What Does Equal Mean?

IMG_8076.JPGSo often we provide students with mathematics instruction that “makes sense” to us and we wonder why they don’t develop understanding.  Often this is due to the fact that they have not been given an opportunity to MAKE sense of what we are teaching them; to discover it on their own.

An example of this is teaching an addition algorithm that asks students to complete a number sentence that uses two addends and a sum.  We tell students that “equal” means that they are the same on both sides.  Imagine yourself as a 6-year-old, you look at the equation 5 + 3 = 8.  There is nothing the same on both sides of the equal sign.  On the left side, there is a 5 and 3 and the right has an 8.  We must stop assuming that students make sense of their world in the way that we do now that we have had countless experiences that have shaped our perceptions;  We forget that we have access to a much larger schema.  We must allow them to “make sense” and not assume that what we teach “makes sense.” IMG_8075.JPG

One of the best sense-making tools for students to use to discover the meaning of the equal sign is to use a pan balance.  Put a post-it in the middle of the balance that has the equal sign on it and let students explore different number combinations that will balance the scale.

After they have time to discover and play (yes play is an essential learning opportunity), ask questions like, “if I have 5 cubes in this side, and I have 3 in the other, how many cubes will I need to add to make the sides balance (or make the cubes equal)?”

Here is an example of the pan-balance-sheets I made for students to record their thinking.  We start out with the mat, and place cubes on the mat to model the problem, then we use the recording sheet to write numbers that correspond to the problem we modeled.  Following this lesson, we can introduce the formal number sentence that students should know in first grade, but only after they have been allowed time to make sense of why the equation is set up that way.  This allows students to experience the task in a concrete, visual and abstract way.  Weighted numbers are a great way to connect this as well!

The other great thing about connecting the concept in this way is that missing addend problems are already embedded in the sense making and students see addition as interconnected.  The same goes for equations that are written with the sum first and the addends on the right side of the equation.

When we begin talking about subtraction, we again take out the pan balance and play with numbers.  This is a great way to connect the operations of addition and subtraction and show the reciprocity between the two.

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