Tag Archives: equality

Equality: Part 2

In my last post, I suggested a way to build on what students already know to introduce the equal sign.

Let’s say you followed that plan and your students are now proficient in using symbols to describe a quantity as equal to, greater than or less than another quantity.

What do you do next?

Use numbers. Instead of writing “length of line A=length of line B,” have them measure the quantities and then write number sentences like 8=8 or 7<8.

After enough practice using numbers, give them tasks that invite them to use different operations to create equal quantities.

What might this look like?

Let’s start with addition.

Give students Unifix cubes and ask them to complete the following tasks:

  • If you have a bar that is 5 units long and you join it to a bar that is 10 units long, is it more than, less than or the same as a bar that is 15 units long?  Show with the cubes how you know.
  • Is a 13 unit long bar more than, less than or the same as a 6 unit long bar joined to 7 unit long bar?  How do you know?
  • Is a 2 unit long bar joined to a 6 unit long bar, more than, less than or the same as a 3 unit bar joined to a 5 unit bar?  How do you know?

In order to solve the tasks, students can join the bars, determine how large the new bar they created is and compare it to the other bar.   Also include tasks where the quantities aren’t equal.

You can then introduce notation to help students record these relationships in various ways:

  • 5 +10=15
  • 13=6+7
  • 2 +6 =3+5.

Eventually you can move to doing these tasks without the cubes and using larger numbers.

Can you see how this plan could be adapted for other operations?

The key is providing situations that allow students to create different quantities, explore what is the same about them and then record this relationship using mathematical notations.  It builds on what students know and formalizes it using symbols.

I’d love to hear what happens when you try it with students.

This plan was based on the Measure Up research that Barb Dougherty and her colleagues conducted in Hawaii.  It’s a really interesting project and I’ll write more about it some other time, but if you want to know more now, check out the research:

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

Dougherty, B. J., & Venenciano, L. C. (2007). Measure Up for Understanding: Reflect and Discuss. Teaching Children Mathematics13(9), 452-456.

Equality: How elementary school teachers can help algebra teachers

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.


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

Two Different Worlds

I started my Ph.D. program thinking I knew a lot about teaching and learning.  After all, I spent 7 years in a classroom, went to many professional development sessions, attended courses, and read a lot of books and articles on education.   However, having the luxury to step away from the classroom and really immerse myself in thinking about teaching and learning exposed me to a whole new world.    I realized that there was all this research out there that would have helped me when I was teaching.

Here’s an example.  Recently, I was looking through the research on algebra.  One of the things I learned after reviewing the literature was that researchers have known since the 80’s that elementary students develop misconceptions about equality that cause major problems later on in algebra.  For example, when most elementary school students see the equal sign, they view it as a signal of where to write an answer or as a direction to work out an equation.  So “5 + 3=__” becomes a direction to add 5 and 3.  They don’t see the equal sign as relating two equivalent quantities or amounts.  Therefore, when they see a number sentence like:  5 + 4 = __ + 2, they often write 9 as their answer.   Various researchers have offered teaching interventions, such as using balance beams when first introducing the equal sign, to avoid this misconception.

What amazed me was that the research community documented this problem years ago and yet, I never heard about it when I was teaching.  Now maybe my fellow teachers were secretly reading lots of journal articles and not telling me about what they were learning, but I have a feeling that I was not the only one who was unaware of the research.

Why was this work that was being conducted in universities by people who had the time and money to study these things not being relayed to the teachers who needed it?  This seemed ridiculous to me.  Imagine if doctors in hospitals weren’t using the research that scientists did in the 80s.

I’ll mention this in other posts, but I want to make it clear that this lack of communication goes both ways.  Some researchers are completely disconnected from what goes on in classrooms today.   Just as I was surprised to learn about the research world, I would imagine some of these researchers would be surprised at what they would learn if they went into a classroom today and had to teach full-time.

So why does this happen?    Why is there this gap between these two worlds?  I’d love to hear your thoughts.