Threshold Concepts Vs. Tricky Topics

Exploring the Causes of Student´s Misunderstandings with the Problem Distiller

Tool

Sara Cruz

, José Alberto Lencastre

, Clara Coutinho

, Gill Clough

and Anne Adams

Institute of Education, University of Minho, Campus de Gualtar, 4710-057, Braga, Portugal

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

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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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

“Students have more difficulties in what we call the basic prerequisites, this is, the level of

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

“Students come in different primary schools accustomed to different methods, some learn

through successive subtractions others by adding the reverse” (T2.m1)

Reasoning

“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

“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).

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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 the “use 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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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

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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

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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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