I’m certainly not the first person to talk about the importance of the equal sign. Researchers have been talking about it since the 80’s and discussions about the equal sign have made the rounds on the blogs and twitter (most recently Justin Aion posted about strings of equal signs here)

Often, the concept of equality comes up because high school algebra teachers notice all sorts of wacky things going on when students start working with equations.

Why?

It all comes back to what students understand about the equal sign. The research on equality shows that understanding that **two sides of an equation have the same value and are interchangeable **is linked to students’ later abilities to solve equations and reason algebraically.

However, young students are often only introduced to the equal sign as a signal to “do something” or “put the answer here.” We know that this leads to misconceptions such as answering 12 or 17 to the following problem 8 +4=___+5 or finding the equation 8=8 to be nonsensical. This causes big problems later on in algebra.

Now, if you’re a high school teacher, check out the comments section of Justin’s post. In particular, check out what Christopher Danielson and Sadie Estrella suggest for how to assess if your students have this misconception and then what you can do to address it.

But I want to talk about what we can do as elementary school teachers that will really help out high school algebra teachers.

Here’s my advice: **Build on what students already know about equal and unequal quantities to introduce the equal sign. **

Students have an intuitive notion of more than, less than and the same. For example, young students know whether their brother is taller than their sister or if two people have the same amount of juice.

Use what already makes sense to students to introduce the equal sign.

I’d suggest starting with quantities first and not introducing numbers at all. For example:

- Show students two glasses full of juice and ask if glass A has more juice than B, less juice than B or the same amount as B.
- Show them lines of various lengths and ask them if line A is longer than B, shorter than B or the same as B.
- Show them a balance beam and ask them if A weighs more than B, less than B or the same as B.

After a lot of experience examining equal and unequal quantities, formalize the relationships *they already know *about equal amounts with the equal sign. Introduce the equal sign as a way to symbolize these situations that involve same amounts. Then do the same with the inequality signs.

Next, have students write statements such as “length A < length B” or “container B =container A.” Switch the order they write these statements and ask them to justify whether the statement is still true.

Later, you can do tasks that involve adding or subtracting quantities. Then you can bring in numbers. Make sure they understand what the equal sign means first. Trust me, high school algebra teachers will thank you.

I’ll talk more about tasks that incorporate numbers and operations later this week, but if you have ideas now, please share them below.

If you want to know more, check out the research on equality:

Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. *Mathematical Thinking and learning*, *9*(3), 221-247.

Dougherty, B. (2007). Measure up: A quantitative view of early algebra. *Algebra in the early grades*, 389-412.

Falkner, K.P., Levi, L., & Carpenter, T.P. (1999). Children’s understanding of equality: A foundation for teaching algebra. *Teaching Children Mathematics, *6, 56-60.

Kieran, C. (1981). Concepts associated with the equality symbol. *Educational studies in mathematics*, *12*(3), 317-326

Nicora PlacaPost authorChristopher Danielson (http://christopherdanielson.wordpress.com/) brought up some points about this post on twitter that I want to capture here.

He reminded me that teachers should focus students on the quantities involved and what is equal about them. In particular, he noted that Container A is not equal to Container B. The volume of juice is equal. That’s something that is important to highlight in conversations.

Second, there can be some confusion when we talk about Container A and Container B if the students begin to think about A and B as labels for the containers. We don’t want students thinking that a variable labels a specific quantity. This can lead to problems with the concept of variable later on. I’m not sure how to get around this, but maybe saying “the amount of juice in the blue container=the amount of juice in the red container” will help. I’m not sure what the best plan is. Please share any suggestions you might have

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