Sunday, July 12, 2015

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. 


  1. Kathy and Andrew were equally efficient, I'd say. And much more efficient than you were. Theirs are strategies that draw on fewer external resources. Yes, Kathy used pencil and paper but she could find the nearest sandpile and flat surface, or burnt sharpened stick, to accomplish the same strategy if she needed to. She was probably faster than Andrew (was she?) on this particularly challenging problem, so I'd give her points there. Andrew required no external resources at all, so he gets a couple points there.

    I found myself wanting a pen and paper on this one. I tried distributing (10+1)•18.75 mentally and struggled. So I ended up getting 206.25 by doing (10+1)•19 – 11 quarters.

    Can you check that for me on your calculator, please?

  2. "Efficiently" is relative, but the way math is presented in most classrooms assumes one efficient method for students, usually the standard algorithm or a trick that was devised to subvert understanding in favor of time. I'd think you'd need to investigate the reasoning for each method deployed and will probably find that each is the most "effective" for that particular person based on their current understanding.

    1. Replace "effective" with "efficient" in my comment. Darn autocorrect.

  3. I find myself working with 10 * 18 and 10 * .75, then adding another 18.75. I too get 206.25. Working with 10 makes it easier to add the little bit that's left. I believe that all ways are efficient in their own way, even using the calculator, but it's definitely good to be able to do it without a calculator.