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Have you ever learnt something ‘by heart’?
Take a moment to stop and consider what is was. A poem? Song lyrics? Lines from a play?
What about your times tables?
I have always loved the phrase ‘to learn by heart’, with its (delightful, if biologically flawed) connotations of committing something to memory via that most emotional of lovely, fluttering, bloody organs beating steadily in our chest cavities. The connection with emotion comes via the original Greek use of the phrase, where the heart was not only the seat of irrational feeling, but also the rational intellect. The Latinate and Greek roots for heart are ‘card’ and ‘cor/cour’, giving us all sorts of interesting etymological connections between minds and hearts – for example, to ‘record’ is to ‘put back in the heart’, or to remember. We also have concord and discord, the harmonising or grating of two or more hearts and consequently minds. (Many of these themes are also musical: the ‘recorder’ is easy to play and practise tunes on, as a memory aid.) Feelings and intellect are all bound up together, along with the crucial idea that we can only be successful as humans (intellectually or otherwise) by finding emotional sustenance from others. We take heart when others tell us to have courage; we collaborate with others by being cordial; if we get disheartened, others encourage us.
How do you feel about your times tables?
Do you have memories of being drilled, of practising out loud, of being tested on them, perhaps in a timed fashion? Are they pleasant memories? Anxious ones?
These two phrases may feel somewhat aligned with the important ideas of conceptual knowledge and procedural knowledge in maths education. ‘The term conceptual knowledge has come to encompass not only what is known (knowledge of concepts) but also one way that concepts can be known (e.g. deeply and with rich connections)’ (Star, 2005). ‘Procedural knowledge … is ‘knowing how’, or the knowledge of the steps required to attain various goals. Procedures have been characterized using such constructs as skills, strategies, productions, and interiorized actions’ (Byrnes & Wasik, 1991). Fierce debates, particularly over the learning of ‘fluency facts’ such as times tables, have often suggested tussles over which comes first, or which is more important when. But research suggests ‘there is general consensus that the relations between conceptual and procedural knowledge are often bi-directional and iterative’ (Rittle-Johnson & Schneider, 2015).
Quickly, what is 7 x 8?
How did you get there?
What about 17 x 18?
What kind of process did you go through?
Increasingly, maths education research recognises that the affective and the cognitive domain are not only both significantly implicated in maths learning, but are linked in all kinds of complex ways. There is also another factor here, often called the ‘conative’:
One way to gain insight into how their learners feel, think, and act, about and toward mathematics is to examine their psychological domains of functioning: the affective, the cognitive, and the conative… Conation refers to the act of striving, of focusing attention and energy, and purposeful actions. Conation is about staying power, and survival. The conative domain includes students’ intentions and dispositions to learn, their approach to monitoring their own learning and to self-assessment. Conation includes students’ dispositions to strive to learn and the strategies they employ in support of their learning. It includes their inclination to plan, monitor, and evaluate their work and their predilection to mindfulness and reflection.
Although their biology may have been somewhat suspect, the Ancient Greeks may have had something right: thinking and feeling are inextricably linked. When we ‘do’ maths, the way we position ourselves with regards to what we consider maths is, the physical and psychological reactions we anticipate, experience and memory are all bound up in the act of doing. When we consider ourselves conatively, we connect them even further by trying to step outside ourselves and think about thinking, or think about feeling. Getting stuck, and moving on from being stuck, often has a deeply emotional resonance to it. Professor Mike Askew recently recalled a teacher he worked with who used to encourage students to shout ‘I’M STUCK!’ and punch the air when they felt a mathematical blockage, in order to help ‘discharge some of the powerful emotion’.
So, far from the false (if neat) dichotomies of thinking and feeling, learning by heart or by rote, or procedural or conceptual knowledge, what actually happens when we learn maths is likely to be much messier and more interesting.
Byrnes, J. P. & Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural learning. Developmental Psychology, 27, 777–786
Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. In R. C. Kadosh & A. Dowker (Eds.), Oxford library of psychology. The Oxford handbook of numerical cognition (pp. 1118-1134). New York, NY, US: Oxford University Press
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411
Tait-McCutcheon, S. L. (2008). Self-Efficacy in Mathematics: Affective, Cognitive, and Conative Domains of Functioning. Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia, Brisbane, 28 June-1 July 2008, 507-513
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