Can students learn fluency through inquiry?
or How drill obstructs mathematical learning


Currently in mathematics teaching, there’s an idea that the subject cannot be taught through inquiry. More even, it is a dereliction of teachers’ duty not to drill students to become fluent. This claim is normally accompanied by reference to a contentious theory about cognitive load and to research on memory that is in its infancy. Of course, the meaning of ‘fluency’ itself is contentious. To some, fluency is developed through repetitive practice and demonstrated by the immediate recall of basic number facts and the accurate application of procedures. To others, fluency means something different (and more). The NCTM, for example, expect students who are mathematically fluent to demonstrate flexibility by transferring procedures to different contexts, building or modifying procedures from other procedures and recognising when one strategy or procedure is more appropriate than another.
 
In this post, I will argue the following: firstly, using drill and recall to promote fluency in classrooms rests on flimsy scientific arguments and does not work; secondly, we have to view fluency as encompassing both procedural and conceptual understanding (although, as I will go on to say, this distinction is not helpful); and, thirdly, fluency can be developed through inquiry
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The arguments for drilling rest on shaky foundations. Even if the Cognitive Load Theory is not at “an impasse, and dissatisfaction with it is growing” as this post claims, the idea that a ‘limited working memory’ should dictate how we teach completely ignores the social side of classrooms. Teachers have been supporting learning for years by using proxies for working memory, such as ‘holding’ provisional results during a multi-step calculation in their own memory. Furthermore, to devise teaching methods from a science that is under continual revision would suggest that the rudimentary techniques of drill and recall are out-of-date before the teacher arrives in the classroom. If the latest finding on how memory works “may force some revision of the dominant models of how memory consolidation occurs”, then will it also force a revision in teaching methods?
 
However, the main problem with drill comes later. Once students have memorized and practiced procedures, they have less motivation to understand their meaning or the reasoning behind them (NCTM). Drilling in facts and procedures interferes with conceptual development in three ways (Pesek and Kirshner): (a) cognitive interference results from the development of such strong routines that students block subsequent learning; (b) attitudinal interference occurs when they see no point in attempting to connect well-practised and successful rules with other representations that might give them a deeper meaning; and (c) metacognitive interference arises when conceptual learning threatens to draw away mental resources required to maintain a procedural competence. In light of these conclusions, the fluency aim of the National Curriculum (England), which requires frequent practice “so that pupils develop conceptual understanding” (my italics), is ill-conceived. Frequent practice potentially blocks conceptual understanding.
 
The information processing model of the brain in which ‘facts’ are banked in long-term memory is only one way of understanding how we think and learn. An alternative model focuses on concept formation and emphasises the growth of concepts and their relationship to other concepts in a connected network or ‘schema’ (Skemp, The Psychology of Learning Mathematics). In the classroom, the model leads teachers to prioritise opportunities to make links between facts, propositions and principles. The degree to which a student understands mathematical ideas or procedures is determined by the number, strength and richness of the connections in the network. For example, students drilled about types of triangles and other polygons bank them as disconnected facts. In a conceptual approach, students broaden the concept of a triangle through categorising different types and deepen the concept by, for example, linking the triangle to the construction of other polygons.
 
Conceptual learning is important because, by developing relationships and links, students have a wider repertoire of approaches to solve a problem. Drilling might work if the structure of problems does not change; the student simply applies the same procedure each time. However, faced with a novel situation, the student needs to identify properties of the problem and their links to other mathematical ideas. By linking concepts, the student can generate a new procedure. Conceptual knowledge becomes a pre-condition for “adaptive” or flexible procedural expertise (Baroody et al.
). 
   
Having promoted conceptual learning over the drilling of facts or procedures, it is nevertheless the case that researchers have become increasingly uneasy with the separation of different forms of mathematical knowledge. Making a distinction between conceptual and procedural knowledge has been described as limiting and an impediment to the study of mathematical learning (Star). Rittle-Johnson et al. argue that “conceptual and procedural knowledge develop iteratively, with increases in one type of knowledge leading to increases in the other type of knowledge, which trigger new increases in the first” (p. 346). Conceptual knowledge allows for a deeper structural analysis of a mathematical situation, which leads to more flexible procedural approaches; and correct procedural knowledge helps students represent key aspects of situations, which underlies advances in conceptual understanding. While we might take issue with the idea that there is only one (iterative) relationship between the two types of knowledge, the idea of a relationship is important in the development of mathematical fluency.
  
Different types of knowledge develop hand-in-hand in inquiry classrooms. In the inquiry on fractions and decimals that O’Connor observed, students discussed mathematical ideas directly, but conceptual understanding also developed from computational activity. The same occurs in Inquiry Maths lessons. The prompt 24 x 21 = 42 x 12 motivates students to practice multiplication facts, requires students to multiply accurately and also encourages them to reason about the structure of the equation. The time spent on each aspect and the teacher’s actions to support each one vary from class to class. The teacher makes the decision based on the students’ questions and observations in the initial phase of the inquiry and on their selection of regulatory cards. As the inquiry develops, students devise (or co-construct with the teacher) new pathways to which they transfer their learning. They evaluate the relevance of the facts, procedures and concepts from the original pathway to the new situation and, if necessary, modify (or seek the teacher’s help to modify) them. In this way inquiry combines all forms of mathematical thinking and relates them to each other. 
 
Drilling, on the one hand, obstructs conceptual learning. It leaves facts and procedures isolated and unconnected and, furthermore, discourages students from developing a deeper understanding. Inquiry, on the other hand, links different forms of mathematical thinking in a unified process. It promotes the NCTM’s idea of an enhanced fluency.

Andrew Blair
August 2017