# What’s the Difference?

I was helping my daughter, who is in kindergarten, with her homework. She claimed the homework was too hard. She was working on subtraction in her class and the instructions for her homework were: FIND THE DIFFERENCE. I thought to myself, does she even understand subtraction as a comparison situation? Was that the thinking her teacher wanted her to do?

Here is an example of the type of problem on her homework:

# 9 – 6 = ____

I think most parents would simply have their kids use their fingers to find the answer because that is how they were taught. There is no shame in that; as a parent myself, I say we have to do the best given what we know. But technically, the homework asked my daughter to “find the difference.” So let me quickly explain the difference between the two situations with the help of Mathematics Learning in Early childhood (pages 45-46) from the National Academies Press (Click here to go to their website and then click on the Download the Free PDF below the book).

**Situation #1 Change Minus (Take Away)**

This situation will always have a start, change, and result.

**Situation #2 Comparison (Find the Difference.)**

This situation will have two quantities that can be compared. In the beginning, students compare numbers and find out which number is bigger or smaller, or they will find out if the numbers are equal. After a lot of matching and counting, students begin to see the third quantity: the amount more or less (the difference).

When kids use their fingers, they are doing a change minus situation. Kids will understand change minus first because it is so concrete. But in order to understand subtraction, kids need to understand all of the situations. Here is how I made my daughter’s homework an opportunity to understand subtraction as a comparison situation:

I started by drawing dots with a pencil, but she struggled a bit with it. She preferred to draw a circle, but her circles were all different sizes and the paper was getting cluttered. “Markers!” I thought. The markers allowed her to make simple dots and then connect what was the same with her pencil. On a side note, she loved being able to use markers for homework. (Sometimes the simple pleasures in life keep us going.) Visually, my daughter was finding the difference, but interestingly enough she kept saying after drawing her lines, “There are _3_ left.” Yes, technically there were three dots left unmarked, but really there were three more dots in one of the groups.

Here is how I explained it to Bethany.

It is so crucial for students to understand that there are different situations for subtraction. All it took was a marker to build a visual for my daughter. If all she had done was use her fingers to subtract, she would be isolating her understanding of subtraction to the “take away” situation. Then later, when she reads a story problem that asks her to find the difference, “finding the difference” wouldn’t be meaningful, and she might not know what to do.

Let’s work together to help students have meaning tied to math! Leave a comment below if you have had a similar experience.

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Great illustration, but was thinking that the dots for 9 should be lined up vertically instead of 9 being separated into groups of 3 and 6. What do you think?

That is a great thought. I chose 6 and 3 because it looked the best given the space on the slide. When I did it with Bethany I had her make one line if they would fit because it made drawing the line more efficient.

My experience with kids has been that how they line up the dots often shows me how they are thinking about the number. Most of the time I like to make a group of 5 and 4, because having 5 as an anchor number can be helpful. If students have to make a line of 9 sometimes they lose track of how many dots they have. So it is really a matter of preference and what works best for the child.

Looking back at how I learned subtraction, I think Situation #2 would have helped me understand sooner than the way I learned (Situation 1).

Thanks Jesse. I don’t think you are alone. I am certain that one of the reasons I struggled with word problems was because I learned subtraction with beads and only knew the “take away” model (Situation #1). Comparison Subtraction is another way for students to really think about subtraction and understand the concept. Now you can help your kids understand.