12th International Congress on Mathematical Education
Program Name XX-YY-zz (pp. abcde-fghij)
8 July – 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not change this part.)
DESIGN AND ANALYSIS OF MATHEMATICAL TASKS USING
THE ONTO-SEMIOTIC APPROACH
Neto, Teresa B. and Xuhua Sun
Univ. of Aveiro, Portugal; Univ. of Macau, China
[email protected]; [email protected]
Mathematical tasks are central to students’ learning because “tasks convey messages about what
mathematics is and what doing mathematics entails” (NCTM, 1991, p. 24). Accordingly, task design
and analysis is an important basis for a mathematical teacher to understand content knowledge and
develop his/her pedagogical content knowledge. The systematic approach to design and analyze task
play critical roles in knowledge development of a teacher. In this paper we present part of a study
that investigates prospective teachers when solving a mathematical task using the onto-semiotic
approach (OSA). This is, a systemic tool developed by (Godino, Batanero and Font, 2007) to develop/
assess prospective teachers` content knowledge and pedagogical content knowledge.
Key words: task design, onto-semiotic approach, prospective teachers
INTRODUCTION
Mathematical tasks are central to students’ learning because “tasks convey messages about
what mathematics is and what doing mathematics entails” (NCTM, 1991, p. 24). Different
tasks may place different cognitive demands on students (Hiebert & Wearne, 1993). Different
tasks may structure different ways students think and can serve to limit or to broaden their
views of the subject matter and their actual experiences with mathematics (Schoenfeld,
1994). Accordingly, task design and analysis is an important basis for a mathematical teacher
to understand mathematical subjects and develop his/her pedagogical understanding of these.
Godino (2009) had previously made the analysis of the Shulman’s teacher knowledge model
and proposed an alternative model that includes more detailed categories of analysis for
mathematics and didactics teachers’ knowledge, based on the application of the onto-semiotic
approach to mathematical knowledge and instruction. He also described a guideline to
formulate questions to assess such knowledge based on the proposed model. These new
categories can be used by the teachers as tools to analyse and reflect on their practices.
In this paper we present part of a broad study with the main objective of providing new
knowledge, instructional units and methodological tools for improving the initial training of
elementary school teachers in mathematics, considering the European Space for Higher
Education frame. In this context, we shall focus mainly in one specific aspect of this study: to
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design, implement and assess didactical situations and study processes that promote the
professional development of prospective teachers.
Godino, Batanero and Font (2007) develop a systemic and integrative approach to research in
mathematical education, the onto-semiotic approach (OSA), which takes into account the
three basic dimensions of teaching and learning processes: epistemic dimension (concerning
the nature of subject knowledge), cognitive dimension (concerning subjective knowledge)
and instructional dimension (related to interaction patterns between the teacher and the
students in the classroom). The starting point for the onto-semiotic approach was an ontology
of mathematical objects that takes into account: mathematics as socially shared; as
problem-solving activity; as a symbolic language and as a logically organized conceptual
system. Taking the problem-situation as the primitive notion, they defined the theoretical
concepts of practice (personal and institutional) and of object and meaning, with the purpose
of making visible and operative, both the mentioned triple character of mathematics and the
personal and institutional genesis of mathematical knowledge, as well as their mutual
interdependence. In this study, we will focus on the epistemic and cognitive dimensions.
Task design and analysis involves the configurations of objects and mathematical processes.
These configurations can be epistemic (networks of institutional objects) or cognitive
(network of personal objects), and constitute the basic theoretical tools to describe
mathematical knowledge. These relationships (configurations) emerge through, both, their
personal and institutional facet, by means of mathematical processes, which we interpret as
sequences of practices. The emergence of linguistic objects, problems, definitions,
propositions, procedures and arguments take place throughout the respective primary
mathematical processes of communication, problem posing, definition, enunciation,
elaboration of procedures (algorithms, routines…) and argumentation. We must take into
account (in the sense of Godino et al.)contextual factors to which the meanings of
mathematical objects are relative and which attribute a functional nature to them:
personal/institutional - The personal cognition is the result of thought and action of the
individual subject confronted by a class of problems, whereas institutional cognition is the
result of dialogue, understanding and regulation within a group of individuals who make up a
community of practices; Ostensive / non-ostensive – The ostensive attribute refers to the
representation of a non-ostensive object, that is to say, of an object that cannot be shown to
another. The classification between ostensive and non-ostensive depends on the contexts of
use. Diagram, graphics and symbols are examples of objects with ostensive attributes,
perforated cubes and plane sections are examples of objects with non-ostensive attributes;
Expression / content (antecedent and consequent of any semiotic function) - The relationship
is established by means of semiotic functions, understood as a relationship between an
antecedent (expression, signifier) and a consequent (content, signified or meaning)
established by a subject (person or institution) according to a specific criterion or code of
correspondence; Extensive / intensive (specific / general) – This duality is used to explain one
of the basic characteristics of mathematical activity, namely generalization. This duality
allows for the center of attention to be the dialectics between the particular and the general,
which is undoubtedly a key issue in the construction and application of mathematical
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knowledge; Unitary / systemic – In certain circumstances, mathematical objects participate as
unitary entities, in others, they should be taken as the decomposition of others so that they can
be studied.
The primary processes are complementary and interrelated. The configuration of objects and
primary processes is the principal focal point in this contribution to ICME-12. The adoption
of an integrationist notion of objective and pragmatic meaning (content of semiotic functions)
articulates a coherent manner of anthropological concept (Wittgenstein, 1998), with realistic
positions (not platonic) of mathematics. The various means of expression (language) play the
dual role of instruments and performing mathematical work of other mathematical objects.
METHODOLOGY
Twenty prospective elementary school teachers participated in this study. They are students
in the 2nd cycle degree (master) of the Primary School Teacher Education Program. During a
class of Didactics of Mathematics in Basic Education, they presented solutions for a task that
consisted in identifying and justifying an isometry. This context was chosen due to the
familiarity of the related concepts to these students, prospective elementary school teachers,
because during the 1st cycle degree of the Primary School Teacher Education Program, they
had the following mathematical subjects: Mathematical Concepts I (1 year, 1 semester);
Mathematical Concepts II (1 year, 2 semester); Elements of Geometry (2 year, 2 semester);
Mathematics and Education (3 year, 1 semester); Didactics and Technology of Mathematics
(3 year, 2 semester). Therefore, they had knowledge of Geogebra and mathematical concepts
necessary for the development of the envisioned task.
The task
Observe the following figure
B
A
C
E
D
B''
A''
C''
D''
E''
Figure 1. Pentagon [ABCDE], pentagon [A´´B´´C´´D´´E´´] and a line l
The pentagon [A´´B´´C´´D´´E´´] is obtained from pentagon [ABCDE].
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Identify the isometry that transforms de pentagon [ABCDE] (pentagon object) in the
pentagon [A´´B´´C´´D´´E´´] (pentagon image). Justify your answer.
Predicted solution 1:
1) Translation of the [ABCDE] associated with BB´´ vector, obtaining pentagon
[A´B´C´,D´E´];
2) Reflection of the [A´B´C´,D´E´] across de line, obtaining [A´´B´´C´´D´´E´´]
Table 1: Epistemic configuration of solution 1
Type
Mathematics objects
Meanings in the task.
Language
Verbal (terms and expressions)
Translation; reflection; glide
reflection; vector; reflection
across a line
Iconic/ Graphics
A parallel displacement
Concepts / Properties
Previous:
Reflection; Translation
The translation is a direct
isometry; The reflection is a
opposite
isometry;
An
isometry preserve distance
Emerging: A glide along a line
A glide along a line l will
A glide along a line is an consist of a translation along l,
isometry
followed by a reflection
across l; The glide along a line
is an opposite isometry.
Procedures
Arguments
With the tools of GeoGebra, A glide along a line l
construction of the pentagon
[ABCDE] and acquisition of the
pentagon
image
([A´´B´´C´´D´´E´´])
A glide along a line l will consist Definition of a glide along a
of a translation along l followed line l
by a reflection across l
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Predicted solution 2:
Apply isometric properties that define the glide along a line as a composition of reflections distinct
from the definition of glide reflection. A double reflection φ is an isometry which can be written as
the composition of two distinct reflections: φ =  l  m with l≠m. φ is a rotation with center A if l
∩m ={A}. φ is a translation along n if n is a common perpendicular of l and m.
The mathematical objects involved in the development of this task are shown in table below.
Table 2: Epistemic configuration of solution 2
Type
Mathematics objects
Meanings in the task
Language
Verbal
Rotation, reflection, glide
reflection, vector; axis of
reflection; angle of a rotation.
Iconic/ Graphics
Apply a rotation around a
fixed point; apply a reflection
across an axis
Previous:
Reflection across an axis.
Reflection; Rotation
A rotation around a fixed
point known as the centre of
rotation and an angle
associated; reflection and
rotation, are examples of
isometry; the rotation is a
direct isometry;the reflection
is a opposite isometry; an
isometry preserve distance.
Concepts / Properties
Emerging:
Glide along a line l
Glide along a line l is an isometry
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A glide along a line l will
consist of the application of
the isometric properties.
The glide along a line is a
opposite isometry.
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Procedures
Trace a pentagon and
transformed this geometric
object using transform menu
(rotation and reflection).
Arguments
Isometrics properties
Isometric properties; function
composition.
The isometry that transforms the
pentagon [ABCDE] in the
pentagon [A''B''C''D''E''] is a
composition of rotation followed
of a reflection
Results
The solution 1 was thought to be the main considered by the students because the figure 1
provides a direct visual procedure for the task. Effectively, it was the one used by most
students, sixteen of the twenty students, given that two students didn’t accomplish the task,
used solution 1. On the other hand, solution 2 was predict as being the most unforeseen by the
students, because it requires more sophisticated visualization skills and deeper knowledge of
isometries and their properties in the Euclidian plane. This was proven by the fact that only
two students used it.
Cognitive configuration – Expected solution (solution 1)
Figure 2. Construction
accomplished in GeoGebra by
the student
After the construction using GeoGebra software, most students presented the solution 1. The
following is an example of the cognitive configuration students exhibited:
The isometry that transforms the pentagon [ABCDE] in the pentagon [A''B''C''D''E''] is a glide
along a line f. First, the pentagon [ABCDE] becomes the pentagon [A'B'C'D'E '] through
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reflection referent in line f, after this step, vector u must be defined, which will allow the
translation of the pentagon [A'B'C'D'E '] to the pentagon [A''B''C''D''E''].
Within the responses that correspond to this type of configuration we find the following
primary relationships:




Language: Graphic (as we can see in figure 2) and verbal (“glide reflection”; “vector”,
“allow the translation”);
Concepts: Glide reflection (Reflection across an axis f; Translation associated with a
vector);
Procedures: Construct, using GeoGebra, a pentagon and transform it using transform
menu (reflection and translation);
Arguments: “The isometry that transforms the pentagon [ABCDE] in the pentagon
[A''B''C''D''E''] is a glide reflection (…)”.
Cognitive configuration – Unexpected solution (solution 2)
Figure 3.
Construction
accomplished in
GeoGebra by the
student
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Rotation of 180º around point J - to draw the line g, we can draw line f that intersect a point
equidistant to the points D and D '. This form allows you to rotate the figure 180 degrees from
the point J, keeping the same distance equivalent points of f. Let’s go from figure [ABCDE]
to [A'B'C'D'E '].
Reflection - There was a point equidistant from B and B ‘and others A and A', linking these
points and scoring a point equidistant to them. Drawing a straight line passing the midpoint
(M and N) and reflected [A'B'C'D'E '], making the figure [A''B''C''D''E ''].
Within the responses to correspond to this type of configuration we find the following
primary relationships:




Language: Graphic (as we can see in figure 3) and verbal;
Proprieties: Isometric properties;
Procedures: Construct using GeoGebra, a pentagon and transform this using transform
menu (rotation and reflection);
Arguments: Describe the construction of the image obtained using isometric properties
The following figure (figure 4) illustrates both expected and unexpected solutions, the first
and de second diagram, respectively. Initially the student wrote:
“Transform polygon ABCDE in the polygon A’’B’’C’’D’’E’’ through isometry
1. Draw a polygon ABCDE;
2. Draw vertical axis f for the reflection of ABCDE;
3. Draw vector u with a vertical direction and from top to bottom, to make the polygon
translation.”
In second we can read:
“Resume the position ABCDE from A’’B’’C’’D’’E’’ by an isometry
4. Draw an axis of reflection g;
5. Perform the reflection of the polygon A’’B’’C’’D’’E’’;
6. Draw a rotation axis h joining the points BB'';
7. Define the midpoint (k) of the segment BB'' (h-axis);
8. Rotate A’’B’’C’’D’’E’’ around point (k) with the amplitude of 180º (clockwise).”
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Figure 4.Procedures adopted in cognitive configuration of another student
If our focal point is the contextual attributes present in the solutions, we can state that: the
ostensive objects brought forward in presenting the solution to the problem were the
representation of a pentagon, lines and vectors in the drawing, that satisfy the conditions of
the problem-situation (Ostensive – non-ostensive); the solutions put forward consisted in
writing that a glide along a line l is a translation along l, followed by a reflection across l or,
consisted in a composition of reflections across lines (Extensive – Intensive); visualization is
revealed to be a means of providing the solution to the problem. The initial picture has a line
that induces a reflection across it, followed by a translation, but, it’s relevant to note that some
students presented other solutions (Institutional – personal); the notions of translation,
rotation and refection are considered to be previously known. Their composition is
understood as a more complex object to be learnt (Unitary – systemic); the problem-situation
served as motivation (induces) for the study of the subject of isometry (Expression – content).
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CONCLUSION
In this paper we presented the design of a task and analysis from an onto-semiotic approach
(OSA). This theoretical perspective can be discussed as a tool to develop/assess prospective
teachers’ understanding of mathematical subjects and their didactic character; connected with
the primary and secondary objects (language, situations, concepts, propositions or properties,
procedures and arguments used to validate and explain the propositions and procedures) and
contextual attributes defined in the onto-semiotic focus of mathematical cognition.
The study suggests that a diversified geometric approach, through various epistemic and
cognitive configurations, promoted different ways for students to think and can serve to
increase their views of the matter and their experiences with mathematics (Schoenfeld, 1994),
promoting the development of skills in prospective elementary school teacher (Godino,
2009).
The present paper goes on the way to justify a larger study on the potential connections
between the onto-semiotic perspective and task design. In addition the study highlights the
importance of the study of geometric transformations in Euclidian plane model for the
development of visualization skills in prospective teachers.
References
Godino, J. D., Batanero, C. & Font, V. (2007). The onto-semiotic approach to research in
mathematics education. ZDM. The International Journal on Mathematics Education,
39 (1-2), 127-135.
Godino, J. D. (2009).Categorias de analisis de los conocimientos del professor de
matemáticas. UNION, Revista Iberoamericana de Educacion Matematica, 20,13-31.
Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students
‘learning in second grade arithmetic. American Educational Research Journal,30,
393-425.
Millmann, R. S. & Parker, G. D. (1991). Geometry- A Metric Approach with Models,
Springer-Verlag.
National Council of Teachers of Mathematics. (1991). Professional standards for teaching
mathematics. Reston, VA: Author.
Schoenfeld, A. H. (1994). Reflections on doing and teaching mathematics. In A. H.
Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 53–70). Hillsdale,
NJ: Erlbaum.
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