This is the introduction for the next four (or more) posts, as we will explore learning and understanding math facts for all four operations, focusing on fluency as well as memorization.
I was the District Math Specialist when my son was in first grade. So imagine my horror when the teacher (who was highly respected) called me up to discuss some concerns regarding his math skills. I mean, this was MY KID! The one I worked with and guinea pigged ideas on for me to learn and grow in math strategies. Ugh. So in I went, and persuaded the teacher to allow my son to be a part of the conversation. She proceeded to explain how the class (and the first grade in general) practiced their addition and subtraction facts. They would take a timed test at the beginning of the class, and to be considered “proficient” you had to get so many right. My son was no where near the cutoff.
I didn’t get it. This kid loved math. How could this be? I took a deep breath, and asked her if she had timed him orally. Sometimes, the problem isn’t that they don’t know the facts, but their motor skills are lacking and it just takes them too long to write down their answers. Or they get lost or overwhelmed with the sea of drill and kill presented to them. Or nervous, because they know they are being timed. She said she had (she really was a conscientious teacher), and it didn’t help.
I turned to my son and asked him, “What is 6 + 7?” Not an easy one. My son just sat there. FOR TWENTY SECONDS! (Research shows that a child should have 3 full seconds to find the value. If your child’s teacher isn’t giving them that amount of time per fact, have a chat with him/her and send them to this blog.)
The teacher just looked at me, as if saying, “You see???”
The next question I asked was critical. “How are you thinking about the problem?”
My son’s response. “Well, I was trying to figure out how you wanted me to solve it. Did you want 6 + 6 + 1? Or 7 + 7 – 1? Or 6 + 4 + 3 more? I mean, how to you want me to find the 13?” (Insert annoying sigh from my very drama-king son.)
So the question is; what is more important, to have facts memorized or to be fluent with the numbers so you can manipulate them and create an easier problem to solve? (Yeah, that is a totally loaded question, so I will rephrase.)
Fluency does not simply mean that you have your facts memorized.
Think of a young toddler who knows their alphabet. Just because my 3-year old son can say the alphabet does not mean he can manipulate the letters to create sounds and words. Just because a young child knows all of their math facts (because they are good at memorizing) does not mean s/he has a sound understanding of what those facts represent.
Here’s a perfect example (try it on your kids!). If I know that 12 x 3 is 36, how does that help me get the value for 13 x 3? (It is one more group of 3, so add 3 to 36 to get a value of 39.)What does the actual 12 x 3 represent (You could think of it as 12 groups of 3, but answers may vary.), and how is it related to 3 x 13? (You have one more group of 3 than you did in the previous fact.) What stayed the same? (They are all groups of 3.) What changed? (The number of groups changed. I have one more group.) How does this change affect the value? (It is just 3 more than the value before.) Most students will not go into the relationship between the two facts, and simply calculate the fact as if it has nothing to do with the first one.
Now I am not saying that memorizing facts is not important. If you can memorize the facts into long term memory, you now are saving space to learn more difficult concepts and try new ideas. But it shouldn’t be the goal. The goal should be that students are so comfortable with these facts that they serve as a basic foundation for more difficult (bigger or smaller number systems and algebraic expressions) work. And by the way, if they truly have math fluency, they will have the facts memorized. They just have the bonus of seeing relationships, conserving numbers, and using properties to help find equivalent expressions that are easier to solve.
To finish my initial story, once we told my son that he could choose just one way to solve the fact, he exceeded the amount for proficiency. Sometimes, all we need to do is ask what they are thinking to get an idea of how to move them forward.
Want to read more regarding fluency? Read this article from NCTM President Linda Gojak!