Math Fact Fluency: What you need to know and avoid
Many common misconceptions float around in educational circles about math fact fluency. These misconceptions are mostly based on an incorrect reading of the research. Some common misconceptions are about:
- what math fact fluency is,
- if it is important,
- how to teach it,
- how to understand it,
- and finally, how to assess it.
These misconceptions lead to wasting time on incorrect strategies to teach math facts and math games that don’t work. More importantly, these misconceptions have led to an epidemic of students lacking fluency in math facts. Teachers in the upper elementary grades still see students counting on their fingers or using multiplication fact charts. Having math fact fluency is a key foundation for future success in mathematics. Wasting time on these misconceptions is the main impediment to developing this in all students. Let’s look at the correct conceptions and contrast those with the misconceptions as we go.
Conceptual learning must precede committing math facts to memory
Students must be taught the conceptual meaning of operations and of course, know their numbers, before embarking on the task of committing facts to memory. What do we mean by the conceptual meaning of an operation? When you give a basic fact problem to a student, such as 9 + 6, the student can represent the problem and figure out the answer. Whether they use manipulatives, or draw lines, is unimportant as long as they understand the process and can get the right answer. The same goes for subtraction, multiplication and division. Students must be able to show the problem and derive the correct answer to have a conceptual understanding of the operation. Students need to develop a conceptual understanding before committing facts to memory.
The misconception: Continuing to require students to figure out math facts is all you need to do. It will automatically lead to math fact fluency. What is true is that students need structured teaching to develop fluency.
What is Math Fact Fluency?
Fluency is the second stage in learning. In the first stage of learning, a learner develops accuracy, the ability to answer correctly given time. Imagine not just math facts, but other learning, such as a musical piece or a dance step. After some work you can do it, but slowly with a lot of concentration. You are accurate only. However, with a lot of practice you can do it correctly and quickly. That is fluency or its synonym, mastery. So fluency is accurate and quick, or efficient.
Math fact fluency is committing facts to memory and answering them by direct recall. We want students to just remember the answer.
The misconception: Using “flexible strategies,” like the student in this picture, to figure out math facts is precisely NOT fluency. It takes time to figure out math facts and so you are not quick. You’re still at the accuracy stage. What is true is that to develop fluency students must use “recall” to get the answers, rather than “flexible strategies.” Time spent playing games or developing a variety of strategies is time that should be spent committing facts to memory.
Misreading the research
The research on students using strategies to figure out math facts came from following what students do when left to their own devices to learn math facts. These students were not being taught correctly. Their teachers did not help to commit a small number of facts to memory at a time. Breaking this down into bite-sized pieces is necessary for memorization. With no help memorizing facts, children were seen to get started by using various tricks to help themselves remember.
However, all evidence shows that such strategies for remembering are an intermediate stage in learning that is replaced with direct recall by proficient students and adults.
The misconception: Students must spend a lot of time developing various strategies for remembering math facts. There is no research showing this is a necessary stage. We know that students who memorize facts directly have no problem if learning is structured correctly. Common sense tells you that an intermediate strategy, that is later abandoned in favor of direct recall, cannot be necessary in the first place. In other words, if you aren’t going to keep using these flexible strategies, why learn them at all? Instead, students should be helped to memorize the facts by systematically giving them a small number of facts to commit to memory at a time.
However, being able to answer fluently is not enough. Students need to keep practicing and learning so they can develop automaticity with math facts.
Math Fact Fluency should be developed into Math Fact Automaticity
Automaticity is the third stage of learning. It only comes after fluency is developed and only with additional practice. Not only can the learner do the task fluently (correctly and quickly), but does so without much, if any, conscious thought. Imagine a member of a marching band who has to play a piece of music quickly enough to keep time, but also has to think about marching in step with the rest of the band. That musical piece must be learned to automaticity so the band member doesn’t have to put much attention into playing the right notes.
Decoding in reading and math facts in math are both tool skills. These skills are but a tool to do a more complex task. Tool skills need to be learned to the level of automaticity so the learner can focus on the bigger task. Automaticity in decoding (reading words) is essential so that students can focus on the author’s meaning rather than figuring out the words. Students who read slowly, puzzling out words one at a time, lose the gist of the passage. Students use math facts to do higher order computation. Therefore math facts need to become automatic so that the student has cognitive capacity left to focus on the larger problem or procedure.
The misconception: Playing games and knowing a variety of strategies for deriving math facts are essential for developing fluency. The truth is that math fact fluency and automaticity are related to simple recall of facts and are developed through practice recalling facts. On the other hand, after facts can be recalled instantly, lots of games and “number sense” type activities are easy and fun for students.
Why is Math Fact Automaticity Important?
Automaticity in recall of math facts is important because it enables students to…
- focus on the processes in which they are using math facts rather than on deriving the facts as they go.
- better follow instruction in the classroom without being distracted by trying to figure out math facts during the lesson.
- solve difficult problems and to complete math assignments quickly and easily.
- have more confidence in their math abilities.
- have more success in their future math classes and careers.
How to assess Math Fact Fluency and Automaticity
I know a piano player who can play “The Flight of the Bumblebee” almost faster than I can hear it. He has clearly learned this piece to the level of automaticity. That being said, he doesn’t have to play it fast and it’s better when he slows it down some.
Unfortunately, there’s no better way to measure the development of fluency and automaticity than by measuring the rate at which the person can perform the task. There is a limit after which more speed makes no difference. But there is also a lower limit below which you know the person does not have automaticity. Research shows that direct recall of math facts happens in a little less than a second. So if a student is reading a fact off a flashcard or on the screen, once they have finished reading more than a second has passed and their answer should be instantaneous. If it takes two or three seconds after someone reads aloud the equation for the answer to come to mind, then that fact is not yet fluent or automatic.
The misconception: It shows fluency if, after reading a fact off the card or a screen, a student has to think for 2 seconds to come up with the answer. The truth is, we want direct recall which, after reading the fact, is instantaneous, less than 1 second. Having to stop and think about facts is not automaticity and it means that students need more practice recalling facts, not figuring them out.
Best ways to assess math fact fluency
Interestingly, the fact that an individual may automatically recall the answer to one math fact does not tell you about their recall of other math facts. Students memorize math facts better in small handfuls, not all at once. As they are learning we would expect students to be able to answer some facts instantly, but need more time to learn the rest. The ideal way to assess math fact fluency is with flashcards or a computer display. These tools help keep track of the ones that are answered instantly and which are not. So, it is not an all-or-nothing result, but determining which facts students know at a fluent level and which ones they still need to learn.
Giving students a sheet of 100 math facts to answer, some of which they know and some which they don’t, gives you a mixed result. On top of that issue, there is the issue of how fast a student can write. Most elementary students cannot write as fast as they should be able to answer math fact problems. Expectations for fluency would be for students to answer math facts at between 66% and 80% of their writing speed.
The misconception: Students are either fluent or not. We know that learning math facts is not an all or nothing proposition. Students learn facts individually by committing each fact to memory individually. Students can have memorized some of the facts but still need to learn others.
How to Memorize Math Facts to the level of Automaticity
Students must first learn the concept of the operation, such as addition or multiplication, before they begin memorization. Once students can represent and figure out any fact in the operation, then they understand the concept. Then they are ready to begin memorization.
A few at a time
Also as mentioned above, the only way to memorize the many facts in an operation is a few at a time. One memorizes the words to a song one stanza at a time, or your lines in a play one response at a time. This requires organization and a system to work through all the facts in some sequence so that gradually the students learn all of them.
With corrective feedback
When learning facts it is essential that there is corrective feedback, either from a partner or from a computer. Someone or something needs to give the learner the correct answer when the learner is uncertain. And someone needs to give the learner additional practice when there is a hesitation. Computers can reliably do this. Student partners can, too, but they need some training, which takes little effort to learn.
Bring facts to mastery before teaching more
Next, there has to be a way to ensure that each small batch of facts has to be learned to mastery before the next batch is introduced to be learned. This is the principle of feeding mush to the baby. One spoonful at a time, making sure the baby swallows the last one before giving them more. It is important to base the decision on when a student goes on to learn the next set when they’ve mastered the last set and not some pre-set schedule.
The misconception: You can push a class of students through the facts at the same pace. Truthfully, if you place learners in a position to try to memorize more before they have digested the previous sets you will cause proactive and retroactive inhibition. The student will begin getting more confused and lose ground.
After memorizing math facts, use them daily.
After students have learned math facts to automaticity, they enjoy using them in computation, which they can do very easily now. Students who are automatic in math facts are happy to race through math computation. They can also do mental math, which they now find fun. Math games can be used to practice using math, not as a method to learn, but as a method to practice what has already been learned.
All of the interesting relationships among numbers (that are incorrectly touted as a method to learn math facts), can be engaged in after committing these facts to memory. Second grade students who were in the process of learning subtraction facts, volunteered to their teacher, “These are easy because they are just the opposite of adding.” Because they had previously memorized their addition facts, this aspect of number sense was perfectly obvious to them, without any instruction. Memorizing math facts does not hinder number sense, it just makes it easy.
Find out more about the Rocket Math Worksheet Program for peer partner math facts learning
or the Rocket Math Online Tutor for computer-assisted math facts learning.