## 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.

3. It is exciting when we step back and allow students to figure things out for themselves and see that, in fact, they have many different ways of thinking about computation problems. When we expose students to powerful mathematical models such as open number lines but don’t insist that they use the number line, and instead allow them to use whatever representation or tool they choose, we are given insight into the way they think about numbers. For me this is the most powerful form of assessment. When we take the time to cultivate student discussions about math and really listen to our students we are often impressed by their ideas, conjectures, and willingness to play with numbers. For more than two decades now, I have been studying ways to improve math discourse in the classroom and among teachers as they study student work and have yet to find a more powerful pedagogy than well facilitated teacher and classroom discussions for developing teacher understanding of student understanding of math. In my talk toolkit, Adding Talk to the Equation: Discussion and Discovery in Math, I’ve used video examples to show how to take math talks even deeper to connect the underpinning math concepts to the strategies and models at play. Deepening the discourse aligns well with the 8 Math Common Core Practices. It is clear that you and your colleagues are into analyzing the student work and it sounds like you are using math talks to have kids explain their thinking, which is terrific. I wonder if you are also asking your students to not only notice the different approaches, but to look for how they are similar, to compare and contrast two or more strategies and maybe make conjectures about them or to state an underpinning idea (e.g. equivalence). For example, how many of your students can say something like, “we can break the numbers apart any way we want and then add them in any order we want, and we will always get the same total as long as we include all the parts.” I noticed, you said that one of the students who created a picture representation was transitioning to a more abstract understanding. What if the class looked at the base ten picture-like representations and searched the number line and/or numeric student approaches for one that matched or was similar to the pictorial approaches? Might that help those students who are in transition take the leap to a more abstract approach? Simultaneously, it might develop the habit of mind in the other students to look for commonalities among approaches?

-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.