Language aspects of algebra

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Language aspects of algebra

• Beginning algebra
• - first encounters with algebraic language
• Algebraic notations
• - variables, equations, functions
• Language aspects of algebra
• - in classroom practice and in theories of
  learning
• Language aspects of algebra
• - from a semiotic/linguistic perspective

• Jean-Philippe Drouhard, Desmond Fearnley-Sander
• Bernadette Barker, Nadine Bednarz, Dave Hewitt,
  Brenda Menzel,
• Jarmila Novotná, Mabel Panizza, Cyril Quinlan,
  Anne Teppo, Maria Trigueros

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Beginning algebra first encounters with
algebraic language
Language aspects of algebra
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• Distinguishing an algebraic activity versus the
  use of symbols
• Symbolisation as a process and not seeing algebra
  as the formal use of symbols
• Taking account of the influence of students
  previous or simultaneous experience in
  mathematics and/or in other subjects (science
  etc.)

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Distinguishing an algebraic activity versus the
use of symbols
Examples of the use of symbols
in a non-algebraic activity
Geometry labeling objects and using it
e.g. in describing or proving geometrical
properties

• Symbols with the role of labels
  ?ABC

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Examples of the use of symbols
in a non-algebraic activity
Geometry labeling objects and using it
e.g. in describing or proving geometrical
properties

• Manipulations with symbols but non-algebraic
  activity

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Examples of the use of symbols
in a non-algebraic activity
Geometry labeling objects and using it

• Algebraic activity

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Examples of the use of symbols
in a non-algebraic activity
Solving of word problems The solution is purely
arithmetical but students use letters to record
the information described in the assignment.
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Examples of algebraic activities
where formal symbols are not used
Problem solving in natural language (history)
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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)
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Symbolisation as a process and not seeing algebra
as the formal use of symbols

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

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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,

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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)

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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.

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Modelling
Example Flexibility in interpreting symbols
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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?
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Taking account of the influence of students
previous or simultaneous experience in
mathematics and/or in other subjects (science
etc.)

• Conflict with previous conventions used in
  arithmetic

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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.)
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Possible conflict with previous or simultaneous
use of letters in other fields
It seems necessary to have more research in this
domain