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Creating examples as a way to examine mathematical concepts’
definitions
Rachel Hess Green and Shai Olsher
University of Haifa, rachely.hg@gmail.com
Using examples supports the construction of concept images and concept definitions
of mathematical concepts. Examples can also serve to examine relationships and
connections between different mathematical concepts that learners sometimes make
during the learning process of different mathematical concepts. This paper focuses on
the interactions between teachers and definitions of mathematical concepts, as they
manifested when working with online mathematical tasks that require learners to
create examples that meet a specified set of conditions. These interactions required the
teachers to re-examine the definitions of mathematical concepts and their boundaries.
We will describe how the automatic formative assessment platform enables, and
sometimes forces, reiteration and fine tuning of mathematical concepts' definitions,
and examine possible impact on the learning process of both teachers and students.
Keywords: definitions, concepts, automatic formative assessment.
BACKGROUND
"I can’t understand anything in general unless I’m carrying along in my mind a specific
example and watching it go" (Feynman, 1985, p. 244). In mathematics, examples are
an essential part of many theories of learning processes. The connection between
examples and concepts was described in Vinner (1983) that conceptualized concept
image as a mental image that is connected to the concepts in the mind, determined by
the examples that are connected to the concept. Examples are also used to illustrate and
communicate concepts between teachers and learners, and offer some insight about
mathematical concepts and relations between concepts. A key feature of examples is
that they are chosen from a range of possibilities (Watson & Mason 2005, p. 238) and
it is vital that learners appreciate that range. Various mathematicians have written about
the importance of examples in appreciating and understanding mathematical ideas and
in solving mathematical problems (e.g. Pólya, Hilbert, Halmos, Davis, & Feynman).
Whenever a mathematician encounters a statement that is not immediately obvious, he
thinks of a particular example. When a conjecture arises, one practice is to seek a
counter example or to use an example perceived as generic to see how the conjecture
can be proven. The relationship between definitions and examples in mathematics is
complex. Sometimes an example is given when a concept is used, while on other
occasions a formal definition is required and afterwards the example is given as an
illustration of the definition. Research indicates that students’ comprehension of the
concept is made up of a collection of examples, which form a concept image. Vinner
(1983) indicates the connections between understanding the concept image and the
concept definition. It seems that having a wide variety of examples for a concept allows
for a better understanding of its definition. With the development of technology,
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students have the opportunity to create multiple examples with the aid of dynamic
geometry environments and other types of technological tools. The development of
technology presents new opportunities to exemplify using different means. With
technology we can exemplify quicker and easier and can access the exemplification
automatically (Sweller, 2013). Submissions of various sample sizes to geometrical
questions with multiple solutions were analysed to study the loci constructed by their
solutions (Leung & Lee, 2013). Other studies suggest analyzing mathematical and
didactic characteristics in the presentation of examples by students (Olsher,
Yerushalmy, & Chazan, 2016), suggesting this automatics predefined analysis could
assist teachers in performing real-time decisions in the classroom based upon student
data, thus performing formative assessment (Black & Wiliam, 1998). When we assess
an example, we want to assess if the example fits the conditions of the task. In most
cases, we can assess other features that the student was not explicitly asked to meet.
For example, in geometry tasks, we sometimes want to relate the orientation of the
shapes, or we want to recognize some extreme cases of submissions. With the
development of technology, we can assess this feature automatically (Olsher et al.,
2016). The research questions in this paper are: (1) How does the use of a system for
automatic formative assessment enable and encourage discourse on definitions of
mathematical concepts? And (2) How do tasks invite a discussion about the definitions
of concepts?
METHODOLOGY
Research setting
The setting for this research was a professional development program (PD) for in-
service teachers aimed at instructing and supporting the implementation of the STEP
platform (Olsher et al., 2016) in classrooms. The PD included four face to face
meetings (total of 30 hours), that included theoretical representation of formative
assessment in mathematics, and use of the platform as students and teachers. The
participants were also expected to use the platform in their classrooms between the
meetings. Each classroom enactment was documented with a questionnaire. In
addition, following the first enactments, each teacher had a discussion about their
implementation in the classroom with the PD instructor (first author).
Participants
The participants in the PD were 22 teachers, teaching mathematics in Israeli secondary
and high-schools. The teachers teaching experience ranges between 2 and 25 years.
The teachers are all certified teachers; some hold MA degrees in Mathematics
education, while others studied for BSc. or BEng. Degree in computer engineering or
electrical engineering, worked in that profession for several years, and then participated
in programs for career retraining aimed at enhancing the number and abilities of Israeli
Mathematics teachers.
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Data sources and analysis
The data sources include: (1) the students’ submissions of solutions. (2) Teachers
answers to the questionnaire, (3) Reflections, that were collected in discussions with
the PD instructor post-enactment, (4) Field notes made by the two researchers during
the PD meetings, and (5) Video recordings of the PD meetings. Analysis of the
enactment of the assessment tasks in the classroom focused on discussions about
mathematical definitions. First, we identified the episodes in which mathematical
definitions were addressed by the teacher. These episodes occurred either in the
classroom or during the PD sessions. Next, we categorized the episodes. Trouche
(2004) uses the term "instrumental orchestration" to describe didactic configurations
and the way that they are being exploited in the classroom, and also suggests them as
a construct that could "give birth to new instrument systems" (ibid, p. 304). In the
reflected lessons in this study, this framework is suitable to describe the way the teacher
works with the students' answers, and suggest "new instrument systems" that help to
the teacher and the student to think again about mathematical definitions of concepts.
The “new instrument systems” embodies the practice of mathematical definition and
concepts as they were enacted by the teachers. Our analysis process was iterative,
fitting each relevant episode into a specific category that have specific characteristics
of teachers practice with the definitions of mathematical concepts. The four different
categories that were identified will be described in the following section.
FINDINGS
We identified four different instrumental systems of dealing with mathematical
concepts’ definitions: (1) Addressing the definition of a mathematical concept during
the activity, (2) Resolving conflicts between definitions of a mathematical concept, (3)
Establishing an inclusion relation between mathematical concepts through their
definitions, and (4) Differentiation of characteristics into subcategories. In this section,
we will describe each of these new systems that we categorized, and give an example
of one of the episodes that were identified from this category.
Addressing the definition of a mathematical concept during the activity
Definitions of mathematical concepts lie at the heart of many tasks that require giving
an example that fits certain constraints. Yet, sometimes the definitions that are
available to learners are not appropriate for the task at hand. These cases, in which a
definition that was taught, or the general definition, is not sufficient to determine
whether the examples fits the definitions, requires the teacher to address the definition
of the mathematical concept during the activity. The instrumental system is that of the
teacher revisiting the definition of a concept, which is part of the task at hand. The
learners dealing with the task demonstrate uncertainty in terms of the definition, and
the teacher addresses this uncertainty in referring the students to or leading the student's
way towards the definition of the relevant mathematical concept. For example, the
definition of a tangent is that is parallel to the Y axis. This does not fit to the regular
definition of tangent, since the function has no derivative at the point in which the
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tangent is parallel to Y axis. As manifested during the classwork on the following task
(figure 1). One of the teachers, Sapir (pseudonym) gave this task (Figure 1) to her 12th
grade students as homework. While preparing the next lesson, Sapir encountered a
wide variety of answers submitted by her students (Figure 2), demonstrating various
instances of tangents, not necessarily aligned with the definition (e.g. Figure 2a where
tangent and asymptote might be mixed up by the student).
Figure 1. Task requiring student examples of functions and tangents meeting
certain conditions
In the following lesson, Sapir presented the student answers to the class, and initiated
a discussion with the students. One of the students asked, “How we can calculate the
tangent if we cannot find the derivative of the function in this point?” Other students
attempted to think how this could be performed and to find a concrete definition for
this case. Sapir asked the students to find an appropriate definition of a tangent that
would include this case, thus addressing the finer points of the definition, not
necessarily clear before the task addressed them. Later, during the PD, Sapir stated that
she also thought about it when she assigned the task to the students, but she could not
find a good definition and she performed an online search, to be better prepared for
such.
Figure 2. Example of submission of students
This example presents an episode in which the students completed the task, while not
emphasizing whether their constructed tangents fit the familiar definition of the
concept. The definition of the concept at hand, the tangent, was not fully clear to both
the teacher and the students. The collective example space of the students in the class
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as shown by the STEP platform, followed by the description in the following lesson
raised the question about the need for a better, clearer, definition.
Resolving conflicts between definitions of a mathematical concept
Mathematical concepts could sometimes be defined in different ways. Having different
definitions for one mathematical concept, might lead to inconsistencies when students
deal with the concept. These cases, in which there are multiple definitions for a concept,
require the teacher to negotiate between the different definitions in order to clarify the
relevant characteristics of the concept that they wish to emphasize. The instrumental
system is that of the teacher choosing between the different definitions in order to elicit
and treat the inconsistencies that might rise. The learners dealing with the task follow
a certain definition, while not attending to characteristics that derive from a different
definition. For example, the definition of an extremum point of a function. As
manifested in Anna’s classwork on the following task (Figure 3). The task requires
students to draw a graph of a function, which has one line that is tangent to the function
at two different points. Anna, the teacher, focused on examples in which the tangent
was a horizontal line, but the tangency point was not an extremum in her opinion. In
the PD session following the enactment, we discussed this mathematical concept. Anna
said that the red dot (which was asked for in the question as a tangent point (Figure 3)
is not an extremum. The PD instructor asked her to specify the reason for her opinion,
and Anna answered that the derivative after the point is zero (for ), so for
this point, the function does not have an extremum.
Figure 3: example of concept of extremum point
This led Anna to explore the definition of an extreme point. According to the definition
in one mathematics book local extremum exists if there is an environment of so that
all . On the other hand, in the book that Anna taught from
there was an examination of extremum points according to the variance of the
derivative (if the previous derivative rises and then decreases). A closer examination
of the mathematical definition of extremum points in the textbooks revealed an
interesting phenomenon: Different definitions appeared in different books. This could
change the decision that a horizontal line does have extrema points. The first setting
allows a maximum for a fixed function, while the second setting does not allow this.
This example presents an episode in which the students completed a task, but the
constructed mathematical object did not fit with the definition that the teacher was
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familiar with. Furthermore, the constructed object did fit an alternative definition that
appeared in another mathematical textbook. The use of an automatic analysis of student
answers could easily make these conflicting definitions episodes more accessible, as
the technological platform requires an algorithm to automatically categorize the
different answers: and this algorithm would comply with the chosen definition, when
they are conflicting.
Establishing an inclusion relation between mathematical concepts
Inclusion relations between mathematical concepts appears in various mathematical
strands, such as numbers, functions and geometry. These relations lay out the hierarchy
between different mathematical concepts: which concept is a specific, special case of
a more general concept. Unfolding these hierarchies requires the teacher to differentiate
between definitions, clarifying what are the relevant characteristics that induced the
inclusion relation between the concepts. The instrumental system is that of the teacher
choosing tasks that reveal incorrect inclusion relations assumed by the students, and
then sorting out the different concepts and the relations between them. For example,
regarding inclusion relations of special types of quadrilaterals, the teacher Pnina,
assigned the following task to Primary school students. The teacher did not introduce
the task to the students, rather she let the students discuss and ask each other questions
regarding the task.
Figure 4: a task in inclusion relations between types of quadrilaterals
This task is part of a longer activity regarding inclusion relations between
quadrilaterals. The students had to choose as many statements as they could, and drag
the quadrilaterals points so that it matched the chosen statements. Some students began
to ask questions about the relations between the statements in the task and definitions
of types of quadrilaterals. For example, one student asked “if all the sides are equal
than it is square, right?” Pnina did not answer the question and waited until the students
submitted the task. Afterwards she reviewed all the examples in the task and the
students had the opportunity to ask about definitions and the relation between the
definition and the statements that appeared in the task. Some students revealed their
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thinking about some of the mathematical objects, such as “a square is not a
parallelogram”, or “if all the angles are equal it must be a square”. The students
discussed each other's claims, and with subtle guidance of Pnina, concluded the
definitions of these mathematical concepts and the relations between them. This
example presents an episode in which the students completed a task, and in the
discussion after the task, they concentrated on definitions and on inclusion relations.
The use of an automatic analysis of students answers, could easily connect between the
statements in the task and the student's’ submissions, the discussion in the class raised
question and statement about inclusion relation between quadrilateral that reflect their
concept images about these concepts.
Differentiation of characteristics into subcategories
Differentiation of characteristics into subcategories is a process that splits a category
into subcategories. In these cases, a general definition or category is too wide and
contains more than one subcategory that emphasizes some important criteria. The
instrumental system is that of the teacher revisiting the category of a concept and
splitting the examples that fit into this category to sub-categories. For example, Alon’s
use of the task “Claim: There are functions that have one line tangent to their graph at
two different points, If you think this claim is true, provide three examples by sketching
a graph of a function and a tangent line at two points If not, explain” (Figure 1). Alon
assigned this task to his students as homework.
Figure 5: three different submissions that fit the automatic characteristics
This led to a discussion about the difference between the right and left submissions and
between two different types of infinite points of tangent to line. In figure, 5 A, there is
a symbolic function f(x)=sin(x). The tangent line represents the line “y=1”. This
tangent has infinite countable points at every point x=/2 +k, k Z. In the figure windows,
only four of them appear. In figure 5 C we can see a linear function in a symbolic
representation in which the function and the tangents converge. In this example, there
are non-countable, infinite tangents points. These three different categories are all
using the filter “more than 2 tangent points”, which is the general category. The
differentiation between them raises two questions about definitions. The first was about
the definition of tangents, in which one student claimed that in figure 5 C this is not a
tangent point because the lines (of the function and the tangent line) are convergent
and so they cannot be tangential.
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DISCUSSION
In the findings, we presented some examples of discourse about definitions of concepts
during the use of the formative assessment platform. The focus of the tasks on
examples, and specifically the opportunity to interact with extreme cases facilitates the
opportunity to address the definitions of mathematical concepts, thus enhancing the
concept image (Vinner, 1983). The linkage to automatic assessment of the examples
and the requirement to specifically program an algorithm for the identification of the
mathematical concepts in a precise fashion adds another point of interaction with the
definitions, especially when there are several conflicting definitions for a single
concept. These new instrument systems (Trouche, 2004) enable teachers and students
to communicate and interact with mathematical definitions of concepts. Finally, it is
not enough to address the student answers and define the algorithm for the automatic
assessment. The tasks should also be open ended, enabling different possible correct
and incorrect answers in order show different exemplifications, possibly challenging
the definition, as suggested by Olsher et al. (2016). The tasks included the automatic
assessments in them, i.e. there are characteristics that were used to classify the
submissions. This classification was the basis for discussions challenging or
elaborating about the definitions of the mathematical concepts.
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