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 There is a history of deep wisdom about the multiple of the existing expertise of learners, thus having roles of representations in learning, expressing and both motivational and emotional benefits, as well as communicating mathematics (see Duval, 2006) that indicating the distributive law. avoids superficial attention to learning styles or dual- As well as optimising cognitive load and respecting coding (combining two representations to harness working memory limitations, we find “spacing” and visual and verbal attention, see Paivio, 1971). In “interleaving” are currently advocated by those who mathematics, visual and verbal representations have taken up CL theories as their guide. While these usually arise together and since 1500 BC special might be new ideas in some school subjects, they ways have been devised to do this. are automatically embedded in the organisation and Some applications of CLT to mathematics are content of mathematical subject matter. Spacing is about getting new knowledge; learners without about how often, and at what time intervals, practice prior experience of an idea might need worked needs to take place to ensure retention in long term examples and direct information to get started but, memory. We do not see spacing (the spread of as expertise develops, these approaches become reminder episodes after initial learning) as an extra less effective (Ward and Sweller, 1990). Mathematics concern in planning mathematics, still less as a magic in a well-planned curriculum builds new ideas on bullet, because new learning will always include more elementary concepts through extension and earlier facts, concepts and techniques. If thinking combination as the curriculum progresses. Learners about spacing helps to remind teachers about this, are seldom novices when they meet a new idea but that is good, but to treat spacing as a planning usually have some expertise of a more elementary goal, without thinking about the development of kind because of the internal connections in mathematical understanding for which spacing might mathematics. For example, solving linear equations help, puts the cart before the horse. Thinking about depends on understanding: the necessary conceptual construction takes care of spacing. Why would anyone give learners a task • The roles of variables and coefficients. that depends on knowing multiplication facts, such as • The ways symbols are combined;. factorising, without first ensuring these are not going • The meaning of the equals sign. to be stumbling block? A similar question arises in column subtraction – learners need fluent access • The additive relationship in its various forms. to number bonds otherwise the teaching endeavour It would obviously be foolish to teach ‘solving linear is going to fail. Some of the writing about spacing equations’ to learners without these skills being seems to assume that learners are always and only familiar or readily available in some form (for example, being reared for some random test or examination. computer algebra systems or graphing software). Our focus is the goal of becoming and being However, fluency might arise while using these ideas mathematical. in the context of solving linear equations, just as Interleaving is used in two very different ways, one fluency of sports skills develops in the game. If this being mixed questions after a few weeks of study, is done, learners can suggest methods themselves a very old idea in mathematics textbooks and tests, to identify the values of the variables. Methods may the other being the juxtaposition of examples and not need to first be demonstrated by a teacher if the questions that are similar in most respects but different components and meanings on which they build are in critical aspects, such as whether linear graphs go understood. In several high-performing countries through the origin or not, or whether a difference the development of methods for new problem-types between two numbers is positive or negative. takes place by comparing learners’ suggestions, or comparing alternative but mathematically-equivalent Variation theory claims we can only learn concepts methods with which a class might be partly familiar. For from contrasts (Marton, 2015), so this second kind example, balancing, graphical, concrete or flowchart of interleaving draws on what learners naturally models can be compared for solving equations; for do. Working with, and then comparing, carefully

34 February 2020 www.atm.org.uk Inquisitive about inquiry? Loaded with cognitive load? Part 1

structured examples to understand relationships is a tell the whole story of how school learning takes different process from doing lots of similar examples place (Niss, 2003). Mogens Niss points out that to get answers quickly. Variation principles are international comparisons show the importance for currently used to support a mastery discourse as successful teaching of: a way to make it more likely that learners will “get” • Classrooms that support mathematical the method or concept of a sequence of lessons by learning habits. limiting the changes to those that make the concept accessible. Variation is therefore a tool for noticing • Interactions and how they take account of whilst inquiring into mathematics, since it depends learners’ emotions. on our natural processes when faced with the • Explanations that relate everyday ideas to features and outcomes of mathematical situations. mathematical ideas. Reflecting on the effects of what we do is natural, • but too often in mathematics this has been reduced Teachers directing attention to particular to, “Did I get it right or not?”, rather than, “How do features using words, gestures, and materials. these two situations compare?”. Exercises described as “intelligent practice” are therefore, we believe, Each of these is fundamentally important in teaching only intelligent if used in intelligent ways in which and learning; none of them is described in teaching the learner’s own curiosity and experimentation can methods derived mainly from CL theories, yet all lead to a new conceptual insight. To learn through influence learning. CLT describes how people can applying inquiry skills involves reflection to identify learn and remember those facts and methods that what has been learnt and what to do next. We will give cannot be easily constructed from fragments, such an example of that in part 2 of this article. Inquiring as names and symbols for classified objects; formats learners often deliberately generate examples for that make manipulation possible; facts that support themselves to explore relations between variables in calculation. In international comparisons, PISA- a mathematical situation. OECD (2016) highlights the paucity of teaching and Some aspects of CLT are embedded in mathematical learning approaches that focus mainly on memory inquiry. These include: and performance. UK students report massive reliance on memory when compared to other OECD • Controlling variables to focus on one thing at countries. Also, to our shame, they report very low a time. reliance on ‘elaboration’ approaches such as: • Using multiple representations to present and • Relating unfamiliar ideas to what they already express relationships. know. • Extending the application of ideas (such as • Trying several approaches to a problem. from two-digits to three-digits, from positive to negative numbers, from linear to polynomials, • Analogical reasoning. from triangles to other polygons). • Using examples. • Seeking relationships; recognise similarities. Analysis of PISA data shows that, while memory • Making conjectures and generalising. is connected to success in simple problems as we would expect, it is not particularly helpful for harder Inquiry approaches to mathematics explicitly problems for which elaboration strategies turn out encourage, nurture, develop and value these to be more valuable. For example, current ratio methods of thinking and reasoning mathematically questions set for 16-year-olds, or reasoning questions whereas CLT informed approaches, such as some for 11-year-olds, make all four of these strategies interpretations of direct instruction, assume that clearly relevant. Ultimately, PISA concludes that what learners have to be told or shown how to do the is needed is: necessary thinking in a particular mathematical context. … a combination of learning strategies, particularly control and elaboration strategies. This provides Why cognitive science does not tell the whole students with enough direction and strategic story thinking for easier mathematics problems and International mathematics education researchers enough motivation and creativity for the most have long recognised that cognitive science cannot complex problems. (PISA, 2016, p.55).

February 2020 www.atm.org.uk 35 Inquisitive about inquiry? Loaded with cognitive load? Part 1

From a teaching perspective, PISA deduces teaching Mike Ollerton is nearly retired. He believes in the that ensures “cognitive-activation” is associated with social justice and of not setting by ʻabilityʼ. He higher PISA scores. This means that learners must is still passionate about teaching mathematics whenever opportunities arise. have opportunities to:

• Extend thinking. Jude Stratton enjoys working with mathematics teachers in schools. • Decide procedures.

• Reflect on problems. Anne Watson has retired from the University of Oxford but is still active in curriculum and • Compare different methods. professional development work. • Explain solutions. References • Learn from mistakes. ACME (2015) http://www.acme-uk.org/media/33228/ • Apply ideas in new contexts. beginningteachingbestinclass2015.pdf . Is CLT wrong then? Most studies that contribute to Davis, B. and Renert, M. (2013). The math teachers know: Profound understanding of emergent CLT involve memorising novel ideas in short-term or mathematics. London: Routledge. laboratory contexts, seldom in authentic educational contexts over time and often with willing adult learners Dewey (1910) How we think. Boston: D.C.Heath. rather than, for example, unwilling adolescents. It tells Duval, R. (2006). A cognitive analysis of problems us something about how memory might be improved of comprehension in a learning of mathematics. and brand-new concepts grasped but little about the Educational studies in mathematics, 61(1-2), 103-131. development of flexible mathematicians. We need to Geary, D. C. (2005). The origin of mind: Evolution of look at successful professional practice to see how brain, cognition, and general intelligence. Washington, regular use of elaborative and cognitive-activation DC: American Psychological Association. strategies are developed over time. These include Marton, F. (2015) Necessary Conditions of Learning. familiarity with structures in different forms and London: Routledge. contexts; accumulation of strategies that inform future Niss, M. (2003) Mathematical competencies and the actions; the skills of constructing new concepts from learning of mathematics: The Danish KOM project. simpler elements and much more besides, including In 3rd Mediterranean conference on mathematical education (pp. 115-124). Available at www.math. the development of independent mathematical chalmers.se/Math/Grundutb/CTH/mve375/1112/ curiosity. Such strategies could be ingredients of all docs/KOMkompetenser.pdf . kinds of teaching and learning but have a particularly Paivio, A. (1971). Imagery and language. In Imagery high profile in inquiry methods, with the added (pp. 7-32). Academic Press. ingredient that the associated mathematical thinking arises from personal purpose and mathematical PISA-OECD (2016) Ten questions for mathematics teachers. www.oecd.org/publications/ten-questions- need, rather than as instructed action. for-mathematics-teachers-and-how-pisa-can-help- In part 2 of this article we shall write about current answer-them-9789264265387-en.htm . policy and advice about pedagogy and present Vygotsky, L. S. (1987). Thinking and speech. The examples of lessons for which inquiry is an obvious collected works of LS Vygotsky, 1, 39-285. Springer. description, yet that have deep compatibilities with Ward, M. and Sweller, J. (1990). Structuring effective the aims and structures of lessons more overtly worked examples. Cognition and instruction, 7(1), influenced by CLT. 1-39.

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