How much do we owe Andrew?

Andrew getting ready to add some doors
to a utility room at our camp,

Our son, Andrew, has been doing some work for us this summer. The other day was payday, and he let us know that he had put in 11 hours of work the past week. We are paying him $18.75 per hour. (He's 27 and has a degree in Building Technology from NMU, so these are not simple chores.)

As we did the math to pay Andrew for his services, I was interested in the different approaches we picked to determine what we owed him. Kathy grabbed a pencil and paper to do the standard algorithm. Andrew looked at me and asked how I would do it. "Honestly," I said, "when there's money involved, I'd grab a calculator." Andrew proceed to talk through how he would calculate 11 x 18.75 mentally. (He has always had an affinity for numbers, though he struggled with school math that relied on "rules without reasons".)

That same week, I participated with a group of about three dozen elementary teachers in training for Math Recovery. When it came to supporting students' multi-digit multiplication and division strategies, several of the teachers discussed how kids' mental math needed to lead to more efficient strategies. This seems reasonable; it's even in the Standards for Mathematical Practice(emphasis mine):

... procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), ... 

But what does "efficiently" mean when it comes to multi-digit computation? Who calculated 11 x 18.75 efficiently: Kathy, Andrew, or me? What criteria are you using for efficiently? This is not a rhetorical question - I really want to know. 

Don't you want math to be better for your kids?

That's not the way I learned it! And if it was good enough for me, then it's good enough for my kid! (Along with either: "I was bad at math." or "I was good at math.")

That's how I interpret some of the posts trying to pass themselves off as examples of "bad Common Core math problems" (Google it and take a look at some of the images). Justin Aion has a great post that points out the problem with associating these examples with the Common Core State Standards in Mathematics (CCSSM). However, even if these examples are decoupled from the CCSSM, there's still the sentiment that these new math approaches are flawed.

Take this post, for example. The parent's letter says it all:

From Jeff Severt (some context)

"simplification is valued over complication" writes the Frustrated Parent. But is the parent's approach the simplest way to compute the difference between 4,000,002 and 3,999,999? As math educators, we encourage young mathematicians to build up a variety of computational tools so that they can attack any problem with confidence and phronesis.

Recently, my class explored the thinking inherent in the work of these third grade girls.

From The Big Dinner 

This was a Big Idea on the Multiplication and Division Landscape, Proportional Reasoning, that was new to nearly all of my preservice elementary teachers. Consequently, I followed up with a Think Aloud to reinforce the Big Idea and connected it to the CCSSM 3.OA.B.5.

Afterwards, one of the preservice teachers said, "I've never seen this before. Why?" Why, indeed. 

Not so easy, is it?

Deborah Ball, Dean of the University of Michigan's School of Education, chairs the Michigan Council for Educator Effectiveness (MCEE) which was tasked to make recommendations to the Legislature about teacher evaluation. As part of her testimony before a joint meeting of the House and Senate Education Committees, she took some time to make the point that the work of teaching is often misunderstood and harder than most non-educators imagine.

The example she gives comes from fourth grade mathematics - multiplying two two-digit numbers (4.NBT.B.5). First, the teacher must be competent in the content. For example, what is the product of 49 and 25?

From Dr. Ball:

Obviously one wouldn't want anyone teaching third or fourth grade who couldn't do that. But, in fact, what skilled teaching involves is responding when students don't understand the material...

Knowing the mathematics is not enough. Teaching requires more than simply marking students' answers right or wrong. To be effective, teachers must be able to diagnose students' misconceptions in order to provide the support necessary for students to develop mathematically. Dr. Ball provides three examples of possible errors teachers might encounter from students multiplying 49 and 25:

When she asks the legislators how the students might have arrived at these answers, ... watch for yourself (it starts at about 2:30).

Dr. Ball does an effective job of pointing out that teaching is more than knowing and sharing content. However, this example focuses almost entirely on the "upper-half" of the Teaching-Learning Cycle: Assessment and Evaluation. Developing a plan for addressing the student-misconceptions and implementing that plan with a class of fourth graders raises the level of difficulty even higher. I hope these lawmakers now understand that there is more to teaching than what they experienced from the student-side of the desk.