Inquisitive about inquiry? Loaded with cognitive load? Part 1 Mike Ollerton, Jude Stratton and Anne Watson share their thoughts on cognitive load theory (CLT). here have been major discussions and mathematical perspective on the world. Traditional arguments among bloggers, tweeters and has tended towards direct instruction (with various T other pundits in mathematics education meanings), learning facts and procedures, exercises about the value of inquiry approaches to teaching and the passing of tests involving procedures. mathematics and approaches that claim to be Literature from US sources, therefore, has to be read informed by cognitive science. We believe that these against this background. There is absolutely no need are not in opposition to each other. Brent Davis and to import these divisive arguments into professional Moshe Renert describe the issue as, “[theories of discussion in the UK because we already have a learning are] discussed in terms of their superficial shared curriculum; shared goals; shared descriptions differences rather than their deep compatibilities.” of desirable knowledge and capabilities and shared (2014, p. 19). An example of superficiality in current assessments. So why have similar arguments debates is that “direct instruction” is interpreted developed in the UK? as telling learners everything they need to know The limited time for educational input on many (and we see many examples of this when we visit routes into teaching mean that generic teaching classrooms) while “inquiry” is interpreted as leaving skills are often the major focus for new teachers them to construct all methods and knowledge for themselves (which we seldom see, although we do rather than pedagogic knowledge for mathematics, often see inquiry). In two articles we hope to braid that is the specific knowledge about how children these different strands of commitment, belief and learn mathematics and how our concepts build up in experience. sequences of lessons (see ACME, 2015). In most of Europe new teachers study the conceptual structure The context of, say, multiplication or axiomatic reasoning as part First, some history. Much of the literature about of their training, as well as experiencing mathematical using cognitive science to shape mathematics inquiry as a feature of being mathematical. They teaching comes from the US. In the US there has therefore understand the role of inquiry in learning. been no National Curriculum nor national high stakes In England, by contrast, many teachers enter the testing, nor a common school leaving examination. profession with only generic theories about managing This means that until recently there has been no behaviour, presenting information, understanding national shaping of expectations, educational goals cognitive architecture and so on. While the generic or curriculum. Indeed, the word “curriculum” in the context is important for teachers to know, it is not US has meant a textbook scheme having its own the whole story of how mathematics can be, and principles and goals. During at least three decades has been, taught successfully. In the second of our in the US there have been opposing views of how two articles we say more about this. Mathematical mathematics should be taught, and these are conceptualisation is not achieved only by memory and aligned with arguments about the goals of school fluency, but also by constructing webs of meaning. mathematics. However, the content goals of school Mathematics has specific methods of inquiry that are mathematical learning are broadly agreed, as they not merely facts and methods, but are habits of mind. are everywhere, and there are moves towards more For example, learning mathematics includes learning national uniformity in the US with a Common Core how to make and use examples, how to make curriculum. The arguments have been divisive and, conjectures, how and when to use mathematical occasionally, harsh and damaging; two camps symbol systems (for example, numbers, algebra, dichotomise mathematics teaching: “reform” and geometry, diagrams, graphs and so on). Mathematics “traditional”. Reform has tended to mean the use also has several distinct types of reasoning: of everyday thinking, inquiry learning, groupwork, • focus on language and interaction, extended tasks, Exact arithmetical. portfolio assessment and the development of a • Approximate quantitative. 32 February 2020 www.atm.org.uk Inquisitive about inquiry? Loaded with cognitive load? Part 1 • Relational. • Generalising. • Algebraic. • Naming and communicating. • Axiomatic. These are often implicit in mathematics lessons and can be treated as redundant skills when everything is • Statistical and probabilistic. given and told to students by a teacher, yet they are None of these characteristics of mathematics can the basis for developing mathematical reasoning and bend to fit generic pedagogical advice, any more than inquiry and also include some of the cognitive actions the specifics of art, or history, or modern languages, necessary for grasping a current fact or procedure. might do so. The special forms of mathematical ATM publications such as Thinkers and Questions reasoning and methods of inquiry are just as much and Prompts for Mathematical Thinking describe an entitlement for all students as are factual and ways of keeping these explicit. procedural knowledge and competencies. So, how can results from cognitive science be Such entitlement is a social justice issue. Recent coordinated with doing mathematics in its fullest PISA results claim that the poorest children in many sense? Below we present some of the deep countries only get factual and procedural knowledge, compatibilities. while mathematical inquiry and reasoning are of much How ideas from cognitive science relate to school more value for employment and social empowerment. mathematical practice Reasoning, inquiry, creativity and adaptive applications depend on the active mathematical An idea currently being promulgated is that working mind and such a mind supports future learning. Of memory is limited so teachers need to control cognitive course, there are facts and procedures to be learnt load for learners. In practice this is sometimes and automatised, but a list of those necessary for interpreted as a need to simplify everything into future learning would be shorter than the current small steps. In an extreme interpretation, teachers curriculum. For example, laying multiplication out in are being told to remove wall displays since they a particular way is less important than recognising distract learners. We have no room to discuss it, but possible multiples; naming different types of triangle this latter instruction is a gross misuse of a study in is less important than knowing how to analyse an unrealistic context. While the metaphor of working shapes in general. However, while we are shackled memory does indeed describe the limitations we by the current tests, the problem for teachers is how have in dealing with lots of information at once, to make the whole of “being mathematical” available turning it into an argument for controlling cognitive for all students while also helping them get good test load must be carefully considered. Mathematics results. Talk about memory and performance focuses already contains methods for keeping complicated on the latter. ideas within what we can work on. This is why written tools such as numbers, diagrams and symbols have Cognitive science focuses on a limited range of what been developed over centuries. Working memory David Geary (2005) calls ‘secondary knowledge’ limitations are traditionally managed in mathematics - those kinds of knowledge that can only be met in by writing and drawing, whether this takes the form of a deliberate educational environment. But learners idiosyncratic sketching and jotting or the use of formal also need to use and develop their natural powers symbol systems, whose use we have to learn. The (as John Dewey called them, 1910) and “primary role of symbol systems is to record and communicate knowledge” (as Geary calls it) to develop the thinking chunked knowledge. Chunked knowledge is the way that helps them do mathematics in its fullest sense. cognitive science describes structured groupings of Primary knowledge includes the natural ways of familiar units of knowledge. For example, knowing the thinking that enable us to survive in the world: number system means we do not have to remember • Engaging physically with objects. every separate number; knowing the important features of linear expressions means we do not have to • Seeing wholes and parts. remember every possible linear expression; knowing • Discerning. the properties of triangles and how to analyse shapes • Comparing. means we do not have to remember every separate triangle. Algebra allows us to express relationships • Classifying. in equivalent formats; dynamic geometry systems February 2020 www.atm.org.uk 33 Inquisitive about inquiry? Loaded with cognitive load? Part 1 enable spatial relationships to be explored. Rather multi-digit multiplication the area model, grid layout, than worry about cognitive load as if it is a new idea (it column layout or decomposition layout (e.g. 37 x 23 was first described technically in the 1950s) teachers = (30 + 7) x (20 + 3) = 30 x 20 + … etc.) can all can ask, “What representations are learners familiar be compared. Not only does this approach relate with? What representations ought we to use?”. methods to concepts, it also confirms the usefulness