Fact Fluency: Not Memorization Part 3

In fact fluency part 1 I discussed why I don’t think that students need to memorize their basic facts. In part 2 I discussed the strategies for gaining fluency in addition and subtraction. Now in part 3 we will discuss fluency strategies that can be used to develop multiplication and division fact fluency.
As we discussed in my earlier posts, fact fluency consists of 3 parts. Students need accuracy, efficiency, and flexibility to achieve fluency. It is my belief that the easiest way to gain fluency in multiplication and division is skip counting. (Read my post on the benefits of skip counting.  It’s not just for multiplication and division.) If you can get buy in from 2nd and 3rd grade teachers, students can come into 4th grade with skip counting in place. Since students are already familiar with skip counting by 2’s, 5’s, and 10’s, it is not difficult to extend that to all the numbers from 1-10. Most 2nd graders can learn their 3’s and 4’s rather easily and move on from there. Third grade teachers should work on mastery of skip counting and helping students new to the school begin the process.
In Virginia, third graders learn the concept of multiplication and division by looking at arrays and groups and skip counting, but are not required to master their facts. This does not mean that they shouldn’t be encouraged to use the most efficient way of getting an answer. Even if they are given groups or arrays, students can use their skip counting knowledge to quickly answer a multiplication or division fact.
To use skip counting to solve a multiplication problem such as 7 x 5=, you choose one of the factors to skip by and use the other to count. For this fact, most students would choose to skip count by 5 and do it 7 times. 5, 10, 15, 20, 25, 30, 35. I encourage my student to use their fingers to make sure they have counted 7 times until they are more comfortable. This does not affect the efficiency of the strategy. Students can choose to count by 7’s if they have become fluent in skip counting. 7, 14, 21, 28, 35 is going to get them to the same place. This is where flexibility of a strategy comes in.
Division is a similar process and can be easily mastered by students fluent with skip counting. 36 ÷ 4= can be solved by skip counting by the divisor until reaching the dividend. 4, 8, 12, 16, 20, 24, 28, 32, 36. As with multiplication, I encourage students to start this strategy by counting the number of jumps on their fingers as they skip count by 4. They will then see that it takes 9 jumps of 4 to reach 36, so 9 is the answer. If students are more comfortable drawing the number line for either of these strategies until they are more comfortable, that is fine. They will lose some efficiency, but only until they become familiar with the process.

 

If you inherit a student without skip counting in place, you can use what they know to help them with fluency. Refer to the multiplication chart below to use the common skip counting numbers to develop their multiplication and division fact fluency.They should be able to learn their 3’s and 4’s rather quickly and use the 5’s and 10’s to get the rest.

Just as with addition and subtraction, memorization will come the more they use their facts. You don’t have to “drill and kill” and frustrate your students. Good Luck!!

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.