Traditionally, math algorithms were treated as isolated systems to memorize. After introducing a procedure, teachers would model several example problems before providing students with massed, rote practice, hoping to burn the concepts into memory. Once children mastered these processes, they could apply the acquired skill to solve future problems (see Algorithm Examples).
Over the past several decades, elementary math education has undergone a philosophical shift. Today’s elementary school students are expected to accurately perform traditional algorithms, but also understand why the systems work. Emphasizing the latter leads children to connect prior learning to the present, while laying a foundation to access more complex topics in later grades (see Algorithm Connections).
Concrete, pictorial, abstract (CPA) pedagogy is the most common method for teaching children to execute algorithm procedures and understand why they work. Because students comprehend and master each stage in its continuum before moving on, abstraction levels become natural extensions that follow more concrete learning phases. Both in theory and practice, some students remain in the less complex stages indefinitely (seeCPA Example).
The opposite of CPA instruction is Memorization Before Conceptualization (MBC). The latter is used exclusively to teach algorithms and is far less common than the former. MBC blends traditional and progressive learning, beginning with abstraction and later segueing into pictorial and/or concrete representations.
CPA learning follows an evolutionary trajectory in which students discover mathematical truths before gradually finding more efficient and abstract methods to represent them. MBC advocates believe that when learning algorithms, many children struggle to draw connections along the CPA continuum. Instead, they compartmentalize the stages, learning each in isolation, but not threading the concept from start to finish. In the end, these students learn the same topic three or more times without ever understanding its symmetry. The work, therefore, is inefficient and less meaningful.
Constructivist educators often stigmatize procedural teaching as rote and mindless. They marvel at CPA algorithm progressions, never considering that procedural learning could lead to greater conceptual understanding. Consciously or subconsciously, they fail to acknowledge that having already memorized the standard process is one reason they see beauty in it.
MBC advocates are willing to teach procedurally, but strongly support conceptual learning. They think that students who have memorized algorithmic rules are better positioned to appreciate and understand them on a concrete and/or pictorial level. Internalizing a standard process, they believe, frees up working memory to conceptualize why systems work.
Possessing a sense of urgency, MBC practitioners sacrifice meaningful learning in the present for long term efficiency. They know that students who struggle with calculations are at a disadvantage in future learning, so they begin algorithm topics like interventions, i.e. short procedural lessons followed by rote student practice.
Their goals – memorizing procedures and conceptual understanding – don’t change, but they maintain a practical outlook on student learning. Computational mastery without conceptual understanding, they rationalize, is better than comprehending neither. At worst, children never understand the algorithm, but gain a reliable strategy to solve problems. At best, students achieve both goals in less time.
Although they are philosophically antithetical, CPA and MBC instruction possess identical objectives. Both aim for students to master algorithms and conceptually understand why they work. Two mountain climbers can take separate routes to reach a common summit, and different pedagogical techniques can lead to the same mathematical destination.