Mastering the Basic Facts, Part 1Mastering the Basic Facts, Part 1

Computational Fluency

 

Computational fluency is an essential goal for students in elementary school mathematics. It is an integral part of learning mathematics with depth and rigor about numbers and operations. So what exactly is computational fluency and how can we support students during this developmental period? The guidelines for computational fluency were set forth in 2000 by The National Council of Teachers of Mathematics (NCTM) in Principles and Standards for School Mathematics (PSSM).

  • Fluency involves 3 ideas: efficiency, accuracy and flexibility (Russell, 2000)
    • Efficiency means that the student is able to carry out a strategy easily, without losing track of the many parts, while maintaining the logic of the strategy
    • Accuracy involves careful recording, the knowledge of the basic number combinations and other number relationships and concern for double-checking one’s results
    • Flexibility requires the knowledge of more than one approach and the student’s flexibility at choosing an appropriate solution pathway for the problem at hand and then choosing a different pathway to double-check one’s results. 
  • Fluency is a product of a firmly developed mathematical foundation that contains 3 parts:
    • a deep understanding of the meaning of the operations and their relationships to each other
    • the knowledge of a large bank of number relationships including facts, relative magnitude and hierarchical inclusion to name a few
    • a deep understanding of the base-ten number system, the structure of numbers and how the place value system behaves differently in each operation
    • being flexible is a key indicator of a child’s command of this mathematical foundation
      • In Knowing and Teaching Elementary Mathematics (1999) Liping Ma wrote, “Being able to calculate in multiple ways means that one has transcended the formality of the algorithm and reached the essence of the numerical operations-the underlying mathematical ideas and principles. The reason that one problem can be solved in multiple ways is that mathematics does not consist of isolated rules, but connected ideas. Being able to and tending to solve a problem in more than one way, therefore, reveals the ability and the predilection to make connections between and among mathematical areas and topics”.

 

Mastering the Basic Facts

 

Mastery of the basic addition and subtraction facts should be attained by the end of second grade, and basic multiplication and division facts by the end of fourth grade. Mastery means that your child can give a quick and accurate response without resorting to inefficient strategies such as counting. Work toward mastery should begin in preschool and continue until your child can recall their facts from long-term memory. Children must master basic facts because those who struggle will be challenged to understand higher mathematical concepts because their cognitive energy will be spent working on the basic computation and not focused on developing understanding and relationships with the new concept.

 

rote Memorization versus recall from long-term memory

 

Mathematics is a deeply connected discipline. Developing computational fluency takes time because rote memorization is not effective. Instead, we need to allow our children time to discover and develop relationships between numbers and concepts. The goal for mastery is not to memorize but to the development of mathematical memory. These types of strong mathematical understandings and relationships cannot be forgotten. Many adults have learned mathematics as a set of rigid, disconnected rules, facts and procedures. It may be difficult for adults to recognize the importance of building understanding of mathematical principles and the relationships that underlie all of mathematics.

 

algorithms versus guided invention

 

What is an mathematical algorithm? An algorithm is a step by step procedure for solving a mathematical problem that was “invented” by someone trying to solve a problem and later decided upon as the
standard procedure for solving that type of mathematical problem.

What is the problem with going right to learning the steps of an algorithm? Well, not all algorithms are the most efficient method for solving certain math problems. The other problem involves students memorizing the steps of an algorithm without any understanding of how the place value system behaves or understanding of number and operational relationships. Research suggests that once students have memorized and practiced procedures and algorithms without understanding, they have difficulty learning to bring reasoning and meaning to their work. The standard algorithm for addition and subtraction should be understood by the end of 4th grade, and for multiplication and division by the end of 6th grade.

Guided invention emphasizes strategies and reasoning. Children generate strategies based on their own knowledge and understanding of number relationships. Some children need guidance with developing strategies on their own which makes it imperative that classmates share, explain and analyze a variety of utilized strategies to solve problems. Just because a strategy has been shown or taught to a child does not mean the child understands and will use that strategy. Children need time to invent their own strategies and practice strategies they have been introduced to in order to build their own understandings of number relationships.

 

how to support mastery

 

Children move through stages on their journey toward mastery of basic facts. Those stages are counting, efficient reasoning strategies and finally quick recall from long-term memory. Support of learning facts should focus on the building of number relationships through efficient reasoning strategies.

So, if we shouldn’t memorize, than what should we do?

  • notice and build relationships with numbers and operations
  • allow children time and space to invent their own strategies, make connections and build understandings
  • play math games with your child such as dice or cards

 

Mastering basic facts at Forest View

 

Teachers at Forest View provide a variety of opportunities for students to master their basic facts. They use CGI word problems, number talks, number strings, math games, and technology to give students plenty of time to notice and build relationships and deepen understandings.

 

Look for Part 2 “Experiences that support reasoning and mastering the basic facts” coming soon…

 

Professional articles related to mastering the basic facts:

 

Fluency without Fear

Fluency: Simply fast and accurate? I think not!

Computational Fluency

 

Computational fluency is an essential goal for students in elementary school mathematics. It is an integral part of learning mathematics with depth and rigor about numbers and operations. So what exactly is computational fluency and how can we support students during this developmental period? The guidelines for computational fluency were set forth in 2000 by The National Council of Teachers of Mathematics (NCTM) in Principles and Standards for School Mathematics (PSSM).

  • Fluency involves 3 ideas: efficiency, accuracy and flexibility (Russell, 2000)
    • Efficiency means that the student is able to carry out a strategy easily, without losing track of the many parts, while maintaining the logic of the strategy
    • Accuracy involves careful recording, the knowledge of the basic number combinations and other number relationships and concern for double-checking one’s results
    • Flexibility requires the knowledge of more than one approach and the student’s flexibility at choosing an appropriate solution pathway for the problem at hand and then choosing a different pathway to double-check one’s results. 
  • Fluency is a product of a firmly developed mathematical foundation that contains 3 parts:
    • a deep understanding of the meaning of the operations and their relationships to each other
    • the knowledge of a large bank of number relationships including facts, relative magnitude and hierarchical inclusion to name a few
    • a deep understanding of the base-ten number system, the structure of numbers and how the place value system behaves differently in each operation
    • being flexible is a key indicator of a child’s command of this mathematical foundation
      • In Knowing and Teaching Elementary Mathematics (1999) Liping Ma wrote, “Being able to calculate in multiple ways means that one has transcended the formality of the algorithm and reached the essence of the numerical operations-the underlying mathematical ideas and principles. The reason that one problem can be solved in multiple ways is that mathematics does not consist of isolated rules, but connected ideas. Being able to and tending to solve a problem in more than one way, therefore, reveals the ability and the predilection to make connections between and among mathematical areas and topics”.

 

Mastering the Basic Facts

 

Mastery of the basic addition and subtraction facts should be attained by the end of second grade, and basic multiplication and division facts by the end of fourth grade. Mastery means that your child can give a quick and accurate response without resorting to inefficient strategies such as counting. Work toward mastery should begin in preschool and continue until your child can recall their facts from long-term memory. Children must master basic facts because those who struggle will be challenged to understand higher mathematical concepts because their cognitive energy will be spent working on the basic computation and not focused on developing understanding and relationships with the new concept.

 

rote Memorization versus recall from long-term memory

 

Mathematics is a deeply connected discipline. Developing computational fluency takes time because rote memorization is not effective. Instead, we need to allow our children time to discover and develop relationships between numbers and concepts. The goal for mastery is not to memorize but to the development of mathematical memory. These types of strong mathematical understandings and relationships cannot be forgotten. Many adults have learned mathematics as a set of rigid, disconnected rules, facts and procedures. It may be difficult for adults to recognize the importance of building understanding of mathematical principles and the relationships that underlie all of mathematics.

 

algorithms versus guided invention

 

What is an mathematical algorithm? An algorithm is a step by step procedure for solving a mathematical problem that was “invented” by someone trying to solve a problem and later decided upon as the
standard procedure for solving that type of mathematical problem.

What is the problem with going right to learning the steps of an algorithm? Well, not all algorithms are the most efficient method for solving certain math problems. The other problem involves students memorizing the steps of an algorithm without any understanding of how the place value system behaves or understanding of number and operational relationships. Research suggests that once students have memorized and practiced procedures and algorithms without understanding, they have difficulty learning to bring reasoning and meaning to their work. The standard algorithm for addition and subtraction should be understood by the end of 4th grade, and for multiplication and division by the end of 6th grade.

Guided invention emphasizes strategies and reasoning. Children generate strategies based on their own knowledge and understanding of number relationships. Some children need guidance with developing strategies on their own which makes it imperative that classmates share, explain and analyze a variety of utilized strategies to solve problems. Just because a strategy has been shown or taught to a child does not mean the child understands and will use that strategy. Children need time to invent their own strategies and practice strategies they have been introduced to in order to build their own understandings of number relationships.

 

how to support mastery

 

Children move through stages on their journey toward mastery of basic facts. Those stages are counting, efficient reasoning strategies and finally quick recall from long-term memory. Support of learning facts should focus on the building of number relationships through efficient reasoning strategies.

So, if we shouldn’t memorize, than what should we do?

  • notice and build relationships with numbers and operations
  • allow children time and space to invent their own strategies, make connections and build understandings
  • play math games with your child such as dice or cards

 

Mastering basic facts at Forest View

 

Teachers at Forest View provide a variety of opportunities for students to master their basic facts. They use CGI word problems, number talks, number strings, math games, and technology to give students plenty of time to notice and build relationships and deepen understandings.

 

Look for Part 2 “Experiences that support reasoning and mastering the basic facts” coming soon…

 

Professional articles related to mastering the basic facts:

 

Fluency without Fear

Fluency: Simply fast and accurate? I think not!

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