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!

3yellowsandpailsKristen–so interested in having some conversations with colleagues about the question you posed–what do students think about when we ask what is the same? And, asking kids as well. Thanks for pushing us to think about this more deeply and carefully.

Judy

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Michael Paul GoldenbergCoincidentally, taught adult algebra students about = today. Here’s the thing: after kids are exposed to countless arithmetic problems that boil down to (number) [operator] (number) = ? (or a box, a circle, a blank, etc.), the come to read “=” as “here comes the answer now.” So later, they have no trouble writing absurdities like “2 + 3 = 5 + 1= 6” even though equality is a transitive operator.

You might start with something that is less symbolic and requires no particular mastery of “=,” like the relation “is the same height as” or “is the same age as.” Then you can ask, “If Meg is as old as Hoa, and Hoa is as old as Avi, what can we say about Meg and Avi’s ages?” (or, more directive, perhaps, “Who is older, Meg or Avi?”) After exploring a number of verbal questions about familiar quantities, whether more concrete or not (age being more abstract than height since you can’t “see” it or measure it without “inside” information unless two people are drastically different in age), students might be ready to see “=” as shorthand for “the thing on the left and the thing on the right are “the same” (in terms of some quality/quantity).” Books will say that “=” means that the quantities or values of the expressions on either side of “=” are the same, but that is perhaps too restrictive to numbers or symbols for numbers and I think for younger kids, that restriction can be a problem when you want to help them develop understanding based on experience and “common sense.”

You will know kids have a decent mastery of “=” when they are repulsed by expressions like “2 + 3 = 5 + 1= 6” and recognize that it makes no sense at all.

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Nili PearlmutterI love the way you build the equations with cubes. Could kids do that and discover if there are the same number of cubes on each side. I’m imagining a big piece of paper with an equals sign in the middle and a deck of cards that have either single numbers or an expression. Students could turn over two cards, put one on either side of the equal sign, use cubes to represent each side, and then check and see if it’s true or false. Maybe they could have a recording sheet where they write the true ones on one side and the false ones on another.

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howardat58Going back a bit it seems that “apple 1 is a golden delicious and apple 2 is a cox’s orange pippin” is rather abstract, and is simpler and better put as “the first apple is a golden delicious and the second apple is a cox’s orange pippin”.

It is twenty times more natural to use first, second, third as they are adjectives, describing numerical features. Only later can we use quantity, as in three apples, and still later isolate the abstract 3 as a noun.

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Michael Paul Goldenberghttp://www.adjectivenounmath.com/id4.html

Herb Gross has been writing about many of these issues for a while, unbeknownst to me until a couple of years ago. I have been, too, from a slightly different angle, but not in print.

I view algebraic terms and fractions as two good, related examples where there are adjectives (coefficients, numerators) and nouns (variables, denominators). And when the operation is addition/subtraction, you better be sure you’ve got the same noun in all the numbers you wish to add or subtract. Otherwise, you get nonsense for an answer.

On the other hand, you can, in general, multiply or divide two numbers without regard to whether they are the same noun. And I think that is part of why it is not correct to claim that “multiplication IS repeated addition.” There is something fundamentally different about what it means to multiply two numbers than what it means to repeatedly add some number. Units of measure also reveal some of the differences between addition and multiplication, and again because of the vital issue of having the same noun for the former, but no such restriction for the latter.

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Michael PershanLove the post and your questions!

Sometimes I think it’s fun to think about these sorts of questions backwards. We could always just tell the kids that 5 = 5 is true because ‘=’ means ‘the same as.’

What would go wrong if we did that?

Here’s what I was able to come up with:

(1) They wouldn’t believe you and would immediately forget it, because they aren’t able to connect it to any other equations that they know.

(2) They would believe you but think that it’s stupid and arbitrary. [Is it stupid and arbitrary? Is the meaning of the equals sign just a convention? Or is it logically forced?]

At this point in the comment, I have two thoughts. The first is that the meaning of the equals sign might be logically forced by the transitivity of equations. If we want to say that 5 + 3 = 8 and 2 + 6 = 5 + 3 and that you can make a chain of equations, then we need to say that 8 = 8. I don’t know if that idea would matter much to kids, but it’s true I think.

Second, I wonder what kids would say to 3 = 8. Would they say that this is false because there is no operation? Or would they say that 3 = 8 is false because three is, in fact, not equal to eight. Continuing this thought, maybe part of the way kids could get used to the idea that 8 = 8 can be seen as true is by contrasting it with things that are EVEN FALSER. Maybe the contrast would be helpful.

So…what about a string of problems that was, like, “I think that one of these equations is true and one of them is false. You might disagree, but I’m curious what you think.”

3 = 1 and 3 = 3

3 = 1 + 1 + 1 and 3 + 1 = 4 + 1

0 + 0 = 0 – 0 x 0 and 3 = 1 + 1 + 1 = 6

Or something.

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Fran De La RosaThis is such an important concept for students to learn. As a fourth grade teacher I find that many students don’t understand equality. Often students cannot solve problems if not written in the standard 5+4=9 format. I generally spend the first week or so working on equality and think I will use your task with cubes to help students visualize the concept.

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Kari A AugustineI think this bullet point needs some editing. “Realize there the equal sign in the first one is not a plus sign so their is no answer of 10?” What are you hoping students will understand?

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Pam Jones (@teacherpj83)I always begin equality discussions in my second grade classroom with seesaws. I wish we actually had one on the playground at my school it would be far more experiential, but we don’t so we have to settle for a representation. Various combinations of students are placed on either side of the designated seesaw. If the sides balance students on both sides remain standing. If one side is greater than the other side the students on that side stand while the other students sit on the floor. We try various story problem combinations representing numerous expressions to see if we can balance the seesaw. Once they understand the concept of balancing the seesaw we connect numbers and symbols. Students who are not “on ” the seesaw are given a large number or symbol card to stand behind the representation of that card on the seesaw. From here we move onto replicating this experience with a pan balance, number balance and then a paper seesaw and cubes. Given these experiences, my students now first look for the equal sign in a given expression and then seek to balance it.

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DeniseI sometimes question that we go to the abstract too soon. Providing students with MULTIPLE experiences to see a quantity represented in multiple ways before we ever put written symbols to them may help with some of the confusion later on. Dot images or “show me 4 with your fingers, now show it to me a different way, now a different way” are just a couple of examples of how your experiences can begin to develop the language that a number can be represented many ways. Eventually you attach the numbers to the experiences and maybe begin by writing it as 4 = 3 + 1, 4 = 2+2, 4 = 2 + 1 + 1.

I agree that the word “same as” sometimes causes confusion, although I have used it in my classrooms. When kids first explore 5 x 3 = 3 x 5 and they are thinking in terms of a context, 3 baskets of 5 apples is NOT the same as 5 baskets of 3 apples although the quantity is the same.

I look forward to hearing from others on this issue.

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CassytI’ve always used “the same value a”… they are worth the same amount … with young students. Oddly enough, I am finding that fewer students have the problem understanding the = sign than even 5 years ago. Common Core success?

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Simon Gregg (@Simon_Gregg)I feel not a bit predictable recommending Cuisenaire rods here. When a “train” (some rods end-to-end) is side by side with another train of rods, young students can so clearly see whether the lengths are equal or not. In fact, they find it hard not to make unequal ones equal! You’ve had experience of this, Kristin. And as there’s not really an ‘order’ to which one is the same as which, if students write it down, it’s natural to write 1 + 3 = 2 + 2, or w + g = r + r, or whatever. I had this great moment when Blanca wrote g = g and showed it to me with a big smile; I showed it to the class, and since then I get, for instance, 10 = 10 regularly when I’m asking for different ways of composing ten.

This does involve asking the students to use the rods, but it has the advantage of the true nature of the equal sine emerging without explanation. This and other ways they put the finding out into students’ hands makes the trade off worth it for me.

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Simon Gregg (@Simon_Gregg)*sign (working with my son on his trig a bit too much!)

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Clara Maxcy (Cleargrace)I teach Alg 1 right now, and expect to next year (I can teach thru Pre Calculus, our school mixes it up from year to year) to kids with learning issues. Your blog is incredibly helpful! It allows me to think about and observe my students ideas about the numbers. I often use your activities for starters and number talks, reverting to the concrete (cubes, tiles, etc) to expose those foundations, and to strengthen or build new/missing foundations. And I love to hear what the little ones are thinking and accomplishing!

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