True or False?
5 = 5
5 = 4 + 2
2 + 3 = 1 + 4
After reading so much about the meaning of the equal sign in books such as Thinking Mathematically and About Teaching Mathematics , I anticipated students may think each was false for different reasons…..
5 = 5: There is no operation on the left side.
5 = 4 + 2: The sum comes first or 4+2 is not 5.
2 + 3 = 1 + 4: there is an operation on both sides or because 2+3 =5 (and ignore the 4) or because 2+3+1 ≠ 4.
While I anticipated how students may respond, I was so intrigued by the number of students (probably about 75%) that said false for 5=5. They were about split on the second one, but for many reasons – not many of them being that 5 ≠ 6. The final one left many confused, in fact one student said, “Well now you are just trying to confuse people by putting two plus signs.” So cute.
As they explained their reasoning, my mind was reeling….
- What questions do I ask to get them to:
- Think about what the symbols mean?
- Talk about what is the same?
- Realize the equal sign in the first one is not a plus sign, so there is no answer of 10?
- See the equal sign to not mean “the answer is next”?
- What wording do I use for the equal sign?
- “The same as” felt wrong because the sides do not look the same in both cases….so, is “Is the same amount” a helpful way for them to think about it?
I got back to my room and starting thinking about what learning experiences would be helpful for students in building their understanding of the equal sign? I talked through it with some colleagues at school and reached out to those outside of school, I needed some serious help!
I started playing around with some cubes and realized how interestingly my thinking changed with each one. I didn’t take a pic of those cubes so I recreated them virtually to talk thru my thinking here.
The first set represents 5 = 5. I can see here where “the same as” works for the equal sign because there are 5 and they are all yellow. But what if I put 5 yellows on the left and 5 red on the right? Then they are the same quantity, but do not look the same.
The second set represents 2+3=5 and is definitely the one students are most comfortable seeing and representing as an equation. It looks and feels like composition to me so I can definitely see why student think the equal sign means “makes” or “the total is.” It looks like 2 and 3 more combine to make 5.
Something interesting happened with the green set. I made two sets of 5 and then broke one set to make the right side – felt like decomposition. I can see why it would feel differently to students. I also realized that when I look at them, I look left to right and much of that lends itself to the way I was thinking about what was happening.
The last set I made by taking my 2 sets of 5 connected cubes and breaking each set differently. Again, “the same as” doesn’t work for me here really well either because they don’t look the same.
Still thinking of next steps because I always like to put context into play with these types of things, but I am finding that very difficult without forcing the way students represent their thinking which I don’t want to do.
Right now, things I am left thinking about before planning forward:
- What do students attend to when we ask if things are the same?
- Our language and recording is SO incredibly important.
- How can these ideas build in K-1 to be helpful in later grades?
- If I am thinking of moving students from a concrete to more abstract understanding, how does that happen? Is it already a bit abstract in the way the numbers are represented?
- Do we take enough time with teachers digging into these ideas? [rhetorical]
I look forward to any thoughts! So much learning to do!