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Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
Enhancing Pre-Service Teachers’ Noticing: A Learning
Environment about Fraction Concept
Pedro Ivars
Departamento de Innovación y Formación Didáctica, University of Alicante, Alicante, Spain
Pere.ivars@ua.es
Received: January 26th, 2021 Revised: February 14th, 2021 Accepted: February 17th, 2021
Abstract
Professional noticing allows teachers to recognise important events in a classroom and
give effective responses using their knowledge. Hence developing this competence in
teacher training programs is an issue in the Mathematics Education field. In this study, we
present the design of a learning environment about the part-whole meaning of fraction to
develop pre-service primary school teachers’ noticing of students’ mathematical thinking.
The learning environment is designed around three tasks (vignettes) that pre-service
teachers have to analyse using knowledge from research on mathematics education
provided as a students’ hypothetical learning trajectory. Eighty-five pre-service primary
school teachers participated in this learning environment. Pre-service teachers’ written
answers to the three tasks are the data of this study. Results allow us to characterise the
enhancement of pre-service teacher noticing through looking at the changes in the
discourse generated in the three tasks.
Keywords: fraction; hypothetical learning trajectory; pre-service teacher; representations of
practice; teacher noticing.
1. INTRODUCTION
During a mathematics lesson, a mathematical understanding and use it
teacher faces overlapping situations and as the basis for making instructional
interactions simultaneously, hindering decisions” (p. 53). From this
their attention to all of them. In this perspective, effective teaching implies
context, teachers should focus their observing students, listening attentively
attention on those classroom situations to their ideas and explanations, planning
or interactions that could potentially objectives and using the information to
enrich students’ mathematical learning make instructional decisions.
(Mason, 2002; van Es, & Sherin, 2002). Therefore, teachers must develop
In fact, NCTM (2014) points out that greater flexibility in recognising students’
teaching effectively implies that teachers mathematical thinking while teaching
“elicit evidence of students’ current (van Es, & Sherin, 2002) and must be
P. Ivars, Enhancing pre-service teachers noticing
Page | 1
Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
aware of what happens in their (Brown, Fernández, Helliwell, & Llinares
classrooms and how to manage it 2020; Seidel, Stürmer, Prenzel, Jahn, &
(Mason, 2002; 2020). In this context, Schäfer, 2017).
teacher noticing has been conceptualised Jacobs, Lamb, and Philipp (2010)
as a competence that allows teachers to particularised professional noticing of
recognise important events in a children’s mathematical thinking as a set
classroom and give effective responses of three interrelated skills: attending to
(Mason, 2002). Hence, a line of research children’s strategies, interpreting
has emerged trying to identify tools and children’s understanding, and deciding
contexts to develop teacher noticing how to respond on the basis of children’s
(Fernández & Choy, 2020). understanding. This competence is
understood as a knowledge-based
a. Professional noticing reasoning since teachers must attend to
A professional vision is the ability to a classroom situation and then interpret
see certain phenomena in a particular the situation, considering their available
way which distinguishes a professional or knowledge (Sherin, 2007) to decide what
expert in a certain area from someone to do next. Therefore, this competence
who is not (Goodwin, 1994). In the case highlights the need of specialised
of mathematics teaching, a professional knowledge (Thomas, Jong, Fisher, &
vision (or professional noticing) allows Schack, 2017).
teachers to identify relevant aspects in a Professional noticing of children’s
teaching-learning situation that non- mathematical thinking implies the
professional people would not be able to teachers’ ability to use their knowledge
identify (Roller, 2016) and reason about (mathematical content knowledge and
the situation using mathematical pedagogical content knowledge) to
knowledge and knowledge of the attend to, interpret and decide what to
teaching and learning of mathematics. do next (Fernández & Choy, 2020;
Sherin (2007) characterised this Thomas et al., 2017). When pre-service
competence as two sub-processes: teachers notice children’s mathematical
selective attention (noticing) and thinking (attending to, interpreting and
knowledge-based reasoning. Selective deciding), they have to use their
attention is linked to the teachers’ ability knowledge (subject matter knowledge
to focus their attention on a particular and pedagogical content knowledge).
classroom situation relevant to students’
learning. Knowledge-based reasoning is b. Professional noticing as a
linked to the teachers’ ability to use their knowledge-based reasoning
available knowledge to make sense of competence
this classroom situation. Therefore, this Brown et al., (2020) linked the
competence helps teachers connect different domains of the Mathematical
theoretical knowledge with practice Knowledge for Teaching framework
P. Ivars, Enhancing pre-service teachers noticing
Page | 2Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
(MKT; Ball, Thames, & Phelps, 2008) their understanding (Knowledge of
with the three skills of noticing; attending Content and Curriculum, KCC).
to children’s strategies implies that
teachers identify important mathematical c. Developing teacher noticing
details in students’ common or In recent years, an important line of
uncommon procedures and the roots of research has emerged examining
their mistakes (Specialized Content contexts and tools for pre-service
Knowledge, SCK). To interpret students’ teachers noticing development in teacher
understanding, teachers must coordinate training programs (Fernández & Choy,
what has been attended with what is 2020; Shack et al., 2017). These studies
known about children’s mathematical have shown that using representations of
understanding. Therefore, in interpreting, practice, such as videos of classroom
knowledge for explaining procedures, interactions or transcriptions of students'
understanding common and uncommon written responses, is a favourable
strategies and explaining the origin of context for its development (Ivars,
their errors (SCK) is required. Fernández, & Llinares, 2020; Ivars,
Furthermore, knowledge about which Fernández, Llinares, & Choy, 2018;
aspects of the concept are the easiest or Sánchez-Matamoros, Fernández &
the most difficult ones for students, Llinares, 2015; 2019; Schack et al.,
which are the most common errors 2013; van Es & Sherin, 2008). Other
related to a concept and how a contexts that can also support its
mathematical concept develops over time development are writing narratives
(Knowledge of Content and Students, during their period of practice at schools
KCS and Horizon Content Knowledge, (Fernández, Llinares, & Rojas, 2020;
HCK) is also needed to interpret different Ivars & Fernández, 2018), online
levels of understanding. Finally, deciding discussions and tutor feedback
how to respond involves taking into (Fernández, Llinares, & Valls 2012; Ivars
account which aspects of the concept are & Fernández, 2018; Llinares & Valls,
the easiest or the most difficult ones for 2010). Results from these studies have
students; which are the most common shown that this competence can be
errors related to the concept and how a developed in teacher training
concept develops over time (KCS); and programmes although its development is
which are the strategies or not an easy task without a frame that
representations more adequate for guides pre-service teachers noticing
introducing the concept (Knowledge of (Fernández & Choy, 2020), such as the
Content and Teaching, KCT). students’ learning trajectories ( Sztajn &
Furthermore, teachers should use their Wilson, 2019).
knowledge about the best sources and
materials to help students progress in
P. Ivars, Enhancing pre-service teachers noticing
Page | 3Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
d. Hypothetical Learning design learning environments to provide
Trajectories pre-service teachers with opportunities to
Learning trajectories have been learn about and for practice. (Fernández
conceptualised from different Sánchez-Matamoros, Valls, & Callejo
perspectives (Lobato & Walters, 2017). 2018). In our research group (GIDIMAT-
Nevertheless, a shared assumption is UA), these learning environments are
that a learning trajectory articulates the designed using vignettes. A vignette
students’ conceptual progress from includes a representation of practice,
informal thinking to a more sophisticated some questions to guide its analysis and
mathematical reasoning. Following Simon information from previous research on
(1995), a Hypothetical learning trajectory students’ mathematical understanding of
(HLT) is made up of three components: a the mathematics concept. This
learning goal, learning activities, and a information can be provided as a
hypothetical learning process (levels of student’s hypothetical learning trajectory.
understanding). This information provides pre-service
Previous research has shown that teachers with the theoretical knowledge
students learning trajectories could required to analyse the representation of
provide pre-service teachers with a practice.
structured framework to focus their Representations of practice are
attention on students’ thinking understood as depicting a classroom
(Edgington, 2014; Edgington, Wilson, situation (e.g. a transcription of students’
Sztajn, & Webb, 2016), since they can answers to an activity, or a cartoon
support pre-service teachers in showing an interaction teacher-student)
identifying learning goals in the to promote pre-service teachers’
instructional activities, in interpreting reflection and discussion of real-life
students’ mathematical thinking and in contexts. A representation of practice can
responding with appropriate instruction ( represent one aspect or several aspects
Ivars et al., 2018; Sztajn, Confrey, of a class situation, but not all the
Wilson, & Edgington, 2012). characteristics of a class (Buchbinder &
Furthermore, learning trajectories can Kuntze, 2018). This reduction of
provide pre-service teachers with a information makes the vignettes useful
mathematical language to describe instruments in the professional
students’ thinking (Edgington et al., development of teachers in multiple ways
2016). (Skilling & Stylianides, 2019) since they
allow teachers to focus the attention on
e. Using vignettes in teacher those aspects of the practice object of
training programs learning. Furthermore, they can be
To connect theoretical knowledge designed in different formats (Friesen &
with mathematics teaching practice in Kuntze, 2018): video recordings of real
teacher training programs, researchers classroom situations (van Es & Sherin,
P. Ivars, Enhancing pre-service teachers noticing
Page | 4Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
2008), student responses to different 2018; Ivars et al. 2020; Sánchez-
problems or dialogues between the Matamoros et al., 2019). Following, we
teacher and the students solving various will present an example of a learning
problems (Fernández et al., 2018; Ivars environment related to the part-whole
et al., 2020) or animations or cartoons meaning of fraction, which is part of a
(Herbst & Kosko, 2014). pre-service primary school teachers’
Research has shown that vignettes training programme.
provide teachers with real contexts to
analyse and interpret aspects of the f. A part-whole meaning of fraction
teaching and learning of mathematics learning environment
and provide them with opportunities to This learning environment is
relate theoretical ideas about the organised around six sessions lasting two
teaching and learning of mathematics hours each (Figure 1, Ivars, 2018). In
with examples from practice (Buchbinder the first two sessions, we introduced the
& Kuntze, 2018; Fernández et al., 2018). mathematical elements related to the
Results from our research group have part-whole meaning of fraction to pre-
shown that these learning environments service teachers. They had to solve some
using vignettes support pre-service fraction activities and analysed video-
teachers’ attention on important aspects clips of students solving fraction
of students’ understanding, and therefore activities. In the last four sessions, we
help them develop the competence of introduced the learning trajectory of the
noticing (Buforn, Llinares, Fernández, part-whole meaning of the fraction
Coles & Brown, 2020; Fernández et al., concept. Pre-service teachers had to
2012; Fernández, Sánchez-Matamoros, analyse three vignettes, using the
Moreno, & Callejo, 2018; Ivars et al., theoretical information provided in a HLT.
Figure 1. Learning environment of the part-whole meaning of fraction
P. Ivars, Enhancing pre-service teachers noticing
Page | 5Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
The HLT is provided as a theoretical considering the previous research on
document and includes: a primary school how students’ reasoning about fractions
students’ hypothetical learning process develops over time (Battista, 2012;
(levels of understanding, Figure 2), Steffe & Olive, 2010). This theoretical
examples of primary school students' document provides pre-service teachers
answers reflecting characteristics of each with the theoretical knowledge (about
of the levels of understanding (Figure 3) fractions and the teaching and learning
and examples of activities that could help of fractions) they need to identify
students progress in their understanding noteworthy mathematical aspects of the
of the concept of fraction as a part-whole representation of practice, interpret them
(Figure 4). The HLT is designed by and support their teaching decisions.
Level 1. Students Level 2. Students can identify Level 3. Students can
cannot identify and and represent proper fractions identify and represent
represent fractions fractions
Recognising that the parts
Not recognising into which the whole is Recognising that a part
that the parts into partitioned must be of can be divided into
which the whole is equal size (but not other parts
partitioned must necessarily of the same Using fractions as
be of equal size shape) iterative units to
Not keeping the Using unit fractions as construct fractions
same whole when iterative units to construct Recognising that the
comparing proper fractions size of a part decreases
fractions Keeping the same whole to when the number of
compare fractions parts increases
Not recognising that a part
can be divided into other
parts
Figure 2. Students’ hypothetical learning process
Figure 3 shows some examples of same whole when comparing. Student
primary school students' answers 2's answer reflects difficulties in
included in the HLT reflecting representing the improper fraction by not
characteristics of level 2. Student 1's identifying the unit fraction 1/4 as an
answer shows difficulties in comparing iterative unit. These are characteristics of
fractions since he/she does not keep the level 3.
P. Ivars, Enhancing pre-service teachers noticing
Page | 6Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
Activity. Student 1 answer
I draw it and 2/3 is
Which is greater greater
2/3 or 2/5?
Activity. Student 2 answer
Represent 5/4 of I divide the figure in 5
this figure parts
Figure 3. Examples of primary school students' answers included in the HLT
The HLT also includes examples of The following section details the
activities that could help primary school characteristics of the vignettes used in
students progress in their understanding this learning environment and shows
of the part-whole meaning of fraction. vignette 2 as an example.
Figure 4 shows an activity that can help
students progress from level 2 to level 3 g. A vignette of comparing fractions
of understanding according to the HLT). The vignettes of the learning
The activity aims at using the unit environment include three elements: i)
fraction as an iterative unit to construct transcriptions of primary school
fractions. students' answers to a fraction activity
showing different levels of
understanding (students’ names are
pseudonyms), ii) guiding questions
related to the three skills that articulate
the competence of noticing: identifying
mathematical elements, interpreting
students' understanding and making
teaching decisions to support students’
progress in their understanding, and iii)
the theoretical documents with the HTL
explained before.
Vignette 2 consists of an activity of
comparing fractions, the answer of three
primary school students (showing
different characteristics of the levels of
Figure 4.Examples of activities included in understanding shown in the HLT), the
the HLT
P. Ivars, Enhancing pre-service teachers noticing
Page | 7Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
HLT as a theoretical document and four And, then we have ¾ which also
guiding questions: represents three out of four that is…
(She draws the following figure):
1. Which is greater 4/5 or 3/4?
Explain it with a picture or
your words
(Answer of Ana and Iván) Teacher: What do you think? Has
Iván: Well we think 4/5 is anyone done it in another way? No one?
greater than 3/4 Can someone else explain, differently,
Teacher: And how do you know? that 4/5 is greater than ¾?
Ana: Because we have drawn four
fifths, which is… and three quarters that (Answer of Núria and Louis)
is… (while she was drawing on the Louis: Yes, of course ... we can
blackboard the following images): but… we have not drawn it
Teacher: What have you done?
Núria: Well we thought that 4/5 needs
1/5 to complete the whole and ¾ needs
¼ to complete it. Therefore... as 1/5 is
smaller than ¼, then 4/5 is greater than
3/4 because it needs less to complete
Teacher: And? the whole than ¾.
Iván: Well, so you can see that Louis: That's it!
4/5 is greater than ¾
Teacher: Do you all agree? ...
Vicent? What do you think? h. Guiding questions
Q1- Describe the activity
(Answer of Marta and Vicent) considering the learning
Vicent: Well, we agree, but objective: what are the
we've done it differently.
mathematical elements that the
Teacher: Could you show us how
have you done it? student needs to solve it?
Marta: Yes, look, here we Q2- Describe how each pair
have 4/5 that represents four out of five of students has solved the
(while she was drawing the following activity identifying how they
figure): have used the mathematical
elements involved and the
difficulties they have had with
them.
P. Ivars, Enhancing pre-service teachers noticing
Page | 8Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
Q3- What are the relationship. This couple provides a
characteristics of students’ correct answer but using an incorrect
understanding (levels of reasoning. Finally, Núria and Louis
answer focuses on the inverse
understanding of the
relationship between the number of the
Hypothetical Learning parts and the size of each part as they
Trajectory) that can be inferred stated that “[…] 4/5 needs 1/5 to
from their responses? Justify complete the whole and ¾ needs ¼ to
your answer. complete it”. Therefore they show
Q4- How could you respond to characteristics of level 3.
these students? Propose a
learning objective and a 2. METHODOLOGY
a. Participants
new activity to help students
In this learning environment
progress in their understanding participated 85 pre-service primary
of fractions. school teachers (PTs) enrolled in a
university course related to the teaching
The mathematical elements that and learning of mathematics during their
should be considered to solve this third out of the four years of the degree
activity of comparing proper fraction are to become a primary school teacher.
the wholes must be the same to
compare and the inverse relationship b. Instrument and data collection
between the number of the parts and During the learning environment PTs
the size of each part (a bigger number solved the three vignettes individually.
of parts makes smaller parts). The Data of this research were PTs’ written
answer of Ana and Iván show answers to questions Q2, Q3 and, Q4 to
characteristics of level 2 since they the three vignettes. To maintain
identify that the wholes must be the participants’ anonymity we named PTs
same to compare fractions (they as: PT1, PT2…. PT85.
represent both fractions using the same
whole). Nevertheless, their answer does c. Analysis
not show characteristics of We made a qualitative analysis of
understanding the inverse relationship PTs’ answers to the three vignettes in
between the number of the parts and two phases. In the first phase, three
the size of each part (since their answer researchers analysed PTs' answers,
relies on the graphical representation). individually, according to whether they
The second couple, Marta and Vicent, (i) identified the mathematical elements
shows characteristics of the level 1 since in the student's answers; (ii) interpreted
they do not keep the same whole to the student's understanding considering
compare both fractions nor show the HLT and the mathematical elements
understanding of the inverse
P. Ivars, Enhancing pre-service teachers noticing
Page | 9Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
identified; and (iii) provided activities (Evidencers) were more able to provide
that helped students progress in their activities to support students' conceptual
understanding. We then compared our progression in the three vignettes. We
results and discussed our discrepancies are going to show this relationship in the
(triangulation process) until we reached vignette of comparing fractions as an
an agreement. From the analysis of (i) instance of this result.
and (ii), we identified three categories In the task of comparing fractions, 71
(groups of PTs) regarding the details out of the 85 PTs interpreted students’
identified and how they interpreted understanding, while 14 did not
students' understanding: interpret it since they only provided
Non-evidencers: PTs who descriptive answers. Fifty-four out of the
interpreted students' 71 PTs who interpreted students’
understanding but did not provide understanding provided details from
details from students' answers to students' answers to support their
support their interpretations. interpretations (Evidencers) and 17 out
Adders: PTs who interpreted of the 71 PTs did not provide details to
students' understanding providing support their interpretations (Non-
details from students' answers, evidencers).
but they added information that For instance, PT23 interpreted
cannot be inferred from students' students’ understanding as follows
answers. (emphasis added to the mathematical
Evidencers: PTs who interpreted elements identified):
students' understanding providing Ana and Ivan They are at level 2
details from students' answers to since when they compare fractions, they
support their interpretations. recognise that the wholes must be the
Then, we analysed whether each same.
Marta and Vicent They are at
group of PTs provided activities focused
level 1 since they do not keep the same
on students' conceptual progression. In whole when they compare fractions
the second phase, we analysed changes Louis and Núria They are at level
in PTs' answers along the three 3 since when comparing fractions they
vignettes. These changes provided us recognise that the wholes must be the
with information to characterise PTs same and establish the inverse
noticing enhancement. relationship between the number of the
parts and the size of each part.
3. RESULTS This PT identified the mathematical
a. Interpreting and deciding elements in students’ answers, (e.g.
relationship “when they compare fractions recognise
Our results highlight that those PTs that the wholes must be the same” or
who provided a more detailed discourse “They establish the inverse relationship
to support their interpretations between the number of the parts and
P. Ivars, Enhancing pre-service teachers noticing
Page | 10Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
the size of each part”) and interpreted wrote that Marta and Vicent do not keep
students’ mathematical thinking the same whole when comparing
recognising the relationship between fractions and they “[…] only notice four
those mathematical elements and the shaded parts in 4/5 and three shaded
different levels of the HLT. Nevertheless, parts in ¾. Therefore, although they
he did not provide details from students' provide a correct answer the reasoning
answers to support their interpretations is not correct […]”. She also interpreted
(Non-evidencer). students’ understanding of the inverse
On the other hand, PT73 interpreted relationship providing details from
students’ understanding as follows students’ answer when she stated that
(emphasis added to the mathematical Louis and Nuria use this mathematical
elements identified): element to “justify their answer since
Ana and Ivan They are at level 2 they stated “…4/5 is greater than 3/4
since they keep the same whole when because it needs less to complete the
they represent both fractions. Then they whole than ¾needs”.
compare both areas to provide the Excerpts from the answers, such as
answer.
the ones from PT73 (Evidencer) and
Marta and Vicent They are at level
PT23 (Non-evidencer) show the different
1. When they represent both fractions,
they only notice four shaded parts in 4/5 ways in which PTs interpreted students
and three shaded parts in ¾. Therefore, understanding in this task.
although they provide a correct answer Regarding the activities provided, 26
the reasoning is not correct because they out of the 54 PTs, who interpreted
do not keep the same whole when students' thinking providing details
comparing fractions. (Evidencers), proposed at least one
Louis and Núria Level 3 activity to help students progress in their
These students use the mathematical understanding of fractions (48%). On
element inverse relationship between the the other hand, five of the 17 PTs who
number of the parts and the size of each
did not provide details in their
part, to justify their answer since they
interpretations (Nonevidencers),
stated “…4/5 is greater than 3/4 because
it needs less to complete the whole than proposed at least one activity (29%).
¾needs”. In this context, the Non-evidencer
PT73 identified the mathematical PTs provided activities that were not
elements in students’ answers and related to the objective proposed or
interpreted students’ mathematical without sense. For instance, with the
thinking recognising the relationship objective of understanding that the
between those mathematical elements wholes must be the same to compare
and the different levels of HLT. fractions, PT23 provided an activity of
Moreover, she provided details from representing fractions and partitioning
students’ answers to support her the whole that was not related with the
interpretations. For instance when she objective proposed:
P. Ivars, Enhancing pre-service teachers noticing
Page | 11Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
Objective: Recognise that the whole Rosa eats ¼ of her cake and Carlos eats
must be the same when comparing 1/5 of his cake. Who do eat more cake?
fractions. Explain it.
Activity: We have two equal paper
In the next section, we present how
sheets and we want to divide one of
them in ¼ and the other in ½. these different ways of interpreting
On the other hand, PTs in the changed through the learning
Evidencer group provided activities environment.
connected with the learning objective
that could help students progress in b. Noticing enhancement
their understanding. For instance, PT73 Through the three vignettes (Table
provided the following activity, focused 1), 42 PTs consistently provided a
on comparing unitary fractions, to help detailed discourse (Evidencers), and 10
Ana and Iván understand the inverse PTs provided a non-detailed discourse
relationship between the number of (Non-evidencers). Twenty-eight PTs
parts and its size. This activity provides improved their discourse since they
students various unitary fractions of the shifted from Non-evidencers or Adders
same whole. Hence, it requires PTs to Evidencers. This discourse'
comparing the number of the parts of improvement was followed by an
the denominator and realising that a improvement of PTs' ability to provide
bigger number of parts in the activities according to students'
denominator makes each part smaller: understanding (from 34% in the first
Objective: To understand the inverse vignette to 63% in the last, see Table
relationship between the number of the 2). However, five PTs, showed a
parts and the size of each parts; a bigger regression since they changed from
number of parts makes smaller parts. Evidencers or Adders to Non-evidencers.
Activity: Jose, Rosa and Carlos buy
the same cake. Jose eats 1/3 of his cake,
Table 1. Pre-service teachers discourse progress in the learning environment
Discourse progress
From Non- From
Ways of interpreting
evidencer Consistently Evidencer or Consistently
students’ mathematical TOTAL
or Adder to Non-evidencer Adder to Non- Evidencer
thinking
Evidencer evidencer
Interpreting through the
12 1 2 19 34
three tasks
Difficulties in at least
16 9 3 23 51
one task
Total 28 10 5 42
P. Ivars, Enhancing pre-service teachers noticing
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Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
Table 2. Number of activities provided in each task considering the discourse provided
Number of Number of Number of
activities activities in activities in
PTs in Task 1 Task 2 Task 3
From Nonevidencer/Adder to Evidencer (28) 19 (34%) 14 (25%) 35 (63%)
Always Non-evidencer (10) 4 (20%) 2 (10%) 9 (45%)
From Evidencer/Adder to Nonevidencer (5) 4 (40%) 2 (20%) 2 (20%)
Always Evidencer (42) 45 (54%) 21 (25%) 49 (58%)
4. DISCUSSION AND CONCLUSION attentive focus on students’
Our results allowed us to characterise understanding and prepares them to
the enhancement of teacher noticing make instructional decisions based on
through PTs changes in the discourse students’ mathematical understanding.
generated, since the progress in the The regression of some PTs can be
discourse, considering the details used explained by the difficulties interpreting
to support their interpretations, some of the mathematical elements of
influenced PTs ability to propose the fraction concept in the vignettes.
activities. This highlights the critical role played by
These results suggest a relationship mathematical content knowledge in the
between the discourse provided by PTs enhancement of teacher noticing.
interpretations of students’ Furthermore, our results show that
mathematical understanding and the the characteristics of the vignettes used
activities provided; PTs who produced a and the HLT supported PTs to link
more mathematical detailed discourse to theoretical knowledge and practice, and
interpret students’ understanding helped them improve their professional
proposed more activities according to discourse. This improvement can be
students’ understanding. Therefore, the seen as evidence of PTs noticing
value of a detailed mathematical enhancement since PTs who generated
discourse can be seen “as a major a more detailed discourse to interpret
learning outcome in its own right” students' understanding, were more able
(Clarke, 2013, p. 22). In this sense, “the to provide an activity to support
more sensitive you are to noticing students' conceptual progression.
details, the more tempted you are likely Nevertheless, the enhancement of
to be to act responsively” (Mason, 2002, noticing is challenging and remains
p. 248). Consequently, it seems that dependent on PTs' mathematical content
producing a detailed discourse may be knowledge, since the mathematical
distinctive in enhancing noticing. Our elements linked to each vignette
results seem to suggest that a detailed- generated different difficulties in PTs.
sensitive discourse leads PTs to a more
P. Ivars, Enhancing pre-service teachers noticing
Page | 13Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
We are aware that we cannot Teachers Education: Volume 3.
generalise our results, and more Participants in Mathematics
research is necessary to analyse how Teacher Education (vol. 13, pp.
PTs’ noticing develops during teacher 343-366). Koninklijke Brill NV,
training programs. In future projects, it Leiden.
will be interesting to analyse how PTs https://doi.org/10.1163/97890044
use the knowledge learned during the 19230_014
university courses in their internship
Buchbinder, O, & Kuntze, S. (2018).
period at the schools to notice children’s
Representations of practice in
mathematical thinking.
teacher education and research –
Spotlights on Different
ACKNOWLEDGEMENTS
Approaches. In O. Buchbinder &
The project I+D+i EDU2017-87411-R
S. Kuntze (Eds.), Mathematics
from Ministry of Economy and
teachers engaging with
Competitiveness (MINECO, Spain)
representations of practice. A
supported this research.
dynamically evolving field (pp. 1–
8). Cham, Switzerland: Springer.
REFERENCES
https://doi.org/10.1007/978-3-
319-70594-1_1
Ball, D. L., Thames, M. H., & Phelps, G.
(2008). Content knowledge for Buforn, A., Llinares, S., Fernández, C.,
teaching what makes it Coles, A., & Brown, L. (2020).
special? Journal of Teacher Pre-service teachers’ knowledge
Education, 59(5), 389-407. of the unitizing process in
https://doi.org/10.1177/00224871 recognizing students’ reasoning to
08324554 propose teaching decisions.
International Journal of
Battista, M. T. (2012). Cognition-based
Mathematical Education in
assessment & teaching of
Science and Technology.
fractions: Building on students'
https://doi.org/10.1080/0020739
reasoning. Portsmouth:
X.2020.1777333
Heinemann.
Clarke, D. J. (2013). Contingent
Brown, L., Fernández, C., Helliwell, T., &
conceptions of accomplished
Llinares, S. (2020). Prospective
practice: the cultural specificity of
mathematics teachers as learners
discourse in and about the
in university and school contexts.
mathematics classroom. ZDM
From university-based activities to
Mathematics Education, 45(1),
classroom practice. In G. M. Lloyd
21-33.
& O. Chapman (Eds) International
https://doi.org/10.1007/s11858-
Handbook of Mathematics
012-0452-8
P. Ivars, Enhancing pre-service teachers noticing
Page | 14Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
Edgington, C. (2014). Teachers’ uses of Fernández, C., Llinares, S., & Valls, J.
a learning trajectory as a tool for (2012). Learning to notice
mathematics lesson planning. In students’ mathematical thinking
J. J. Lo, K. R. Leatham & L. R. through on-line discussions. ZDM.
Van Zoest (Eds.), Research trends Mathematics Education, 44, 747-
in mathematics teacher education 759.
(pp. 261-284). Cham: Springer. https://doi.org/10.1007/s11858-
https://doi.org/10.1007/978-3- 012-0425-y
319-02562-9_14
Fernández, C., Sánchez-Matamoros, G.,
Edgington, C., Wilson, P.H., Sztajn, P., & Valls, J., & Callejo, M. L. (2018).
Webb, J. (2016). Translating Noticing students’ mathematical
learning trajectories into useable thinking: characterization,
tools for teachers. Mathematics development and contexts. AIEM.
Teacher Educator, 5(1), 65-80. Avances en Investigación
https://doi.org/10.5951/mathteac Matemática, 13, 39-61.
educ.5.1.0065 https://doi.org/10.35763/aiem.v0i
13.229
Fernández, C., & Choy, B. H. (2020).
Theoretical lenses to develop Fernández, C., Sánchez-Matamoros, G.,
mathematics teacher noticing. Moreno, M., & Callejo, M. L.
Learning, Teaching, (2018). La coordinación de las
Psychological, and social aproximaciones en la
perspectives. In S. Llinares & O. comprensión del concepto de
Chapman (Eds) International límite cuando los estudiantes para
Handbook of Mathematics profesor anticipan respuestas de
Teachers Education: Volume 2. estudiantes [Coordination of
Tools and Processes in approximations in the
Mathematics Teacher Education understanding of the limit concept
(vol. 12, pp. 337-360). Koninklijke when prospective teachers
Brill NV, Leiden. anticipate students’ answers].
https://doi.org/10.1163/97890044 Enseñanza de las Ciencias, 36(1),
18967_013 143-
162. https://doi.org/10.5565/rev/
Fernández, C., Llinares, S, & Rojas, Y.
ensciencias.2291
(2020). Prospective mathematics
teachers’ development of noticing Friesen, M., & Kuntze, S. (2018).
in an online teacher education Competence assessment with
program. ZDM Mathematics representations of practice in
Education. text, comic and video format. In
https://doi.org/10.1007/s11858- O. Buchbinder & S. Kuntze (Eds.),
020-01149-7 Mathematics teachers engaging
P. Ivars, Enhancing pre-service teachers noticing
Page | 15Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
with representations of practice. international perspective (pp.
A dynamically evolving field (pp. 245-260). Cham: Springer.
113–130). Cham, Switzerland: https://doi.org/10.1007/978-3-
Springer. 319-68342-3_17
https://doi.org/10.1007/978-3-
Ivars, P., Fernández, C., & Llinares, S.
319-70594-1_7
(2018). Características del
Goodwin, C. (1994). Professional desarrollo de la competencia
vision. American Anthropologist, mirar profesionalmente el
96, 606-633. pensamiento de los estudiantes
https://doi.org/10.1525/aa.1994. sobre fracciones [Characteristics
96.3.02a00100 of the development of noticing
students’ mathematical thinking
Herbst, P., & Kosko, K.W. (2014). Using
related to fraction] . In L. J.
representations of practice to
Rodríguez-Muñiz, L. Muñiz-
elicit mathematics teachers’ tacit
Rodríguez, A. Aguilar-González, P.
knowledge of practice: a
Alonso, F. J. García García, & A.
comparison of responses to
Bruno (Eds.), Investigación en
animations and videos. Journal of
Educación Matemática XXII (pp.
Mathematics Teacher Education,
270-279). Gijón: SEIEM.
17(6), 515-537.
https://doi.org/10.1007/s10857- Ivars, P., Fernández, C., & Llinares, S.
013-9267-y (2020). A Learning Trajectory as
a scaffold for pre-service
Ivars, P. (2018). Mirar profesionalmente
teachers’ noticing of students’
el pensamiento matemático de los
mathematical understanding.
estudiantes: El esquema
International Journal of Science
fraccionario [Professional Noticing
and Mathematics Education, 18,
of students mathematical
529-548.
thinking: the fractional schema]
https://doi.org/10.1007/s10763-
(unpublished Doctoral
019-09973-4
dissertation). University of
Alicante, Spain. Ivars, P., Fernández, C., Llinares, S., &
Choy, B. H. (2018). Enhancing
Ivars, P., & Fernández, C. (2018). The
noticing: Using a hypothetical
role of writing narratives in
learning trajectory to improve
developing pre-service primary
pre-service primary teachers’
teachers noticing. In G.
professional discourse. Eurasia.
Stylianides & K. Hino (Eds.),
Journal of Mathematics, Science
Research advances in the
and Technology Education,
mathematical education of pre-
14(11), em1599.
service elementary teachers. An
P. Ivars, Enhancing pre-service teachers noticing
Page | 16Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
https://doi.org/10.29333/ejmste/ .https://doi.org/10.1007/s11858-
93421 020-01192-4
Jacobs, V. R., Lamb, L. L., & Philipp, R. National Council of Teachers of
A. (2010). Professional noticing of Mathematics. (2014). Principles to
children's mathematical actions: Ensuring mathematics
thinking. Journal for Research in success for all. Reston, VA:
Mathematics Education, 41(2), NCTM.
169-202.
Roller, S. A. (2016). What they notice in
https://doi.org/10.5951/jresemat
video: A study of prospective
heduc.41.2.0169
secondary mathematics teachers
Llinares, S., & Valls, J. (2010). learning to teach. Journal of
Prospective primary mathematics Mathematics Teacher
teachers’ learning from on-line Education, 19(5), 477-498.
discussions in a virtual video- https://doi.org/10.1007/s10857-
based environment. Journal of 015-9307-x
Mathematics Teacher Education,
Sánchez-Matamoros, G., Fernández, C.,
13, 177-196.
& Llinares, S. (2015). Developing
https://doi.org/10.1007/s10857-
pre-service teachers' noticing of
009-9133-0
students’ understanding of the
Lobato, J., & Walters, C. D. (2017). A derivative concept. International
taxonomy of approaches to Journal of Science and
learning trajectories and Mathematics Education, 13(6),
progressions. In J. Cai 1305-1329.
(Ed.), Compendium for research https://doi.org/10.1007/s10763-
in mathematics education (pp.74- 014-9544-y
101). Reston, VA: National
Sánchez-Matamoros, G., Fernández, C.,
Council of Teachers of & Llinares, S. (2019).
Mathematics. Relationships among prospective
Mason, J. (2002). Researching your own secondary mathematics teachers’
practice. The discipline of skills of attending, interpreting
noticing. London: Routledge- and responding to students’
Falmer. understanding. Educational
https://doi.org/10.4324/97802034 Studies in Mathematics, 100(1),
71876 83-99.
https://doi.org/10.1007/s10649-
Mason, J. (2020). Learning about 018-9855-y
noticing, by, and through,
noticing. ZDM, Mathematics Schack, E. O., Fisher, M. H., & Wilhelm,
Education. J. A. (Eds.). (2017). Teacher
P. Ivars, Enhancing pre-service teachers noticing
Page | 17Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
noticing: Bridging and broadening https://doi.org/10.1080/1743727
perspectives, contexts, and X.2019.1704243
frameworks. Cham, Switzerland: Steffe, L. P., & Olive, J.
Springer. (2010). Children's fractional
https://doi.org/10.1007/978-3- knowledge. Springer Science &
319-46753-5 Business Media.
Seidel, T., Stürmer, K., Prenzel, M., https://doi.org/10.1007/978-1-
Jahn, G., & Schäfer, S. (2017). 4419-0591-8
Investigating pre-service
Sztajn, P., Confrey, J., Wilson, P.H., &
teachers’ professional vision
Edgington, C. (2012). Learning
within university-based teacher
trajectory based instruction:
education. In D. Leutner, J.
Toward a theory of teaching.
Fleischer, J. Grünkorn, & E.
Educational Researcher, 41(5),
Klieme (Eds.),Competence
147–156.
Assessment in Education (pp. 93-
https://doi.org/10.3102/0013189
109). Cham, Switzerland:
X12442801
Springer.
https://doi.org/10.1007/978-3- Sztajn, P., & Wilson, P. H. (2019).
319-50030-0_7 Learning trajectories for teachers:
Designing effective professional
Sherin, M. G. (2007). The development
development for math instruction.
of teachers’ professional vision in
Nueva York, NY: Teachers’
video clubs. In R. Goldman, R.
College Press.
Pea, B. Barron, & S. J. Denny
(Eds.), Video research in the Thomas, J., Jong, C., Fisher, M. H., &
learning sciences, 383-395. Schack, E. O. (2017). Noticing
and knowledge. Exploring
Simon, M. A. (1995). Reconstructing
Theoretical connections between
mathematics pedagogy from a
professional noticing and
constructivist perspective. Journal
mathematical knowledge for
for Research in Mathematics
teaching. The Mathematics
Education, 114-145.
Educator, 26(2), 3-25.
https://doi.org/10.5951/jresemat
heduc.26.2.0114 van Es, E. A., & Sherin, M. G. (2002).
Learning to notice: Scaffolding
Skilling, K., & Stylianides, G. (2019).
new teachers' interpretations of
Using vignettes in educational
classroom interactions. Journal of
research: a framework for
Technology and Teacher
vignette construction.
Education, 10(4), 571-595.
International Journal of Research
and Method in Education, 541- van Es, E. A., & Sherin, M. G. (2008).
556. Mathematics teachers’ “learning
P. Ivars, Enhancing pre-service teachers noticing
Page | 18Journal of Teaching and Learning in Elementary Education (JTLEE) ISSN (Print) : 2615-4528
Vol. 4 No. 1, February 2021 ISSN (Online) : 2622-3023
©All rights reserved DOI: http://dx.doi.org/10.33578/jtlee.v4i1.7876
Printed in Indonesia
to notice” in the context of a https://doi.org/10.1016/j.tate.200
video club. Teaching and Teacher 6.11.005
Education, 24(2), 244-276.
P. Ivars, Enhancing pre-service teachers noticing
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