*This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.*

Lampert spends this chapter talking about the decisions she makes while students are doing math each day in class. Students are both working on their own and collaborating with others in their group.

I’m fascinated by how many times Lampert uses the word “teach” in this chapter. Some examples:

But I was also teaching Varouna that making a certain kind of picture is a strategy for finding the solution to this kind of problem (125).

…

I worked on interpreting his responses, both in the interaction and as I thought later about how to teach him to be more mathematically assertive (127).

…

Perhaps one of them has “taught” the other that it is better to have the boxes filled in with

somethingthan to leave them empty, even if there is no clear reason for the numbers chosen, or perhaps they did not interpret the task as having to do with multiplication (130).…

His actions presented me with an occasion to teach the task structure of collaboration explicitly (130).

She uses the word “teach” in many contexts that I often don’t think to use the word — in interactions when she redirects student attention, or answers a question students have, or gives a hint to a student who is stuck.

Lampert is incredibly thoughtful with respect to the impact of each action — or inaction — in the classroom, and what they may or may not teach students. I think it’s worth categorizing the types of things that are worth teaching in these situations, based on her examples:

**Teaching content**. Lampert does this both explicitly, by taking moments to summarize and make points about the problems students are working, and implicitly, through the connections she helps students make in their independent work.

**Teaching strategy.** Lampert does not do this explicitly, where I think it is less likely to be successful. Instead, she works to reinforce effective strategies that she sees, and model useful strategies that students have not thought of.

**Teaching beliefs.** Lampert works to teach students that they are capable of making sense of mathematics, that they have ideas worth sharing, and that finding new approaches to problems is more useful than finding a single straightforward solution.

**Teaching norms and routines. **Lampert pays attention to repeated student actions that are productive or counterproductive for future learning, and works to build norms and routines that will facilitate learning.

**Acknowledging that students teach each other. **And that this can be both productive and counterproductive, in any of the above realms, and that Lampert’s teaching influences its utility.

**An Instance of Teaching Practice**

I taught this Illustrative Math task today in a Pre-Calc class:

It’s an interesting one. There’s a great deal of content here. I taught arithmetic with complex numbers, the complex plane, and geometric interpretation of complex numbers.

But while I think there was a great deal of content learning happening, the way I facilitated the task may have been counterproductive for learning in other areas. I let students try the task independently first. We had worked through this task previously, and most students were comfortable operating with complex numbers, but those operations were an obstacle for a few students and slowed them down significantly.

In addition, no students realized that the geometric operation was a 45 degree rotation with a dilation. The majority of student work was simple computation and placement of complex numbers in the complex plane. I did not provide effective scaffolding to move students individually or in partners toward this realization; instead, we did it on the board, after partner work, and I was doing much of the work. I implicitly taught students that they were not able to make big realizations about the structure of mathematics — that was the role of the teacher in this lesson.

This task also provides strategy for students; they do not choose what operations to use or what representations may be useful. Instead, the problem tells them what to do, they follow directions, and they arrive at an answer. I taught them that, often, problems provide strategy for them, and finding new strategies is not their role as a student.

Students worked in partners on this task, but the result of that structure was that much of their collaboration was around checking their work and fixing arithmetic errors, rather than doing rich, divergent mathematical thinking and sharing ideas. I taught that partner work is primarily about getting right answers, not sharing new strategies.

**Doing Better**

I think a simple change might be useful for using this task in the future: remove the direction to graph the powers of *z* in the complex plane, and have students compute through higher powers of *z*. With more examples, students are better provided with the resources to make a generalization, and share strategies for how they might extrapolate it forward. Students would be unlikely to draw on the representation of the complex plane in this instance, but it is possible to make valuable conjectures about the pattern without it. I can then provide the representation of the complex plane, teaching students that alternate representations can often make a pattern or conjecture more clear, and that these choices should not be arbitrary or dictated by forces outside of their control.

I’m looking forward to teaching this again, and I’m bummed I won’t have the chance for a while. But I hope this perspective on teaching through independent work — teaching content, strategy, beliefs, norms and routines, and cooperative learning — will help me better analyze and reflect on more of my teaching in the future.

howardat58My view is that a number has no meaning in itself, only in relation to other numbers, in particular a base or origin number, So with real numbers 2 is 2 more than zero, a sensible origin number for counting, and also as twice as big as one, which is a suitable base number for measuring or proportion. These both work for complex numbers.

Now picture it. Join each (complex) point to zero, or join each complex point to the previous one in a power sequence.

What you are trying to achieve should become apparent from the picture, but even more so if you had a smaller imaginary bit, say 1 + 0.5i or 1+0.2i

This should then do as a nice lead-in to cos(theta) + i*sin(theta) as the points on a circle.