## Thursday, May 12, 2016

### Second Grade for the Win

One morning, a few weeks after publishing this post detailing the struggles some of our third and fourth graders were experiencing with the traditional algorithm, I found myself in Maggie's second grade classroom.  A unit progress check was looming, and the kids were hard at work on a review page.  As I walked around observing their efforts, I found myself admiring the varied ways they were solving this addition problem:

I asked the three other teachers on the team to save the work their students had done.  Here's a representative sample of what I saw:

Number lines

 Starting with 12 + 8 was common.

 This student decomposed the 50 into 5 jumps of 10...

 ...while this one took the entire 50 in one jump.

 Although it came last in the sequence of addends, this student started at 54.  A great question to ask him would be, "Why did you do that?"  He also broke the 12 up into a 10 and a 2 with the 8 in between.
Partial Sums

 This student added the tens, then the ones, then put them together.

 This student also started by adding the tens, then put the 8 and the 2 from 12 together to make another 10.  She added that ten to get 70, then added the 4 left from 54.

 This student started by putting the 12 and 8 together to get 20.  Then added 20 and 50, and tacked the 4 on at the end.

 Same as above, but he used words to describe what he did.

 Can the traditional algorithm be far behind?  It's close, and when it comes it will have a conceptual underpinning.

Base 10 Block Representations

 This student is transitioning from the representational to the abstract...

 ...while this one is not quite ready yet.

 This example would provide a nice opportunity to discuss equations.

 Another student in transition.

One after another, these beautiful examples of student thinking passed before my eyes.  And out of about 70, I saw only one traditional algorithm.
I have not been shy about calling out our curriculum when I feel it's been lacking (see here and here.)  But I have to give credit where credit is due.  The work I saw was a result of the focus that the new, Everyday Mathematics 4 has placed on computation strategies that include finding number bonds for 10, decomposing multi-digit numbers into tens and ones, and using open number lines and base 10 block representations.  The teachers have maintained fidelity to the program, and in addition have started to explore number talk routines.  The results are plain to see.  I've been teaching elementary math for 30 years, and I've never seen anything like this.  The second grade teachers haven't been at it that long, but neither have they.  Does it represent a sea change, or is this an isolated phenomenon?  What will the third grade teachers make of these students?  Will they notice a difference?  How will they bring them forward?  It's questions like this that keep me excited to come to work every day.  Stay tuned and we'll find out together.

1. I teach 2nd grade and over the past few years have changed the way I teach add and sub. This year I am floored by some of the thinking of my students and I attribute it to the fact that we do NOT use algorithms at all. We have just started multiplication so I used ideas from your post from Amanda Bean's Amazing Dream. I was so surprised to see the many different ways my students solved 9 x 12. some of them drew the array, some decomposed it to 10 x 12 then subtracted the extra... some were just counting squares by 1's or 2's or 5's or 10's. It was lovely to see all the different approaches. When they shared their solutions with each other, they were also amazed at how many different ways they could all reach the same answer. I am having a great time with math challenges, and so are the kids! Thanks for all the great ideas and inspiration!

2. Thanks so much for your comments. I agree, it is amazing to see how varied and creative student thinking can be when we give them the opportunity and don't force a one-size-fits-all procedure. I'm impressed that your second graders did so well with the picture from Amanda Bean's Amazing Dream! Please continue to share your successes.

-Lucy West
MetamorphosisTLC.com

4. Lucy, Thanks so much for your observations and insights. "Expose, but don't insist" is something we are just now getting used to. Sometimes we get mixed messages, because we still see questions in our journals and assessments that require students to use a particular tool, representation, or strategy. Allowing students to choose their own tools and representations, cultivating student discussions; again, something new. In the past we have exerted much more control. Opening things up has been an exhilarating but also somewhat scary experience for teachers, because now they know they might see or hear something come out of a student that's unexpected, something they themselves might not fully understand.
Comparing strategies, making conjectures, matching strategies across representations; we're getting there, slowly, but still have a long way to go. Building teacher capacity in the area is hard, especially in the primary grades where teachers are responsible for every subject, and reading and writing suck all the oxygen out of the room.

1. Thanks for sharing your experiences Mark. I will definitely check out Alex Lawson's book. You're right, these are good problems to have. The vertical alignment is really important here. Schedules permit for teachers to meet in grade level teams, but not always in vertical teams. When they do, they can share the strategies and practices they've used and build on what the kids already know, instead of reinventing the wheel every year. It's a powerful experience for a teacher to see kids coming in with that type of number sense. It only reinforces that promoting that kind of thinking (via number talks, for example) is the way to go.
I also agree with your insight that the less time the teachers have to spend on computation the more they can spend of having kids engage in rich problem solving tasks where the computation is a means to an end, not an end in itself. So the challenge for the student who can do the multi-digit addition in his head isn't giving him bigger numbers to add, but richer tasks within which that skill is embedded.
Great conversation and thanks again!