Pattern analysis as entrance to algebraic proof situations at c-level

MSc-thesis defense by Aske Henriksen

Censor: Prof. Morten Misfeldt, AAU

Supervisor: Prof. Carl Winsløw, IND

Abstract

Algebraic work often appears as isolated problems in High School mathematics, for which reason students find it difficult to comprehend the different meanings of algebraic symbols, as these appear as unknowns, variables and parameters. This affects the students’ ability to use algebra as a language to explain a mathematical phenomenon.

This master’s thesis seeks to examine danish gymnasium students’ opportunities of applying algebra in proving situations. Based on the Theory of Didactical Situations and an analysis of six selected didactical variables, a course of study was constructed and implemented in December 2015 in a first year STX class. The desired effect of the didactical variables was to make the students examine patterns in number tables in order for them to formulate a conjecture and use this as a catalyst in a proving situation. This was sought fulfilled through an a priori analysis of the exercises, which was handed to the students during the course of study.

The a posteriori analysis focused on the affect of the didactical variables on the students’ work and the students’ transition between three intended phases: examination of pattern, formulation of conjecture and proving of conjecture. The analysis was based on audio recordings, pictures taken during the course of study and the students’ written
work. The a posteriori analysis showed the importance of completing the three phases successfully in chronological order, as an incomplete phase hampers the work in the next. The analysis also showed that the didactical variable the Algebraic Prerequisites needs much attention, as this has the ability to hamper the work of formulating a proof. Lastly the a posteriori analysis showed the importance of creating a pattern, that diminishes the use of verbal proofs and that the didactical variables the Calculations Needed and the Number Table play an important role in this phase. The Sum Table undermined the use of algebraic proofs, while the Calendar and the Multiplication Table fostered the need for algebraic proofs, especially when the calculations
involved multiplication.

The teaching experiment showed that if the conjecture was based on numerical examination and the use of verbal arguments were diminished, the students engaged in meaningful proving situations by the use of algebra.