Teachers’ procedures when introducing algebraic
expression in two Norwegian grade 8 classrooms
Unni Wathne
To cite this version:
Unni Wathne. Teachers’ procedures when introducing algebraic expression in two Norwegian
grade 8 classrooms. Konrad Krainer; Naďa Vondrová. CERME 9 - Ninth Congress of the
European Society for Research in Mathematics Education, Feb 2015, Prague, Czech Republic. pp.508-509, Proceedings of the Ninth Congress of the European Society for Research in
Mathematics Education. <hal-01286985>
HAL Id: hal-01286985
https://hal.archives-ouvertes.fr/hal-01286985
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Teachers’ procedures when introducing algebraic
expression in two Norwegian grade 8 classrooms
Unni Wathne
University of Agder, Kristiansand, Norway, [email protected]
We investigate similarities and differences in two
teachers’ way of introducing algebraic expressions by
designed examples. One teacher moves from the specific
to the general, and the other moves from the general to
the specific. They both mediate the passage from the
students’ real world and the school mathematics they
know, to algebra.
Keywords: The introduction of algebra, designed
examples, mediating tools.
Two teachers, Kari and Ola, are introducing algebra
in two Norwegian grade eight classrooms. Kari holds
up a set of large playing cards and writes on the blackboard what is written in the corner of the cards. Ola
starts walking in one direction in the classroom, asking the students to describe what he is doing. These
are the starting points of two examples that each
teacher has designed as a tool for communicating and
explaining new algebraic ideas in her/his classroom.
Kari makes the students familiar with using letters
for numbers with the help of playing cards. Ola makes
the students familiar with algebraic expressions and
variables with the help of body movement.
The aim of this study is to investigate the introductory
lesson when two teachers mediate algebraic concepts
through their examples. The purpose of the analysis
is to capture how the teachers approach the complexity students meet in such learning situations. More
specifically: Which procedures do the teachers’ use
to introduce the concept of algebraic expressions?
THE THEORETICAL FRAMEWORK
The shift in school mathematics from arithmetic to
algebra is known to be challenging for students. The
learning of algebra includes new symbols, new concepts and also new ways of thinking. Examples play
CERME9 (2015) – TWG03
a central role in the teaching and learning of mathematics as described among others by Rowland (2008). The concepts of mediation and mediating tool
(Wertsch, 1991) have emerged from our empirical
material. The theoretical term of mediating tools facilitates our analysis in making a distinction between
the goals of the lessons (including the mathematical
objects of variables and algebraic expressions) and
the tools (designed examples, concretes and semiotic
items) that the teachers employ in their interaction
with the students. THE METHOD
In order to accomplish the aims of the study we use
a qualitative approach to collect and analyze the empirical data grounded in a sociocultural theoretical
perspective of learning. The data has been collected
after the VIDEOMAT design (Kilhamn & Röj-Lindberg,
2013); we observed the first five algebra lessons in each
classroom (videotaping), interviewed the teachers after the fifth lesson (audiotaping) and collected written
material used in the classrooms (teacher and student
material). As a first approach to the collected data,
lesson graphs for each lesson were elaborated, and
the first lesson in all classrooms was transcribed.
The two examples were chosen through a process involving the lesson graphs and several viewings of the
video material. They comprise the introductory part
of the first lesson in each classroom. The designed
examples stood out as unique in the international
video material. The examples are also referred to in
later lessons by the teachers and therefore play an
important role in the introduction of algebra in these
two classrooms.
508
Teachers’ procedures when introducing algebraic expression in two Norwegian grade 8 classrooms (Derek Pilous and David Janda)
RESEARCH RESULTS
Wertsch, J. V. (1991). Voices of the Mind: A Sociocultural
Approach to Mediated Action. Cambridge, MA: Harvard
There are similarities and differences in these teachers’ way of introducing algebra. The two teachers both
design introductory examples which are used as their
central means for explaining the same concepts (variable and algebraic expression). The examples are
easily distinguishable in their use of concrete materials, cards versus the body. However, there are more
fundamental differences in the examples’ structures.
University Press.
Kari starts with numbers, number operations, and
arithmetic expressions, and she makes generalizations introducing algebraic expressions. She continually connects the arithmetic and the algebraic
elements, and sees variables as numbers. Kari moves
from the specific to the general in her approach to
introduce algebraic expressions.
Ola, on the other hand, establishes an algebraic expression directly from the imaginary number line
with given direction and units (first step, then foot)
without using numbers. He builds the algebraic expression through a transformation chain following
this path: bodily movement – words – abbreviations
– variables, and sees variables as quantities, numbers
included. Ola moves from the general to the specific
in his approach to introduce algebraic expressions.
The aim of this study has not been to propose how
algebra should be introduced in the classroom. The
analysis illuminates the complexity students meet
when facing introductory algebra in school, and the
challenge it is for teachers to make algebra accessible for all students. The main procedure of the teachers in this study has been to use examples designed by
themselves, mediating the passage from the students’
real world and the school mathematics they know to
algebra. REFERENCES
Kilhamn, C., & Röj-Lindberg, A.-S. (2013). Seeking hidden dimensions of algebra teaching through video analysis. In
B. Grevholm, P.S. Hundeland, K. Juter, K. Kislenko & P.-E.
Persson (Eds.), Nordic research in mathematics education,
past, present and future (pp. 299–328). Oslo: Cappelen
Damm.
Rowland, T. (2008). The purpose, design and use of examples
in the teaching of elementary mathematics. Educational
Studies in Mathematics, 69(2), 149–163
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