A teacher recently handed me a second grade interim assessment with some true or false and fill in the blank problems:
She asked me if I could develop some number strings to help her students prepare for this assessment. It called to mind some equivalency number strings that are developed in the Trades, Jumps and Stops algebra unit (Contexts for Learning Mathematics) as well as some problems in the Thinking Mathematically book by Thomas Carpenter, Megan Franke, and Linda Levi (2003). Carpenter et al. (2003) develop the concept of relational thinking in their book, which also discusses in depth the misconceptions kids can have about the equal sign. For example, when presented with 8 + 7 = 15 + 1, many kids think that the expressions are equivalent because they believe that the equal sign means the answer comes next. Carpenter et al. (2003) found that when presented with problems based on equivalency, kids first need to struggle with what the equal sign means, then they generally compute each side, then they gradually see the relationships between the numbers which can make computation unnecessary, or relational thinking.
Carpenter and his colleagues are the originators of Cognitively Guided Instruction, otherwise known as CGI. I have recently been lucky enough to collaborate with CGI teachers and researchers, including using number strings in CGI classrooms. After I drafted some strings, I sent my work to Nick Johnson, a CGI nonbeginner ( a CGI joke) and doctoral student at UCLA. I preserved his comments below each string, demonstrating how an elementary math educator trained in RME (Realistic Mathematics Education) can collaborate with one trained in CGI.
First, Nick notes:
I did want to clarify that these all seem to assume a relational understanding of the =. I find it hard to work on both relational thinking and the meaning of = at the same time. I’m guessing you’re already thinking this, but if there were students who were answering “false” to the first number sentence in any of these that the teacher would want to go in a different direction and just work on the meaning of the =. Some nice examples of these in Ch. 2 of thinking mathematically and in the video, but I think the big idea would be that I think that the teacher needs to choose the next number sentence differently depending on what particular idea (or misconception) is voiced.
For the strings you have here, it seems like we’re assuming that kids can mostly compute things mentally, and the discussion will primarily be around different ways of finding the correct answer (especially computing versus making use of relationships) rather than kids having different ideas of what the answer should be (though you never know).
String 1: Nick wrote this one. This one is a beginning string, to make sure that the kids understand the equal sign and equivalence.
5 + 7 = 12
12 = 5 + 7
12 = 12
5 + 7 = 5 + 7
5 + 7 = 7 + 5
String 2: 5 and 10 structure, decomposing numbers and getting used to the true false format
5 + 7 = 7 + 5
5 + 5 = 5 + 2 + 3
5 + 7 = 5 + 5 +1
4 + 5 + 5 = 9 + 5
If they are familiar with the format, an open number sentence might be useful here too. Something like 4 + 5 + 5 = 4 + __
String 3: Fill in the missing number, compensation, making 10s
9 + 6 = 5 + 10 (true or false)
9 + 5 = ___ + 10
8 + 6 = ___ + 10
8 + 7 = ___ + 10
really nice. I might introduce the open number sentences earlier though.
String 4: Extending thinking with doubles, decomposing numbers, practice with a single number on the left in a equation (which can be challenging for kids).
10 = ___ + ___
12 = 6 + ____ + _____
14 = 7 + ____ + _____
18 = ___ + ____ + 9
20 = 10 + ____ + _____
String 5: Relational thinking, compensation (moving to a ten), decomposing numbers
10 + 7 = 7 + 10
10 + 5 + 3 = 3 + 5 + 10
9 + 8 = 8 + 10
10 + 6 + 1 = 2 + 7 + 9
nice. If they struggled here you could even go 10 + 3 = 10 + 1 + 1 + 1 or something like that.
String 6: Relational thinking, compensation, decomposing numbers
9 + 1 + 4 = 10 + 5
10 + 6 = 9 + 5
10 + 6 = 9 + 7
So I think a lot of kids are still going to be likely to compute here, and there’s a chance you might not get the relationship on the table. If relational thinking is a primary goal here, I’d either a) go to bigger numbers that make the computation a pain, b) go to an open number sentence like 7 + 9 = 10 + ___, or c) do both: 29 + 56 = ___ + 55
8 + 5 = 9 + 6
8 + 5 = 9 + 4
String 7: Relational thinking, decomposing numbers
6 + 3 = 3 + 3 + 3
11 + 6 + 3 = 3 + 3 + 3 + 11
5 + 4 + 10 = 10 + 5 + 5
5 + 4 + 1 + 100 = 100 + 5 + 5
String 8: Relational thinking, compensation, bigger numbers, based on Nick’s comment above. Here the fill in the blanks are helper problems. The final problem asks them to think about compensation without the help.
19 + 26 = 18 + ____
19 + 26 = 20 + ____
34 + 39 = ___ + 40
35 + 39 = ___ + 40
29 + 43 =
The teacher tried all the problems above, except for String 8, which I just wrote. She reported that the kids loved them, and that the work they had just finished on the rekenrek, which emphasizes five-structures was very supportive for this kind of thinking. They ended up breaking String 4 into multiple days in which kids presented multiple solutions for each. Only a few could be modeled their thinking on the rekenrek because the numbers were too big. We both wondered if a bead string (maybe two?) would be another great way to model strategies because you could move it like a rekenrek, but you would not be limited to the 10 on top and the 10 on the bottom.
I also want to write more like String 8. I am interested in this idea of using fill in the blank problems as helper problems in a number string . . .
I sometimes use the analogy of a balance scale to clarify (and sometimes to totally correct) student’s ideas about the equals sign. Surprisingly, many of them know what a balance scale is, especially after I describe or draw one for them. I wonder if this is useful to the discussion here. Are there manipulatives for this purpose, ones that don’t require hauling around an actual scale (which is usually quite delicate) and a set of weights (like I don’t have enough to haul around already!)?