers to expand thinking while spending that time saved taking more biology classes.

 

[LegionOnomaMoi]

I have a brain for words. I can do math, but I'm very slow at it, I have to mess it up a bunch of times, and if it's not taught to me in a certain way I will not get it.

Until I took logic and statistics in college, I believed that I was bad at math and that I hated it. For one thing, I am pretty terrible when it comes to mental calculations. For another, I am also the type of person who will forget to include a minus sign or make some other trivial mistake which gives me the wrong answer. One reason for this and other posts as well as my intent to write one or more books (textbooks or "teach yourself" books) is because I found out that there is very little in common with all of mathematics through calculus, and most undergraduate mathematics (let alone graduate). Not only that, but much of what I had to learn about math was so that I could solve calculus 1 problems (and the reason these problems exist is because calculus is usually taught in 3 semesters instead of 1. Why? Because the way calculus is taught is by using a lot of pre-calculus to avoid having to require students to understand calculus concepts).

I felt cheated. I had to learn how to manipulate algebraic expressions in order to do things like rationalize the denominator just so I could solve problems involving limits if I went on to calculus, and I had to do this because the concept of a limit is hard so it is largely ignored except in the ways that students are given rules for computing them. I felt cheated because it turns out that I love math, but I had thought it to be something it isn't.

There are many who, it turns out, suffered the reverse. They were great at math all through calculus, but then took abstract, upper-level math courses and all of a sudden they had to understand the logic underlying the math. They couldn't rely on rote calculations. The problems in the OP are on topics that any AP calculus student or student who has passed calculus I have covered. I doubt many could answer any of them. This is because they require understanding calculus concepts that are not taught.

It would be like requiring a biology major to write a thesis about Nietzsche.

In the thread I linked to that this is a continuation of (at least in some sense), I was surprised to find a biologist who had to take calculus but never used it in research or anywhere outside of math classes taken to become a biologist.

I am hoping to post portions of drafts of my text(s) to get feedback in the future (not there yet), but the idea is to increase mathematical literacy while increasing interest in mathematics and increasing the number of topics covered at a high school level.

Have you by any chance taken logic (by that I mean symbolic logic or propositional/predicate logic; i.e., logic in which you do proofs/derivations using symbols, truth tables, etc.)? One of my thoughts is starting with logic, as symbolic logic is naturally related to language and begins by learning how to represent sentences (well, propositions) using letters. There is a book I've used parts of to teach titled Sets, Functions, and Logic: An Introduction to Abstract Mathematics which has a chapter "Math speak." The first mathematical "operator" (by that I mean something like the symbols used to represent multiplication, addition, division, etc.) is "and". In some math/logic texts, the symbol is one we use to mean "and" anyway: &. To me, this is both more intuitive (at least at the beginning) and far more valuable than learning how to deal with logarithms (particularly because you then have to learn that in real mathematics, starting with calculus I, logarithms are defined differently anyway and you've spent a lot of time learning that "log" refers to something it no longer does).

 

[Shadow Wolf]

Have you by any chance taken logic (by that I mean symbolic logic or propositional/predicate logic; i.e., logic in which you do proofs/derivations using symbols, truth tables, etc.)?

No. The highest I've had is a 300-level statistics course. I can impress philosophy and humanities teachers with my knowledge of those subjects, but when it comes to math I really struggle with it until it "clicks." I'm probably one of the few odd balls that likes MyMathLab because I can keep doing the problems over and over and over and over until I finally figure them out.

 

[LegionOnomaMoi]

No. The highest I've had is a 300-level statistics course. I can impress philosophy and humanities teachers with my knowledge of those subjects, but when it comes to math I really struggle with it until it "clicks." I'm probably one of the few odd balls that likes MyMathLab because I can keep doing the problems over and over and over and over until I finally figure them out.

You majored in psychology, correct? Did you enjoy the statistics class at all? I ask because I am thinking of including topics from statistics as I think that, if presented the right way, statistics is one of those math topics that has immediate relevancy and utility and the mathematics is "simple" (at least relatively).

My first job working for the university I did my undergrad majors in was tutoring, and the first class I tutored for was behavioral statistics, so I recall the kinds of problems students had (sigma notation being one), but I also recall showing a student how to get an answer by using some pretty simple algebra. All of a sudden something clicked and the idea behind the math made sense. I learned later that the teacher told him not to do the problems this way and to only use the formulae that were in the book. I don't know her reasoning, but I can't imagine what would be bad about understanding the formulae enough to know how you can "play" with the terms to make the problem simpler for you or easier for you (after all, I'm pretty sure every statistics textbook for students of sociology, psychology, and a slew of other disciplines learns e.g., the straightforward mathematical representation of the sum of squares/sum of squared deviation from the mean and then the computationally equivalent but easier form).

 

[Laika]

That is one thing that I am interested in. These are problems designed for students studying for a doctoral examination for a PhD in mathematics, but the questions I chose are deliberately from elementary calculus. That is, there is nothing in the symbols or language that a student who has passed AP calculus or a first semester of university calculus wouldn't recognize. Unfortunately for my purposes, many members haven't and many who have doubtless don't recall much of what they learned. I had hoped, though, that a few who use mathematics in their work or who do remember might recognize the terms and symbols used enough to at least comment on the difference between the nature of the questions and what they recall or know of "elementary calculus".

That said, as I am really interested in mathematics education more generally and in particular that it should be reformed drastically (it is mainly geared towards preparing students for a calculus course they might not take and if they do, will be useless anyway). So I am interested in any and all comments to do with mathematics education, and would like to thank you for your comments!


It was independently "invented" (or "discovered"; the difference gets into a philosophical debate of no relevancy here so we'll stick with invented) by Newton and Leibniz. However, British mathematics suffered drastically after Newton because Leibniz' notation was superior to Newton's. Like you said, being clear what is being communicated is vital, and Leibniz' notation did a better job, so German, French, and Italian mathematicians progressed much more than their British counterparts.



This is my belief and much of my motivation. High school (and the European equivalents to high school) teach math as a bunch of rules. At a certain age, of course, this necessary (you can't explain the logic of algebra or the mathematical structure of algebra, probability, statistics, etc., to a 10 year old). But once students turn 14 or 15 they are beginning to be able to learn the concepts that make up the foundations of mathematics. Instead, they are taught how to solve standard questions using route application of rules mostly in preparation for a calculus course that, if they take it, will not teach them calculus so much as it will allow them to solve calculus route calculus problems using mostly pre-calculus skills (which turns out to hinder them enormously if they go on to take courses in linear algebra, complex analysis, abstract algebra, etc.).

As I recall, Newton had a pathological desire to take credit, so I can well imagine that.

Yeah, what you say about teaching maths as a bunch of rules sounds spot on. I did not realize that maths education was intended as a progression towards understanding calculus or "pre-calculus skills" as you put it.
I remember having to use Pythagoras to calculate the maximum area of a right-angled triangle capable for a fixed fence (about 1000m or something like that) as that was a piece of coursework. this was probably tens years ago- so having something to grasp it's application meant it stuck for a bit. The only time I used it outside of maths class was when my design tech teacher wanted us to build a model bridge- so I was trying to figure out the dimensions for the frame on it (as triangles are structurally strong shapes).
I did a bit of stuff to do with Matrices at degree level (I remember that, because it can be reduced down to addition and subtraction if you read them right; which was kind of rewarding). I was doing Economics for a year and still don't have a clue what Matrices were going to be used for.
There was a class on Econometrics where there was an equation of the board and it had "U" at the end; so I asked what the "U" stood for and the lecturer said 'unknown'. it was a tutorial I think, so the lecturer got diverted onto Dick Cheney's "known unknowns". I felt kind of uncomfortable with it because I felt we were supposed to figure out what the "U" stood for by using actual data, rather than an equation. Economics adopted mathematics in the late 19th century because it helped advance its pretensions as a science but without any methodology to check whether it was actually right.
What I realized later was it was precisely a formula like that which had screwed up the financial system; as they have given a value to estimate market volatility (the "U") and of course the market exceeded it. In the late 1970's NASA scientists were laid off and got hired by wall street and they used maths as a way to sell their financial products (precisely because no-one understood them but it looks convincing). A bizarre thing was they way maths was applied to sub-prime mortgages; Banks sold mortgages in bundles, and mixed good mortgages (where people could afford to repay) in with the bad (those that couldn't repay) as a way to "average" out their value. Of course, the bad mortgages stay bad and eventually the banks had trouble. so when the formula "broke" because of the volatility, the banks had the credit crunch because no-one knew the value of the assets or liabilities they were holding or where the bad assets were. So, in this instance, the failure to relate equations to real-world applications had some really drastic consequences.

it's a little more scarey, when you realize that most financial transactions are now done by Computers who calculate risks purely on the mathematics. They periodically "crash" the stock market because they into a cycle of selling. I'm not sure how the maths would work for that one.

 

[Wirey]

No I have the answers, they're not particularly difficult. I can post them if you are curious, but as the answers are one of the things I was interested in, I would prefer not to post all the answers at one. Pick any 3 you wish and I'll provide the answers to those now, and the rest later.

I was joking. You math nerds have no sense of humour.

 

[LegionOnomaMoi]

I was joking. You math nerds have no sense of humour.

Don't be so quick to judge. It's not necessarily true that I didn't get it because I have no sense of humor. I could just be an utter moron.

 

[Wirey]

I've occasionally suffered a similar fate.

 

[Yerda]

I'll apologise to the non-mathematicians here because this is going to get technical.

I'd say that those problems look really hard. As in pure dead hard. As in if I was hit with that stuff in my calculus class I'd have soiled myself.

Again my apologies.

 

[Drolefille]

As someone who passed AP calculus relatively easily but is 12 years away from it, I couldn't begin to do those problems anymore. It isn't something I've used since.

I've even forgotten a lot of my psych stats class which was far more useful, but is also far easier for me to pick back up when I've needed to. (So yes, include stats stuff, applied, particularly. My college stats class was high school level, my college psych stats class was very educational.)

 

[Wirey]

I use calculus daily. The text book holds my door open.

 

[Draka]

As someone who passed AP calculus relatively easily but is 12 years away from it, I couldn't begin to do those problems anymore. It isn't something I've used since.

See, this is my take on it as well. Now, I didn't take any calculus in college or anything, but when I went to A-school in the Navy the math I had to take was pretty much what I see in the OP. I don't think I even realized that that was calculus, it was just advanced math to me. I recognize pretty much most of that. Most people in that school didn't even make it through the math portion. I didn't have any problem with it, but to be honest, I've forgotten the vast majority of it as, even though a requirement to learn for my rating, I rarely ever had to actually use it. So, as a person whose job it was to fix the electronics on military aircraft, it wasn't all that needed. At least not to such an extent to need on a daily basis. In fact, I think the most I saw it while in was on rating exams. Being as I've been out for many years now I've lost much of it.

I do have to say I am curious as to why learning calculus is pushed as much as it is. Working in such a field that would perhaps actually have cause to need to know such I can say it isn't used that much, or at least as much as it is made out that it is needed. Why would someone who wasn't going to go into a technical field of some kind need to know how to solve such problems?

 

[LegionOnomaMoi]

I use calculus daily. The text book holds my door open.

And the solutions manuals make good paperweights.

 

[Drolefille]

See, this is my take on it as well. Now, I didn't take any calculus in college or anything, but when I went to A-school in the Navy the math I had to take was pretty much what I see in the OP. I don't think I even realized that that was calculus, it was just advanced math to me. I recognize pretty much most of that. Most people in that school didn't even make it through the math portion. I didn't have any problem with it, but to be honest, I've forgotten the vast majority of it as, even though a requirement to learn for my rating, I rarely ever had to actually use it. So, as a person whose job it was to fix the electronics on military aircraft, it wasn't all that needed. At least not to such an extent to need on a daily basis. In fact, I think the most I saw it while in was on rating exams. Being as I've been out for many years now I've lost much of it.

I do have to say I am curious as to why learning calculus is pushed as much as it is. Working in such a field that would perhaps actually have cause to need to know such I can say it isn't used that much, or at least as much as it is made out that it is needed. Why would someone who wasn't going to go into a technical field of some kind need to know how to solve such problems?

Personally I was on a pre-med track, and open to the idea of going into the hard sciences as well. It was good for me to take AP calc in high school, but when I got to college I was told I needed 1 math class for my bio/pre-med major and my options were "Calc 2" or "Stats." Guess which I took. Despite that Stats class being mind numbing as it was mostly returning adult students and I was an 18 year old know-it-all.

I think it's reasonable to encourage and teach, but also reasonable to lose, much as I have forgotten many of the rules of chemistry since my soph. year of college.

 

[LegionOnomaMoi]

So, as a person whose job it was to fix the electronics on military aircraft, it wasn't all that needed.

Interesting! Thanks!

Why would someone who wasn't going to go into a technical field of some kind need to know how to solve such problems?

If the word problems from calculus textbooks are any indication, then it is required knowledge if you need to figure out to minimize how much you spend on a length of fence for your pig pen that is bordered on one side by a river, or to estimate the error in hundredths of an inch when constructing a box.

There as a time (pretty much from Newton & Leibniz up to the early 20th century) that calculus as at the heart of all the natural and life sciences. Of course, most people during this interval weren't educated and it wasn't until around the mid-twentieth century that the population of college students changed from consisting entirely of those devoted to learning or from rich families that made their sons obtain a degree, and while the first high school was founded in the early 1800s, it wasn't until the 20th century that they started to become common (by high school I do mean actual public high school, the first of which was founded in Boston). Even then, it wasn't until decades later that it started to become expected for students to go from high school to college.

Meanwhile, the importance of calculus was drastically reduced (or rather, new developments in mathematics and computing became prominent and more important). One of the early formulations of quantum physics as Heisenberg's "matrix mechanics" (matrices are mathematical "things" like series, sequences, etc., and are a core part of linear algebra, multivariate statistics, multivariable calculus, etc.). Interestingly, though, Heisenberg didn't know what matrices were. Max Born had to tell him, and neither of them could figure out what to make of there appearance. Physics had been the primary motivation for the development of calculus, and suddenly the new mechanics (quantum mechanics) involved more math from linear algebra than it did calculus.

That's nothing compared to statistics. All the sciences use statistics, and the underlying logic & concepts of statistics barely relates to calculus but is heavily based on linear algebra. Finally, now that we not only have computers but mathematical & statistical software packages lie MATLAB, SAS, Mathematica, Maple, etc., the power of linear algebra and the scope of its applications simply dwarf that of calculus (in all fairness, this becomes less true of higher level calculus, as multivariable mathematics is a combination of linear algebra and calculus). For those who don't know what linear algebra is, the reason it required computers to really be as powerful and useful as it is has to do with the incredible amount of computations. This is why most of the calculations students have to perform in a linear algebra class are simple addition and subtraction. Unlike calculus, it's impossible to reduce linear algebra to a bunch of rules so that students can use high school math to solve problems. Rather, one has to understand the concepts, so the calculations are made simple, as they are barely related to what is important (and in real life application humans never do the calculations).

So thanks to the apparent inability for the educational system to update its math curriculum to reflect a 21st century world rather than a 19th century world, students aren't exposed to a lot of important, practical, and (at least relatively) interesting mathematics and are instead required to learn a lot of math that most will never use even if they go into the sciences.