Twenty years on since he published his famous result, Sir Andrew Wiles has been awarded the prestigious Abel prize for his “stunning proof” of Fermat’s Last Theorem. Why such a delay? While the proof put to bed a notorious problem that had been taunting the mathematical community for 300 years, perhaps the main reason for the award is that the results Wiles proved had significance in “opening a new era in number theory”, implications that have taken time to be realised. In particular he proved results that connected two quite different areas of mathematics, number theory and modular forms.
As Alex Bellos explained at the announcement of the prize in Norway, Wiles’s proof demonstrated equivalence between the integer solutions of the equation , the rational coordinates on elliptic curves (curves given by the form ) and modular forms – highly symmetric mappings related to the geometry of complex numbers. This facility for making connections and shifting between multiple representations of mathematical ideas, so powerfully demonstrated in Wiles’s proof, is built into the aims of the new National Curriculum at Key Stages 3 and 4:
Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas… pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems
In particular they need to be able to “move freely between different numerical, algebraic, graphical and diagrammatic representations” (Department for Education, 2014).
Research into multiple representations in teaching and learning algebra has been undertaken since at least the 1980s (p. 12, Kieran, 2006; summary of research pp. 18 – 25). Liz Bills (2001) examined shifts in meanings of literal symbols amongst 16 – 17 year olds to identify “the kind of thinking required to move between different uses” of letters in algebra. Building on previous work (e.g. Küchemann, 1981) she considered questions where letters took on different roles within the same problem. “My study led me to the conclusion that very many of the problems that my students were tackling involved a subtle shift in the role played by the literal symbols; moreover it was this shift which in each case provided the power that made the solutions to these problems ‘standard methods’.” Her first shift considered the question: Solve the simultaneous equations and . In moving from the perspective of two isolated equations to two related simultaneous equations, the roles of and change. They move from those of “variable” (“a quantity whose importance is entirely in its relationship with another quantity”) to “unknown-to-be-found” ( and take on the roles of unknowns, whose numerical values are to be found). In a second problem, “What is the equation of a straight line with gradient 3 which passes through the point (2, 8)?”, use of the standard form involves making a shift for the letter from “placeholder-in-a-form” to “unknown-to-be-found”. For after substituting and , becomes the unknown-to-be-found in the equation .
Bills drew the conclusion that the subsequent improvement in student performance indicated that “it may be that drawing their attention to the shifts in meanings involved in some standard problems and their routine solutions would be an effective way of helping students to improve such understanding.”
Amit and Fried (2005), however, found difficulties with drawing students’ attention to multiple representations. In particular they found a disparity “to a great degree” between “teachers’ and students’ interpretations of the meaning and intent of the classroom activity”, represented schematically below:
|from Amit and Fried, 2005
In a sequence of 15 lessons on linear equations the teacher wanted “the students to know what representations and the act of representations are all about” and in particular for students to see “equations in a different light.” But to the students, while solving equations carried weight, drawing graphs was just another task, “a redundant exercise”. The students “do not appear to understand them as showing different mutually reinforcing views of the linear equation.” The authors suggest that the absence of mediating elements was a possible contributing factor, that the presence of “connectors” is needed as well as different representations. They conclude that “it may be that we have to challenge a multiple representations approach as a framework to begin with in teaching and think of (it) as a distant goal that may not be achieved until the learner has had considerable experience in kinds of thinking that potentially link representations.”
Focussing on representing processes rather than objects, Davis and McGowen (2002) identified significant success amongst a group of 87 college students who had to revisit functions as part of their “developmental algebra” classes. The research assessed the students’ “flexibility” of thought - which they defined as encompassing Krutetskii’s (1969) “reversibility” (for example, recognising and applying the idea of the inverse of a function), Gray and Tall’s (1994) “proceptual thinking” (see From the Library in Issue 126) as well as “connections between various representations, including tables, graphs and algebraic syntax.” Through use of “function machines” as representations, the pre- and post-test responses as well as the student self-evaluations indicated a significant improvement in understanding of functions as processes and a “dramatic change in flexibility of algebraic thinking.”
Coles (2014), again rather than emphasising concepts, focusses on relationships. He makes a distinction between “absolute” and “relational” representations, terminology he attributes to a suggestion by Tim Rowland. In examining the teaching approaches of Caleb Gattegno and Bob Davis, Coles suggests that it is the relationship between symbols that is important and which provides meaning. The approach is to “set up contexts or structures in which symbols can be introduced with a limited number of dimensions or variations” (following Marton and Booth (1997); Mason (2011)) so that “symbols represent relationships between the objects, or actions on those objects.” Coles illustrates this with the example of Gattegno using Cuisenaire rods to develop the concept of “number” as a relationship to a “unit” (see the video of the Gattegno’s lesson) and Davis working with negative numbers and developing them through the idea of relationships between numbers. “Symbols soon become meaningful in their connections to each other, and not linked directly to particular objects.”
Enabling students to make connections may be problematic but it is one of the keys to the power and pleasure of mathematics. As Sir Andrew Wiles said of the expanding role played by modular forms - “objects that really come out of geometry” - in number theory: “It’s a very surprising connection and I think it shows something very, very deep in mathematics and the more we study it the more surprised (we become) … and the more beautiful it seems.”
Activities developed by the Standards Unit for Improving Learning in Mathematics (ILIM) provide tasks for linking representations and are available to download at the National STEM Centre eLibrary:
- A1 Interpreting algebraic expressions
- A7 Interpreting functions, graphs and tables
- A11 Factorising cubics
- A14 Exploring equations in parametric form
and Mostly Calculus
- C1 Linking the properties and forms of quadratic functions
- C3 Matching functions and derivatives
- C4 Differentiating and integrating fractional and negative powers
While the Singapore Bar Method is often aimed at primary school children, this explanation of the representation leads to multiplicative problems suitable for Key Stage 3. Also this research by Spencer and Fielding on using the Singapore Bar for word problems may be of interest.
For exploring understanding of mean average: An idea for the classroom - multiple representations.
Realistic Mathematics Education (RME), described in From the Library in Issue 131, uses increasingly abstract “models” to represent and mathematise “realistic” contexts.
Amit, M., & Fried, M. N. (2005). Multiple representations in 8th grade algebra lessons: Are learners really getting it? In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th PME International Conference, 2, 57–64.
Bills, L. (2001) Shifts in the meanings of literal symbols. In M. van den Heuvel-Panhuizen (Ed.) Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 161-168). Utrecht, The Netherlands: PME.
Coles, A. (2014) Absolute and relational representations: the challenge of Caleb Gattegno and Bob Davis in Barmby. P. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 34(1) March 2014.
Davis, G.E. and McGowen, M.A., (2002) Function Machines & Flexible Algebraic Thought. Proceedings of the 26th PME International Conference.
Department for Education (2014) Statutory guidance National curriculum in England: mathematics programmes of study.
Gray, E.M. and Tall, D.O. (1994). Duality, ambiguity, and flexibility: A “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25 (2), 116-140.
Kieran, C. (2006) Research on the Teaching and Learning of Algebra in Gutierrez, A. and Boero, P. (eds.) Handbook of Research on the Psychology of Mathematics Education. Past, Present and Future. PME 1979-2006. Sense Publishers.
Krutetskii, V.A. (1969) An investigation of mathematical abilities in schoolchildren. In Kilpatrick, J. and Wirszup, I. (Eds.) Soviet Studies in the Psychology of Learning and Teaching Mathematics, vol I, pp. 5 – 57. Chicago: University of Chicago Press.
Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: Lawrence Erlbaum Associates.
Mason, J. (2011). Explicit and implicit pedagogy: variation theory as a case study. Proceedings of the British Society for Research into Learning Mathematics, 31(3), 107-112