The Mid-Class Launch

I've been thinking lately about how formative assessment and feedback can sometimes feel overwhelming. Maybe that's just the lay of the land, or maybe there's something we can do about that. Can formative assessment be more manageable?

I think that there are some easy shifts to make. A small change that makes a big difference is to increase the number of activities that you launch half-way through your class period. That way, activities don't wrap up at the end of a single class period and you have the possibility of responding in some way to your students' work.

The bare-bones version of this can still make a big difference. Here's what it might look like:

• MONDAY: You do something in the first half of class to prepare kids for an activity. Then they spend 20 minutes working on the activity. You collect it.
• MONDAY NIGHT: You read their work. You notice where they had trouble.
• TUESDAY: Relaunch the activity the next day. Give the whole class some feedback about common issues that came up, and then hand them their work. Have them continue for 20-30 minutes. 

This is the indispensable core of formative assessment: learning what your kids are thinking and using that to make informed decisions. Even if you don't have time for comments or if you feel the need to slap a grade on everything your kids work on, this routine will still help.

Questions for y'all:

1. Do you agree that this routine make a big difference?
2. Are there other easy changes that you'd recommend for increasing the amount or quality of formative assessment we give?
3. Which is more important for learning: knowing what your students think, or giving them feedback?
4. Is it important for teaching techniques to be easy?

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. 

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.

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.]

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. 

Endnotes:

• 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.