Sunday, January 25, 2015

Thinking About Complex Numbers Geometrically

There are two different ways to think about complex numbers -- algebraically or geometrically. Most math students only learn how to think algebraically about complex numbers. This is a shame. It’s like playing piano with one hand tied behind your back. Without the geometric perspective, many complex numbers problems become harder.

(I also think that without the geometric perspective complex numbers hardly make any sense at all.)

Here's a fairly typical complex numbers problem (from a PARCC Algebra II sample test):

The vast majority of our students are going to attempt to solve this problem algebraically, which means lots of multiplying.

A fundamentally different approach involves understanding how multiplication by a complex numbers encodes rotations and scalings.

Most multiple choice problems expect your students to take an algebraic approach, and to make algebraic mistakes. Here’s a great example of that tendency (from a past NY Regents exam).

From an algebraic perspective, yes, 8-6i and 8+6i are awfully similar options. But if you see things geometrically, 8 +6i is a fairly silly option. It makes no sense! How could we have ended up back in the first quadrant by performing a modest rotation or two in the clockwise direction?

The vast majority of mistakes that students currently make with complex numbers are algebraic. This speaks volumes about how our kids are generally taught and the limitations of doing so. To really understand complex numbers with any depth is to understand them both algebraically and geometrically. 

Thursday, January 15, 2015

Maybe It's OK To Prove Obvious Things

I was poking around one of my favorite textbooks for higher math, and I came across a problem that I thought was obvious. 

I felt, in myself, a weird combination of reactions:
  • "Of course this is true!"
  • "I have no idea how to prove this."
What a funny pair of things to think! 

I thought about the problem for a bit, and I realized how I might prove this pretty obvious fact. And that felt satisfying also, like I had cracked a puzzle or made a connection.

I see a strong parallel between my reaction and what students of geometry proof often feel. They'll see a problem and it just seems impossible to prove because it's so. damn. obvious.

As Dan puts it
To motivate a proof, students need to experience that “Wait. What?!” moment of perplexity, the moment where the seemingly-true has revealed itself, a perturbing moment experienced by so many mathematicians before them.
But maybe that's not the whole story. There are at least two ways to mess with the "too obvious to prove" dilemma. The first is to wear away at our certainty in the to-be-proved, to introduce doubt and confusion. The second might be to ensure that students have reasons to give. 

Maybe there's room for proving obvious things, just as there might be room for working on easy problems. Our job, partially, is to make sure that students have reasons to give in their arguments. 

In sum, I think there might be two big things that teachers of proof can do to help students enjoy and understand it:
  • Create genuine moments of confusion that lead to natural, informal mathematical arguments and debates that become increasingly formal in the course of time.
  • Make explicit for students the reasons  (e.g. "vertical angles are congruent") and patterns of argument (e.g. area-equivalence arguments) that one can use in mathematical contexts, and help them make connections between these and their informal tendencies.
Hypothesis: One won't work without the other.

Monday, January 12, 2015

The Extended Family of Pythagorean Theorem Proofs

In 4th Grade today, I previewed a proof of the Pythagorean Theorem. Sort of.

This activity (TERC) has nothing to do with right triangles, but it has a lot to do with this proof of the Pythagorean Theorem. (Shell Center)

The deep structure of this proof of the Pythagorean Theorem is identical to that of the array problem above. The same area is to be described in two different ways. One of those ways is obtained by describing the area of the shape as a whole and the second comes the sum of its parts. An equation (an identity) is then derived by equating these two descriptions.

One reason why my 9th Graders have a hard time with proofs of the Pythagorean Theorem is because they aren't familiar with this type of proof. And it is a type of proof, one that shows up throughout mathematics. For instance, it shows up in the study of algebra when studying visual patterns. (Shell Center)

These types of proofs continue to show up throughout mathematics. The "Proofs Without Words" genre is littered with them. Consider the following, which is a proof that the a square is a sum of odd numbers. (Wikipedia)

Why does this matter? I have two takeaways, though I'm curious to know what you think.
  • The justification for teaching proofs of the Pythagorean Theorem might have nothing to do with understanding the Pythagorean Theorem. Instead, there is a genre of proof that shows up throughout mathematics that competent students need to be able to grasp.
  • I think that students find proofs of the Pythagorean Theorem difficult in the main because they do not understand how this genre of proof works. This goes against the typical analysis which would say that students have trouble thinking logically or that they lack persistence.  (Previously: Top 4 Reasons Students Struggle With Proof
The deep pedagogy of proofs of the Pythagorean Theorem differs from its surface appearance. What other areas of k-12 math are like this?

[This post follows up on this.]

Saturday, January 3, 2015

Proofs of the Pythagorean Theorem - What Am I Even Trying To Teach?

My students have always struggled to make sense of any sort of proof of the Pythagorean Theorem. It's hard math, but this year I have been trying to push myself to get clearer about why the hard things are hard. 

I started making some progress on this when I started asking myself a series of questions during my planning time.
  • Why am I good at reconstructing these proofs? What do I know that my students don't?
  • If you were good at making sense of these Pythagorean Theorem proofs, what else would you be good at?
After thinking about these two questions, I realized that these visual proofs of the Pythagorean Theorem are part of an entire family of proofs. There are lots of proofs that require the same sort of analysis as these proofs, though they have nothing to do with the Pythagorean Theorem.

This quickly gave my lessons a new life. Here were some implications I drew from this realization:
  • In these types of proof, we almost always make progress by describing the same area in two different ways.
  • We usually get one of the ways of describing area from thinking about the shape as a whole and a second way by adding up the area of each of its parts.
(By the way, we were working on this activity.)

Figuring out where Pythagorean Theorem proofs exist in the mathematical family tree helped me clarify what I was trying to teach, and that in turn gave me ways to help kids along. I made these worksheets to draw out the connections between the Pythagorean Theorem proofs and other proofs in the "Visual Area Proofs" family.

If you want the files, they're here. I lifted the quadrilateral area problems from here and the dot problems from here.

The takeaway from all this, I think, is that it pays off to get specific about what mathematical knowledge we want our kids to have. 

  • Danielson has a similar moment where a student question prompts him to reconsider what's involved in determining the range of a function.
  • One of the joys of teaching elementary math is that there's actually a fairly decent specification of what we mean by "fluency with arithmetic."
  • This was another instance when directly telling came in helpful. Questioning was definitely important, but it was also important for me to emphasize the whole area/sum of partials parallel. And it was also important for me to encourage students to use this framework when they worked on other Pythagorean Theorem proofs. 
  • So, what do kids need to know to successfully make sense of these proofs? (1) a geometric interpretation of "a-squared" and so on; (2) a working understanding of the Pythagorean Theorem; (3) knowledge of how these "visual area proofs" work, as detailed above; (4) how to expand (a+b)^2 and do other binomial algebra; (5) how to find the area of triangles and various quadrilaterals. That's why it's hard math -- there's a lot of stuff you need to know, all converging in one problem.

Tuesday, December 30, 2014

Year In Review

Here's what I wrote about in 2014:

Making Sense of Complex Numbers - I wrote a series of posts developing an argument against the idea that complex numbers were introduced for "whimsical" purposes. (this, then this) Teaching and reading about elementary school math helped push me to the conclusion that rotations might be the best way into imaginary numbers. At the highest level of sophistication, though, complex numbers are nothing more than points. So: start with rotations, end up with points.

Feedback and Revision - Before this year I was pretty pessimistic about feedback. If not pessimistic, then confused. I started making progress when I realized some potential drawbacks of immediate feedback. The next step was understanding how feedback could come in the form of questions and that feedback didn't even need to be individualized. (Though it could.) It was hard to square these conclusions with SBG-as-feedback, and I spent some time worrying about that. But I also ended up worrying about how we talk about feedback, and this lead to my series of posts about feedback and revision.

Exponents as Numbers - Past research made me very familiar with the sorts of mistakes that kids make with exponents, but I didn't really have a prescription. I still don't, but a few posts this year brought me closer. I argued -- and I don't know if I still agree with this -- that a sophisticated understanding of exponents is closer to seeing exponents as a number and not as an operation and certainly not as an abbreviation. How do we help students see the sorts of numbers that exponents represent? I think geometric series are key. I argued that we could use this to give kids a sense of what a "power" is. I then wrote about two lessons that show how I tried this in class last year. (here and here)

Teaching Proof - Much of my writing about proof this year has a focused thesis: proof is hard because geometric reasoning is hard, not because logic is hard. In fact, to the extent that logical reasoning can be called an independent skill, children (even young children) already have it. To show this I asked parents to try an experiment with their kids and the results were clear. Rebecca's children can reason logically, so can these kids, and so can your's. This goes against a popular teacher account of why students struggle with proof, and it has implications for instruction. We need to spend less time teaching "everyday" logic and more time scaffolding geometric proof with the full range of proof activities. In general, I came to think that we need to get more specific about the proof-knowledge our kids are missing.

Researchers and Teachers - Some of the most fun I had this year was reading and writing about From The Ivory Tower to the Schoolhouse with Raymond. (Thanks, Raymond!) The book is all about the ways university research does and doesn't make its influence felt in classrooms, and our posts dug into these ideas. The trouble is that good ideas aren't always popular ones, and a lot has to do with the popularizer and the message. These sorts of concerns popped up when I wrote about feedback and generally caused me to be anxious about my own career.


This list is disparate. Does anything unify these concerns?

Besides for a pain-in-the-ass contrarian streak (but isn't every argument contrarian?) I think that my writing this year struggled mightily with the theory/practice divide. Teachers that I know (myself included) tend to seek activities and easily usable answers and resources. But on the topics that I've thought the most about -- proof, exponents, feedback, complex numbers -- I see the existing answers as inadequate. Teachers aren't theorists, though, and the way that we communicate is through easily usable activities and resources. (That is, sharing resources is teacher discourse.) 

What will it be: essays or resources? This next year I'd like to do a better job with both. 

Saturday, December 20, 2014

In Conclusion (Post 10 of 10)

In conclusion, teaching has a writing problem.

Part of the problem, I've been arguing, is that in education we talk about the wrong things. We talk about concepts that don't get at the heart of classroom teaching -- concepts like feedback -- and so we ask questions that are impossible to answer. As a consequence, these questions -- like how do you give effective feedback? -- are very difficult to respond to. They require the ability to split hairs and to generalize. Academic researchers and consultants are very well-suited to these tasks. Teachers aren't.

And so almost all the writing about teaching comes from researchers or consultants, people who are no longer k-12 classroom teachers. This would be a fine state of affairs if it worked. But writing about teaching rarely does.

My favorite writer about feedback is Dylan Wiliam. His work is as smart as Edutopia's is silly and reductive. Here's a picture that makes him look like a super-villain:

I heartily recommend Embedded Formative Assessment. There's a great passage in there where Wiliam realizes how difficult it is to communicate about feedback.
In 1998, when Paul Black and I published "Inside the Black Box," we recommended that feedback during learning be in the form of comments rather than grades, and many teachers took this to heart. Unfortunately, in many cases, the feedback was not particularly helpful. Typically, the feedback would focus on what was deficient about the work submitted, which the students were not able to resubmit, rather than on what to do to improve their future learning.
Got it! Focus on what the kid can do to improve, right?
I remember talking to a middle school student who was looking at the feedback his teacher had given him on a science assignment. The teacher had written, "You need to be more systematic in planning your scientific inquiries." I asked the student what that meant to him, and he said, "I don't know. If I knew how to be more systematical, I would have been more systematic the first time."
So Wiliam needs a way to say exactly how feedback can tell a kid how to improve. What does he land on?
To be effective, feedback needs to direct attention to what's next rather than focusing on how well or badly the student did on the work.
Feedback should cause thinking. 
If, however, we embrace the idea of feedback as a recipe for future action, then it is easy to see how to make feedback work constructively.
It seems to me that Mr. "Be More Systematic" probably thought that his feedback focused on what's next, caused thinking, and gave a recipe for future action. Wiliam's metaphors and slogans are not enough.


It's not Wiliam's fault. He's trying to tackle an incredibly broad subject. (The chapter is titled Feedback That Moves Learning Forward. All of it??)

Here's an embarrassing belief that I have: classroom teachers have a chance to make things better. Nobody in education is better equipped to find the most helpful way to think or write about classroom teaching than classroom teachers. We have the stories. We have the teachers -- novice and experienced -- that we talk to daily. We know what our colleagues find helpful. We know our kids.

When we write about teaching, we have to play to our advantage. We have experience, stories and students to work with. What we lack is precisely what researchers and consultants have -- wide-ranging perspective and time. I think a lot of the silliness we see in education writing comes from writers trying to hit the scope and abstraction of university research. (It's like writing an op-ed in binary.)

There is so much fantastic story-telling from teachers that I read, but just telling stories isn't going to be enough. If we want to contribute to our common knowledge about teaching we need to try to do something with our stories. To analyze them, to collect them, to point out patterns and to generalize from them. That's one of the challenges that teachers writing about teaching face.

Maybe. I think.


  1. I've argued that revision is a central concept in teaching. No way this covers the entire (but wildly abstract) formative assessment landscape. What other core concepts could we break formative assessment/feedback in to?
  2. It's easy to help other teachers improve their repertoire of activities or teaching moves. (Just tell us, and we add it to the mental list.) But can we also share our decision-making, how we navigate through our repertoire?
  3. Is there an intellectually respectable way for teachers to write about teaching that other teachers will want to read?
  4. Is there an intellectually respectable way for classroom teachers to write about teaching that university researchers will want to read?
This is the last post in a series about feedback. To read the rest of the series, click here.

Tuesday, December 16, 2014

How To Write About Teaching... (Post 9 of 10)

...question mark? This is really an open question for me. I am far less confident about how to write about teaching than I am about how to teach.

Here's what we know: teachers don't like educational research.

Another thing we know: there is a lot of educational research about feedback.

Look: if you're reading this then you're probably a math teacher. And we've already established that math teachers don't really care for educational research. You might have a faint interest in what the research says, but in all likelihood you don't find that research compelling in any way.

Q: What does compel a teacher, then?

A: Twitter.
Where is the middle ground here? Social media doesn't offer much argument, but academic research is toxic for teachers. Is there any intellectually respectable way to talk about teaching that teachers will care about?

Like I said, to me it's really an open question. I don't know how to do it. I'm struggling here.

My recent writing about feedback is an attempt to find a place to land. Here are the (loose) guidelines I had for this series of posts on feedback:
  • Talk about a kid, not "the kids." Whenever possible, I tried to focus in on a specific student. That helped me give enough context about the kid's specific needs (social, cognitive, motivation, etc.) that you could understand my situation better.
  • Talk about decisions instead of what happened. I tried to avoid the "Here's what I used to's what I did worked!" cliche of teacher-writing. It's hard to know what's really happening in someone else's classroom. I can't help you see that my feedback is working, and you shouldn't just take my word that it is. Instead, I tried to argue for the way that I was thinking about teaching.
  • "Enough Context" = The math goals + kid's thinking + kid's social status. It was important to me that I give you enough detail so that you could be capable of skepticism. I found that three things often felt important to offer: what I was trying to do, what the kid thinks about the math, and how he/she interacts with math and her peers.
Here's what we know: teachers don't like reading research, but we have no good substitute for it. In a world like ours, the only sensible thing to do seems to be to read research but write from experience. But experience is famously unreliable. We need to find some way to raise the bar on writing from experience. 

But I feel very unsure about this whole thing, so here are some questions.
  1. The nutty thing is that sometimes teachers -- even the very teachers who poopoo research! -- cite or rely heavily on the research world. "Research shows that white boards work," or "Science shows that timed tests are bad for kids." I mean, what's the deal here? (See Schenider, to start?) 
  2. Is blogging the solution?
  3. Forget research for a second. What do you like/hate about non-academic writing about teaching?
  4. The test for teachers often seems to be usability. "It has lots of practical, simple suggestions that I can use in class the next day." Is this what's missing from writing about teaching? Maybe a mix of instantly usable ideas with longer-form arguments could help?
This is the ninth post in a series on feedback. To read the rest of the posts click here.