Traditionally, computational fluency has meant speed and accuracy.  But in order for children to use a skill fluently and flexibly in everyday life, computational fluency means much more.  Refer to Adding It Up  (National Academy Press, 2001) for an extended discussion. 

For purposes of this summary, in order to develop the number sense required for flexible and fluent use of a skill, students need to make sense of

• different ways to use models to represent the related mathematical concepts, and
• different thinking strategies that can be used to solve computation problems related to that skill.

Many students need four types of experiences in order to develop computational fluency.  They include (1) distributed problem solving to develop number sense and flexible thinking, (2) conceptual previews to make sense of the needed thinking, (3) lessons to learn to record the skill or perform the skill mentally, without the use of a model, and (4) conceptual reviews to practice these skills.

Distributed problem solving  Fluency and flexibility are developed as children make sense of and become comfortable "mentally manipulating" these different representations and thinking strategies throughout the school year. 

Implications for curriculum include posing tasks

• repeatedly over an extended period of time,
• that provide a variety of opportunities to make sense of different representations or models,
• that encourage the use of a variety of thinking strategies to solve problems related to the skill, and
• with a wide variety of contexts in which the skill is useful.

Implications for teaching include encouraging children to

• choose their own ways to represent and their own strategies to solve these problems,
• explain their representations and solution strategies to each other and to the whole class,
• try to make sense of and use new representations and new strategies,
• reflect on when representations and strategies can efficiently be used, and
• use flexible and strategic choices for their representations and solution strategies.

Expected student outcomes include to

• begin to make sense of when and how to use a skill,
• learn multiple representations and multiple solution strategies,
• develop mathematical vocabulary to clearly explain their representations and solution strategies, and begin to extend their thinking to mental computation and estimation strategies.

  

Thinking With Numbers is designed to provide those experiences through the five-minute lessons.  Children, who have solved and discussed these problems daily, have much better opportunities to develop number sense and flexibility related to skills than they would in the traditional approach of teaching each skill within a separate unit of instruction. 

Conceptual previews  However, while all children will begin to develop number sense, unfortunately not all will be prepared to make the connections they need in order to learn a skill with understanding.  Skills are typically illustrated concretely by performing actions on manipulatives.  These students must understand the specific representations or models and the specific thinking they will use as they perform the “actions” symbolically or mentally. 

It simply takes some students longer to make sense of these new representations and thinking than our traditional curriculum provides.  A two-week unit is not sufficient for many students.  By using brief two- to three-minute conceptual previews for about two weeks prior to teaching a skill, students are given an extended amount of time to understand.

Implications for curriculum include to

• pose daily problems involving the skill to keep the thinking "alive,"
• concretely represent problems with some concrete model (manipulative or drawing), and
• provide opportunities for students so solve and explain their solutions.

Implications for teaching include to

• choose a model, which can be used to solve problems related to the skill, by using actions that correspond directly with the thinking that is used with the symbolic or mental skill,
• pose problems which the model can be used to represent,
• help students learn to solve the problems concretely by performing these actions,
• help the students make sense of these actions,
• encourage the students to explain these actions, and
• model any new language to help students connect the concrete actions to symbolic thinking.

Consider the following examples: 

Grade one example:  Put some objects in a plastic cup.  Tell the students how many are there.  Then drop in one, two, or three more.  Let the students watch and listen as each object drops into the cup.  This model will naturally encourage students to count on because they can’t see the counters in the cup.

Grade three example:  Give each student a rectangular piece of graph paper that has 6 rows of squares with 8 squares in each row.  Ask what multiplication problem this graph paper illustrates.  (6 rows of 8 or 8 rows of 6 depending on which way the paper is oriented)  Have the students fold their graph paper one time, on a line, so that the problem is split into two parts.  Then ask a few students to tell what parts they got and ask them to explain how they could add these parts mentally.  For each explanation, draw a rectangle, split it into two parts to illustrate what the student did, and label to parts to be added.  This will encourage students to use splitting into parts to help them solve multiplication problems.

Expected student outcomes include learning to

• use a model to concretely solve problems related to the skill,
• use actions with the model that correspond directly to the thinking that is used with the symbolic or mental skill, and
• clearly explain that thinking.

  

For a key skill that needs to be taught at this grade level, a unit of instruction will be a part of the curriculum sometime during the year.  That skill is represented by the dark line above.  Immediately preceding that, for about two weeks, brief daily previews can be used to help the student develop the thinking that will be needed for that skill.

Thinking With Numbers includes suggestions for concrete activities to help students make sense of different representations and different thinking strategies.  They should be used for very brief activities, perhaps only two to three minutes, each day for about two weeks prior to teaching a skill.

Recording the skill or performing the skill mentally  The previews provide students some extra time to make sense of the skill at the concrete level.  The next stage is to learn to record the symbolic skill or perform it mentally, without the model.  Since the students already understand how to perform the concrete actions to solve problems with the same thinking that is needed symbolically, all that remains is to learn how to record these concrete actions or how to perform the actions mentally, without the model.  This is usually taught in a traditional unit of about two weeks.  However, by using previews for about two weeks, students have a much better opportunity for success. 

Implications for curriculum include to

• provide opportunities for students to make connections between the concrete actions and the symbolic or mental skill.

Implications for teaching include to

• help students make these connections, and
• encourage students to explain these connections.

Expected student outcomes include learning to

• perform the written skill or mental skill, without the use of a concrete model,
• explain what each written symbol means, in terms of the concrete model.

Conceptual reviews   After students have made sense of the concrete actions to solve problems related to the skill and learned to record these skills symbolically or solve them mentally, some students still require a limited amount of practice.  Traditionally, we have provided pages and pages of worksheets. 

With the understanding that has been developed prior to this stage, two or three problems a day for about two or three weeks is sufficient for most students.  Each day, students should explain at least one of the problems in terms of the concrete models that were used.  Additional practice may be provided for students who need it.  With the use of previews and reviews, students now have about six weeks to make sense of the skill, not the usual two weeks for a traditional unit of instruction.

Implications for curriculum include to

• provide opportunities for students to practice the skill symbolically or mentally, and
• provide opportunities for students to explain the skill in terms of the concrete models that were used.

Implications for teaching include to

• encourage students to become proficient, but with an emphasis on fluency, and
• encourage students to explain the skill in terms of actions on the concrete models that were used.

Expected student outcomes include learning to

• correctly solve problems using written or mental skills,
• proficiently solve problems using written or mental skills, but with an emphasis on fluency, and
• clearly explain the skill in terms of the concrete models that were used.

  

Thinking With Numbers provides strategy practice worksheets to provide students opportunities to practice specific thinking strategies.  These activities may be used as conceptual reviews--that is, they provide an opportunity to practice that specific thinking and daily opportunities for students to explain that thinking. 

It is important to note that these strategy practice pages should not be used until students have already made sense of the thinking needed to quickly solve those problems.  Without this thinking, drill will not be effective.  In fact, without this thinking, drill will encourage students to guess.  This thinking also assures that students will have success, which is an important factor for students to develop confidence in their abilities.  These experiences will help them begin to develop number power.

Organizing the curriculum  What is described above has proven to be an effective way to help all students develop computational fluency for a skill.  Teachers cannot do this with every topic they teach, but they can for four or five key skills at each grade level.  The distributed problem solving needs to include problems related to each key skill.  The preview--skill unit--review activities need to be spread out throughout the school year.  Each will take about six weeks.

National Research Council (2001). Adding It Up:  Helping Children Learn Mathematics.   Jeremy Kilpatrick, Jane Swafford, and Bradford Findell, eds.  Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education.  Washington D.C.: National Academy Press.