Fact Fluency: Not Memorization Part 2

In Part 1 of Fact Fluency, I talked about the fact that math facts should not be “kill and drill” until they are memorized. I wrote that fact fluency has 3 parts: accuracy, efficiency, and flexibility. We should be encouraging students to use strategies that will help them quickly get to the answers. In this post we will look at the strategies for addition facts to 20.
After doing a lot of research and reading lots of information about addition facts, I created my own addition chart based off several I studied in different books and websites. I believe that there are 4 basic types of facts: Zero, Counting on, Doubles, and adding 10.

The first type of fact is “adding zero.” Students should be able to see 0+6 or 6+0 and automatically know that the answer is 6. This is the identity property of addition and students should be comfortable with it.
The second type is “counting on.” This is the strategy that most students without fact fluency fall back on for all their facts. I encourage my students to only use this strategy for 1’s, 2’s, and 3’s. If you try numbers bigger than that, you lose efficiency. Again, students should be able to use the commutative property to know that 9+2 and 2+9 are the same fact and that they can use the same strategy to get the answer.
The third type of fact is Doubles. Many students learn their doubles quickly, but many times we fail to use this as a stepping stone to those Doubles =/- 1 facts. If they know that 6+6 =12, they should be able to quickly figure that 6+7=13 because 7 is one more than 6 and 13 is one more than 12. If you have been using the number rack (or rekenrek) and 10 frames, this is easy for student to visualize.

The fourth type if fact is making and adding 10’s. Most students are quick to learn their 10+ facts. Again, if students are familiar with 10 frames and math racks, this strategy will be easy for them to visualize. If they are comfortable with making 10 in all its combinations, they should be able to easily break apart numbers to make 10 and add to 10. For example, 8+6=, students know that 8 and 2 are 10. They break 2 off the six to make the 8 into a 10, leaving 4 behind. They already know that 10+4 =14.

If students can become comfortable with these strategies they will be able to develop fact fluency. This will also help them with subtraction as you talk about fact families and use addition facts to learn subtraction facts. Eventually, as students become more and more fluent, many of the facts will be memorized without the “kill and drill” nightmare.

Part 3 of Fact fluency will focus on Multiplication and division facts.

Math Fact Fluency: Not Memorization part 1

When I was in elementary school fact memorization was ingrained into our thinking. First it was our addition and subtraction facts; later it was multiplication and division. We were victims of “drill and kill” and teachers did it without remorse. This was what educators thought was best. In fact, I still hear it among many teachers still today.

I do not subscribe to this theory and I know that many of the biggest names in math education don’t either. I believe that as long as a student has a quick, comfortable strategy to get to the answers, memorization is not necessary. If instant answers are what you are looking for (although I don’t think they are needed); I will say that they more they use their strategies to answer the facts the more they will memorize the ones they use the most.
In this post I will just cover the basics of developing fact fluency. True fact fluency is composed of 3 things: Accuracy, Efficiency, and Flexibility. (Explained in the following article: http://investigations.terc.edu/library/bookpapers/comp_fluency.cfm) Students who just have the answers memorized may have accuracy and efficiency, but they don’t have flexibility. Our students who are still counting on their fingers or using pictures or manipulatives only have accuracy. Our students need all 3 in order to have true fact fluency.
Before we can start with facts our students need to have subitizing down. It is important that students recognize numbers 1-10 in different patterns; such as, dice/domino patterns, finger patterns, 10 frame and/or rekenrek patterns. This will help students with conservation of number that will go a long way to helping them with addition and subtraction facts. Let’s be honest, we have all had those students that had to count their fingers every time they put them up. For example, the problem 4+2= may go something like this. They count the 4 on one hand, the 2 on the other hand and then they go back and start at one and count the total. It takes kids much longer than if they could just put up 4 fingers and 2 fingers without counting them and then start with 4 and count 2 more to get the answer.

Let’s start with the addition and subtraction facts. Students start learning facts to 10 in kindergarten and early first grade. Most of the time they are using some sort of manipulative to solve the problems whether it is their fingers or counting cubes, bears, etc… I would encourage you to use 10 frames and rekenreks along with any other manipulatives you are letting the students use. This will help students in the long run by letting them see how numbers can be broken apart and how they relate to the benchmarks of 5 and 10. If you start teaching addition and subtraction facts using 10 frames and rekenreks it will be much easier for students as they move into facts to 20.

Part 2 of this series will focus on specific strategies that will help students become fluent with their addition and subtraction facts to 20. Part 3 will focus on multiplication and division fact.

Counting: Not as simple as it seems

The more and more I work with struggling math students the more it becomes clear to me that part of their number sense issues are that they can’t count. I’m not referring to starting at 1 and counting, but really understanding numbers enough to start at any number and count on or backwards.
I was working with a fourth grade student last year and the following problem came up: 43,046 – 42,958=. She wrote down the problem vertically and started regrouping. Because following steps and procedures is very difficult for her, she made a careless mistake and ended up with the wrong answer.

As a teacher, I’m sure you can quickly figure out what she did wrong. I know that you have seen the same mistake time and time again.
Instead of going back over the procedure with her I asked her if she thought the 2 numbers were close together and she answered yes. Then I offered the alternative of counting up like she was taught to do in first grade. Although she remembered the strategy, she said that they numbers were too big to count. I simply asked her what number would come next after 42,958 and she couldn’t tell me.
I was amazed and started asking some of my other 3rd, 4th, and 5th grade students if they could count on from larger numbers and most of them struggled with it. The few who could do it confidently could only go forward and not backwards. I couldn’t believe it. I went back in the Virginia Math Standards and saw that counting forward and backwards is only mentioned in K, 1, and 2; forward to 100 and backwards from 30.
Because it’s never mentioned in the standards again, we forget that it is an essential skill that should be reviewed and expanded upon all the way through the elementary grades and beyond. As students learn numbers with more places they should be counting those kinds of numbers. They should be able to count forward and backwards from any number that is given to them.
Once activity I started doing with my students is writing random numbers (up to the same number of digits they are required to work with in their grade) on index cards. As they were getting ready for group or packing up I would pick a card and a student. They would have to start counting forward from that number until I said stop. I would then call on another student to start counting backwards from that same number. I would ask another student for 10/100/1000/etc… more or less than the number. It was a simple, quick activity that helped my students develop a better sense of numbers and counting.
I continued working with the same student using open number lines to solve addition and subtraction problems.

As she became a more confident counter she stopped drawing a number line and started counting on using her fingers to keep track of how many jumps she was making. A few months later I brought up the same subtraction problem from the original story and asked her to solve it. This was her answer, “ 42,958. (while keeping track of her jumps on her fingers) 68….78…88….98…43,008…18…28…38….48, nope, too far. 38 is 80 (and she wrote down 80 on her board); (again keeping track on her fingers) 38..39…40…41…42…43…44…45…46… that’s 8 more. 80 plus 8 is 88. The answer to the problem is 88.”

No regrouping, no struggling to remember steps. She could confidently count the numbers and find the answer. This helped all my students with their mental math skills. Keep on counting with your students, don’t assume that they can do it.