Combining the theory of didactical situations and semiotic theory to investigate students enterprise of representing a relationship in algebraic notation MEC Annual Symposium LOUGHBOROUGH UNIVERSITY 25 MAY 2017 Heidi Strmskag Norwegian University of Science and Technology TDS: THE THEORY OF DIDACTICAL SITUATIONS IN MATHEMATICS

Systemic framework investigating mathematics teaching and learning supporting didactical design in mathematics Particularity of the knowledge taught Applicability - Intention Methodology

Didactical engineering Ordinary teaching situations Brousseau, G. (1997). The theory of didactical situations in mathematics: Didactique des mathmatiques, 1970-1990. Dordrecht: Kluwer. 2 A DIDACTICAL SITUATION (design and implementation) Target knowledge

Didactical contract SITUATION that preserves meaning for the target knowledge 3 SEMIOTIC THEORY Four registers of semiotic representation: -

Natural language Notation systems Geometric figures Cartesian graphs Two types of transformations of semiotic representations: treatments and conversions Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103-131. 4 TWO TYPES OF

TRANSFORMATIONS TRANSFORMATION from one semiotic representation to another CHANGING REPR. REGISTER BEING IN THE SAME REPRESENTATION REGISTER but keeping reference to the same mathematical object TREATMENT CONVERSION Duval, R. (2006). A cognitive analysis of problems of

comprehension in a 5 CLAIM Changing representation register is the threshold of mathematical comprehension for learners at each stage of the curriculum. (Duval, 2006, p. 128) 6 THE STUDY Teacher education for primary and lower

secondary education in Norway (four-year undergraduate programme) Data collected as part of a case study (Strmskag Msval, 2011) Research question: What conditions enable or hinder three students opportunity to represent a general relationship between percentage growth of length and area when looking at the enlargement of a square? 7 METHODS Research participants:

Three female student teachers: Alice, Ida and Sophie (first year on the programme) A male teacher educator of mathematics: Thomas (long experience) Data sources: A mathematical task on generalisation Video-recording of the student teachers collaborative work on the task, with teacher interaction (at the university campus) Data analysis: Task: with respect to the mathematical knowledge it aims at Transcript: Thematical coding (Robson, 2011) Robson, C. (2011). Real world research: A resource for social scientists

and practitioner-researchers (3rd ed.). Oxford: Blackwell. 8 REASONS FOR CHOICE OF THE EPISODE it provides an example of an evolution of the milieu which enabled the students to develop the knowledge aimed at it shows the utility and complexity of changing representation register when solving a generalization task 9

THE TASK SIMILAR figures Relationships (scaling laws) between length, area and volume: 10 THE TASK Percentage growth --- growth factor 11

STUDENTS ENGAGEMENT WITH THE TASK Particular case: 50 % increase of side 125 % increase of area 12 REPRESENTING A QUANTITY ENLARGED BY P % First conjecture (Ida): Correct representation of growth factor up by the group. Second conjecture (Sophie, turn 160): not successful.

13 REPRESENTING A QUANTITY ENLARGED BY P % Third conjecture: 2 + p % : Fails to represent that it is p percent of the original length (two plus p percent of two). Conversion is not successful. 14 ADIDACTICAL SITUATION

DIDACTICAL SITUATION Conjecture (2.5 %) %) fails to be true for %) = 25. Adidactical situation breaks down 15 THE MILIEU CHANGED BY THE TEACHER 50 % increase on particular cases: squares of sizes 4 x 4, 6 x 6, 8 x 8 (cm2) Leads to students conclusion: The original square can be a unit square (1 x 1) Seeing structure leads to students invention of manipulatives (paper cut-outs).

16 NEW MATERIAL MILIEU SHAPED BY ALICE Enabling enlargements to be calculated 17 THE GENERAL CASE Enlargement of side

Enlargement of area 50 % 25 % 10 % 125 % 56.25 % 21 % p% ?

They find out about the two congruent rectangles in each case. The small square in the upper corner is more complicated 18 DIDACTIC CONSTRAINT DUE TO A CHOSEN EXAMPLE Relationship between increase of side length and the area of the small square in upper corner in fraction notation

19 DIDACTIC CONSTRAINT DUE TO A CHOSEN EXAMPLE Relationship between increase of side length and the area of the small square in upper corner in fractional notation ENLARGE MENT OF SIDE AREA OF SMALL SQUARE IN UPPER CORNER

Alice and Idas model (squaring) Sophies model (halving) 1/2 1/4 1/4 1/4

1/16 1/8 1/5 1/25 1/10 1/100 20 CONSTRAINT BY DIFFERENT NOTATION

SYSTEMS (FRACTIONS PERCENTAGES) Alice (838): increase by one fifth is mixed with five percent increase 21 CONVERSIONS Arithmetic notation Natural language Geometrical figures

Algebraic notation 22 STUDENTS SOLUTION Formula for the area of a 1 x 1 square as a consequence of its side length being enlarged by p %: Justification by a generic example: 1 p/ 100

23 RESULTS Conditions that hinder the students solution process: - Lacking a technique for representation of growth factor - Various notation systems (percentages, decimals, fractions) and various concepts are at stake (length, area, enlargements). Conditions that enable the students solution process: - Teacher encouraging several empirical examples:

- specialising, conjecturing, generalising seeing structure Realizing the utility of a 1 x 1 square Inventing paper cut-outs change of semiotic register Arithmetic expressions enabling algebraic thinking Generic example (manipulatives) used to justify the formula 24 RELEVANCE Fine-grained analysis of transcripts of classroom communication

a detailed analysis of the functioning of knowledge and exploration of didactic variables that can lead to its modification - What figures to be used? What numbers to be used? What should the material milieu look like? What semiotic representations to be used intended conversions? 25

FORMULATION CONVERSION Explaining to someone else how to operate on the material milieu CONVERSIONS From action: Implicit model of solution explained to someone else. Operating on the material milieu using natural language and other representations Result: Explicit model of

solution Representations from other registers (notation26 systems, geometric figures, NEUROSCIENCE Symbols and spatial information different areas of the brain Mathematics learning and performance is optimized when the two areas of the brain are communicating (Park & Brannon, 2013) J., & Brannon, E. (2013). Training the approximate number system ves math proficiency. Psychological Science, 24(10), 17.

r, J. (2015). Mathematical mindsets. Unleashing students' potential th 27 ve math, inspiring messages and innovative teaching. New York: Pengu Thank you for your attention!