Research about Fluency

(Cited from the Book Fluency Doesn’t Just Happen)

“A rich and thorough development of number relationships is a critical foundation for mastering of basic facts. Without number relationships, facts must be rotely memorized. With number understanding, facts for addition and subtraction are relatively simple extensions” (p.120).

Fluency doesn’t just happen! It is a well-planned journey. This community is meant to help you navigate that journey. Fluency is a multi-dimensional concept. We like to think of it as a 4-legged stool: accuracy, flexibility, efficiency and instant recall. Ann-Elise once said that “Automaticity has hijacked fluency. “ We love this because it is so true. Students need to be able to instantly recall their facts at some point so that they don’t get bogged down in the little stuff when they are working on multi-digit operations and fractions, decimals, and integers. BUT, students must learn their facts through a variety of engaging, ongoing, interactive, rigorous, student-friendly activities that build a fundamental understanding of how numbers are in relationship with each other. The research resoundingly states that computational fluency is multi-dimensional (speed and accuracy, flexibility and efficiency) (Brownell & Chazal, 1935; Brownell, 1956/1987; Kilpatrick, Swafford, & Findell, 2001; National Council of Teachers of Mathematics, 2000).

Dolch Words of Math

Students should learn their facts rather than memorize them. If you just memorize them, then you can easily forget them. If you learn them, then you can always do them through a variety of strategies, based in place value, properties and the relationships between the operations. There is a continuum for learning basic facts. Baroody (2006) calls it the Phases of Mastery. Batista (2012) calls it the Levels of Sophistication. This continuum has been discussed by many researchers. Basic facts for Addition and Subtraction are sums and differences within 20.

Addition

Subtraction

Plus 1

Minus 1

Plus 0

Minus 0

Count On 1,2,3

Take a number away from itself

Add within 5

Count Back 1,2,3

Make 5

Subtract within 5

Add within 10

Subtract from 5

Make 10

Subtract within 10

Add 10

Subtract from 10

Doubles

Subtraction 10 from a teen number

Doubles Plus 1

Subtract 1’s from a teen number

Doubles Plus 2

Subtracting differences of 1 or 2

Adding 7,8 9

Subtract by bridging ten

Adding within 20

Fact Families

Make 20

Subtract from 20 

Multiplication

Division

Multiplying by 0

Dividing by 1

Multiplying by 1

Dividing by 0 by a number

Multiplying by 10

Dividing by 10

Multiplying by 5

Dividing by 5

 

Dividing a number by itself

Multiplying by 2

Dividing by 2

Multiplying by 4

Dividing by 4

Multiplying by 8

Dividing by 8

Multiplying by 3

Dividing by 3

Multiplying by 6

Dividing by 6

Multiplying by 9

Dividing by 9

Multiplying by 7

Dividing by 7

Multiplying Square Numbers

 

Multiplying by 11

Dividing by 11

Multiplying by 12

Dividing by 12

Strategy Talk

As students are learning their facts, there are different approaches to working with numbers. These strategies have names.

5 + 7

Counted AllCounted onKnown FactsDerived FactsAutomatic Facts

Students count out the first addend, then they count out the second addend, then they count the total.

Sounds like:  1,2,3,4,5  then 1,2,3,4,5,6,7

then 1,2,3,4,5,6,7,8,9,10,11,12

There are different types of counting on… one is when students start at whatever addend comes first, the next level is when students consistently start with the higher addend.

Sounds like:  7 – 8,9,10,11,12

These are facts that students just know… oftentimes they are intermittent, with really no rhyme or reason… they could be random facts.  But they could be things like doubles… many students know that 5 + 5 is 10 without thinking about it.These are facts where students use what they know to figure out new facts.  So here they might say well 5 + 5 is 10 and 2 more is 12.

This is when students know their facts without having to think about them.  Logan calls this the “instant popping into of mind.”

(    )

Cycle of Engagement: Concrete, Representational, Abstract

We strongly believe that students should go through the cycle of engagement so that they have several opportunities to learn about the ways in which numbers are related and work together.  Many researchers maintain that this cycle gives students access to a deeper understanding of mathematical concepts (Anstrom,2017; Bender, 2009;Devlin, 2000; Van de Walle, 2001; Maccini & Gagnon, 2000). This cycle is a 3-step instructional process that allows students to gain conceptual understanding of a strategy by working with manipulatives.  The second part of the cycle is for students to do pictorial representations of the math.  The third part of the cycle is for students to work at an abstract level with the concepts.  Each stage builds on the previous stage.  It is important to give all students an opportunity to work through the stages because otherwise some students can get the answer but don’t understand the concept.  For example, if we were teaching students how to double a number, we would give them plenty of opportunities to actually pull objects and make the doubles fact.  Next, we would have students draw out doubles facts and work with them on scaffolded flashcards that have visual supports.  Finally, we would have students play a variety of dice, domino, card and board games where they just have to recall the facts.

So Math Running Records help us to know where students are on their learning journey so that we may provide an instructional program that meets their needs. When we know what they can do, we can then scaffold their learning in targeted, engaging, purposeful ways. The research lays out the learning trajectories, discussing the sophistication of strategy levels and the different types of strategies that students learn on their way to basic fact fluency.