There’s a scene in the award-winning film A Beautiful Mind where the real-life math genius John Nash (played masterfully by Russell Crowe) stands before two massive screens full of cryptic numbers formulated by an enemy government and computes in his mind for hours until he demystifies the code. Numbers are illuminated on the screens, indicating the highly complex mental math he’s performing. This iconic, albeit exaggerated, scene exemplifies computational fluency cranked to the nth degree.
Nobody expects elementary or middle school students to decode cryptic messages for the highest levels of government, but computational fluency is a skill that every math student should continuously develop throughout their education. It is defined as the ability to calculate accurately, flexibly, and efficiently. Computational fluency contrasts with fact fluency—the ability to recall the basic rules and tenets of mathematical processes. Fact fluency makes computational fluency possible, but they are not the same. We’ll explore fact fluency more in another blog.
For now, let’s take a closer look at the hallmarks of computational fluency and how STEMscopes Math helps students cultivate them.
Flexibility is an important component of computational fluency because it indicates that the student is moving beyond rote memorization. Knowing one computational method may be enough to pass an exam, but knowing just enough for testing purposes stifles students’ mathematical imagination. Worse still, it hinders them from becoming mathematical thinkers capable of solving real-world problems.
One of the focuses of STEMscopes Math is developing flexible problem solvers. Every scope (lesson) includes open-ended, real-world problems. Students may be asked to use a particular method they have been learning, but they are given leeway in how they implement that method, and always have the option to employ a method of their choosing. Just like in real life, they will have to choose the most appropriate method.
Flexibility and efficiency go hand-in-hand. To be efficient, students must be flexible—they must know different computational methods. Sometimes teachers and students misunderstand efficiency to mean speed. Efficiency may lead to speedier calculations, but speed is not the objective. Rather, the objective is comprehension. When students comprehend a method of computation they avoid unnecessary steps—the hallmark of efficiency.
STEMscopes Math uses the concrete-representational-abstract (CRA) method as a learning aid for developing comprehension and efficiency. This sequential approach gradually introduces new math concepts with tangible objects (called manipulatives) and moves to diagrams and pictures. By the time students enter the abstract phase, they have a basic grasp of the ideas and methods so that working with numbers and formulas seems less daunting. During all three phases, they practice new processes and learn which computational steps are essential.
Popular opinion suggests that accuracy means computing the correct answer. This misunderstanding may come from decades of curriculum that reduced math to “getting the right answer.” However, getting the right answer is just one aspect of accuracy. If we simplify accuracy to this one aspect, we hamper students from deepening their learning and moving beyond rote memorization.
Accuracy, instead, encompasses the ability both to communicate the why and how of an operation and to select the best methodology and solution. In other words, students must know how to reason and think mathematically.
STEMscopes Math dedicates an entire section to developing computational fluency. This section includes space for students to demonstrate their knowledge (Show What You Know), practice fluency (Fluency Builder), solve word problems couched in engaging stories (Math Story), work on open-ended problems (Problem-Based Task), and complete refresher exercises that review grade-level and past grade-level content.
Students do not learn computational thinking and then move on. They spend their entire mathematical education honing this skill. Before they do so, they must first know the basic facts and rules of the operations they are learning. In other words, they must have developed fact fluency. We explore the ins and outs of fact fluency in our blog “Fact Fluency: Beyond Rote Learning.”