I TA'd for an undergrad ODE course for two semesters. I'm a grad student at a public university with very good undergrad engineering students. Nonetheless, with the course I taught, at least, the main problem was that WAY too much was packed into a single semester. I suspect that this is the same everywhere. For example, about a month and a half of the course was devoted to systems of linear equations. These were students who, a priori, knew basically no linear algebra, being asked to understand an entire linear algebra course, with some extra stuff thrown in (you know, the differential equations), in six weeks. In order to solve a general system of constant coefficient linear odes, you have to take a matrix and compute its Jordan canonical form. The students I taught were performing this calculation by the end of the unit, but it was of course a joke. On their exam, they were asked to do this calculation (in differential equations language), and most of them were able to do it because they were good students and had memorized the procedure. Then, in another question, they were presented a 5x5 matrix, told that its only eigenvalues were 2 and -2, and asked if it was invertable. I don't think anyone gave a reasonable answer.
What is the point of that?
Edit: Everyone should be aware of this amazing Gian-Carlo Rota quote, the entirety of a book review on contemporary philosophers: "When pygmies cast such long shadows, it must be very late in the day."
>The students I taught were performing this calculation by the end of the unit, but it was of course a joke. On their exam, they were asked to do this calculation (in differential equations language), and most of them were able to do it because they were good students and had memorized the procedure.
This is a prime example of what I believe is wrong with traditional education (i'm not judging your teaching ability of course), as you are obviously aware, this is more of a box ticking activity than learning in the sense of having any actual understanding.
This has always personally caused me a great deal of trouble, for whatever reason I can only learn substantially through understanding - attempting to memorise procedures when I don't have time to understand the underlying workings is like trying to eat cardboard for me, in that sense I am a bad student and am exiled to purely self taught methods.
Although I am quite content learning on my own, it makes me feel like much of my formal education was a big waste of time, If I could go back in time knowing this I would avoid it altogether.
My experience is hopefully a more dated view by now, I also hold out hope that the new exposure the internet has given courses through MOCs who provide no kind of certification will by proxy refocus traditional education more on the value of learning than certification.
Also in the UK at least there are some new ranking systems emerging for universities that are more focused on learning like TEF, although indirect that is at least a step in the right direction.
> These were students who, a priori, knew basically no linear algebra, being asked to understand an entire linear algebra course, with some extra stuff thrown in (you know, the differential equations), in six weeks.
Yep, Diff Eq 1 (ODEs) and Diff Eq 2 (PDEs) for us as undergrads were exercises in memorization, even as a math major. I didn't really understand them until after I took a (well, several) linear algebra classes, after which it clicked that it was basically all the same theory.
You don't make students take Linear Algebra before differential equations? I thought students were supposed to take Lin-Alg alongside their Calculus 2 or Multivariable Calculus course, then have all that in their heads for DiffEqu.
Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.
Interestingly, children as young as 5 show an aptitude for understanding overarching concepts of calculus.[1]
This makes sense: it is much easier to talk about "rates of change" and "accumulation" in simple terms and show how they are related using models that appeal to children. We don't need to dive right in to the notation and algebraic manipulations to get across the basic idea. That can come later when children can handle the rigor. For now, let them play with it. That makes math a lot more fun and less draining, esoteric, impenetrable.
This makes total sense. I remember learning the ideas of calculus for the first time and thinking "wow, that makes sense" and also realizing how incomplete my picture of the world was without that coherent thought framework.
And I love that article. She really captures the damage that my early math education did (which I've been working the last year to overcome).
"Unfortunately a lot of what little children are offered is simple but hard—primitive ideas that are hard for humans to implement,” because they readily tax the limits of working memory, attention, precision and other cognitive functions. Examples of activities that fall into the “simple but hard” quadrant: Building a trench with a spoon... or memorizing multiplication tables as individual facts rather than patterns."
My oldest, while in second grade, learned enough calculus to determine the _location_ of a train given it's acceleration and since it started. That's because she was interested, and asking questions, and luckily I had explained to her the slope of a graph just a few days before the train ride. She didn't learn the formula to memorize, but rather the concepts. Only then did we do the calculations the long way, on paper, pen in _her_ hand.
Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later.
One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick (EE 120 Linear Systems) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to linear problems.
Maybe Sheldon Axler will do Calculus Done Right.
I agree that calculus sans trig is a good idea if you aren't going to use complex numbers.
I feel like we're still only scratching the surface with complex numbers. I didn't learn Euler's Identity nor most of the interesting parts of i until recently on Youtube.
We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?
While we're talking about turning school math on its head... I suggest teaching vectors before trig. You can get the cosine and sine from the dot and cross products.
Learning about vectors in physics finally got me comfortable with trig.
Now, one thing we did learn in trig was to get much more proficient with algebraic manipulation.
Amusingly, with complex numbers, proving trig identities becomes a trivial algebra problem.
To elaborate, “angle measure” (i.e. circular arclength corresponding to a particular rotation) is a derived quantity, not the primary abstraction we should be thinking about.
The way I think about it is that angle measure is the logarithm of a rotation, with the information about the orientation of the plane of rotation stripped out. Composition of rotations is an inherently multiplicative kind of structure, and for something a computer can understand the best representation is usually a unit-magnitude complex number. We can treat it additively by taking the logarithm, in precisely the same way we can treat scaling additively by taking the logarithm.
Symbolically, iθ = log(z), where z = x + yi is a complex number with x^2 + y^2 = 1.
This tool is very useful if you want to e.g. smoothly interpolate between rotations, but often dramatic overkill. For many problems it’s better to deal with rotations in pure vector terms, and never bother with angle measure whatsoever.
Let’s take that up a notch - try both and actually measure the results.
It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching?
So many people (myself included) have stories of, if only I had been exposed to such and such concept in a different way it would have had a much bigger impact.
Why not measure multiple aspects? Efficiency of learning, motivation, inspiration, relevance...
Maybe it’s being done and I don’t see it. Maybe it’s not being done, because companies will pay six figures to have people A/B test a different button location on a web site, but the business case for optimizing learning curriculum at traditional institutions is piss poor.
>It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching?
Yes and...of course it is done and tried. The problem is that most parents (especially those whose kids are most likely not to have the support to get past these struggles) are not exactly ecstatic for their children to be used as experiments. They want concrete answers, fixes, solutions...they don't want permutations.
Also, measure success as what? Learning? at a conceptual level or an execution level? Do you want a standardized test? Do you want to train teachers to effectively measure these concepts? Do the teachers really understand these concepts?
Educational systems are incredibly difficult and complex. Blackboxing them is not easy.You nailed the business case...we can't agree on who will pay for this, can't agree on how to measure it, and can't even agree on what should be taught. One experiment in this way was gasp the evil Common Core curriculum.
As an education researcher myself, I wouldn't fault anyone who works in K-12 policy and development to just phone it in and spend the days day drinking. They are underpaid, poorly treated/respected, and everyone thinks that they are equally qualified as those experts to have an opinion (e.g., parts of this thread) because they experienced education themselves.
The question I always ask people when they propose really concrete fixes to educational issues analogized to their personal technical field of expertise is this...Can you define and support from research your definition of what learning is?
Most students will benefit from already understanding calculus, when they are learning trigonometry.
The argument cuts both ways: not only can one strongly benefit from a basic understanding of trigonometry when learning calculus (how the derivatives of sin/cos/etc all related to one another) - it really isn't possible to get far with basic integration techniques without learning the fundamental trig-based substitutions. And ultimately you can use any set of subjects, in just about any order. What matters is the thought processes, and that what you're really trying to teach is not a set of facts or techniques - but the underlying mathematical essence of these techniques.
So at the end of the day, what it really comes to is: "It doesn't matter so much what you teach, but how you teach".
I suppose what you say makes sense for the general population.
As a scientist/engineer who used trigonometry a lot, it was extremely useful to have a dedicated course on trigonometry early on. The reality is many will need to use basic trigonometry before they need to use calculus (e.g. in physics).
OK, but the algebra of complex numbers is so much easier than trig -- less to memorize -- that I think it must be inertia that's keeping our trig courses around, not inherently better pedagogy. Has anyone tried teaching this way in high school or middle school?
Certainly integration of trigonometric functions, requiring trigonometric identities, seem to get more time allotted to them than the pedagogical value they provide AFAICS, but I'd be interested to hear someone pointing out what I'm missing there.
Trigonometry was important (presumably much more than calculus) for navigation/military applications at some point in history, so maybe it just stuck that way?
This right here! Calculus (the first semester anyway) is really not hard to understand at all. Things like limits also demonstrate why math is done the way it is which makes it much more enjoyable, but most people get hung up with trig and don't see any of this.
Part of the reason may be that trig gets used heavily for a lot of stuff after calc1.
And 20 years later, this essay is still as relevant as the day it was written. I agree with pretty much everything that's in the essay, except for a few small points.
> There is nothing wrong with keeping the functional notation for density functions – as physicists and engineers always did – as long as one bears in mind that density functions cannot be evaluated, but only integrated.
This always bothered me, since, as noted in the very next section, distributions don't have an analogue to pointwise multiplication. Even worse, there is a perfectly servicable notation for such "dual functions/vectors" that physicists have been using throughout the second half of the 20th century. We could just use a consistent notation and not confuse new students, but no. "It's always been done this way" is a terrible argument and leads us to the confusing mess of notations that people still use for integrals and integral transforms...
---
Apart from that I would teach people recurrence equations/stream calculus before going into the limiting case of differential equations. It's true that differential equations are sometimes easier to handle analytically, but this is neither relevant (as the article notes) nor a great point in their favor, since we just end up teaching students a bag of tricks instead of explaining why something works...
Why is it that no one has undertaken the task of cleaning the Augean stables of elementary differential equations? I will hazard an answer: for the same reason why we see so little change anywhere today, whether in society, in politics, or in science. Vested interests dominate every nook and cranny of our society, even the society of mathematicians.
Truly we live in a decadent age. With this much fuel piled up, who will be surprised by the conflagration?
I wish I came across the so called proof-based math before calculus and trigonometry. It would have grabbed me instantly. High school math (and especially physics) classes would leave me with very uneasy feeling that something crucial is left unmentioned, something important is swept under the rug and something important is hidden for whatever reason. Turns out I was wanting for proofs, but couldn't articulate it - I just sensed that something was off :)
Brewster Kahle (of Alexa and Internet Archive fame) went this way: he and his wife decided that school wasn't right for their younger son so the two of them (father and son) worked through Euclid together.
My kids are in high school right now. I was getting my daughter psyched up for geometry, by promising her that she'd get to do proofs.
The geometry class completely glossed over proofs. It was much more oriented towards solving problems. I don't know if it was because of standardized testing, but I have my suspicions. Fortunately, my daughter worked on the proofs herself, outside of class.
I was saddened for many reasons, one of which is that lots of people I've talked to -- especially women -- loved high school geometry because of the proofs. That was where math came alive.
Regarding item 10 in that list, I learned plenty of Laplace transforms, partial fraction expansions, stability, phase planes, etc. It's the core of control systems theory. It was just not taught in differential equations class. The DE class was a bag of useless tricks. All the other EE classes were very useful tricks of how not to directly solving DEs.
Going to drink a beer now for Oliver Heaviside. The invention of the Laplace transform is one of the greatest contributions to engineering.
I clearly remember his class on, basically, Mathematics for Philosophers. As an engineering student, I thought my pal from Urban Studies who suggested taking it was a way to pad one's GPA, which wasn't my style. Nonetheless, I agreed, and, now, I regard it as one of the most important courses I've ever taken. I have never met anyone with that level of enthusiasm, wonder, and love of Mathematics - who also knew how to partially impart it to his students! Mainly, it's the Wonder of it all that remains with me. What a privilege.
I believe this is a huge shortcoming of how math is taught. You can bet your last dollar that the teacher doesn't think it's about tricks. But the students are convinced that it is. Students and teachers are both exposed to the exact same material but end up with diametrically opposing conclusions.
Disclosure: I taught college freshman math for one semester, long ago. It was a course where I was supplied with a syllabus and exams, and the students could buy a packet of exams from previous years.
The tricks are what you remember from doing problems over and over, and recognizing patterns. There is also a higher level pattern that isn't mentioned in class, but is vital to solving problems: You learn to identify each problem with a particular chapter or section in the textbook, and then solve the problem by recalling the methods in that section. This is of course a grotesque distortion of what math is, but will get you through the lower level college math courses with good grades.
The other skill is being able to perform the manipulations quickly enough that you can try one or two before hitting on one that works.
Disclosure: I taught college freshman math for one semester.
My high school calculus teacher was really great at this. He didn't just teach a bunch of transformations to memorize. That was part of it, naturally, as you aren't going to use the definition of limits and the FToC in all your problem solutions. He actually made us construct volumes from pieces of poster board, measure the segments and calculate the Riemann sum. When we did function analysis, he didn't allow us to use the Cartesian plane at first. We had to show visually how a function deformed the one-dimensional real line. How x^2 squished values between -1 and 1 toward 0 and stretched the other values toward +infinity.
It gave me a good "visual" grasp of the concepts and made most of my higher math classes much easier.
I do agree diff eq instruction sucks. I got an A in that class and didn't understand a thing. "This equation has this form; this is the canned solution."
Readit News
What is the point of that?
Edit: Everyone should be aware of this amazing Gian-Carlo Rota quote, the entirety of a book review on contemporary philosophers: "When pygmies cast such long shadows, it must be very late in the day."
This is a prime example of what I believe is wrong with traditional education (i'm not judging your teaching ability of course), as you are obviously aware, this is more of a box ticking activity than learning in the sense of having any actual understanding.
This has always personally caused me a great deal of trouble, for whatever reason I can only learn substantially through understanding - attempting to memorise procedures when I don't have time to understand the underlying workings is like trying to eat cardboard for me, in that sense I am a bad student and am exiled to purely self taught methods.
Although I am quite content learning on my own, it makes me feel like much of my formal education was a big waste of time, If I could go back in time knowing this I would avoid it altogether.
My experience is hopefully a more dated view by now, I also hold out hope that the new exposure the internet has given courses through MOCs who provide no kind of certification will by proxy refocus traditional education more on the value of learning than certification.
Also in the UK at least there are some new ranking systems emerging for universities that are more focused on learning like TEF, although indirect that is at least a step in the right direction.
Sundays are study days from now on. Hopefully I can catch up.
Yep, Diff Eq 1 (ODEs) and Diff Eq 2 (PDEs) for us as undergrads were exercises in memorization, even as a math major. I didn't really understand them until after I took a (well, several) linear algebra classes, after which it clicked that it was basically all the same theory.
In my university curriculum, linear algebra was taught after.
Perhaps my ODE course was watered down, but that wasn't a problem at all for us.
This makes sense: it is much easier to talk about "rates of change" and "accumulation" in simple terms and show how they are related using models that appeal to children. We don't need to dive right in to the notation and algebraic manipulations to get across the basic idea. That can come later when children can handle the rigor. For now, let them play with it. That makes math a lot more fun and less draining, esoteric, impenetrable.
[1] https://www.theatlantic.com/education/archive/2014/03/5-year...
And I love that article. She really captures the damage that my early math education did (which I've been working the last year to overcome).
"Unfortunately a lot of what little children are offered is simple but hard—primitive ideas that are hard for humans to implement,” because they readily tax the limits of working memory, attention, precision and other cognitive functions. Examples of activities that fall into the “simple but hard” quadrant: Building a trench with a spoon... or memorizing multiplication tables as individual facts rather than patterns."
But I would say that that's not math, any more than speaking means you know grammar, or digesting means you know biochemistry. Math is formalization.
For calculus, it's slopes at a point without dividing by zero (using the cheat of limits).
One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick (EE 120 Linear Systems) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to linear problems.
Maybe Sheldon Axler will do Calculus Done Right.
I agree that calculus sans trig is a good idea if you aren't going to use complex numbers.
And if Clifford algebras were introduced, it would get even easier. But they typically don't even teach that.
We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?
Seeing sine, cosine, etc as merely each other's derivative was astonishing and eye opening. So elegant. It made me love math again.
Learning about vectors in physics finally got me comfortable with trig.
Now, one thing we did learn in trig was to get much more proficient with algebraic manipulation.
Amusingly, with complex numbers, proving trig identities becomes a trivial algebra problem.
The way I think about it is that angle measure is the logarithm of a rotation, with the information about the orientation of the plane of rotation stripped out. Composition of rotations is an inherently multiplicative kind of structure, and for something a computer can understand the best representation is usually a unit-magnitude complex number. We can treat it additively by taking the logarithm, in precisely the same way we can treat scaling additively by taking the logarithm.
Symbolically, iθ = log(z), where z = x + yi is a complex number with x^2 + y^2 = 1.
This tool is very useful if you want to e.g. smoothly interpolate between rotations, but often dramatic overkill. For many problems it’s better to deal with rotations in pure vector terms, and never bother with angle measure whatsoever.
What about geometric algebra instead?
http://www.shapeoperator.com/2016/12/12/sunset-geometry/
It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching?
So many people (myself included) have stories of, if only I had been exposed to such and such concept in a different way it would have had a much bigger impact.
Why not measure multiple aspects? Efficiency of learning, motivation, inspiration, relevance...
Maybe it’s being done and I don’t see it. Maybe it’s not being done, because companies will pay six figures to have people A/B test a different button location on a web site, but the business case for optimizing learning curriculum at traditional institutions is piss poor.
Yes and...of course it is done and tried. The problem is that most parents (especially those whose kids are most likely not to have the support to get past these struggles) are not exactly ecstatic for their children to be used as experiments. They want concrete answers, fixes, solutions...they don't want permutations.
Also, measure success as what? Learning? at a conceptual level or an execution level? Do you want a standardized test? Do you want to train teachers to effectively measure these concepts? Do the teachers really understand these concepts?
Educational systems are incredibly difficult and complex. Blackboxing them is not easy.You nailed the business case...we can't agree on who will pay for this, can't agree on how to measure it, and can't even agree on what should be taught. One experiment in this way was gasp the evil Common Core curriculum.
As an education researcher myself, I wouldn't fault anyone who works in K-12 policy and development to just phone it in and spend the days day drinking. They are underpaid, poorly treated/respected, and everyone thinks that they are equally qualified as those experts to have an opinion (e.g., parts of this thread) because they experienced education themselves.
The question I always ask people when they propose really concrete fixes to educational issues analogized to their personal technical field of expertise is this...Can you define and support from research your definition of what learning is?
Admittedly that is an indictment more of the IRB than of the experiment...
The argument cuts both ways: not only can one strongly benefit from a basic understanding of trigonometry when learning calculus (how the derivatives of sin/cos/etc all related to one another) - it really isn't possible to get far with basic integration techniques without learning the fundamental trig-based substitutions. And ultimately you can use any set of subjects, in just about any order. What matters is the thought processes, and that what you're really trying to teach is not a set of facts or techniques - but the underlying mathematical essence of these techniques.
So at the end of the day, what it really comes to is: "It doesn't matter so much what you teach, but how you teach".
As a scientist/engineer who used trigonometry a lot, it was extremely useful to have a dedicated course on trigonometry early on. The reality is many will need to use basic trigonometry before they need to use calculus (e.g. in physics).
Of course one of the main texts i learned diff eq from was the Mary Boas book i think she mentioned in that rant, so what do i know? :)
Part of the reason may be that trig gets used heavily for a lot of stuff after calc1.
> There is nothing wrong with keeping the functional notation for density functions – as physicists and engineers always did – as long as one bears in mind that density functions cannot be evaluated, but only integrated.
This always bothered me, since, as noted in the very next section, distributions don't have an analogue to pointwise multiplication. Even worse, there is a perfectly servicable notation for such "dual functions/vectors" that physicists have been using throughout the second half of the 20th century. We could just use a consistent notation and not confuse new students, but no. "It's always been done this way" is a terrible argument and leads us to the confusing mess of notations that people still use for integrals and integral transforms...
---
Apart from that I would teach people recurrence equations/stream calculus before going into the limiting case of differential equations. It's true that differential equations are sometimes easier to handle analytically, but this is neither relevant (as the article notes) nor a great point in their favor, since we just end up teaching students a bag of tricks instead of explaining why something works...
Truly we live in a decadent age. With this much fuel piled up, who will be surprised by the conflagration?
The geometry class completely glossed over proofs. It was much more oriented towards solving problems. I don't know if it was because of standardized testing, but I have my suspicions. Fortunately, my daughter worked on the proofs herself, outside of class.
I was saddened for many reasons, one of which is that lots of people I've talked to -- especially women -- loved high school geometry because of the proofs. That was where math came alive.
Going to drink a beer now for Oliver Heaviside. The invention of the Laplace transform is one of the greatest contributions to engineering.
https://en.wikipedia.org/wiki/Gian-Carlo_Rota
Even worse is when you have all the proof based and concept oriented course and are tested on trickeries on exams.
Disclosure: I taught college freshman math for one semester, long ago. It was a course where I was supplied with a syllabus and exams, and the students could buy a packet of exams from previous years.
The tricks are what you remember from doing problems over and over, and recognizing patterns. There is also a higher level pattern that isn't mentioned in class, but is vital to solving problems: You learn to identify each problem with a particular chapter or section in the textbook, and then solve the problem by recalling the methods in that section. This is of course a grotesque distortion of what math is, but will get you through the lower level college math courses with good grades.
The other skill is being able to perform the manipulations quickly enough that you can try one or two before hitting on one that works.
Disclosure: I taught college freshman math for one semester.
It gave me a good "visual" grasp of the concepts and made most of my higher math classes much easier.
I do agree diff eq instruction sucks. I got an A in that class and didn't understand a thing. "This equation has this form; this is the canned solution."