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
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| Starting with 12 + 8 was common. |
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| This student decomposed the 50 into 5 jumps of 10... |
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| ...while this one took the entire 50 in one jump. |
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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.
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Partial Sums
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| This student added the tens, then the ones, then put them together. |
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| 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. |
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| 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. |
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| Same as above, but he used words to describe what he did. |
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| Can the traditional algorithm be far behind? It's close, and when it comes it will have a conceptual underpinning. |
Base 10 Block Representations
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| This student is transitioning from the representational to the abstract... |
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| ...while this one is not quite ready yet. |
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| This example would provide a nice opportunity to discuss equations. |
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| 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.
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!
ReplyDeleteThanks 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.
ReplyDeleteIt 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?
ReplyDelete-Lucy West
MetamorphosisTLC.com
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.
ReplyDeleteComparing 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.
Thanks for sharing this post Joe. In Ontario we are lucky to have a curriculum document that states that students should add and subtract using different mental strategies, along with student made strategies. Unfortunately it still adds the traditional algorithm in there too. Hopefully that will be taken out the next time it is revised. In the schools I am the IL for and our board in general number talks have really taken off. Students are getting a chance to use different strategies and see other students thinking while building off each others thinking. It is amazing to see, similar to what you are seeing. To add into your conversation about the grade three teachers coming up for these students, I think they will be very pleased. I am seeing it here, teachers who started the number talks almost three years ago have allowed the following grade teachers to reap the benefits (who are also mostly doing number talks). The students are moving forward with incredible number sense. Here are two examples, one third grade teacher I work with was blown away and came to me with this question. "Mark what do I do, these students are adding and subtracting 3 digit by 3 digit number in their heads this year. Sometimes they are whipping through the addition problems and are telling me answers before they even sit back at their seats!" First of all I said that is great problem to have now you can challenge them and make them into even better problem solvers because the computations part is becoming much easier for them. In the Ontario curriculum they need to add and subtract 3 by 3 digit numbers using student created algorithms and traditional algorithms by the end of grade 3. Now they are coming with such good number sense and strategies they can do these in their heads. So the effect is rippling up! I hear the same thing from grade 6 teachers who now have students that have been doing number talks and using flexible strategies for two years now saying the same thing with regards to multiplying. Like you said it makes me excited to come to school and see it continue to grow and allowing teachers to go deeper with more challenging tasks and problem solving. Seeing your posts makes me happy to know its spreading to other places and that hopefully the people that demand back to basics memorization don't get any more of a grip or can see the light! LOL. I am still seeing traditional algorithm teaching going on in early grades but we are working on moving everyone away from that into teaching more flexible number sense so I just keep plugging away! If you are looking for a excellent resource that has a developmental continuum for how students develop these strategies look into Alex Lawson's book "What to Look For". Its an amazing resource and has continuums for early operations development for addition/subtraction and multiplying/dividing. It also provides videos of students using the strategies as they develop over time with next steps/games/activities to move them along in development. Once again I thoroughly enjoyed your post keep them coming!
ReplyDeleteThanks 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.
DeleteI 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!