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."
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.
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?
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.
Some interesting algebras:
* you can have some fun making the imaginary unit i square to 0 instead of -1. The resulting algebra is called the dual numbers, and has some surprising properties. You can gain a bit of understanding of it by using analogies with the complex numbers.
* You can make i square to +1 without being equal to +1 or -1. Again you can gain some understanding using analogies with the complex numbers. It's perhaps not the most useful examples because it's isomorphic to R \oplus R.
* Another family of examples, a very important one actually, is the algebra of 2x2 or 3x3 or nxn matrices over the real numbers or the complex numbers.
The first two examples are useful for understanding the general concept of a quotient ring. They're hardly exhaustive, but they are easy to picture.
Now I can return to your claim about CD: What makes the Cayley-Dickson family significant is that it produces all the division algebras over the real numbers, which are the algebras for which division by nonzero elements is always possible.
[edit]
Needed to escape an asterisk. Third time lucky.
Corrected grammar.
This is a very formal/technical and not very enlightening (and often quite misleading to novices) way of interpreting what the complex numbers “are”. It is basically just a declaration “we have this particular set of symbolic manipulations which we are making up; deal with it.” It’s also kind of ironic because historically the invention of linear algebra came out of attempts to generalize complex numbers.
For me, the key point is that the complex numbers are isomorphic to the transformations of rotation and scaling centered at a point in a plane, with composition of such transformations corresponding to complex multiplication. That is, a complex number can be associated 1:1 with such a transformation. So to understand complex arithmetic, what we really need to understand is the composition of plane rotation and scaling.
In other words, a complex number is not a vector (in the sense of the word “vector” used in physics; it is a “vector” in the sense of an abstract mathematical object satisfying certain axioms).
Personally I would say that the complex numbers “are” quotients of two-dimensional vectors (using the Clifford product, and with non-zero divisor) which include a given planar orientation (the bivector i, with i^2 = –1) but are otherwise unitless and need not be defined in terms of any particular basis though splitting them into the sum of a scalar and bivector part or into the product of a rotation and a scaling is often convenient. But doing this explanation justice requires spending a considerable amount of time explaining the difference between affine points and vectors, talking about how two-dimensional vectors work and how to think about vector multiplication, etc.
This makes generalization to multiplication of vectors in higher dimensional Euclidean (or pseudo-Euclidean) spaces very natural without bothering with any discussion of classifying Lie groups or whatever.
But you could also say that complex numbers are linear combinations of the two matrices [1 0; 0 1] and [0 –1; 1 0]; which when right-multiplied by arbitrary column vectors are respectively the identity and a 90° rotation.
Or you could say that complex numbers are the quotient ring R[X]/(X^2 + 1). [This is closest to the historical approach, where we just declare by fiat that √–1 will be a meaningful symbol.]
Or you could define complex numbers purely in terms of classical trigonometry and analytic geometry.
Etc.
No we can't. What digit represents 1 + 1j?
But it seems clear that the grandparent poster was using the wrong word (“digit”) and likely meant something like “we can conceive of a 2-dimensional quantity as a single number-like entity”.
Seeing sine, cosine, etc as merely each other's derivative was astonishing and eye opening. So elegant. It made me love math again.
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
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.
The class of C(1) functions is quite easy. The class of intergrable functions is much more difficult. All we know is that it is larger. Consider this: to prove a function isn't differentiable, you need only give a single point where the derivative as a limit doesn't converge. To prove a function has no integral, you need to consider all possible partitions of that function's domain. (You also need to specify what exact measure is being used, etc).
Sure, school != reality, but it's the former we get tortured by...
I see you've never had to do Bayesian inference.
Note the key word there is a function not a function with a closed form that's a tiny subset.
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.
To me, 'harder, more finicky' means exactly that it is of a more constrained scope, so I don't think I interpreted OP wrong.
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?
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?
Research is always difficult, in any field. You don’t get a PhD for courses, when you’re expected to have insights that no one ever has had before, that’s just a hard thing. Same with designing research. It’s so difficult and complex across the board people make big mistakes doing it all the time.
Same with the parent problem. Recruiting humans to study is always a huge mine field, medical/psych/sociology/etc. all deal with it.
So what’s different about education? I don’t know your field, but it sounds like things are too tightly (inherently) coupled to public policy controversy, on top on the money thing.
This is relevant in that I have it in mind to eventually tackle trig-type problems in that program.
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).
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.
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? :)
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.
Part of the reason may be that trig gets used heavily for a lot of stuff after calc1.