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This is backwards. The “proper way” is to start with the right intuition, and then work out the bookkeeping and the generalizations and interconnections afterwa

This is backwards. The “proper way” is to start with the right intuition, and then work out the bookkeeping and the generalizations and interconnections afterward. My version can be explained to 12-year-olds, and your version doesn’t really make sense until grad school. (Anything starting from “real numbers” and “algebraic closure” and “field extensions” is pedagogically horrendous, and calling that version “simple” has to be some kind of sick joke.) All of the “abstract generality” is contained within the original idea of ratios of planar vectors, and this idea can be explored to whatever depth someone likes. > (Same with groups vs. linear maps vs. matrices.) I don’t know where you are going with this, but groups and linear maps are different concepts. Linear transformations (and other kinds of transformations) should be studied geometrically long before anyone learns about matrices (and anyway, the matrix version involves the arbitrary choice of coordinate system). The basic ideas are plenty accessible to children. In a similar way, symmetries of 2- and 3-dimensional shapes, patterns, tilings, arithmetic, etc. should be explored in detail before someone ever sees the definition of a group.

Koshkin 6y ago
Well, one would certainly have a hard time trying to explain to a 12-year-old how to solve a quadratic equation using rotations (let alone division of vectors).
- jacobolus 6y ago
  It’s certainly no harder than any other way of describing complex solutions to quadratic equations. (But you probably want to start with quadratic equations that have scalar solutions.) It’s quite straight-forward to show graphically what it means to square an origin-preserving similarity transformation of the plane: just apply it twice. So if your original complex number transformed an arbitrary chosen vector by scaling it some amount and rotating it some amount, just scale and rotate the result again. This is no mostly the same idea as repeated multiplication by scalars, as taught in primary school, but with planar rotations thrown in. Adding complex numbers can be shown graphically by demonstrating what effect the sum has on a vector (since the geometric product distributes over addition), or where convenient what effect it has on a square grid. We can easily graphically apply some quadratic equation to arbitrary complex numbers, and see what it does to them (again, where we draw the complex numbers using points o + zv for some arbitrarily chosen origin o and “unit vector” v). Showing that when we take a quarter-turn in our plane I, and apply it twice, we get II = –1 is a relatively trivial observation. It relies on the observation that a half-turn rotation is equivalent to a reflection across the axis of rotation. So that gives us I as one of the roots to xx + 1 = 0. The root of a quadratic equation is just the complex number which gets mapped to 0. So try applying your quadratic equation to different complex numbers and see where they go. There will always be two complex numbers mapping to zero (except in the case of a double root). Once we have some intuition for the problem we can try to develop some concrete algorithms or strategies for finding those roots efficiently. We can also try to make other kinds of plots to help us, e.g. using a tool like https://observablehq.com/d/3a81942cd20f81e4 https://observablehq.com/d/3a81942cd20f81e4 I haven’t tried teaching quadratic equations to middle schoolers, but since the graphical version mostly entails drawing pictures, I would guess it to be rather easier than a purely algebraic version. Either way there are quite a lot of tricky concepts to explore and unpack, so the students are going to need months of sustained effort.
  - Koshkin 6y ago
    This post shows clearly that geometric intuition (other, perhaps, than drawing a parabola crossing - or not crossing - the X-axis) is not helpful at all understanding and solving a quadratic equation, while elementary algebra indeed can and does; the imaginary unit, then, appears naturally as the square root of the negative unit (which leads to the need to extend the reals); if anything, the approach that you advocate would make the kid's head spin (no pun intended) with the result being the total loss of the ability to think about things in a simple and natural way.
    - jacobolus 6y ago
      Haha. I said essentially «there are many concepts involved which will take months to unpack, irrespective of which approach is taken, but in good faith here are some sketchy descriptions of what pictures you might draw if you wanted to start with a graphical interpretation ...» To which you responded essentially «your sketch summary consisting of a few picture-free paragraphs aimed at an expert audience doesn’t completely describe all of a complicated set of ideas to a naïve middle school student anywhere near as well as 2 semesters of introductory algebra class does at covering a more limited subset of the same ideas, therefore it must be useless.» This after you earlier proposed an approach to complex numbers that optimistically will first be accessible for typical 3rd year undergraduate math majors. This conversation is absurd. But all the best to you.

Koshkin 6y ago

Well, one would certainly have a hard time trying to explain to a 12-year-old how to solve a quadratic equation using rotations (let alone division of vectors).

jacobolus 6y ago
It’s certainly no harder than any other way of describing complex solutions to quadratic equations. (But you probably want to start with quadratic equations that have scalar solutions.) It’s quite straight-forward to show graphically what it means to square an origin-preserving similarity transformation of the plane: just apply it twice. So if your original complex number transformed an arbitrary chosen vector by scaling it some amount and rotating it some amount, just scale and rotate the result again. This is no mostly the same idea as repeated multiplication by scalars, as taught in primary school, but with planar rotations thrown in. Adding complex numbers can be shown graphically by demonstrating what effect the sum has on a vector (since the geometric product distributes over addition), or where convenient what effect it has on a square grid. We can easily graphically apply some quadratic equation to arbitrary complex numbers, and see what it does to them (again, where we draw the complex numbers using points o + zv for some arbitrarily chosen origin o and “unit vector” v). Showing that when we take a quarter-turn in our plane I, and apply it twice, we get II = –1 is a relatively trivial observation. It relies on the observation that a half-turn rotation is equivalent to a reflection across the axis of rotation. So that gives us I as one of the roots to xx + 1 = 0. The root of a quadratic equation is just the complex number which gets mapped to 0. So try applying your quadratic equation to different complex numbers and see where they go. There will always be two complex numbers mapping to zero (except in the case of a double root). Once we have some intuition for the problem we can try to develop some concrete algorithms or strategies for finding those roots efficiently. We can also try to make other kinds of plots to help us, e.g. using a tool like https://observablehq.com/d/3a81942cd20f81e4 https://observablehq.com/d/3a81942cd20f81e4 I haven’t tried teaching quadratic equations to middle schoolers, but since the graphical version mostly entails drawing pictures, I would guess it to be rather easier than a purely algebraic version. Either way there are quite a lot of tricky concepts to explore and unpack, so the students are going to need months of sustained effort.
- Koshkin 6y ago
  This post shows clearly that geometric intuition (other, perhaps, than drawing a parabola crossing - or not crossing - the X-axis) is not helpful at all understanding and solving a quadratic equation, while elementary algebra indeed can and does; the imaginary unit, then, appears naturally as the square root of the negative unit (which leads to the need to extend the reals); if anything, the approach that you advocate would make the kid's head spin (no pun intended) with the result being the total loss of the ability to think about things in a simple and natural way.
  - jacobolus 6y ago
    Haha. I said essentially «there are many concepts involved which will take months to unpack, irrespective of which approach is taken, but in good faith here are some sketchy descriptions of what pictures you might draw if you wanted to start with a graphical interpretation ...» To which you responded essentially «your sketch summary consisting of a few picture-free paragraphs aimed at an expert audience doesn’t completely describe all of a complicated set of ideas to a naïve middle school student anywhere near as well as 2 semesters of introductory algebra class does at covering a more limited subset of the same ideas, therefore it must be useless.» This after you earlier proposed an approach to complex numbers that optimistically will first be accessible for typical 3rd year undergraduate math majors. This conversation is absurd. But all the best to you.