While visiting classrooms in the Pacific Northwest region, I hear a lot of discussion aboutmathematical fluency.As fate would have it, I was fortunate enough to participate in ayoucubed– sponsored class on "Mindset Mathematics" at Stanford University. The class was facilitated byDr.Jo Boaler, anexpert in the field of mathematical fluency, and it helped to bolster my understanding of the subject. So, just what isfluencyand when should students be fluent in their basic facts?
What I learned in Dr. Boaler's class was that while the Common Core State Mathematics Standards (CCSMS) specifiesmathematics fluencyas part of the kindergarten through eighth grade (K-8) mathematics standards, the word fluently is used sparingly in the K-8 CCSMS. For example, fifth grade students are asked to “fluently multiply multi-digit whole numbers using the standard algorithm.” Mastery of the standard algorithm follows two to three years of developing conceptual understanding of multiplication through repeated exposure to meaningful contexts, use of manipulatives, visual models and alternative algorithms. See theCCSMS Progression Documents-Numbers and Operations in Base Ten, K-5pages 12-15 for alternative algorithms that deepen students’ conceptual understanding of multiplication.Graham Fletcher’s Progressionvideo series includes a multiplication video which supports students' progression of learning multiplication, including how to use manipulatives as a sense-making tool.
Below are some examples of fluency in mathematics by grade level:
1By end of year, know from memory all sums of two one‐digit numbers
2By end of year, know from memory all products of two one‐digit numbers
“Fluentin the Standards means 'efficient and accurate.' It might also help to think of fluency as meaning the same thing as when we say that somebody is fluent in a foreign language: when you are fluent, you flow. Fluent is not halting, stumbling, or reversing oneself. Assessing fluency requires attending to issues of time (and even perhaps rhythm, which could be achieved with technology).” fromEngage NY’s Math Fluency Document
In grades K-5, students work towards an efficient algorithm whereas in grades 6-8 students are asked to use their skills to solve problems. The procedural skills students use depend on the context and the ability for students to choose the mathematical calculation that will highlight the specific mathematics of the context. No matter the grade level, procedural skills should facilitate coherence around mathematical ideas that students build over time.
The CCSMS outlines three key shifts for educators to consider:Focus,Coherence,andRigor.Rigoris defined as conceptual understanding, procedures and skills, and applications in mathematics. Students should experience all three components ofRigorwith equal intensity.
Important mathematical concepts follow a progression in our math standards that consist of all three components ofRigor. Often our educational system focuses on one component ofRigor:procedures and skills in the name of fluency. Adults today often connect fluency with computation and equate mathematics with computation. This view of mathematics is limited and can be harmful when considering how students determine their mathematical identity.
Then what about timed tests used as a tool to build mathematics fluency? When is the use of timed tests appropriate? After a lot of research and discussion with students past and present, I believe that timed tests are never appropriate especially as a measure of students’ mathematical abilities. Jo Boaler’s article Fluency without Fear, calls for the removal of timed tests for all students. Boaler states that “mathematics facts are indeed important but the memorization of math facts through times table repetition, practice and timed testing is unnecessary and damaging.” p. 1.Fluency without Fearprovides research around timed tests and suggest alternative means to support mathematical fluency.
Boaler also claims that number sense and flexibility of number are critical skills for all students and cannot be developed with timed tests.
In closing, I have to remind myself that when students leave our classrooms, they need to be confident mathematical thinkers; they will need to solve problems that currently do not exist. Physicist Conrad Wolfram, in a 2010 Ted Talk, states that students spend eighty percent of class time computing numbers and twenty percent of their time solving meaningful mathematics. Therefore, two components ofRigor: conceptual understanding and application in context do not appear in our mathematics classrooms as often as necessary. As Wolfram points out in hisTED Talk, classrooms mathematics should reflect more problem solving and less computing, especially by hand and students should use technology whenever possible.
Fluency will continue to be a topic of discussion with educators across our region. Our discussions should consider the definition of fluency and what research is best practice around fluency as it serves students mathematical thinking.