Ever hear a teacher tell a child “addition makes numbers bigger and subtraction makes them smaller?” How about, “Put the bigger number on the top when you subtract?” I recently read an article called “12 Math Rules That Expire in the Middle Grades.” The more I thought about it , the more I realized how much confusion it causes kids when something they have previously learned turns out not to be true.
Examples of Rules That Don’t Expire
This concept of never teaching a child something they would later find out not to be true revolutionized and improved my math instruction. When I introduced fractions we included improper fractions from the very beginning. The kids were able to manipulate them accurately. More importantly, they had a much deeper understanding of what fractions actually are.
When I taught subtraction, we talked about how it is the difference between two numbers on the number line. We practiced finding it that way so the kids didn’t see it as only “take away.” I wrote a whole post about the importance of this as it greatly improved their mental math skills. It also prepared the way for them to easily understand operations with negative numbers.
When we worked with the number line, I stressed that what we wrote was a piece of the number line. (I actually had a 2nd grader ask me, “Ms. Amy, you told us a line goes forever in both directions. Would the number line do that too?”)
When I had the freedom to do so, I had kids start adding and subtracting negative numbers by 4th grade. They were able to understand it easily using a number line and physical activities. They were also much less confused than the 6th graders who had to jump straight to remembering rules.
Importance of Correct Vocabulary
Related to this, I make it a point to use correct mathematical vocabulary even with very young children. When playing a triangle musical instrument with a toddler you can point out the “corners” of the triangle. Or you can just as easily refer to them as vertices, and the kids will too. When we danced with a group of 3 year olds around the “perimeter” of the rug and inside the “area” of the rug, a mom asked later, “Did my 3 year old really just accurately use the words area and perimeter?”
Results
By doing these things I have seen an increase in kids’ mathematical curiosity and questions, because they come to know you will give them a real answer. (They also love learning concepts that are “usually for older kids”) Some of the questions I have gotten from real kids:
- How could you measure a circle since it doesn’t have straight sides? (asked by kindergartener)
- What are those little baby numbers you see up in the air next to other numbers? (Exponents, asked by 1st grader, and when I showed him, he said “Oh, it’s like how many times you skip count” )
- “Did you know you can never actually count down to 0? Because you can keep breaking numbers into smaller and smaller parts infinitely.” (highly gifted/talented 2nd grader)
This does not of course mean you go fully into lessons that children do not have the background to understand. The important thing is just to keep in mind the full progression of learning and how the skills you teach now will be applied in later grades.
I realize that combining standards and teaching things in a different order is a luxury I have had since leaving the public school system. This is harder for teachers who must follow a pacing guide.
However, I challenge all teachers to think about the words they use and how the math they teach today is best approached for long term understanding. Even just acknowledging “Later you will learn another way to do this,” or “when you get into more advanced math you will see some different ways this can happen” goes a long way in building the math mind set we want kids to have, which is thinking about HOW and WHY math works instead of memorizing rules.