TY - GEN
T1 - ABC - Actors at the Scene of Mathematics
T2 - An investigation of how students understand mathematical symbols and formulas in upper secondary school
AU - Schou, Marit Hvalsøe
PY - 2018/10/31
Y1 - 2018/10/31
N2 - Mathematics is characterised by the use of the symbolic language.
Symbols are used for developing, storing and communicating
mathematical knowledge (Steinbring, 2006). Formulas are built from
symbols, and they express features of objects that can be abstract or
concrete; for instance, there are formulas for determining the area
under a curve and for determining the volume of a solid.In upper secondary school formulas occur in almost every topic, and
they are a useful tool for generalising mathematical results, making
them applicable in various contexts. This quality of formulas is part of
the problem: “[…] that multiple meanings are comprised within the
same expression or can be derived by transforming it – is what
simultaneously blurs the sense of it (Arzarello, Bazzini, & Chiappini,
2002).The objective of this dissertation is to investigate how students in
upper secondary school understand symbols and formulas thereby also
shedding light on the reasons that hinder this understanding. In the
investigation, a semiotic approach is applied.The study begins with an examination of the instructional practice
around the transition from lower to upper secondary school. Pivoting
on the use of symbols, the instruction students has experienced in lower
secondary school and how they are met in upper secondary are
characterised. The result shows a noticeable gap from a context- to a
concept-based instruction. On this background, the students’
encounter with the mathematical symbolic language can be
characterised. The findings of the dissertation rest on three consecutive
classifications. The first is an identification of the roles symbols can
play in mathematics in general and in school-mathematics in
particular. The second is an operative characterisation of mathematic
understanding. ‘Understanding’ takes on many meanings, but in the
study of how students understand symbols and formulas there is a need
to put forward a characterisation which makes it possible to compare
and contrast various types of understanding. The third and final classification is the major finding of the dissertation. From a teaching
experiment, eight different kinds of ‘understanding formulas’ are
advanced. In order to be a “proficient formula user” students must
possess all of them and be able to combine and shift between them. By
employing the various understandings to describe students’ handling
of formulas, it becomes apparent if, how and not the least why students
are not always capable of dealing with formulas in developing
mathematical knowledge, including solving tasks. It is noticeable how
a single understanding, at certain times, can dominate other
understandings and preventing them from using their full potential.
A main component in the teaching experiment is the use of concrete
materials. The findings of the dissertation suggest that concrete
materials can scaffold students’ work with formulas, especially setting
them up. In some cases, the use of concrete materials leads to a
disadvantageous dominance of a particular understanding, resulting in
the concrete materials acting as a hindrance for learning.
AB - Mathematics is characterised by the use of the symbolic language.
Symbols are used for developing, storing and communicating
mathematical knowledge (Steinbring, 2006). Formulas are built from
symbols, and they express features of objects that can be abstract or
concrete; for instance, there are formulas for determining the area
under a curve and for determining the volume of a solid.In upper secondary school formulas occur in almost every topic, and
they are a useful tool for generalising mathematical results, making
them applicable in various contexts. This quality of formulas is part of
the problem: “[…] that multiple meanings are comprised within the
same expression or can be derived by transforming it – is what
simultaneously blurs the sense of it (Arzarello, Bazzini, & Chiappini,
2002).The objective of this dissertation is to investigate how students in
upper secondary school understand symbols and formulas thereby also
shedding light on the reasons that hinder this understanding. In the
investigation, a semiotic approach is applied.The study begins with an examination of the instructional practice
around the transition from lower to upper secondary school. Pivoting
on the use of symbols, the instruction students has experienced in lower
secondary school and how they are met in upper secondary are
characterised. The result shows a noticeable gap from a context- to a
concept-based instruction. On this background, the students’
encounter with the mathematical symbolic language can be
characterised. The findings of the dissertation rest on three consecutive
classifications. The first is an identification of the roles symbols can
play in mathematics in general and in school-mathematics in
particular. The second is an operative characterisation of mathematic
understanding. ‘Understanding’ takes on many meanings, but in the
study of how students understand symbols and formulas there is a need
to put forward a characterisation which makes it possible to compare
and contrast various types of understanding. The third and final classification is the major finding of the dissertation. From a teaching
experiment, eight different kinds of ‘understanding formulas’ are
advanced. In order to be a “proficient formula user” students must
possess all of them and be able to combine and shift between them. By
employing the various understandings to describe students’ handling
of formulas, it becomes apparent if, how and not the least why students
are not always capable of dealing with formulas in developing
mathematical knowledge, including solving tasks. It is noticeable how
a single understanding, at certain times, can dominate other
understandings and preventing them from using their full potential.
A main component in the teaching experiment is the use of concrete
materials. The findings of the dissertation suggest that concrete
materials can scaffold students’ work with formulas, especially setting
them up. In some cases, the use of concrete materials leads to a
disadvantageous dominance of a particular understanding, resulting in
the concrete materials acting as a hindrance for learning.
U2 - 10.21996/jw6q-s205
DO - 10.21996/jw6q-s205
M3 - Ph.D. thesis
PB - Syddansk Universitet. Det Naturvidenskabelige Fakultet
ER -