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# Dedicated to Teachers

Students do not develop numerical fluency by memorization and regurgitation of rules. Rather, numerical fluency develops over time as students engage in active thinking and doing. They must strategize, reason, justify, and record and report on their thinking. Accordingly, the effective development of numerical fluency involves the use of a set of cognitive processes throughout mathematics, not just in one lesson or introductory lessons. When mathematics lessons are systematically planned and implemented with these six processes at the forefront, teachers maximize the chances of all students becoming numerically fluent and mathematically powerful!

There are six identifiable processes that support the development of numerical fluency. these processes are not unique to numerical fluency−in fact, the same processes are essential for the development of spatial sense, algebraic reasoning, and other big ideas in mathematics.

1. Contextualizing or Storytelling

Understanding that life can be described mathematically is at the foundation of fluency. Equations exist because they are a shortcut to explain situations, look at reality, and make predictions. Too often we present equations without giving them context, leaving children without understanding and causing misconceptions.

2. Physically Constructing or Building

Children need to manipulate materials to develop an understanding of the action of operations, which is then extended to the visual or pictorial level and then to abstraction. Fluency demands multiple models and making connections between and among models.

3. Representing Graphically and Symbolically or Drawing and Using Symbols

Seeing and using models and relationships between models supports visual memory, building relationships, and mental fluency, and enhances long term memory. For many students, fluency depends on being able to visualize concepts in different ways and understand the relationships between these different representations.

4. Visualizing or Seeing

Children learn to visualize quantities and the relationships between them. As children accumulate a visual repertoire, their numerical fluency grows because they are able to “see” the mathematics in which they are engaged.

5. Verbalizing or Describing What and How

Understanding of operations is achieved when students describe and explain what they did with the materials that they manipulate and the pictures they draw. Students need to describe the representations they create and how various representations are similar and different. Students should always be expected to describe the "what" and "how" of the mathematics they are learning, using informal and, gradually, formal mathematical language.

6. Justifying or Discussing Why

Discussing relationships and justifying solutions to problems is fundamental to developing metacognition and crucial to long-term fluency. Justifying answers the question "why?" and is one of the best ways to monitor the development of numerical fluency in students.

Fluency emerges from diverse experiences that link numbers and operations to contexts and familiar situations, and that provide students throughout their mathematical development with opportunities to construct, visualize, verbalize, and justify. These processes are the essence of the differentiation needed to effectively teach mathematics. One student might make sense of subcontracting in a context, while another needs to touch and feel the comparison of two qualities, while a third learns best by "talking through."

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Patsy Kanter is an author, teacher, and international math consultant. She worked as the Lower School Math Coordinator and Assistant Principal at Isidore Newman School in New Orleans, Louisiana, for 13 years. Patsy is the co-author of Every Day Counts: Calendar Math and a consulting author for Math in Focus.