Fluency with Arithmetic Operations and
Automaticity with Basic Arithmetic Facts
Benchmark expectations regarding arithmetic operations have been developed with a hierarchy in mind consisting of three stages: exploration, procedural reliability and procedural fluency.
Students will first explore arithmetic operations with no fluency expectations, then will be able to show procedural reliability and finally they will carry out these operations with procedural fluency. Interwoven into this hierarchy is the development of direct recall of basic arithmetic facts. Basic arithmetic facts are first derived, then utilized while becoming procedurally reliable or fluent and finally recalled with automaticity.
Stage 1: Exploration
The expectation is to develop understanding through the use of manipulatives (math tools), visual models, discussions, estimation and drawings. An example of an “exploration” benchmark is shown
MA.3.NSO.2.2: Explore multiplication of two whole numbers with products from 0 to 100, and related division facts.
Clarification 1: Instruction includes equal groups, arrays, area models and equations.
Clarification 2: Within the benchmark, it is the expectation that one problem can be represented in multiple ways and understanding how the different representations are related to each other.
Stage 2: Procedural reliability
The expectation is to utilize skills from the exploration stage to develop an accurate, reliable method that aligns with the student’s understanding and learning style. Students may need the teacher’s help to choose a method, and they will learn how to use a method without help. An example of a “procedural reliability” benchmark is shown below:
MA.3.NSO.2.4: Multiply two whole numbers from 0 to 10 and divide using related facts with procedural reliability.
Example: The product of 5 and 6 is 30.
Example: The quotient of 27 and 9 is 3.
Clarification 1: Instruction focuses on helping a student choose a method they can use reliably.
Stage 3: Procedural fluency
The expectation is to utilize skills from the procedural reliability stage to become fluent with an efficient and accurate procedure, including a standard algorithm. An example of a “procedural fluency” benchmark is shown below:
MA.4.NSO.2.3: Multiply two whole numbers, each up to two digits, including using a standard algorithm with procedural fluency.
Embedded within Stage 1 – Stage 3: Automaticity
The expectation is to directly recall basic arithmetic facts from memory. Automaticity is the ability to act according to an automatic response which is easily retrieved from long-term memory. It usually results from repetition and practice. An example of an “automaticity” benchmark is shown below:
MA.4.NSO.2.1: Recall multiplication facts with factors up to 10 and related division facts with automaticity.