Preparing Students for Testing
Teacher: I just want to tell a story, just about my own journey and specifically my journey with the standard algorithm for addition and subtraction. Again a lot of my experience has been in second and third grade, and as a second- and third-grade teacher, it’s my—I thought it was my job to be sure that the students used the standard algorithm for addition and subtraction correctly. And … looking back, I wish that I didn’t have to say that my journey toward making a shift there took as long as it actually did, but it did take a long time for me to shift away from that, and, you know, the more I started learning about CGI and the more I started learning from my kids, I began seeing the value of listening to kids and changing what my role as a teacher is. So, I found that … I was feeling some good success with listening to kids and switching my role and putting the power in their hands as mathematicians to allow math to make sense for them, but at the same time, when it came to the standard algorithm, I taught the standard algorithm.
And at first I felt fine with it: You know this is what you need to do, teach a standard algorithm. And the more I started learning about CGI and about kids, the more uncomfortable I felt with this role that I’ve always played of teaching the student this particular procedure for solving 2-digit plus 2-digit or 2-digit minus 2-digit computation. And I had so much encouragement from other people saying, “Trust this process; believe in your kids, your kids actually can do it.” And intellectually I knew that they could, but the logistics …; I felt very accountable for making sure that I sent my students on to the next grade having that particular procedure in place. So, I didn’t do it [stop teaching the algorithms] and I didn’t do it, and then there was a year—this is about 3 years into being a CGI teacher—that I ended up with a class that happened to be a combination class, which is pretty common out in California to have.
So, this year I had second and third graders in my classroom, and it was really more of a logistical issue of How am I going to teach math when I’ve got two different grade levels in my classroom? that caused me to say, “You know what. This is the year that I need to go full force into using CGI … for all of my math that I teach. And that includes teaching … or using CGI instead of teaching the standard algorithm.” And it was really scary for me, and I hate that it was so scary for me to give up that algorithm, but I just decided, “This is the year I’m going to do it. And I may regret this decision.” In fact, I was thinking, “I probably am going to regret it, and I’m going to have teachers next year that are screaming at me because the students didn’t come prepared with that algorithm,” but, I said, “I’ve got to give it a try. Everyone is telling me. Everyone is encouraging me to give this a try. Intellectually I know that I should be teaching this way, and so this is the year I’m going to do it.” And again that was really very, very scary for me, but I did it.
And that particular year with the second and third graders that I had, we just did a lot of story problems that involved larger numbers, and, sure enough, those kids got out those base-ten blocks, and they direct modeled with hundreds, tens, and ones or with the tens and the ones. They made really good sense of numbers, and they moved into using a lot of counting strategies and doing a lot of things just kind of mentally. And we got to the point where whether it was in the form of a story problem or just giving an equation and asking kids, “Today we’re going to do 324 plus 195; solve it however you want to solve it, and we’re going to come back as a class at the end and talk about it.” So, whether in the form of a story problem or just in an equation, the students were using these mental-math strategies or getting out those base-ten blocks and building these numbers and joining them together or separating, and I felt really good. I couldn’t believe it! My kids were actually doing what really I intellectually knew that they’d be able to do. And it was just so exciting to me to see that.
And in the spring I was really concerned or a little nervous about sending my students on—my second graders, sending them onto third grade, and my third graders, sending them onto fourth grade without having that standard algorithm. So, I didn’t know what to do. I’m all, “Here I go … I did … I gave up this algorithm,” and here I am wondering, “Was this the right thing to do?” So, I just decided that we would do some … small-group math centers.
And so I set up some math centers in my classroom, and the center that was with me was just a small group of students, and I decided what I was going to do at the center was to get a dry-erase board and show them a way that students might solve this 2-digit plus 2-digit equation. So, I just said, “Boys and girls, I’m going to do this… 78 plus 45. Let me write that down, so I have to remember to tell the story right, because I don’t remember the exact numbers: 78 plus 45.” So, I wrote it out and said, “I just want you to watch what’s happening here, and, with a partner, see if you can make sense of this.” So, I wrote out 78 plus 45, and I used the standard algorithm. I didn’t say anything, but I just used the algorithm, wrote it all out. And my students were kind of familiar with this because when students were sharing their strategies, there would be lots of times a student would show their work on the overhead or on the board and I would ask the other kid to turn to a partner and try to figure out what this child did. So, asking in this situation, showing something and having the kids reason through it, was not a new experience for them. And I remember putting that up there and the kids looking at it, and I had this little boy, Jordan, in the group, who’s a third grader, and he looked at it. The other kids were talking in partners, but he was just staring at that dry-erase board for the longest time, and I have to act like him because his expression was just so amazing. He looked it and looked at it and said … “That, that, that,” and I’m like, “What Jordan?” “That’s what my teacher was trying to show me last year. That’s what my second-grade teacher was trying to show me last year.”
And I said, “Well what was your second-grade teacher trying to show you last year?”
And he said, “Well, it’s really kind of weird, but you added the 8 plus 5 was 13; so you started over there—started with 8 plus 5, kind of weird, started with 8 plus 5 is 13. So, to make 13, you put the 10 with the other 10s—70 and 40; and then you just have the 3 with the other ones, and then you added up the tens together and that was one …. I don’t remember what the answer was.
And he was able to completely make sense of this, because he’s flexible enough in his thinking—by not only knowing his strategy but making sense of others’ strategies—that he was able to sort through his algorithm. He was able to make complete sense of his algorithm and I said, “Let’s try a few, and what I’d like you to do is, I’d like you to try to use this other strategy.” And that also wasn’t something that was unfamiliar to my kids. There were a lot of times in class that after a class discussion of a particular strategy that I might say, “Boys and girls, What do you think of this strategy. Is this something you might be able to use? Let me give you the same story problem with different numbers. I’d like to see if you can challenge yourself and try this strategy out.”
So, again this wasn’t a new experience for the kids: “Let me just give you another equation and see if you can try out this strategy.” And the students were able to try out the strategy and they were able to do it successfully, with understanding. And I said, “Your teacher next year may want you to do it this way,”—knowing full well that’s exactly what the teacher was going to want to see,—“So your teacher may want you to do it this way. So if she wants you to do it this way or if he wants you to do it this way, you have another strategy you can use.” I said, “If you’re not sure about it, you can always just figure it out mentally in your head, just to check yourself to make sure that this strategy is the same as maybe a mental strategy that you used. Or maybe you want to envision the base-ten blocks and what you’d do with the base ten blocks.”
Another day we did the same thing with the subtraction algorithm, same process. Kids were able to reason through and make sense of the subtraction algorithm also, and I know it’s because they’d spent the entire year both building their confidence that they’re mathematicians being flexible enough to use a strategy but to learn and develop multiple strategies that other kids have been sharing. And their number sense was just so strong that this procedure, it makes mathematical sense. It’s a beautiful algorithm, but it’s not the only way. But just seeing … to me, seeing that in two small-group centers the kids were able to make sense of both the addition algorithm and the subtraction algorithm, to me, was just convincing proof that I never do need to touch this algorithm again.
And since that year that I took the step—as scary as it was and trusting the process and believing that my kids could make sense of math—ever since then and seeing the result of it, the kids taught me that I didn’t have to teach an algorithm. And that’s what’s exciting about CGI.