# What does it really mean to square a number?

May 07, 2024

One day when I was teaching second grade, the math coach suggested that we should do a lesson with the students on square numbers.

“Are you kidding?  These students are only in second grade.  There is no way they are ready for exponents.”

“They can do it, Alison! Just watch.  It’s going to be great,” said the math coach.

I skeptically watched the next morning as the math coach came to my room ready to work with the students.  I sat in a kid-sized chair, just as eager as the students, to see what she was going to do with them.

First she laid out a vertical row of four color tiles.  Then she made a horizontal row of four color tiles.

Next she filled in the array for a total of 16 color tiles.  “What do you notice about this array? “ she asked the students.  “Do you notice how it is four tall and four wide? Since it is four tall and four wide and made into a square, we say it is four squared.  We can write it as a four with a little two at the top to indicate the two dimensions.”

At this moment, I nearly fell out of my little kid-sized chair.  I felt betrayed.  I felt lied to.  I felt cheated.  All of these years of math instruction and I didn’t know that math was supposed to make sense!  In that one small moment, I realized that the way I had been taught had led me to believe that math was a series of procedures.  I had learned that putting a little two at the top was verbalized as “four squared” and it mean 4x4.  Never in a million years did I realize that it meant a square that was four tall and four wide.  One needed to memorize the information about how to do the procedures and I did.  The information was held by teachers who would share it in small amounts.  They did not share WHY.  They only shared HOW.  A student’s job was to memorize and not ask questions.  This math coach showed me that there was a WHY behind math.

So this post is for those of you who perhaps had a similar experience and never learned why you “put a little two at the top and call it squared” or “put a little three at the top and call it cubed.”

So, back to our model of four squared.  We can take any number and build an array that is x squares tall and x squares wide and we will get a square.  Most students are familiar with base ten blocks.  The flat is ten tall and ten wide or ten squared.  Have you and your child played tic tac toe?  The board is three tall and three wide or three squared.

Take a look at what happens when we build the multiplication table.  Do you see where the square numbers are?

Helping students learn through their fingers by building the square numbers, as well as the other numbers in the multiplication table, helps them internalize the concepts.  They build those square numbers and know that they make a square shape and are just as tall as they are wide.  They realize that a product like 3x4 will not make a square shape because the factors are different.

After playing with building the multiplication table visually, we can then build upon this conceptual understanding to build something that is 4 tall, 4 wide, and now 4 deep.  If we do that we have a 4x4x4 cube or 4 cubed.  Take a moment and find some cubes to build 3 cubed or 5 cubed.  How many total cubes are included?  Think of a Rubik's cube which you might have played with as a child.

Then take a moment to ponder what other aspects of math you have accepted as truth without wondering about the why.  “Invert and multiply” is another one I needed to learn from the beginning.  When multiplying I was told to “move the decimal over.”  Why are we doing this?  What does it mean?  I truly hope we are working to build a different version of learning math for today’s children.  I hope for math experiences that build understanding and help them see the “why” behind what they are doing.