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Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles

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.

jacquesm 9y ago
I found trigonometry a lot easer than calculus, which even today usually takes me forever even for what I'm sure are simple problems.
- Nomentatus 9y ago
  Especially that darn three-body problem. (Not poking fun, just agreeing, it's genuinely tough.)
- rocqua 9y ago
  I feel the same, especially with integrals. Integration is way to arbitrary and 'most' integrals cannot be done in a closed form.
  - conistonwater 9y ago
    Funnily enough, from a mathematical point of view, it's the opposite: integration is a really nice operation that can be applied to basically anything, whereas differentiation is really finicky, sometimes derivatives don't exist and you can't even be too sure when, so you need to be extra careful. This carries over to doing numerical computing: integrating an arbitrary function is easy, for smooth 1d functions it's a solved problem, differentiating even a reasonably smooth function numerically is much harder. If there is noise in your function, it doesn't matter so much for integrating, but can completely break the derivative.
    - gaurav_v 9y ago
      More like `from a computational point of view.' Analytically, integration is much more difficult and richer than differentiation!
      - deleted 9y ago
        [deleted]
      - srean 9y ago
        Finding the closed form of the integral of a closed form, yes then what you say is true (this is different from 'analytic' which in mathematics has a different meaning). Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that. Note the key word there is a function not a function with a closed form that's a tiny subset.
        gaurav_v 9y ago
        "Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that." The OP said the opposite, that differentiation is harder 'more finicky.' I agree that the concept of integration is much richer. Also, I didn't mean 'closed form solution' when I said 'analytic.' I also didn't mean 'analytic functions.' I meant that the analytic machinery you have to develop in order to have a theory of integration is far richer than for differentiation - i.e, proving the multivariate change of variable theorem.
        srean 9y ago
        Thanks for clarifying what you meant by 'analytic'. To me, 'harder, more finicky' means exactly that it is of a more constrained scope, so I don't think I interpreted OP wrong.
    - eximius 9y ago
      Only numerically? Analytically, differentiation is much easier and integrals are tough or impossible, unless I'm missing something.
- stephengillie 9y ago
  I found sine and associated concepts to be arcane and Byzantine when learning then through algebra and geometry alone. Seeing sine, cosine, etc as merely each other's derivative was astonishing and eye opening. So elegant. It made me love math again.
  - jacobolus 9y ago
    Byzantine is a good guess: Hipparchus was from a city about 50 miles away (Nicaea), though I think he did most of his work in Rhodes, and lived a few hundred years before the time of the Eastern Roman Empire. ;) But anyway, the etymology helps: “sine” = medieval Latin translation of an Arabic corruption of a word originally from India and meaning “half a bowstring”. “Tangent” = touching. “Secant” = cutting. “Chord” = bowstring. See https://en.wikipedia.org/wiki/Jyā,_koti-jyā_and_utkrama-jyā https://en.wikipedia.org/wiki/Jyā,_koti-jyā_and_utkrama-jyā The reason sine and cosine are each-others derivative is that if you start with uniform circular motion and take the vector derivative, you get another uniform circular motion in velocity space.
  - trentmb 9y ago
    The relationship between sine and cosine is apparent just by looking at the unit circle.
uoaei 9y ago
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. [1] https://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/ https://www.theatlantic.com/education/archive/2014/03/5-year...
- nxc18 9y ago
  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."
- dotancohen 9y ago
  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.
  - g10r 9y ago
    why? The question my three-year-old son asks over and over each day. It's exhausting, and I love it. I do my best to provide the answer instead of simply stating "because" or "just do it" as one of my greatest fears is to suppress his natural desire to understand as much as possible about the world. Also, as a child I hated memorization, yet loved delving into a subject that intrigued me.
WhitneyLand 9y ago
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.
- jessaustin 9y ago
  Would the typical ed-school IRB have a problem with this sort of experiment? Admittedly that is an indictment more of the IRB than of the experiment...
- avs733 9y ago
  >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?
analog31 9y ago
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.
- GregBuchholz 9y ago
  >I suggest teaching vectors before trig. You can get the cosine and sine from the dot and cross products. What about geometric algebra instead? http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/
- jacobolus 9y ago
  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.
CalChris 9y ago
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.
- stephengillie 9y ago
  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?
  - scentoni 9y ago
    https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Clifford_algebra https://en.wikipedia.org/wiki/Clifford_algebra
    - jacobolus 9y ago
      Start here http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/
  - mtzet 9y ago
    > 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? The complex numbers equips the vector space of 2-dimensional real numbers with a multiplication. This structure is known as an algebra. The analogue for 4 dimensions is the Quaternion Algebra, which is no longer commutative (i.e. a * b != b * a). The elements of unit length correspond (essentially) to rotations in 3 dimensions and they are used for this in e.g. computer graphics. The analogue for 8 dimensions is the Octonions which are no longer associative (i.e. (a * b) * c != a * (b * c)). The construction can be continued with a doubling of the dimension through the Cayley-Dickson construction, however these higher dimensional versions are even worse. It is not possible to associate a multiplication to 3-dimensional real space which makes it an algebra. This fact is related to the classification of exceptional Lie groups. The reason is roughly that the units in an algebra form a group, but the 2-sphere is not a Lie group.
yermom 9y ago
Calculus was a lot easier if you paid attention in trigonometry, for me anyway 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? :)
Myrmornis 9y ago
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.
- sannee 9y ago
  Trigonometry was important (presumably much more than calculus) for navigation/military applications at some point in history, so maybe it just stuck that way?
  - jacobolus 9y ago
    Trigonometry† is still super important for all of physics and a great deal of higher mathematics. The problem with it is that before the past 150–200 years, people didn’t have an adequate collectionn of mathematical concepts / tools / number-like objects to work with, so the description is done in an extraordinarily unnatural and cumbersome form. Additionally, the modern lessons are almost entirely anachronistic: the reason that people cared about trigonometric identities was that they did all calculation by hand or using pre-computed lookup tables. In that context, judicious application of some identities could save hundreds of hours of labor by a semi-skilled human computer, by reducing the number of arithmetic calculations and/or table lookups. In an age of computers this is not really a consideration, and modern students don’t have any appreciation for the purpose or context of the tools of classical trigonometry. The classical trigonometry course should have been ripped out and fixed 50 years ago if not before (replaced by courses in vectors, complex arithmetic, and the complex exponential and logarithm). Such deeply entrenched school curricula are very difficult to modify though. †: Note, by trigonometry what we really mean is «the relations between uniform circular motion and/or lengths of circular arcs and a square grid coordinate system»; trigonometry is something of a misnomer as there is only partial overlap between metrical 3-gon (a.k.a. “triangle”) geometry and the study of uniform circular motion, and most of the interesting parts of metrical 3-gon geometry are not covered in a trigonometry course.
swiley 9y ago
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.
BeetleB 9y ago
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).
- abecedarius 9y ago
  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?
  - BeetleB 9y ago
    >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. Let's not forget the geometric aspect of trigonometry, which will be more intuitive to many than complex numbers. I have a force of 10N applied at an angle of 32 degrees. I need the x-component. You still need basic trigonometry.
    - abecedarius 9y ago
      Yes, you need to tie the algebra to the geometry. I had a friend taking a trig class, and when I showed him how all those trig identities just fall out of multiplying complex numbers, he was pissed.
kafkaesq 9y ago
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".

jacquesm 9y ago

I found trigonometry a lot easer than calculus, which even today usually takes me forever even for what I'm sure are simple problems.

Nomentatus 9y ago
Especially that darn three-body problem. (Not poking fun, just agreeing, it's genuinely tough.)
rocqua 9y ago
I feel the same, especially with integrals. Integration is way to arbitrary and 'most' integrals cannot be done in a closed form.
- conistonwater 9y ago
  Funnily enough, from a mathematical point of view, it's the opposite: integration is a really nice operation that can be applied to basically anything, whereas differentiation is really finicky, sometimes derivatives don't exist and you can't even be too sure when, so you need to be extra careful. This carries over to doing numerical computing: integrating an arbitrary function is easy, for smooth 1d functions it's a solved problem, differentiating even a reasonably smooth function numerically is much harder. If there is noise in your function, it doesn't matter so much for integrating, but can completely break the derivative.
  - gaurav_v 9y ago
    More like `from a computational point of view.' Analytically, integration is much more difficult and richer than differentiation!
    - deleted 9y ago
      [deleted]
    - srean 9y ago
      Finding the closed form of the integral of a closed form, yes then what you say is true (this is different from 'analytic' which in mathematics has a different meaning). Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that. Note the key word there is a function not a function with a closed form that's a tiny subset.
      - gaurav_v 9y ago
        "Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that." The OP said the opposite, that differentiation is harder 'more finicky.' I agree that the concept of integration is much richer. Also, I didn't mean 'closed form solution' when I said 'analytic.' I also didn't mean 'analytic functions.' I meant that the analytic machinery you have to develop in order to have a theory of integration is far richer than for differentiation - i.e, proving the multivariate change of variable theorem.
        srean 9y ago
        Thanks for clarifying what you meant by 'analytic'. To me, 'harder, more finicky' means exactly that it is of a more constrained scope, so I don't think I interpreted OP wrong.
  - eximius 9y ago
    Only numerically? Analytically, differentiation is much easier and integrals are tough or impossible, unless I'm missing something.
stephengillie 9y ago
I found sine and associated concepts to be arcane and Byzantine when learning then through algebra and geometry alone. Seeing sine, cosine, etc as merely each other's derivative was astonishing and eye opening. So elegant. It made me love math again.
- jacobolus 9y ago
  Byzantine is a good guess: Hipparchus was from a city about 50 miles away (Nicaea), though I think he did most of his work in Rhodes, and lived a few hundred years before the time of the Eastern Roman Empire. ;) But anyway, the etymology helps: “sine” = medieval Latin translation of an Arabic corruption of a word originally from India and meaning “half a bowstring”. “Tangent” = touching. “Secant” = cutting. “Chord” = bowstring. See https://en.wikipedia.org/wiki/Jyā,_koti-jyā_and_utkrama-jyā https://en.wikipedia.org/wiki/Jyā,_koti-jyā_and_utkrama-jyā The reason sine and cosine are each-others derivative is that if you start with uniform circular motion and take the vector derivative, you get another uniform circular motion in velocity space.
- trentmb 9y ago
  The relationship between sine and cosine is apparent just by looking at the unit circle.

uoaei 9y ago

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. [1] https://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/ https://www.theatlantic.com/education/archive/2014/03/5-year...

nxc18 9y ago
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."
dotancohen 9y ago
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.
- g10r 9y ago
  why? The question my three-year-old son asks over and over each day. It's exhausting, and I love it. I do my best to provide the answer instead of simply stating "because" or "just do it" as one of my greatest fears is to suppress his natural desire to understand as much as possible about the world. Also, as a child I hated memorization, yet loved delving into a subject that intrigued me.

WhitneyLand 9y ago

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.

jessaustin 9y ago
Would the typical ed-school IRB have a problem with this sort of experiment? Admittedly that is an indictment more of the IRB than of the experiment...
avs733 9y ago
>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?

analog31 9y ago

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.

GregBuchholz 9y ago
>I suggest teaching vectors before trig. You can get the cosine and sine from the dot and cross products. What about geometric algebra instead? http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/
jacobolus 9y ago
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.

CalChris 9y ago

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.

stephengillie 9y ago
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?
- scentoni 9y ago
  https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Clifford_algebra https://en.wikipedia.org/wiki/Clifford_algebra
  - jacobolus 9y ago
    Start here http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/
- mtzet 9y ago
  > 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? The complex numbers equips the vector space of 2-dimensional real numbers with a multiplication. This structure is known as an algebra. The analogue for 4 dimensions is the Quaternion Algebra, which is no longer commutative (i.e. a * b != b * a). The elements of unit length correspond (essentially) to rotations in 3 dimensions and they are used for this in e.g. computer graphics. The analogue for 8 dimensions is the Octonions which are no longer associative (i.e. (a * b) * c != a * (b * c)). The construction can be continued with a doubling of the dimension through the Cayley-Dickson construction, however these higher dimensional versions are even worse. It is not possible to associate a multiplication to 3-dimensional real space which makes it an algebra. This fact is related to the classification of exceptional Lie groups. The reason is roughly that the units in an algebra form a group, but the 2-sphere is not a Lie group.

yermom 9y ago

Calculus was a lot easier if you paid attention in trigonometry, for me anyway 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? :)

Myrmornis 9y ago

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.

sannee 9y ago
Trigonometry was important (presumably much more than calculus) for navigation/military applications at some point in history, so maybe it just stuck that way?
- jacobolus 9y ago
  Trigonometry† is still super important for all of physics and a great deal of higher mathematics. The problem with it is that before the past 150–200 years, people didn’t have an adequate collectionn of mathematical concepts / tools / number-like objects to work with, so the description is done in an extraordinarily unnatural and cumbersome form. Additionally, the modern lessons are almost entirely anachronistic: the reason that people cared about trigonometric identities was that they did all calculation by hand or using pre-computed lookup tables. In that context, judicious application of some identities could save hundreds of hours of labor by a semi-skilled human computer, by reducing the number of arithmetic calculations and/or table lookups. In an age of computers this is not really a consideration, and modern students don’t have any appreciation for the purpose or context of the tools of classical trigonometry. The classical trigonometry course should have been ripped out and fixed 50 years ago if not before (replaced by courses in vectors, complex arithmetic, and the complex exponential and logarithm). Such deeply entrenched school curricula are very difficult to modify though. †: Note, by trigonometry what we really mean is «the relations between uniform circular motion and/or lengths of circular arcs and a square grid coordinate system»; trigonometry is something of a misnomer as there is only partial overlap between metrical 3-gon (a.k.a. “triangle”) geometry and the study of uniform circular motion, and most of the interesting parts of metrical 3-gon geometry are not covered in a trigonometry course.

swiley 9y ago

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.

BeetleB 9y ago

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).

abecedarius 9y ago
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?
- BeetleB 9y ago
  >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. Let's not forget the geometric aspect of trigonometry, which will be more intuitive to many than complex numbers. I have a force of 10N applied at an angle of 32 degrees. I need the x-component. You still need basic trigonometry.
  - abecedarius 9y ago
    Yes, you need to tie the algebra to the geometry. I had a friend taking a trig class, and when I showed him how all those trig identities just fall out of multiplying complex numbers, he was pissed.

kafkaesq 9y ago

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".