Saturday, March 29, 2014

Learning Complex Numbers, With One Eye On Elementary School

From Children's Mathematics:
The thesis of CGI is that children enter school with a great deal of informal or intuitive knowledge of mathematics that can serve as the basis for developing understanding of the math of the primary school curriculum. Without formal or direct instruction on specific number facts, algorithms, or procedures, children can construct viable solutions to a variety of problems. Basic operations of addition, subtraction, multiplication and division can be defined in terms of these intuitive problem-solving processes, and symbolic procedures can be developed as extensions of them.
The book Extending Children's Mathematics applies the CGI framework to fractions. They recommend that "Equal Sharing" problems serve as the basis for a kid's fractions knowledge:
In contrast to Equal Sharing, problems that have children identify a fraction based on a given number of parts or divide a shape into a specified number of pieces offer few opportunities for children to make connections between fractions and whole numbers. Even though such problems are concrete in that children can see and maybe even manipulate the shapes, they are not situated in any meaningful context -- such as the process of sharing food with friends -- that would allow children to use their understanding of number to create and make sense of the resulting parts as quantities.
The rest of the book serves as a really masterful show of just how far you can run with "Equal Sharing" problems. The authors turn these Equal Sharing problems inside out. You can change what we know and what we don't -- are we looking for the amount per sharer? the number of sharers? the numbers of shared things? -- and you can change every relevant number. The authors show how these subtle changes in problem design drastically change the way that children see and try to solve them. They show what sorts of numbers and unknowns lead to the most productive insights and conversations about equivalence, ordering, adding, common denominators, decimals, and all things fractions.

I mean it, the book is astounding. And I can't think of any similar treatment of any high school topic.


What would it mean to do something in the spirit of the CGI treatment of fractions for complex numbers?

Skeptic: "Well, you can't really do that. This level of math is entirely different from primary school. Listen. The CGI people say it straight away: children enter school with a great deal of informal or intuitive knowledge of mathematics. That's fine for things like multiplication and fractions. But nobody has informal knowledge of complex multiplication. Why? Because nobody knows that complex numbers exist before we show them that they do!"


Quoth the skeptic, "Nobody knows that complex numbers exist before we show them that they do!"

What are complex numbers?

  • If complex numbers are an algebraic extension of the reals, then students have never pre-formally experienced complex numbers.
  • If imaginary numbers are the square root of negative numbers, then students have never pre-formally experienced complex numbers.
  • If complex numbers are solutions to polynomials, then ditto.
And, of course, complex numbers are all these things, but they are other things as well. 
  • From a geometric perspective, complex numbers are points in the plane
  • From a transformations perspective, complex numbers are rotation/dilations (amplitwists? roliations?)
  • From an algebraic perspective, complex numbers are rules that describe transformations 

What's great about those "Equal Sharing" problems is that they can be solved both efficiently/formally and inefficiently/informally. 

5 kids are sharing 8 bananas. Boom. Everyone gets 8/5. That's how I'd solve it. But a kid might take her time. She'd start assigning individual bananas to kids. And then maybe she'd split the remaining bananas in half. And then maybe do that again. And then maybe she'd think, shoot, how do I share one banana with five kids? And then she'd say that every kid gets 1 banana and 1/2 a banana and a fifth of 1/2 a banana.

We're looking for problems that can be solved both efficiently and inefficiently, both formally and informally. 

Do such problems exist for complex numbers? Problems that are rich, with lots of possibilities, that kids can solve without complex arithmetic but better -- faster, cleaner, more reliably -- with complex arithmetic? And problems that could serve as the basis of a lot of complex arithmetic?

I'm betting that the answer is totally "yes."

This post is already far beyond the attention span you can expect on the internet, so I'll wrap things up with some questions.

  • What other problems fit the bill: questions that can be solved both efficiently with complex arithmetic, and inefficiently with the normal set of skills you can expect a high school student to have?
  • How far can you run with these "rotation rules" questions? What are the different strategies that students will use?
  • What other high school topics are waiting for the elementary school treatment?


  1. Loving this train of thought! I'm curious whether you've seen this slideshow (, and whether it connects with what you are looking for?

    Most textbooks seem to introduce complex numbers without the visual component (why?!). I always show students the complex plane, and how multiplying repeatedly by i takes you around in a circle. (This story might also inspire you:

    I think any good treatment of complex numbers would include what you suggest: problems that can be solved "without complex arithmetic but better -- faster, cleaner, more reliably -- with complex arithmetic".

    1. Hey Sue!

      Thanks for the comments. Truth is, I have seen the John and Betty slideshow, and I actually think that it's very much unlike what I'm looking for.

      I think, actually, that I'm going to write a post explaining what I have against John and Betty...

  2. CGI began with basic arithmetic and was extended to fractions, but apparently attempts to extend CGI much further just got too complicated. CGI relies on a deep understanding of how students construct their mathematical knowledge and we simply don't have the kind of understanding about constructing knowledge of something like complex numbers that compares to our deep knowledge of students' knowledge of arithmetic. To really stay true to CGI, we'd need to not only know about students' understandings, but we'd need to prepare teachers to recognize and respond appropriately to students' construction of knowledge (the "cognitively guided" part). It might be possible, but it will likely take a concentrated research program as big or bigger than the original CGI research to figure it out.

    1. The pushback and context is well-taken, Raymond.

      But what to do? Wait around for a research program or two to emerge?

      There needs to be a role for teachers in the field to make progress on these sorts of issues. I don't really understand what that role is yet, but I'd rather err on the side of doing things.

    2. Oh, don't get me wrong -- there are excellent lessons from CGI to apply elsewhere, namely that good things happen when teachers have an understanding of student learning, look for it, and use what they see to make informed adjustments to their teaching. I'm sure every teacher could read what I just wrote and say, "I do that!" so obviously it's not that simple, and that's where the research needs to come in. A teacher working on his/her own to do this is awesome, but to bring that knowledge to the scale and quality of CGI would need a concentrated effort. Current research in this direction usually goes under the name "learning progressions" or "learning trajectories," and they still tend to be least common at the high school level.

    3. My inner political activist is like, "well let's write to Congress then!" Seriously, might this be one of the big research questions that NCTM asked its members to identify? Could NCTM or the NSF or some consortium of STEM-y groups take the lead on this? How would that work? What's the connection to the Common Core?