If you ask any adult what they remember from elementary math class they will often recall the infamous ‘Timed Test’. Some will recall the staple of math classrooms from the past with fond memories however, for most adults the memory brings memories of anxiety, fear, and tears. Those same adults often will identify with the statement of ‘I am not a math person’. It was once believed that flashcards, repeated practice of a multitude of basic math facts, and timed math tests that included 48 memorized answers were the way to build fact fluency.
The good news is we have a wealth of research dating back a decade and much more recent data to support that the previous assumptions about how children learn math facts were wrong. The bad news is educators have not changed their practices and we are still drilling students with flashcards (although now they appear in an app), Rocket Math, and competitions over how fast a child can produce memorized facts. This madness must stop!
Fluency is NOT about fast!
According to the renowned fluency experts Jennifer Bay Williams and John SanGiovanni, procedural fluency is measured by flexibility, efficiency, and accuracy, not by fast.
Fluency is about flexibility
As educators, we must explicitly teach students to select appropriate math strategies strategically based on the problems they are given. For example, if I ask a student “What is 8 + 6?” we would want the student to select one of the strategies listed below. We want students who are flexible in their thinking about math problems with the 4 operations. We have students we have identified as ‘high achieving’ because they have memorized the math facts however I would ask, ‘How flexible are those students?’. When we only rely on memorized facts and we don’t have strategies at our disposal to build our number sense we are likely to forget those facts. If you ask a 4th grade teacher, ‘Did your current students come to you knowing their math facts?’ Most teachers will say, ‘No, most of my students do not know all of their math facts’. I can guarantee you that the 3rd-grade teacher from last year made sure those same students knew their facts, however, they forgot them! Memorization does not work.
Bay-Williams, Jennifer M.; SanGiovanni, John J.. Figuring Out Fluency in Mathematics Teaching and Learning, Grades K-8: Moving Beyond Basic Facts and Memorization (Corwin Mathematics Series) (p. xii). SAGE Publications. Kindle Edition.
Fluency is about efficiency
Efficiency is in the eye of the beholder. Memorizing math facts is not the most efficient method for a student who forgets them all the time. We must ask students to select the strategy that they understand rather than a strategy we have deemed ‘the best’. Students who have a variety of strategies at their fingertips are better equipped to deal with various math problems and are not tied to just one way. Automaticity with math facts will come along as students practice their various strategies over time. Practice is important but it needs to be directed at the use of a variety of strategies rather than regurgitating memorized facts. Fluency practice should be focused on daily practice through games students play with each other where they explain their strategies. Often we see games being played in classrooms but you don’t hear students justifying their reasoning, that is the key to successful fluency practice. Math centers are a great way for students to spend 10 minutes a day on fluency practice. WARNING: Fluency apps are not a good way to practice math facts as they often ask students to work alone and they do not require students to explain their strategy.
Check out fluency games for all grade levels
Fluency is about accuracy
Of course, we want students to be accurate in finding the solution to a math problem. However, accuracy comes when students are allowed to select strategies that best fit their understanding. So many students struggle to find accuracy when it is coupled with speed because they don’t have the time to work through their selected strategy. If we want students to engage in flexible thinking around math facts we need to eliminate the requirement of speed. Math is not a race to the finish, instead, it is a creative, beautiful, and critical-thinking subject where students can find mastery and build their math confidence. Take the responses from adults who will share their traumatic experiences with timed tests as a cue to remove the barrier of ‘fast’ from students’ ability to truly learn number sense and meaningful mathematics. As a student, if I know that 8 x 7 is 56 but I have no idea why, I do not really have a true sense of numbers. Instead, we want that student to understand that 8 x 7 is 8 groups of 7, 7 groups of 8, 4 groups of 14, 14 groups of 4, or a variety of other equivalent representations. That is the POWER of math.
Assessing fluency
If we don’t assess fluency with timed tests how do we assess them? The best approach is Fact Interviews where students are asked to solve fact-based problems but they are also asked to explain their strategy. Here is a sample interview:
Teacher:
Hi, Alex! Today we’re going to go over some multiplication facts to see how you’re thinking through the problems. I’m interested in hearing your strategies. Ready to start?
Student:
Yep, I’m ready!
Teacher:
Great! Let’s begin with 3 times 4. What do you think?
Student:
That’s 12. I know because 3 times 2 is 6, and then I just double that to get 12.
Teacher:
Ah, I see! So, you broke it down into smaller parts. Can you tell me why you chose to double 3 times 2?
Student:
I guess it just feels easier to do 3 times 2 first. I know my 2’s really well, so I use that sometimes.
Teacher:
That’s a great strategy! Using facts you know to help solve others is a smart approach. Let’s try 6 times 5. What’s your strategy for that one?
Student:
6 times 5 is 30. I just remember that one because we did it in class, and I know 5’s are easy to count by.
Teacher:
Nice! Counting by 5’s is definitely helpful. When you say you “just remember it,” do you have a picture or a pattern in your mind, or is it more like you memorized it?
Student:
It’s kinda both. I can picture the 5’s counting up, but also, I just know that 6 times 5 is 30. It’s stuck in my head now.
Teacher:
That’s fantastic! The more you practice, the more facts will stick like that. Let’s move on to a harder one—7 times 8. What strategy do you use for this?
Student:
Hmm… 7 times 8 is 56. I don’t know it right away, so I think of 7 times 7 first because I know that’s 49, then I just add another 7 to get 56.
Teacher:
I love how you’re using a nearby fact to help you! You used 7 times 7 and added 7 more. Do you think that same strategy could work for other facts? Maybe with 6’s or 9’s?
Student:
Yeah, I think so. Like for 6 times 7, I could think of 6 times 6 first, then add 6.
Teacher:
Exactly! That’s a strong way to approach harder facts.
Fact interviews allow the teacher to hear how the student is thinking through the problem and what strategy they are selecting. The interviews allow for a look inside a student’s thinking which can lead to diagnosed misunderstandings and corrections that might otherwise linger and hinder a student’s number sense. Consider how you are building true number sense for your students while also building their math fluency. There is an approach that can cause long-term damage to your student’s math identity or there are approaches that build their mastery, confidence, and love of math.
Research Basis
Bay-Williams, Jennifer M.; SanGiovanni, John J.. Figuring Out Fluency in Mathematics Teaching and Learning, Grades K-8: Moving Beyond Basic Facts and Memorization (Corwin 2021) (p. xii). SAGE Publications.
Boaler, J., Williams, C., & Confer, A. (2015). Fluency without fear: Research evidence on the best ways to learn math facts. Reflections, 40(2), 7-12.