The Things We Tell Our Kids That Will Mess Them Up Later

Okay, we are only dealing with math things we tell our kids that will mess them up later. You're on your own for the rest of it!

I could have chosen a softer topic for my first post, but why pull punches?

I know we aren't telling our children these falsities with the intent of sabotaging some future understanding. Rather, we don't believe children are developmentally ready to access more complicated math concepts, so we hide them away and pretend they don't exist. Unfortunately, in doing so, we teach kids something that is not true. Something they will need to unlearn later.

This post consists of a few of my favorite comments parents and teachers make that are not mathematically correct and may come back to bite us as our kids advance in their math education.

This is a preschool favorite. My son's preschool teacher was shocked when he replied, "There are numbers that are less than zero. Like negative one." We are shielding our little ones from negative numbers because we think they are too complicated to go into at this age. To avoid negative numbers, we are essentially telling our kids that the only numbers that exist are whole numbers. Negative numbers (which exist within the set of integers) are real numbers (technically!) and our children will eventually need to understand them. So, let's not hide them or deny their existence.

And, if you think the interest is there, there's nothing wrong with talking about a negative in terms of owing someone something. When in doubt, think of a context!

For your reference, here is a visual of the Real Number System. [So as not to hide anything from you, the complex number system encompasses real numbers and imaginary numbers, too.]

These comments usually come into play while teaching the traditional algorithm for subtraction. That will be a whole other blog post! But, for today, let's deal with the accuracy of the above statements.

Once again, we have decided that negative numbers are too sophisticated for our first and second graders, so we are pretending that there are rules governing the order in which the numbers are placed in a subtraction problem. In fact, the bigger number does not always have to be the minuend. If you are like me, you do not remember what a minuend is. I always have to look it up!

We can definitely subtract 5 - 8 = -3. First and second graders don't need to be able to operate with integers, but there's nothing wrong with knowing they exist. Remember, our numbers go on forever in both directions. So, our operations can move up and down the number line.

Note: This is only one way of modeling 5 - 8 = -3. There are many other ways to illustrate this concept! 

So, what does this mean for our primary students? 

This is true again when working with whole numbers. What happens when you multiply by a fraction? What happens when you multiply by a negative number?

Let's find out!

On each of 5 school days, Julia saved 1/2 of her cookie from lunch. How many cookies did she save by the end of the week?

This is even more evident when we multiply a fraction by a fraction.
Nate, William, and Andrew shared a Snickers bar. Nate ate 1/2 of his share and saved the rest. What fraction of a Snickers bar did Nate save?

 

We have clearly ended up with less than when we started. 

What about when we multiply by a negative number? Do we end up with more? 

Danielle has no money in her wallet. Every day she borrows $2 from Erin to buy a coffee. By the end of one week, how much money does Danielle have? 

Once again, we've ended up with less. We usually start teaching multiplication in third grade. And, just three short years later, the students already have to throw out this rule.

It's okay to make conjectures. We just need to be clear about what numbers we are working with and not shut the door on further exploration.

This statement speaks to a misunderstanding of how shapes are classified. Some rectangles have two long and two short sides. Others have four sides the same length.

So, what makes a shape a rectangle? Is a square a rectangle? Is a rectangle a square? And, how do we talk to our primary students about this relationship?

To begin, the figure below illustrates the difference between polygons and non-polygons.

Now let's focus on quadrilaterals, or polygons with four sides.

So, is a square a rectangle? Yes! A square is a rectangle because it is a quadrilateral with four right angles.

Here's where it gets tricky with our primary students. When children first begin to make sense of shapes around them, they recognize a shape by its appearance. That's why we end up telling them that a rectangle has two long sides and two short sides. We want them to be able to distinguish a rectangle from a square by the way the shapes look.

Two Dutch educators, Pierre van Hiele and Dina van Hiele-Geldof, developed the Van Hiele Theory of Geometric Thought in 1957, which has influenced geometry instruction in the US and around the world. See below.

People progress through these levels systematically. This progression is based on maturity and experience rather than age. (See more information here: https://en.wikipedia.org/wiki/Van_Hiele_model.) As you can see by looking at the chart above, children begin by recognizing a shape by its appearance. This is where most of our K-3 students are.

So, what should we say about a rectangle and a square when teaching our little ones who are not yet at a level at which they can classify shapes by properties? Remember, it's okay to use proper math terminology and expose them to shape properties. But, do it in a way that connects to something they already understand. For example, even young children understand that something can be special, which sets it apart in some way from like things. 

My sister is a girl. But, she's a special girl because she's my sister. So, she can both be classified as "girl" and "sister." Sister is a more specific term because it tells us she's a girl and she is a sister. A square is a rectangle, but it's a special rectangle because its sides are all equal. Square, like sister, implies inclusion in another group, rectangles.

Before we wrap up, I have to throw in my pet peeve: the diamond. Diamond is not a mathematical term! What we used to casually call a diamond is now known by its mathematical name: rhombus.
This is my boys' shape puzzle that I defaced when my oldest was a toddler:



And, here's the homework that he defaced as a third grader this year. That's my boy!



I hope you've enjoyed this installment of The Things We Tell Our Kids That Will Mess Them Up Later. There's more to come!

About this Blog

This blog brings together two of my loves, math and motherhood, to serve as a resource for parents of elementary and middle school math students. I am an elementary and middle school mathematics specialist and a mom of three elementary and preschool aged boys.

As I sat at the National Council of Supervisors of Mathematics conference in Washington D.C. and absorbed the latest research in math ed, it occurred to me that so many parents are unaware of the current best practices in math education and can only fall back on how they were taught as they toil together with their children at the kitchen table every night. 

This blog is for you, parents! It will arm you with the latest research and best practices to support you as you work with your child and advocate for his or her education. As parents, our end goal is to foster curiosity, perseverance, and an appreciation for the beauty of mathematics in our children. 

Dig in and please send me any questions!