Language aspects of Language aspects of algebraalgebra

Jean-Philippe Drouhard, Desmond Fearnley-SanderJean-Philippe Drouhard, Desmond Fearnley-Sander

Bernadette Barker, Nadine Bednarz, Dave Hewitt, Brenda Menzel, Bernadette Barker, Nadine Bednarz, Dave Hewitt, Brenda Menzel,

Jarmila NovotnJarmila Novotnáá, Mabel Panizza, Cyril Quinlan, Anne Teppo, Maria , Mabel Panizza, Cyril Quinlan, Anne Teppo, Maria TriguerosTrigueros

Beginning algebraBeginning algebra

- first encounters with algebraic - first encounters with algebraic languagelanguage

Algebraic notationsAlgebraic notations

- variables, equations, functions- variables, equations, functions Language aspects of algebraLanguage aspects of algebra

- in classroom practice and in theories of - in classroom practice and in theories of learninglearning

Language aspects of algebraLanguage aspects of algebra

- from a semiotic/linguistic perspective- from a semiotic/linguistic perspective

Beginning algebra Beginning algebra – first encounters with – first encounters with algebraicalgebraic language language

Language aspects of algebra

Distinguishing an algebraic activity Distinguishing an algebraic activity versus the use of symbolsversus the use of symbols

Symbolisation as a process and not Symbolisation as a process and not seeing algebra as the formal use of seeing algebra as the formal use of symbolssymbols

Taking account of the influenceTaking account of the influence of of students’ previous or simultaneous students’ previous or simultaneous experience in mathematics and/or in experience in mathematics and/or in other subjects (science etc.)other subjects (science etc.)

Examples of the use of symbols in a non-algebraic activity

Distinguishing an algebraic activity Distinguishing an algebraic activity

versus the use of symbolsversus the use of symbols

Geometry: labeling objects and using it

e.g. in describing or proving geometrical properties

Symbols with the role of labels: ABC

Examples of the use of symbols in a non-algebraic activityGeometry: labeling objects and using it

e.g. in describing or proving geometrical properties

Manipulations with symbols but non-algebraic activity

Examples of the use of symbols in a non-algebraic activity

Geometry: labeling objects and using it

Algebraic activity

Examples of the use of symbols in a non-algebraic activitySolving of word problems: The solution is purely arithmetical but students use letters to record the information described in the assignment.

Petr, David and Jirka play marbles. They have 198 marbles altogether. Petr has 6 times more marbles than David and Jirka has 2 times more marbles than David. How many marbles has each boy got?

David ………… dJirka …………. j ………… 2 times as many asPetr ………….p ………… 6 times as many as

198 : 9 = …

Examples of algebraic activities where formal symbols are not used

Problem solving in natural language (history)

Examples of algebraic activities where formal symbols are not used Problem solving using non-standard symbols (using and operating on an unknown, using an arbitrary quantity)

Three levels of the use of language of letters (Hejný, M. et al., 1987): modelling standard manipulations strategic manipulations

Symbolisation as a process and not Symbolisation as a process and not seeing algebra as the formal use of seeing algebra as the formal use of symbolssymbols

Modelling

Methodologically the most important level of the language of algebra in relation to the beginning of algebra

Difficulties related to powerfulness of the general standard algebraic language when students are confronted with it too quickly.

!!! Importance of the intermediate notation and verbalisation to maintain the meaning

Example: In the equation 5x + 15y = 280, 5x may be interpreted as: a certain number of books at $5 each, Peter has 5times more than Luc, a certain number repeated 5times, …

Modelling

Necessary: longer period of transition from non-symbolic to symbolic records

In this longer period of transition it is important:

• to conceive interventions/teaching situations (e.g. the situation of communication – Malara, N. & Navarra, G.; Bednarz, N.) that offer students opportunities for seeing the necessity and the relevance of a symbolisation process. Algebra could appear as a tool of generalisation or modelisation serving to endow symbolism and symbol use with meaning.

• to take account of different intermediate notations (particularly those developed by students): fundamental components of the transition to algebraic reasoning and of the construction of meaning for algebraic notations

• that student see the pertinence of some conventions (e.g. parantheses)

Modelling

Flexibility in using and interpreting different notations at the beginning of algebra

In this longer period of transition it is important:

• (Teppo, A.R., p. 581): “For the mathematics students or his or her instructor, whether in high school or tertiary coursework, coming to grips with this type of flexible abstract thinking is an educational challenge that continues to confront all those involved.”

This idea is important not only for high school or tertiary coursework, but also for the beginning of algebra.

Modelling

Example: Flexibility in interpreting symbols

Modelling

Example: Flexibility in using symbols in relation to modelling

(Bednarz, N; p. 75): There are 3 rackets more than balls and 4 times more hockey sticks than rackets. If there are 255 articles in the warehouse, how many balls, rackets and hockeys?

B + (B+3) + [(B x 4) + 12] = total amount of

articles

Taking account of the influenceTaking account of the influence of of students’ previous or simultaneous students’ previous or simultaneous experience in mathematics and/or experience in mathematics and/or in other subjects (science etc.)in other subjects (science etc.)

Conflict with previous conventions used in arithmetic

Example

Conflict with previous use of letters e.g. in geometry - letters seen as labels or unknowns

Example (Novotná, J. & Kubínová, M.; p.497)

A packing case full of ceramic vases was delivered to a shop. In the case there were 8 boxes, each of the boxes contained 6 smaller boxes with 5 presentation packs in each of the smaller boxes, each presentation pack contained 4 parcels and in each parcel there were v vases. How many vases were there altogether in the packing case?

v = 8 x 6 x 5 x 4 (In most school mathematics situations, letters are only used as labels for something that is to be found by calculations. The amount v is taken as an unknown.)

Possible conflict with previous or simultaneous use of letters in other fields

It seems necessary to have more research in this domain