Rethinking Mathematics Fluency

A common challenge I hear from teachers and parents is that their students do not know their basic math facts, meaning their addition, subtraction, multiplication, and division facts using numbers 0-9. What generally follows are questions of how to support their students with building this mathematical fluency. It’s widely accepted and known among teachers that the drill and kill, rote memorization of facts is not the best way for students to develop automaticity (quick recall) and in fact “the lowest achieving students worldwide [on the PISA tests] were those who used a memorization strategy” (Boaler, 2015). However, we recognize that mastering basic math facts is critical to building a strong foundation to conquer more complex mathematics and that “the past decade [of math instruction] has produced a generation of students who are procedurally competent but cannot think their way out of a box” (Boaler, 2015). This leaves many teachers scratching their heads at what to do and what is the best support to offer parents.  
One common misconception is that because fluency development is now taught differently that it no longer carries as much importance as it previously did. There is also the misconception that with an inquiry based and conceptual understanding approach that fluency will be developed naturally. In reality, fluency must be developed in tandem and partnership with conceptual understanding and “without that skill our students will view even simple math tasks as daunting” (O’Connell, p. 2). Our goal for students regarding math fluency should not just be memorization but automaticity, understanding, and connection making. In order for our students to really develop mathematical fluency with basic facts understanding and memorization must partner.  
A case for understanding 
We cannot expect students to internalize and commit math facts to memory without providing the opportunities to develop understanding first. Compare this to when we are teaching children to read. Asking students to memorize a string of words (such as book in favorite and sat I my chair a read) is much more difficult than when the words are put into context (I sat in my favorite chair and read a book) (O’Connell). Robbing students of the chance to develop understanding in context takes away opportunities for students to be problem solvers and utilize patterns. Teaching students facts with no context lacks authenticity, often leads to confusion and mix-ups, and does not allow students to develop reasonableness about their answers. 
A case for memorization 
On the other hand, any teacher would recognize the importance of students being able to recognize and recall math facts. There are many benefits to students committing facts to memory. When facts are automatic it allows the brain to grapple with more complex components of the task like patterns, fractions, or spatial reasoning. It allows students to answer problems more efficiently, and positions students to solve every day mental math tasks.  
Before we expect students to master and recall facts they must have a solid footing in number sense. First, students should have opportunities to investigate facts through discussions, games, and using concrete materials. Then, students should explore strategies that support their understanding. Finally, students should have targeted and strategic practice with memorization. Below I share some suggestions and routines for how to incorporate more authentic practice that builds students math fluency.  
Tips to develop math fluency appropriately 

  • Consider varying the order with which students learn multiplication math facts. Teaching facts in sequential order takes away the opportunities for additional sense making. A suggested progression is 2, 10, 5, 1, 0, 3, 4, 6, 9, 8, and 7 because the progressions build on pattern making and base ten. 
  • Fluency activities should be short in duration (5-10 minutes at a time) and frequent (a few times a week). 
  • Fluency activities should be varied to maintain engagement. 
  • Speed games and timed drills before students have developed fluency can block working memory and prevents students from accessing facts they know and can lead to develop math anxiety (Boaler, 2015). 
  • Be wary of how Engage New York (ENY) defines fluency. ENY claims that a memorization of facts allows students to successfully complete more complex tasks. This is not supported by research and ignores the importance of number sense (Boaler, 2015).  
  • Progress towards automaticity and fluency should be monitored and shared with students through conferences, data, and goal setting (O’Connell, 2011). 
  • When students are finding answers using manipulatives and visuals always record the corresponding equation.  
  • Ask questions when looking at equations
    • What do the numbers represent?  
    • Can you break apart the factors to help you find the answer? 
    • How is this different from…? 
  • Use several different models (area model, arrays, double number lines, set models, etc.) to show the same fact. 
  • When applicable connect math facts to literature. The Doorbell Rang, Two of Everything, If you Hopped like a Frog, Cloudy with a Chance of Meatballs, Snowmen at Night, Snowflake Bentley, Thunder Cake, and One Hundred Angry Ants all provide points of integration. 

Routines and Lesson ideas to develop math fluency appropriately 

  • Number Talks 
  • Skip counting routines 
  • Quick Images with Dot Cards, ten frames, arrays, dominoes, real life objects (Shumay, 2018) 
  • Encourage students to write their own story contexts to demonstrate their understanding of math facts 
  • Today’s Number routine (Shumay, 2018) 
  • Ways to Make a Number routine (Shumay, 2018) 
  • Ten Frames Routines (Shumay, 2011) 

Grade Level Specific Resources 
Building Conceptual Understanding and Fluency Through Games in Kindergarten 
Building Conceptual Understanding and Fluency Through Games in Grade 1 
Building Conceptual Understanding and Fluency Through Games in Grade 2 
Building Conceptual Understanding and Fluency Through Games in Grade 3 
Building Conceptual Understanding and Fluency Through Games in Grade 4 
Building Conceptual Understanding and Fluency Through Games in Grade 5 
References and additional reading 
Boaler, Jo (2015). Fluency without Fear. 
Boaler, Jo (2015). Memorizers are the lowest achievers and other Common Core math surprises. 
Boaler, Jo (2016). Speed and Time Pressure Blocks Working Memory. 
O’Connell, Susan, and SanGiovanni, John (2011). Mastering the Basic Math Facts in Multiplication and Division.  
Parrish, Sherry (2010). Number Talks. 
Shumay, Jessica (2011). Number Sense Routines Building Numerical Literacy Every Day in Grades K-3. 
Shumay, Jessica (2018). Number Sense Routines Building Mathematical Understanding Every Day in Grades 3-5. 
Van de Walle, John A. et al. (2018). Teaching Student-Centered Mathematics K-2.  
Van de Walle, John A. et al. (2018). Teaching Student-Centered Mathematics 3-5 

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