Perhaps because I’ve watched every step of this saga of Michael’s Journey Into the Complex Plane with hawk-like attention, I’m totally down with what he’s trying to do in his blog post about how he might introduce students to complex numbers. He’s looking for a genuinely perplexing, easy to formulate question that students have the werewithal to begin to answer.

I think he’s nailed easy to formulate, and his 17 pages of work show that students do actually have the arithmetic know-how to answer this. I think students will be perplexed by this, but I do wonder about where are the parts where students need specific mathematical habits of mind & skills to be enabled to persevere.

Some things that need to be in place for this to work:
Students need to be able to hold on to the ambiguity of multiplication as an operation on the plane and the (shorthand) idea of multiplication of a point by a point. Or we need to have a language that unambiguates that. E.g. 3 * -2 is “where does 3 go under the transformation that takes 1 to -2?” [and then commutativity is NOT obvious].

If we keep using the shorthand of multiplication of a point, by a point, students need to be comfortable with having multiple physical representations of the same operation, or we need to train them in one that we want them to use. Again, I’m not sure which is better, but I’m leaning towards really hammering and making both sensible and automatic the idea of a twisting, scaling slide rule kinda thing (i.e. multiplication of real numbers is rotating and dilating the real number line, and adding real numbers is translating the number line left or right).

Also, depending on your definitions of dilation of the number line based on points, you don’t need the rotating idea until you introduce complex numbers, because the signs of your points will take care of that (dilating by a negative ratio includes a rotation in GeoGebra or Sketchpad, but you can define your dilation based on length, not position, and then you do need a rotation. I made a collection to help play with that idea here: It’s not fun and visual, but it is mathematically intriguing to see how the points are defined).

Here’s where the habits of mind really come in. If we ask students to extend their understanding of 1D operations on the real number line to 2D representations, they need to be able to:

  • Understand that generalizing means making a coherent system that doesn’t “break math”
  • Decide on the rules that we want to define not breaking math to be
  • Generate conjectures about what a generalization might look like
  • Test those conjectures
  • Persevere through multiple conjectures and tests
  • Accept a definition of multiplication that is not their initial intuition and may even trouble their 1D understanding of multiplication
  • Persist through defining a generalized multiplication to mastering said multiplication, both geometrically and algebraically.

Most students have never been asked to conjecture possible definitions for an operation, and have never been exposed to the idea that mathematicians posit the existence of objects and operations and then test to see if they break or not. Which is too bad because that’s a lot of what mathematicians do, and something students are capable of, but getting students to the point where they’re willing to define mathematical operations or objects for themselves and then persevere through playing with possibly broken objects long enough to find one that works, is hard.

[It's sort of like giving a kid a huge pile of boxes to open on her birthday, with the caveat that most of the toys she'll find are missing pieces and will never work, but once she's done opening & testing them all she'll have found some AMAZING working toys and learned a lot about how toys work. This is why math class is not a birthday party, it's HARD fun, much more like learning to ride a bike (ouch!) then opening birthday presents.]

Students also need a robust enough understanding of operations that they get what it means to not break math. They need to expect commutativity and associativity and the distributive property (which most kids don’t understand, let alone value!). They need to compute fluently with positive and negative numbers, including distributing. A robust understanding of the geometry of transformations would be nice too.

All images from

Finally, the transformation of the plane that relates closely to complex multiplication is the beautiful Spiral Similarity, which results in lovely spiral tessellations. Could a launch perhaps be based on telling some technology to make spiral tessellations for you, and then making the connection among algebraic and geometric definitions of transformations, and finally generating a robust set of algebraic rules for exploring and defining spiral dilations and 2D translations and then connecting that to the transformation composition that takes 1 to -1. See more about Spiral Similarity here:

PS — on contexts, perplexity, motivation, etc. see Riley Lark:

Update — I’ve been meaning to share this article for a while; it’s not completely relevant here but I like it: Research Mathematicians as Learners And What Mathematics Education Can Learn from Them — it’s about the doing of mathematics as mathematicians see it and the opportunities for students to do that kind of thinking. The second is a book about elementary math concepts and how they relate to higher math, as told through the eyes of what used to befuddle research mathematicians when they were elementary students: Shadows of Truth: Metamathematics of Elementary Mathematics.