A classical dot pattern-based approach to
algebraic expressions in a multicultural classroom
Marc Sauerwein1 and Malte Mink2
Institutional Framework
Selected Educational Publications on ...
... Structure of Algebraic Expressions
International Preparation Class (IVK)
• In February 2016 the independent grammar school Otto-Kühne Schule in Bonn, Germany, established
an International Preparation Class (IVK, Internationale Vorbereitungsklasse) for around 20 foreign
pupils from ages 15 to 18. These pupils came from different conflict and war zones all over the world,
with little to non-existent German language skills, within the last 9 to 15 months. Many of these
children already joined the school in the second half of 2015, and the IVK can be seen as an attempt
to structure their education. The goal of this class is to prepare and help them to participate in the
regular classes (9th or 10th grade) at school within two years, where they can eventually graduate.
There are many publications on structure of algebraic expressions. However, different authors emphasises
various aspects of structure. In the following we briefly describe important publications in this area:
• On the one hand, this help consists of a systematic learning of the German language as foreign language adequate to their skill level (12 hours per week) and on the other hand of other subjects such as
mathematics (3-5 hours per week), English (3 hours per week), politics (2 hours per week) and arts &
PE (4-6 hours per week in regular German classes). Moreover, they participate in exercise classes in
the afternoon where they are supported by older, German students in different subjects.
• Malle [5] defines the recognition of a structure of an algebraic expression differently, namely as the
identification of subexpressions, e.g.
• Hoch & Dreyfuss [3] assign to each algebraic expression a shape which reveals (possibly after a transformation) its internal order (normal form). They refine structure sense (proposed by Linchevski &
Livneh) as the fluent handling of familiar structures. In [4], they suggest a teaching unit in which the
pupils learn to identify five classical structures in highschool algebra.
4 · x + 3 = 11,
• Due to their individual background with respect to their home country, their educational background,
their age and their journey to Germany, the group is very heterogeneous in many different dimensions.
4 · x + 3 = 11.
• Rüede [7]: Contrasting the above mentioned point of views in which structure is an objective property
of the given algebraic expression, Rüede also takes the individual person into account. By structuring,
he refers to the relations in an algebraic expression constructed (resp. perceived) by an individual person. Although not explicitly mentioned, Rüede takes up again Freudenthal’s perspective "mathematics
as a human activity" [2].
• The mathematics class is held by the second author and is usually split into three or more groups where
the topics can cover the curriculum from multiplication tables and negative numbers up to quadratic
equations and functions.
• All the instruction in the mathematics classes takes place in German and only during the introduction
of new terms the pupils are allowed to translate the technical vocabulary into their native language.
This article focuses on one group of around ten pupils in their first lessons about algebraic expressions
and their manipulation. The above mentioned restrictions and conditions led to the necessity to take
a more language-sensitive approach.
4 · x + 3 = 11,
... Variables
• Malle [5] distinguishes tree aspects of a variable: object aspect (Gegenstandsapsekt), substitution
aspect (Einsetzungsaspekt) and the manipulation aspect (Kalkülaspekt).
• A recent survey work of Arcavi, Drijvers and Stacey [1] refines these aspects and shows that it is
widely agreed within the international mathematics education community that one can distinguish five
different aspects of the concept of variables as: "a placeholder for a number, an unknown number, a
varying quantity, a generalized number, and a parameter" [1, p.12].
Goals
• Develop a language-sensible approach for algebraic expressions and their manipulation:
All these different aspects combined constitute the concept of a variable in the end.
– few lingual prerequisites
• Prediger & Krägeloh [6, p.91] point out that "many German textbooks try to support the construction
of the second meaning, the variable as generalizer, by referring to typical linguistic expressions that are
used outside algebra classrooms, ... Assuming these expressions are known by the students, the authors’
intention is to remind students of out-of-school-language resources to help their individual construction
of meaning. However, ...these linguistic resources cannot be taken for granted for all students since
they are part of the language of schooling, not necessarily of students’ everyday register."
– can serve as a catalyst
• Foster the activity of structuring on a beginner level ; structuring of a algebraic expressions
• Broaden the students’ notion of a variable
Conclusions
Bits & Pieces from Practice
• The start with the dot patterns was highly motivating for the students (based on the subjective estimation
of the teacher).
The exercise sheet
The exercise sheet consisted of 10 different patterns, from which the first one was briefly explained at the
blackboard. The patterns are either well-known, taken from [8] or created by the authors.
• Although the students experienced language difficulties in the beginning, they started to argue with the
iconic dot patterns. The iconic representations helped to develop a common language.
• Judging by the passionate discussions, the resulting algebraic expressions were meaningful to them.
Second Implementation (December 2016 - January 2017)
In general, the exercise sheet was too long for the given 4 hours of class and in particular, pattern 7 and
10 were too difficult. The exercise sheet will be revised. Moreover, there will be exercises in which patterns
have to be drawn related to given terms or number sequences.
?
References
[1] A. Arcavi, P. Drijvers, K. Stacey, The Learning and Teaching of Algebra: Ideas, Insights, and
Activities. London, UK; New York, USA: Routledge, 2016.
[2] H. Freudenthal, Mathematik als pädagogische Aufgabe. Band 1, Klett Studienbücher, 1977.
[3] M. Hoch, T. Dreyfus, Structure sense in high school algebra: the effect of brackets. In M.J. Høines,
A.B. Fuglestad (Eds.), Proc. 28th Conf. of the Int. Group for the Psychology of Mathematics Education,
Vol. 3, 2004, pp. 49-56.
[4] M. Hoch, T. Dreyfus, Nicht nur umformen, auch Strukturen erkennen und identifizieren - Ansätze
zur Entwicklung eines algebraischen Struktursinns. Praxis der Mathematik in der Schule, 33-52, 2010, pp.
25-29.
[5] G. Malle, E.Ch. Wittmann, H. Bürger, Didaktische Probleme der elementaren Algebra. Springer,
1993.
Filling Glass
In the following you can see different (algebraic) terms for the pattern "Filling Glass" with a verbal and
iconic representation. These ideas were collected in the plenum with all stuents in the class.
Term
4n + 2 or n · 4 + 2
Verbal
The rectangle is counted (line by line or column by column) and then you add 2.
4(n + 1) − 2
By adding two point, a bigger rectangle is
counted, then you subtract these 2 points.
n + (n + 1) + (n + 1) + n
You count each single column.
2(n + (n + 1))
You can cut the pattern vertically, the left
and right side are the same.
Iconic
[6] S. Prediger, N. Krägeloh. "x-arbitrary means any number, but you do not know which one" - The
epistemic role of languages while constructing meaning for the variable as generalizers. In: Halai, A. &
Clarkson, P. (Eds.), Teaching and Learning Mathematics in Multilingual Classrooms: Issues for Policy,
Practice and Teacher Education. Rotterdam, the Netherlands: Sense, 2005, pp. 89-108.
[7] C. Rüede, Strukturieren eines algebraischen Ausdrucks als Herstellen von Bezügen. Journal for
Mathematik-Didaktik, 33-1, 2012, pp. 113-141.
[8] H. Wellstein. Abzählen von Gitterpunkten als Zugang zu Termen [Counting points in lattices as an
approach to algebraic expressions]. Didaktik der Mathematik, 6(1), 1978, pp. 54-64.
1
University of Bonn, Germany; [email protected]
2
Otto-Kühne-Schule, Bonn, Germany; [email protected]