Threshold Concepts Vs. Tricky Topics
Exploring the Causes of Student´s Misunderstandings with the Problem Distiller
Tool
Sara Cruz
1
, José Alberto Lencastre
1
, Clara Coutinho
1
, Gill Clough
2
and Anne Adams
2
1
Institute of Education, University of Minho, Campus de Gualtar, 4710-057, Braga, Portugal
2
Institute of Educational Technology, Open University, Walton Hall, MK7 6AA, Milton Keynes, U.K.
Keywords: Threshold Concepts, Trick
y Topics, Technology-enhanced Learning, Deeper Understanding.
Abstract: This paper presents a study developed within the international project JuxtaLearn. This project aims to
improve student understanding of threshold concepts by promoting student curiosity and creativity through
video creation. The math concept of 'Division', widely referred in the literature as problematic for students,
was recognised as a 'Tricky Topic' by teachers with the support of the Tricky Topic Tool and the Problem
Distiller tool, two apps developed under the JuxtaLearn project. The methodology was based on qualitative
data collected through Think Aloud protocol from a group of teachers of a public Elementary school as they
used these tools. Results show that the Problem Distiller tool fostered the teachers to reflect more deeply on
the causes of the students’ misunderstandings of that complex math concept. This process enabled them to
develop appropriate strategies to help the students overcome these misunderstandings. The results also
suggest that the stumbling blocks associated to the Tricky Topic ‘Division’ are similar to the difficulties
reported in the literature describing Threshold Concepts. This conclusion is the key issue discussed in this
paper and a contribution to the state of the art.
1 INTRODUCTION
This paper presents a study conducted in the scope
of the JuxtaLearn project. This European project
focuses on helping students understanding ‘threshold
concepts’ in science and technology with the help of
technological tools and collaborative high-level
reflections. The idea of threshold concept came from
a national survey conducted in the United Kingdom
by Meyer and Land in 2003, and since then it has
been a buzz in the Academia (Cousin, 2006).
According to these authors a threshold concept is a
complex concept of high level that the student has
difficulty in understanding and overcoming,
sometimes taking refuge in memorisation without
understanding. Because of this insuperable barrier to
comprehension, the student cannot progress (Meyer
and Land, 2006), and often gives up studying.
Understanding the causes of the students’ barriers
helps the teacher to adopt appropriate teaching
strategies to support the student in overcoming these
barriers to understanding the threshold concept.
This study presents the complex concept
'Division' through the perspective of two Math
teachers and compares that with the related
Academic literature. To support teachers identifying
the barriers to the concept of 'Division' we used a
tool designed and developed in the JuxtaLearn
project entitled 'Problem Distiller'. The Problem
Distiller displays a set of tabbed panes ‘prompting
teachers to reflect on and select possible reasons
why their students might be having a particular
problem, connecting all the information entered to
the appropriate tricky topic and stumbling block or
blocks’ (Adams and Clough, 2015, p. 6). In the
JuxtaLearn project, ‘Tricky Topic’ was the name
suggested by teachers to refer to the threshold
concepts identified by their students (Adams and
Clough, 2015). Teachers said that this term relates
better to their practice, and ‘threshold concept was a
formalised academic term that was a threshold
concept in itself’ (Adams and Clough, 2015, p. 41).
According to these two authors, the tricky topics
identified by teachers in their practice may not
always correspond to the threshold concepts already
documented in the literature. This statement is a key
issue we explore in this paper.
In section 2, we present a framing for threshold
Cruz, S., Lencastre, J., Coutinho, C., Clough, G. and Adams, A.
Threshold Concepts Vs. Tricky Topics - Exploring the Causes of Student´s Misunderstandings with the Problem Distiller Tool.
In Proceedings of the 8th International Conference on Computer Supported Education (CSEDU 2016) - Volume 1, pages 205-215
ISBN: 978-989-758-179-3
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
205
concepts and we introduce the term 'tricky topic'. In
Section 3 we present the methods of the data
collection process. The Section 4, we present the
content analysis of the interviews with the two Math
teachers, the curricular concept of Division and the
Problem Distiller tool. In Section 5, we present our
main results and reflections. We conclude in Section
6 with a synthesis and proposals for future work.
2 BACKGROUND
2.1 What is a “Threshold Concept”
Meyer and Land (2003) introduced the notion of
‘threshold concept’ as learning barriers inhibiting
the students’ deeper understanding of a concept.
They are said to be more than just ‘key’ or ‘core’
concepts (Harlow et al., 2011; Lucas and
Mladenovic, 2007). A threshold concept is able to
create in students a state of uncertainty, anxiety,
confusion, doubt, or even a sense of surprise (Meyer
and Land 2006). The barriers presented by threshold
concept can be so great, they may cause students to
fail or give up a subject altogether (Machiocha, 2014).
According to Meyer and Land (2003), a concept
is likely to be threshold if it has one or more of the
following criteria:
Transformative – once understood, it potentially
causes a significant shift in the perception of a
subject (or part thereof); sometimes it may even
transform one’s personal identity
Irreversible – it is unlikely that a Threshold
Concept is forgotten or unlearned once
acquired due to transformation
Integrative – a Threshold Concept is able to
expose “the previously hidden interrelatedness
of something”
Bounded – a Threshold Concept can have
borders with other Threshold Concept which
help to define disciplinary areas
Troublesome – they may be counterintuitive
(common sense understanding vs. expert
understanding)
Nevertheless, the authors emphasize that once
understood and overcomed, the ‘threshold concept
opens up a new understanding of the concept (Meyer
and Land, 2003), and allows the student to be able to
solve problems with degree of advanced difficulty
(Meyer, Knight, Callaghan & Baldock, 2015).
Loertscher, Green, Lewis, Lin and Minderhout
(2014) conducted a study involving 75 teachers and
50 students, where involved an iterative process
intended to identify threshold concepts in
biochemistry. These authors used a process to
identify threshold concepts that consists of five
phases. Using this process, they were able to identify
threshold concepts that are fundamental to the
deeper understanding of the biochemistry but are
also strongly related to fundamental concepts of the
discipline of chemistry and biology discipline.
Meyer, Knight Callaghan and Baldock (2015)
conducted a case study which used a data
triangulation approach to identify threshold concepts
that students should understand before solving
specific problems of a civil engineering course. For
collection purposes teachers took part in dialogue on
understanding and conceptual capacity enabling
learning for all participants in the process. They
concluded that involving the various course
stakeholders in an analysis about conceptual
understanding and capacity makes learning
achievable to all process participants. It also
provides a basis for pedagogies and evaluations to
facilitate advanced results in students. Also
Barradell and Kennedy-Jones (2013) introduced a
conceptual model that integrates three components:
the students learning, the threshold concepts and
curriculum. According to this holistic model, when
talking about the threshold concepts can meet
various ideas and these ideas when understood as
part of a whole provide a more systematic way of
thinking about how to improve educational practice.
2.2 What is a “Tricky Topic”
The JuxtaLearn project created an interactive online
tool called ‘Tricky Topic Tool' (TTT) to help the
teacher identifying a tricky topic (Figure 1).
Figure 1: Tricky Topic Tool.
Co-developed with teachers, and included in the
CLIPIT - the Web Space for the JuxtaLearn project -
the TTT is an online database with a catalogue of
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206
tricky topics created by teachers from their
perspective and based on their practice. If a tricky
topic does not already exist in the TTT, identified by
other teachers, the teacher can add one that fits their
students’ learning problems.
Once the teacher recognises (or adds) a main
tricky topic, (s)he can link into some ‘stumbling
blocks’ commonly found by the students with
another feature of the TTT: the ‘Problem Distiller’.
The Problem Distiller has a key role in the
JuxtaLearn process (Figure 2).
Figure 2: Problem Distiller Tool.
According to Clough et al. (2015), Problem
Distiller helps the teacher ‘to focus on not just what
the students have problems understanding, but on
why they are having these problems’. Student
problems and their associated stumbling blocks will
be used to give to the teacher guiding cues to create
quizzes that address these specific problems. After
several trials done in the United Kingdom and
Portugal, the CLIPIT has a database of tricky topics,
and their related stumbling blocks, examples of
student problems, quizzes, and teaching materials.
The teacher creates the quizzes in CLIPIT (or
reuse one of the quizzes made previously by another
teacher) to assess whether his students have these
difficulties. As the teacher creates the quiz, they link
each question to one or more related stumbling
blocks, selecting the question type (multiple choices,
checkboxes, true/false or numeric), the possible
options, and the correct answer.
When the students take the quizzes, their results
are presented as a visualisation that shows where the
gaps in their understanding exist. These results
highlight the problem areas and support the teacher
in the design of a proper classroom intervention.
2.3 The Tricky Topic “Division”
According to some authors, many children have
problems on division (Correa, Nunes, & Bryant,
1998; Kornilaki and Nunes, 2005; Nunes et al.,
2015; Fernandes and Martins, 2014), and there is
consensus on the fact that children’s understanding
of division depend on experiences with sharing
(Squire and Bryant, 2002a, 2002b). A global
understanding, in terms of procedures and in
conceptual terms, is essential for the success of the
teaching process and learning of the division
operation. When we add a procedural understanding
to a conceptual understanding, students will be able
to understand the division and use it in their day-to-
day life with ease (Fernandes and Martins, 2014).
The division becomes even more complicated when
the dividend is not evenly divided by the divisor
(Montague, 2003). The multiplication operation and
the division operation are first presented to students
from pre-school. From the 3.º grade to 5.º grade,
students will develop the meaning of multiplication
and division of whole numbers (NCTM, 2008). A
constructivist approach to teaching division uses
problematic situations to develop in students a
conceptual understanding of the process of division
(Montague, 2003). Zhao et al., (2014) in their
research on the differences in the field of the four
basic arithmetic operations (addition, subtraction,
multiplication and division) between Flemish and
Chinese children between 8 and 11 years old, show
that the Chinese students outweigh the Flemish
students in each year analyzed. However, this
difference diminishes as the grade increases. Their
results also indicate that the levels of mastery of the
four skills varies between Chinese and Flemish
students, but that multiplication was easier for
Chinese students. Multiplication is the inverse
operation of division. For Greer (2012) inverting is a
relational fundamental building block in
mathematics and within the purely formal
arithmetic, the inverse relationship between addition
and subtraction, and multiplication and division,
have important implications for the assessment of
conceptual understanding of students. Unlu and
Ertekin (2012) conducted a study to investigate
knowledge of a group of mathematics teachers on
the division in the form of fraction. Their results
showed that the understanding of the problems
Threshold Concepts Vs. Tricky Topics - Exploring the Causes of Student´s Misunderstandings with the Problem Distiller Tool
207
raised by fractions, some students applied the
multiplication of fractions instead of the fractional
division, using reverse algorithm. This was
compelling evidence that the students did not have
an adequate understanding of fractions.
3 METHOD
Data collection involved interviews with two math
teachers from elementary school (5th and 6th
grades). The first teacher (T1) is a male, in his
fifties, and teaches in a school in Marco de
Canaveses, near the city of Porto. The other teacher
(T2) is a female with forty-six years old, and teaches
in a school in the city of Braga. Both teachers
dedicated themselves to teaching for their entire
working career.
Data was collected through structured interviews
(20 minutes each) with the support of the Problem
Distiller tool and Think Aloud protocol (Van et al.,
1994). Based on their teaching practice they
identified the math tricky topics that are problematic
for their students, and checked if the tricky topics
were already listed in the database. Next, we
explained how to generate a new tricky topic and
corresponding stumbling blocks. Then, with the
guidance of the Problem Distiller tool, they divided
each tricky topic into stumbling blocks, and wrote a
brief description of students’ specific problems. The
aim was to ensure that each interview presented the
teachers with exactly the same questions in the same
order (the JuxtaLearn taxonomy). This guarantees
that answers can be reliably aggregated and that
comparisons can be made with confidence between
the two teachers.
For the processing and analysis of the obtained
data, content analysis was performed (Bardin, 2013),
as it allows for logical deductions based on the data
obtained. The teachers’ utterances were recorded
and transcribed for the analysis. During the process,
set of dimensions and categories emerged from data:
(i) algorithm, (ii) basic operations, (iii) teaching
method in the 1st level of education, (iv) reasoning
and (v) use the calculator. It is interesting to notice
that dimensions ii and iv are also reported in the
literature of ‘Division’ (Fernandes and Martins,
2014; Montague, 2003; Zhao et al., 2014). In the
dimension ‘a’ (algorithm), we analysed the
relationship between the difficulty in the division
operation and knowledge that students have the
division algorithm. In this dimension, we represent
the speeches of teachers by “T1.a” or “T2.a”. In
dimension ‘o’ (operations), we analyse the
relationship between the difficulty in the division
operation and the students' knowledge of basic
operations and we represent the speeches of teachers
by “T1.o” or “T2.o”. In dimension ‘m’ (method), we
analyse the relationship between the difficulties in
operating with diagnosed division in students and
the teaching method in the 1st level of education. In
this dimension, we represent the speeches of
teachers by “T1.m” or “T2.m”. In dimension ‘r’
(reasoning), we analyse the relationship between the
difficulties in operating with the division and
thinking capacity demonstrated by students. In this
dimension we represent the speeches of teachers by
“T1.r” or “T2.r”. In dimension ‘c’ (calculator), we
analyse the relationship between the difficulties in
operating with the division and the use of calculators
by students. In this dimension, we represent the
utterances of teachers by “T1.c” or “T2.c”. The
utterances were numbered according to their
occurrences in the text.
4 RESULTS
The content analysis was developed according to the
phases suggested by Bardin (2013). Table 1,
presents teachers´ voices according to the five
categories considered in the analysis.
There appear to be a greater number of evidences
in the dimension “Algorithm" and "Reasoning”.
However, it turns out that there is only one evidence
for the dimension teaching method in the 1st level of
education. Teachers see the lack of knowledge in the
algorithm as a deterrent for students to perform
division operations. They point to students’
difficulty in applying the divide operation
algorithm and the location of elements: divider, rest,
quotient and divisor” (T2.a1), and claim that in “the
division operation students have many difficulties
(T1.a3). In their view, students need to spend more
time learning the algorithm, realizing that they “do
not know the algorithm implementation rules and do
not know decompose a number” (T1.a1). In a subject
such as the Euclidean algorithm, taught in 5.º grade,
teachers recommend “obliging students to do
successive divisions” (T1.a2), pointing out that
students have great difficulties in doing this.
Students also have a lot of difficulties on “the
organization of values in the process of
division”(T2.a2) and on “organization of
calculations”(T2.a3) when they are making the
division operation.
Teachers see the lack of knowledge of basic
operations as an issue that prevents the students
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Table 1: Category of analysis.
Category of analysis N
Evidences
Algorithm 6
“Students do not know the algorithm implementation rules and do not know how to
decompose a number” (T1.a1)
“The Euclidean algorithm requires students to do successive divisions. The difficulties for
them are huge. They can apply the algorithm realize the algorithm because it forces you to do
successive divisions” (T1.a2)
“In the division operation, students have many difficulties, mainly because most students
cannot understand the division by two numbers”(T1.a3).
“Students have difficulty in applying the divide operation algorithm and the location of
elements: divider, rest, quotient and divisor” (T2.a1).
“In the algorithm, students also have difficulty in organizing values in the process of
division”(T2.a2).
“Difficulty in organizing calculations when they are split”(T2.a3).
Basic Operations
5
“Students have more difficulties in what we call the basic prerequisites, this is, the level of
b
asic operations: addition, subtraction, multiplication and division. Of these four operations,
where they appear the greatest difficulties is the division” (T1.o1)
“the main difficulties of them: calculation, basic operations. We may say so, students know
add, they know subtract, but if we multiply there are already great difficulties. So if we are
talking in the room, mainly by two numbers, mainly by two numbers I say that most students
can not do” (T1.o2)
“I think mainly, the great difficulty is their basic operations, they confuse the signs of rules of
multiplication or division. In mathematics master who does not add up, subtract, multiply and
divide, how will dominate powers? how will dominate the other things?” (T1.o3).
“Few can convert fractions to decimals, They have many difficulties” (T1.o4).
“Students need to learn to add, subtract, multiply, are concepts and procedures that have many
difficulties and if they have difficulties, not having the basic knowledge required, these
difficulties still will aggravate”(T2.o1).
Teaching method in the
1st level of education
1
“Students come in different primary schools accustomed to different methods, some learn
through successive subtractions others by adding the reverse” (T2.m1)
Reasoning
6
“They have to use the implicit reasoning in the division operation they fail to do.” (T1.r1)
“Them difficulties appear, for example, conversions of fractions to decimals” (T1.r2)
“Mathematics is a discipline that requires training, this is, students do exercises and give up
the first difficulty of the exercises. And the difficulties begin to be increasing. If the student
fails to follow the matter in 5.º grade, how will you get there ahead? The difficulties are
increasing and not only gets what the student learns in school.” (T1.r3)
Can apply to real life situations and they see that is materializable for them, and with these
real-life situations carrying her later for more complicated mathematical concepts and more
difficult for them to understand" (T1.r4).
They can not perceive, and the difficulty of abstraction combined with the lack of
prerequisites to make the division is a problem that can not overcome this difficulty (T2.r1).
Students have a hard mental calculation, especially in multiplication and division” (T2.r2).
“I notice that students not able to find the successive divisions and do not know the
multiplication table” (T2.r3).
Using Calculator
3
“The problem here is often the use of calculating machine or non-use of the adding machine
(T1.c1).
“If you have difficulties, with the use of the machine, these difficulties will still worsen
because they do not have why not use the calculator.” (T1.c2)
“Then they get used to using the machine and forget what they previously learned” (T2.c1)
from performing division operations. Students
present “difficulties in terms of basic knowledge:
addition, subtraction, multiplication and division
(T1.o1). The development of skills in the basic
operations is seen as essential if the student can
work with division, because “in mathematics, for
students who do not master the add, subtract,
multiply and divide, how will they master powers?,
how will they overcome the other things?” (T1.o3).
Teachers said that students had difficulties to
converting a minute into seconds or to convert an
hour into minutes. They noticed also that if they ask
students to do any form of division “mainly by two
numbers, most students can not”(T1.o2). Students
also have many difficulties in “converting fractions
to decimals” (T1.o4). The competence of using an
algorithm is compulsory according to the Portuguese
educational policies, but students are not prepared or
able to learn them and so difficulties rise: “if they
[the students] have difficulties, not having the basic
knowledge required, these difficulties still will
aggravate” (T2.o1).
Threshold Concepts Vs. Tricky Topics - Exploring the Causes of Student´s Misunderstandings with the Problem Distiller Tool
209
Only in the category “teaching method in the 1st
level of education”, one of the teachers pointed out
that the learning division using didactic methods can
leads to later difficulties when working with
division. Also, the fact that students often come
from “different primary schools, accustomed to
different methods” (T2.m1) are also problems
associated with the Tricky Topic.
Teachers understand that “the difficulty of
abstraction coupled with a lack of basic knowledge
(T2.r1) presents a problem of understanding when
students attempt to acquire new knowledge. The
students “have to use the implicit reasoning in the
division operation and they fail to do so” (T1.r1).
The need for the student to remember the notion of a
multiple number and know how to apply the division
algorithm are factors that hinder students’ ability to
perform the division operation. According to the
teachers, students present “difficulty in mental
calculation, especially in multiplication and
division” (T2.r2) and “are not able to find the
successive divisions” (T2.r3). The fact that the
students “do not know thirr multiplication tables”(
T2.r3) is also a pointed problem for students unable
to do a division. The discipline of Mathematics
requires training, this is, students do exercises and
give up the first difficulty of the exercises. And the
difficulties begin to be increasing. If the student can
not understand the content in 5.º grade, how will
they move forward? The difficulties increase and not
only gets what the student learns in school.” (T1.r3).
To improve understanding and visualization,
teachers call for situations where students: “can
apply maths to real life situations and develop a
sound understand in context, building on this
understanding to learn more complicated
mathematical concepts” (T1.r4).
Teachers see the use of calculators in 5.º grade to
6.º grade as an easier alternative adopted by students
to perform division. They find that theuse of
calculator or non-use of the adding machine
(T1.c1), can lead students to forget the algorithm.
The students that use the calculator a lot “forget
what they previously learned about the algorithm
(T2.c1). According to participant teachers, if
students have difficulties and use the calculator,
their understanding of the fundamental concepts in
division will diminish and their ability to perform
division without the aid of a calculator will get worse.
4.1 Problem Distiller Tool
We used the Problem Distiller tool to help the
teachers reflect on the causes of the student
problems they had identified. When teachers
expressed problems explaining why their students
had difficulty understanding the Tricky Topic, they
were guided by Problem Distiller tool to identify the
Stumbling blocks. To Tricky Topic “division
operation”, T1 identified the following Stumbling
blocks: (1) organize calculations, (2) adding notion,
(3) multiplication and (4) subtraction. We present
below the mindmap created with Tricky Topic and
Stumbling blocks identified by this teacher:
Figure 3: Tricky Topic and their Stumbling blocks to T1.
For the Tricky Topic “division operation”, T2
identified the following Stumbling blocks: (1)
subtraction, (2) multiplication tables and (3)
multiplication. We present below the mindmap
created with Tricky Topic and Stumbling blocks.
Figure 4: Tricky Topic and their Stumbling blocks to T2.
The Problem Distiller tool guides the teacher in
identifying the difficulties of understanding of their
students, adding particular examples of student
problems based on the teacher’s experience with
students.
Figure 5: CLIPIT with info gathered from T1.
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210
Figure 6: CLIPIT with info gathered from T2.
As they made selections from Problem Distiller
tool, teachers were identifying problems that
students typically encounter in understanding the
concept of division and were also able to reflect on
why these problems occur and how they can might
be solved in the classroom.
5 DISCUSSION
Throughout the first years of school, students will
develop a sense of number, but only in their 3rd, 4th
and 5th grade, more emphasis is given on the
development of skills in multiplication and division.
The learning of the division operation and the
calculation of a division is often associated with
several students’ difficulties (Mendes, 2013).
Understanding the implicit thinking in a division
operation, from a mathematical point of view,
involves knowledge of other simple operations such
as addition and multiplication skills. The division
and multiplication operations, although simple,
reveal some complexity at cognitive level when
presented in problematic situations, because the
values have new meanings and the figures presented
are sometimes differently exploited (Montague,
2003). One of the fundamental knowledge in the
teaching of mathematics is the calculation of the
four basic operations: addition, subtraction,
multiplication and division. As the student develops
the sense of number, (s)he should be able to
establish a rationale involving numbers (NCTM,
2008). By using the Tricky Topic Tool we identified
together with these two teachers the concept of
division as complex concept for students.
To work with the division operation at the start
of the 2nd cycle of basic education, it is assumed
that students recall some concepts such as the
concept of multiple of a number, the division
algorithm and algebraic expressions. In general, the
data collected from these teachers demonstrates the
importance of student’s understanding of division in
order to solve problems, knowing how to use the
division algorithm to keep pace with some of the
topics covered in the Curricular Goals for 5th grade.
Students tend to use the existing knowledge or
related concepts when they learn a new concept and
therefore the problems and errors made by the
students tend to be systematic. Thus, when doing
division students often rely on knowledge about
multiplication and division that may well be wrong
(Montague, 2003). This data reinforces the
importance of giving students a solid understanding
of this concept in the 1st cycle.
In Portugal, the concept of division is covered
for the first time in the curricular goals in the 2nd
year of primary school (Bívar, Grosso, Oliveira,
Timóteo, 2012). The concept of division is once
again addressed in the 3rd, 4th and 5th grade where
other concepts will be combined relating to this
operation. According to Professor T1 on the four
operations addressed, "the greatest difficulties arise
in the division, I'm talking about students who are in
the fifth year" (T1). Adding that from his experience
teaching in the 5th year of primary school, "90% of
students have difficulty in the division operation"
(T1) and the "division of two numbers, 99% of
students can't do it" (P1). For the teachers involved
in our study, sometimes the division algorithm "have
difficulty in identify the elements" (T1), the dividend,
the divisor, the quotient, the rest and the
"organization of the elements when making the
division algorithm" (T2). That is, when using the
algorithm to work with the division, sometimes they
"switch between the dividend the divisor" (T2).
According to Professor P1 as the students not always
understand the division, "they do not recognize the
process of division and forget the value that is
carried" (T2). The division algorithm, is a set of
processes that follow the same order in similar
situations (Brocardo and Serrazina, 2008) and it’s
not always understood by the students.
The fact that they do not know their
multiplication tables and are not able to perform a
multiplication limits the students' ability to work on
concepts and procedures (e.g. division) that need
those auxiliary calculations. The poor performance
of students not only in understanding necessary
strategies, but also in using them to solve a problem
Threshold Concepts Vs. Tricky Topics - Exploring the Causes of Student´s Misunderstandings with the Problem Distiller Tool
211
leads them to give up. Therefore it is essential to
teach students these important processes and
strategies that help them solve the problems in a
more effective and efficient way (Montague, 2003).
Zhao et. al. (2012) in a study which looked at
Chinese and Flemish students to know what it takes
to master the four basic arithmetic operations
(addition, subtraction, multiplication and division),
identified that students demonstrated gaps in the four
basic operations.
The division operation involves dividing a given
number of equal parts. During the early years of
school students learn the meaning of the division,
understand the effects of dividing by integers, use
and understand the notion that the division operation
is the inverse operation of multiplication (NCTM,
2008). According to the results, the fact that students
cannot resolve a task or problem involving a
division appears to discourage students and prevent
them from progressing. Also Montague (2003) states
that the division operation is a mathematical
procedure with some complexity and understanding
division therefore involves understanding the other
mathematical operations. Many children have
difficulties in using the traditional division
algorithm. And when the operation is necessary in
mathematical problems, many students give up.
Unlike the addition operation, multiplication or
subtraction, the division algorithm involves the
knowledge and identification of four terms dividend,
divisor, quotient and rest. These terms can also
cause difficulties for the students as the teachers
stated in tricky topic tool when they list the
understanding issues that are commonly found in
students. From the point of view of these teachers
"the great difficulty of the students is the basic
operation" (T1). To develop the competence of
calculation through division operation, students need
to have knowledge in terms of counting and
arithmetic operations such as multiplication tables.
Arguments were put forward by both teachers when
identifying the difficulties that students have when
performing division. They mention that students
sometimes fail to "identify the elements in the
division" (P2) and on the 5th year students are
expected to "work with conversions and the
Euclidean algorithm." (T1). According to Arends
(2008), an effective teacher must in addition to other
duties, be able to list a set of good practice and be
able to think about the process of teaching. The
mathematical knowledge of the teacher is essential
to teach the division operation in order to be able to
identify students' difficulties and realize in which
algorithm stage is this difficulty (Fernandes and
Martins, 2014). The teacher plays a fundamental role
so that students can understand the mathematical
meaning of the division, the procedures involved in
the operation, using the correct terminology and an
appropriate mathematical language. By using Tricky
Topic Tool we promote thinking moments on
teachers around the Tricky Topic, the ability to
recall moments of work between students and
difficulties in the construction of knowledge about
the concept of division.
Students' problems often identified by these
teachers refer to difficulties in terms of successive
subtraction to solve tasks associated with the
division; including "not able to find the successive
divisions" (T2) and "Euclidean's algorithm requires
to do successive divisions." (T1). For Montague
(2003) the use of additions and successive
subtraction is a strategy used by children who learn
division and which is based on pre-existing
knowledge about addition, subtraction and
multiplication. The teachers also mentioned the fact
that students are not aware to the inverse
relationship between multiplication and division, can
also be a problem to the understanding of division
operation. They also report that students usually
manifest difficulty operating between numbers
written in the form of fraction, because "do not
realize the meaning of the elements in the fraction"
(T2), have difficulty to "identify the dividend and the
divisor" (T1). To suit the results obtained by Unlu
and Ertekin (2012) who investigated the knowledge
of a group of mathematics teachers on the division
between numbers written in the form of fraction,
they realized that the knowledge about the division
operation with fractions does not go beyond
functional knowledge. These teachers were able to
apply the rules and the process inherent in the
division, but were unable to explain its meaning.
Through the use of Problem Distiller tool with
teachers, we realized that the understanding of
essential concepts around the Tricky Topic division
"sometimes it depends on a badly learned concept"
(T2). Presuppose the use of "already acquired
knowledge of division" (T1) as new knowledge is
being developed. The lack of essential concepts,
fundamental knowledge that is related to the Tricky
Topic, without which the student can not understand,
was pointed out on Problem Distiller tool as one of
the causes for the difficulties in the division
operation. Teachers mentioned the lack of
knowledge about the scientific method and the lack
of support and understanding prior knowledge that
the student needs to improve to understand the
Tricky Topic. The lack of complementary
CSEDU 2016 - 8th International Conference on Computer Supported Education
212
knowledge to the division operation from the point
of view of these teachers can also be a problem.
They noted also that some imperfect reasoning
around the division and intuitive ways of thinking
about the division process can evenly become an
obstacle to the understanding of division. The
reflection upon the causes for the understanding of
problems detected in students, allowed teachers to
increase the level of awareness about the knowledge
of the student.
The teaching of division operation not only
involves knowing how to use the traditional
algorithm but also understand the division operation
in different situations, understand the relationship
between division and multiplication and
simultaneously develop a network of numerical
relationships around this operation. Even the
teachers who teach mathematics in the 1st and on the
2nd cycles admit that the division is a difficult
operation to teach to their students and their learning
process is sometimes confused with the
mechanization of rules associated with the algorithm
instead of understanding the division operation
(Mendes, 2013). The acquisition of mathematical
knowledge allows us to develop reasoning, structure
thinking and help future students to think and to
decide. Understanding how students learn and how
teachers teach mathematical concepts is of
fundamental importance for the individual student
progress and the organizations to which he belongs.
The Tricky Topic tool guided the teachers in the
identification of the tricky topics, and corresponding
stumbling blocks. The Problem Distiller tool
supports them in thinking through the students’
difficulties, reflecting on possible causes for those
difficulties, and on ways to overcome them. This
was because the connections of each Tricky Topic in
the Problem Distiller tool allowed teachers to dissect
the concept into simpler parts (the stumbling
blocks), and establish a critical and reflective look at
the teaching and learning of division operation based
in the four areas identified as problematical for
students: i) Terminology, ii) Incomplete Pre-
Knowledge, iii) Essential Concepts, and iv) Intuitive
Beliefs. From our perspective, this process was
essential to find ways to enable an effective and
consolidated teaching about the tricky topic. The
difficulties listed by the teachers match the data in
the literature, particularly those obtained by
Montague (2003), by Zhao et. al. (2012) and
Fernandes and Martins (2014). Also the NCTM
(2008) states that from the 3rd to 5th grade, students
need to understand in greater depth the
multiplicative nature of the number system. The
results suggest that the obstacles associated with
Tricky Topic identified by teachers are similar to the
difficulties described in the literature about learning
the division operation. The results also showed that
the thinking achieved among teachers with the use
of Problem Distiller prompted them to think outside
their comfort zone. From the perception of teachers
we can say that the division operation is a Tricky
Topic for the students, and the data obtained so far
allow us to conclude that it is a threshold concept
according to the criteria listed by Meyer and Land
(2003). Linking the perception of teachers with the
criteria listed by Meyer and Land (2003) for which a
concept is a threshold, we found out upon teachers
voices:
Can be seen as Transformative, given that by
understanding the division operation students
will be able to "use in everyday situations"
(T2) and "to make conversions for example"
(T1);
It is Irreversible once learned is difficult to be
forgotten; however teachers recognise that "the
abusive use of calculator" (T1) can lead to loss
of an algorithm learned in the first cycle;
Being the division operation a key operation to
for example "do successive divisions in
Euclidean algorithm" (T1), to respond to
"problematic situations of everyday life" (T2),
it is suggested that it is Integrative;
When the division operation is used to as the
basis for understanding of other mathematical
concepts. The misunderstanding in division can
"compound the difficulties" (T1), because if
students "do not have the necessary base
knowledge, their difficulties in learning related
concepts will increase" (T2), suggesting that
the division operation may be Bounded.
Failure to understand the concept or "confusion
problems with the multiplication operation"
(T2) for example may indicate that it is a
Troublesome, an incorrect understanding can
lead to counterintuitive relations.
6 CONCLUSIONS
This paper compares the process of identifying a
complex math concept ‘Division’ from the
pedagogical practice of two teachers, with the way it
is reported in the literature. The data collected
demonstrates the importance of students acquiring
skills of mental calculation, specifically for
multiplication and division. The data also shows us
that although teachers find it easy to identify the
Threshold Concepts Vs. Tricky Topics - Exploring the Causes of Student´s Misunderstandings with the Problem Distiller Tool
213
Tricky Topics and associated stumbling blocks that
their students have problems with, they benefit from
support in reflecting on ‘why’ the students had these
problems. In this particular, the Problem Distiller
tool proved to be essential in scaffolding teachers
reasoning on students´ difficulties. The technology
supports the teacher's brainstorming process, guiding
them in the identification of the causes for student´s
misunderstandings, once the possible reasons appear
already listed in a catalogue of options provided by
the system. By identifying the roots of student
misunderstandings of a stumbling block, the teachers
became aware of the student's difficulties and could
prepare and adopt appropriate teaching strategies.
The teachers were able to identify the operation of
'Division' as a Tricky Topic. As the teachers used the
Tricky Topic Tool and Problem Distiller to break
down the complexities of division, we were able to
evaluate it against the characteristics of a threshold
concept as specified by Meyer and Land (2003). We
found that the teacher-identified topic ‘Division’
matches the definition of a Threshold Concept as
defined in Meyer and Land (2003).
It also seems appropriate to analyse in future
research if the level of reflection achieved with the
use of the Problem Distiller tool contributes to
change the teachers' professional practice.
ACKNOWLEDGEMENTS
The research leading to these results has received
funding from the European Community's Seventh
Framework Programme (FP7/2007-2013) under
grant agreement no. 317964 JUXTALEARN. We
would like to thank the interviewed teachers for their
collaboration.
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