Lesson Study: Improving Teachers’
Conceptions of Students’
Understanding in Place Value
Emilie Dawson
13
Contents
1.
2
3
4
5
Introduction .................................................................................................................................... 1
1.1
Research Background.............................................................................................................. 1
1.2
Purpose for Research .............................................................................................................. 1
1.2.1
Problem of Practice ......................................................................................................... 2
1.2.2
Research Questions......................................................................................................... 2
1.3
Definitions of Key Terms ......................................................................................................... 2
1.4
Report Outline......................................................................................................................... 3
Literature Review ............................................................................................................................ 4
2.1
Background of Place Value...................................................................................................... 4
2.2
Base Ten Materials .................................................................................................................. 5
2.3
Lesson Study ........................................................................................................................... 7
Methodology................................................................................................................................... 8
3.1
Outline and Justification ......................................................................................................... 8
3.2
Participants ............................................................................................................................. 9
3.3
Data Collection Tools .............................................................................................................. 9
3.4
Lesson Study Stages .............................................................................................................. 11
Results ........................................................................................................................................... 12
4.1
Kylie’s Class ........................................................................................................................... 12
4.2
Miranda’s Class ..................................................................................................................... 13
4.3
Pablo’s Class .......................................................................................................................... 15
Discussion...................................................................................................................................... 17
5.1
Teacher Practice.................................................................................................................... 17
5.1.1
First Implementation .................................................................................................... 17
5.1.2
Second Implementation ................................................................................................ 18
5.1.3
Third Implementation ................................................................................................... 19
5.2
Teachers’ Conceptions of Students’ Understanding in Place Value ..................................... 19
5.3
Student Understanding of Place Value ................................................................................. 21
5.3.1
Kylie’s Class ................................................................................................................... 21
5.3.2
Miranda’s Class ............................................................................................................. 22
5.3.3
Pablo’s Class .................................................................................................................. 23
5.4
Conclusions ........................................................................................................................... 24
5.5
Implications ........................................................................................................................... 24
6
References .................................................................................................................................... 26
7
Appendices .................................................................................................................................... 29
ii
Abstract
This research examines Lesson Study and its potential to improve students’ misconceptions
in place value by targeting teacher Pedagogical Content Knowledge. It aims to test whether a
single, collaboratively planned lesson has the potential to improve teachers’ scaffolding
practices by better understanding students’ misconceptions in place value. Qualitative and
quantitative data were collected as evidence to measure improvement in students’
understanding of place value through the implementation of place value related tasks.
Interviews were used to monitor teachers’ perceptions of their practice. The research
highlighted that Lesson Study enhanced teachers’ conceptions of student understanding of
place value, enabling them to design, implement and review lessons to implement scaffolding
that resulted in improved student outcomes. It was found that Lesson study can be used to
promote shared practice that, if sustained, could create a culture of powerful student
intervention in the development of foundational place value concepts.
1. Introduction
1.1
Research Background
The underperformance of Australian students in mathematics is marked by a decrease in the
number of students achieving high results on PISA and TIMMS testing (Stephens, 2009).
Student disengagement from mathematics is occurring at a younger age as a result of the
development of ‘poor mathematical identities’ (Marshman, Pendergast, & Brimmer, 2011,
p.500). Australian students are increasingly likely to resign from formal mathematical
education in secondary school to pursue less demanding disciplines (Council of Australian
Governments [COAG], 2008).
The issue of disengagement resides in students’ lack of robust foundational knowledge,
which is needed to assist students to transition into abstract thinking (Department of
Education and Early Childhood Development [DEECD], 2012). This highlights the
importance of teachers having the necessary Pedagogical Content Knowledge (PCK) to
effectively scaffold students’ understanding in Mathematics (COAG, 2008).
Place value is an important foundational area of Mathematics that underpins the development
of more complex skills, including addition and subtraction, decimals and multiplicative
thinking (Hiebert & Wearne, 1992; Steinle & Stacey, 2004; Siemon et al., 2011). Students do
not always have the place value knowledge and skills to apply learning to more abstract
contexts (DEECD, 2012).
1.2
Purpose for Research
The research evaluates the effectiveness of Lesson Study for improving students’
understanding of place value by strengthening the Instructional Core of teaching and learning
(City, Elmore, Fiarman, & Teitel, 2009).
Lesson Study is the vehicle for action research and professional learning which supports
teachers to develop and refine their PCK to teach place value with improved mathematics
outcomes for students. Lesson Study was selected due to its ability to provide an ongoing
learning cycle to instil long term change in teachers’ practice (Goos, Dole, & Makar, 2007).
1
Action research such as Lesson Study, can develop teachers’ PCK by ‘deprivatising’
classrooms (City et al., 2009). It provides opportunities for participants to become reflective
practitioners and align teacher and school goals to promote a professional culture (Sagor,
1991).
1.2.1 Problem of Practice
In this research Lesson Study enables teachers to identify a problem of practice and engage in
‘systematic enquiry’ (Sagor, 1991; Thomas, 2011). The focus for the teacher problem of
practice is to use questioning, through systematic enquiry, to scaffold student learning and
improve assessment outcomes.
Evidence is collected to monitor the progress of mathematics outcomes (Askew, Brown,
Rhodes, Wiliam, & Johnson, 1997) to support the development of accountability in
educational practitioners (City et al., 2009).
1.2.2 Research Questions
Through the research we aim to discover whether the provision of opportunities for teachers
to be involved in the Lesson Study increases mathematical understanding for both teachers
and students (Sagor, 1991) by answering these key research questions:

Can Lesson Study improve teachers’ conceptions of students’ understanding of place
value?

What changes in teachers’ classroom practice occur as a result of their engagement in
the Lesson Study process?

As a result of planning, implementing and reviewing three versions of a single lesson,
what improvements can be identified in students’ understanding of place value?
By implementing the Lesson Study, it is expected that students’ understanding of place value
will improve due to teachers’ engagement in collaborative professional learning. It is also
hypothesised that, as teacher pedagogy improves, students’ outcomes based on a deeper
understanding of place value will improve.
1.3
Definitions of Key Terms
Throughout the project there are terms used which are imperative for understanding the
nature of this action research.
Face Value is a term borrowed from Ross (1989), which refers to incorrectly applying
knowledge to make sense of something. This term has been used to describe how students
assign meaning to a digit based on its place value position, rather than understanding the
connection between additive and multiplicative structures of place value.
Pedagogical Content Knowledge (PCK) is defined by Shulman (1987) as ‘blending of content
and pedagogy” (p.8) and an understanding of multiple ways to represent information for
learners.
Instructional Core (City et al., 2009) refers to triangulation of teachers, students and content.
They are interrelated, therefore if one element of the core changes the other two are also
expected to change.
2
1.4
Report Outline
This report will be broken down into sections. Firstly, a review of existing literature
regarding place value and Lesson Study are discussed to provide a context for the research.
The methodology is explained to describe the participants that were involved in the research,
the data collection tools used, and the different stages of the Lesson Study implementation.
Results of student performance are presented in tables, providing the outcomes of each
lesson. Connections between student performance across different tasks are shown in the
Results section.
A discussion of the findings is included in this report to exemplify the links that exist
between Lesson Study and the Instructional Core of the lesson. In the Discussion section,
conclusions are drawn and recommendations are made based on research findings.
3
2 Literature Review
2.1
Background of Place Value
Place value development is dependent on an understanding of our Hindu Arabic numeration
system and the structure that underpins it (Reys et al., 2012). Thomas (2004) and Siemon et
al. (2011) argue that the structure of our base ten numeration system is multiplicative and is
based on countable powers of ten and that this knowledge is vital for developing number
sense (DEECD, 2012; Reys et al., 2012; Thomas, 2004; Thomas & Mulligan, 1999).
Understanding the structure of our number system encompasses knowledge of the following
characteristics; place value, and that the position of a digit represents its value (Reys et al.,
2012; Ross, 1989); base ten elements, which are based on powers of ten that increase and
decrease when shifted to the left or right, and are collections of ten which determine a new
collection (Thomas, 2004); the use of zero to show an absence of a value or to regroup
numbers and lastly, additive structures which can be written in expanded notation (Reys et
al., 2012; Ross, 1989; Steinle & Stacey, 2004).
The difficulty when teaching place value is that development is generally non-linear
(Broadbent, 2004; Thomas & Mulligan, 1999; Thomas, 2004), and when taught beyond their
cognitive capacity, students can develop misconceptions (DEECD, 2012; Siemon et al.,
2011) and view place value as additive only (DEECD, 2012). These factors can potentially
hinder their capacity to access mental and written computation, larger numbers and decimal
fractions as specified in the AusVELS (Victorian Curriculum Assessment Authority
[VCAA], 2012).
The AusVELS (VCAA, 2012) proposes that by the end of Year 3, students should be able to
recognise, model, represent and order numbers to at least 10 000, and apply place value to
partition, rearrange and regroup numbers to 10 000 so that they can solve problems. Whilst
place value is implicit in these areas, it is also applicable to mental computation for addition
and subtraction, multiplicative relationships, number patterns and money (VCAA, 2012).
Literature suggests that before students move onto formal representations of place value, they
must have a solid understanding that whole numbers can be cardinal numbers as well as
composite units; for example, exploring 7 as seven objects or as 4 and 3 (DEECD, 2012;
Reys et al., 2012; Siemon et al., 2011). This can also relate number sense to collections
beyond ‘ten and some more’ to highlight two digit place value patterns, for example, 10 and 4
more, which students are quite often unaware of (DEECD, 2012; Saxton & Cakir, 2006;
Siemon et al., 2011). During this phase of development, students must learn to ‘trust the
count’, which is imperative as they learn to recognise 2, 5 and 10 as countable units. If this
skill is undeveloped it can create further misconceptions (DEECD, 2012; Siemon et al.,
2011), and slow students’ transition into multi-unit counting (Fuson, 1990; Thomas, 2004).
Bartolini Bussi (2011) and Saxton and Cakir (2006) suggest that by grade two, students
should recognise that one unit can be equal to a collection of ten units and can be
interchangeable. DEECD (2012), Kamii (1986) and Siemon et al. (2011) suggest that
students can develop misconceptions around this, which can arise as a result of inadequate
part-part-whole knowledge. This occurs when place value knowledge is developed, by
making numbers by their place value parts, naming and recording (DEECD, 2012; Siemon et
al., 2011).
4
As students’ understanding of numeration becomes more sophisticated, they can begin to
split their ones system of counting into ten parts whilst keeping the first system intact to
develop a ‘part-whole schema’ (Kamii, 1986; Resnick, 1983; Siemon et al., 2011).
The aforementioned multi-unit concept enables students to maintain this part-part-whole
schema, whereby one trusts that once regrouped, the sum of its parts equal the whole (Kamii,
1986, Resnick, 1983; Ross, 1989). A study by Kamii (1986) demonstrated a shift between the
cognitive yield of year one and two students. Year one students tended to push groups of tens
together to ‘restore the whole’ and were counting with one-to-one correspondence but
emphasising the ten, showing a lack of trusting the count. Conversely, year two students were
able to split the group simultaneously to regroup the collections into tens and ones with the
intention of coming back to the tens once the ones were counted. This requires a significant
cognitive shift (Thomas, 2004). Multiunit counting is imperative for understanding place
value (Fuson, 1990; Thomas, 2004).
The study by Kamii (1986) exemplified face value counting, whereby students recognise that
a digit is in the tens column but do not assign a value of ten times as a multiple of ten
(Rogers, 2012; Ross, 1989). For example, some students may recognise that the number 75
has seven in the tens place, and five in the ones place, however this does not necessarily
mean that students are applying a multiplicative base ten structure. In other words, they may
see the 7 in 75 as 7 objects not 70 or 70 objects not (7x10)+5 (Kamii, 1986), which would be
indicative of the misconception that place value is about additive collections of ones
(DEECD, 2012). This is particularly inherent when adding and subtracting by place value
parts. Ross (1989) says that for students to have a solid understanding of place value they
must be able to coordinate the multiple properties of place value.
A common misconception related to place value centres around the way numbers are written,
whereby students cannot maintain or apply knowledge of how a number is recorded if the
representation changes. Ross (1989) conducted an experiment with 26 counters, asking
students to group them into four, creating 6 groups with 2 left over. When asked whether the
number of counters were related to the way 26 was recorded, many students said that the two
were the remainders and 6 represented the number of groups.
In addition, some patterns underpin place value such as ; “10 of these is one of those”, or
“1000 of these is 1 of those” (p. 199). This understanding enforces the idea that trades can be
made and that numbers can be composed and decomposed. This is vital knowledge that can
extend into decimal form (Reys et al., 2012; Rogers, 2012; Siemon et al., 2011).
Siemon et al. (2011) argues that by the end of the third year in school, students need to have
an understanding of the base ten place-value pattern: “10 of these is one of those” (p. 199), so
that they can later work with two digit numbers and beyond.
Broadbent (2004) describe the usefulness in trading to allow students to procedurally and
conceptually build understanding about the number system and these patterns.
2.2
Base Ten Materials
Students are often familiar with place value models which embody base ten. Base ten
materials can become so embedded in the language of place value that provision of pregrouped materials can prevent students from thinking about their meaning (Bartolini Bussi,
2011; Ross, 1989). If the situation task is never realised or they interpret materials at face
value (Ross, 1989) mathematical meaning is lost (Bartolini Bussi, 2011; Broadbent, 2004).
5
Students who rely on unitary counting and/or do not make the connection between materials
and the recursive base ten structure of our number system, which lends itself to repeatable
actions, have difficulty transitioning into abstract reasoning (Bartolini Bussi, 2011;
Broadbent, 2004; Collins & Wright, 2009; Thomas, 2004; Siemon et al., 2011).
Despite the importance of ‘trading’ being highlighted as key to place value, many trading
games have the potential to create or conceal misconceptions (Broadbent, 2004; Steinle &
Stacey, 2004). Saxon and Cakir (2006) suggest that unfortunately, students often fail to
develop an understanding of support materials and the underlying number system, even when
directly taught.
Steinle and Stacey (2004) and Ross (1989) oppose this notion, arguing that explicit teaching
has the potential to uncover misconceptions by making explicit connections between different
aspects of the place value structure of our number system.
Despite the issues associated with the use of materials, many believe that they are useful to
open communication channels. This can aid teachers’ understanding of students’
misconceptions in place value and aid mathematical meaning (Bartolini Bussi, 2011;
Broadbent, 2004; Ross, 1989) when used in appropriate teaching situations (Bartolini Bussi,
2011; Fuson, 1990; Hiebert & Wearne, 1992; Reys et al., 2012; Saxton & Cakir, 2006).
Representing a number with the least number of pieces is critical for understanding place
value. The physical act of grouping by ten reinforces place value, assists counting and, is a
good model to compare the magnitude of numbers (Reys et al., 2012).
Ten frames can aid these key understandings by reinforcing part-part whole knowledge to
emphasise ‘correct language’ and reinforce how different representations of numbers can be
equivalent to ‘ten and some more’ (Siemon et al., 2011). Tens frames can bridge thinking
between concrete representations to abstract representations to extract mathematical meaning
from the material (Siemon et al., 2011). They can also “build a sense of what happens after
each decade” (Siemon et al., 2011, p.293).
Ross (1989) says that teachers need to provide opportunities for students to make their own
meaning and connections with the materials and consolidate place value concepts. Similarly,
Japanese educators highlight the importance of giving students enough opportunities to
problem solve and exploit connections between content to develop alternative solutions (Reys
et al., 2012; Saxton & Cakir, 2006; Isoda, Stephens, Ohara & Miyakawa, 2007) to promote
mathematical meaning, whereas lessons in Australia usually adhere to stringent curriculum
documents and lose their coherence (Reys et al., 2012).
Interestingly, the idea of students making meaning in Mathematics and the importance of
instruction targeting students’ understandings is not new (Hiebert & Wearne, 1992), and is
now named as a key proficiency in the AUSVELS (VCAA, 2012). AusVELS (VCAA, 2012)
also documents that conceptually based curriculum has the potential to yield long term
improvement, particularly when coupled with extra attention for targeted teaching (Steinle &
Stacey, 2004), to promote understanding, reasoning, fluency and problem solving (Reys et
al., 2012).
Uncovering misconceptions and enabling students to make their own connections is heavily
reliant on developing teachers’ knowledge about what students need to understand about our
number system (Broadbent, 2004), and being able to probe students’ understanding
effectively by knowing what they think (Steinle & Stacey, 2004). This cannot occur if
teachers do not have sound content and PCK (Shulman, 1987) because it is difficult to know
6
how and what to teach them which can lead to questioning that conceals misconceptions
(Broadbent, 2004).
2.3
Lesson Study
The Japanese educators employ a professional action research learning cycle called Japanese
Lesson Study which supports teachers to guide a learning situation and encourage deeper
reflection by students.
During a cycle of Lesson Study, teachers collaboratively plan a problem solving lesson,
spending time to support problem solving and meaning making by discussing materials and
talking about ‘problem’ results, which are essential for learning (Tachibana, 2007; Reys et
al., 2012). Additionally problem solving lessons have the potential to encourage ways of
thinking which are conducive to higher order thinking, which leads to better student
performance on tasks (Hiebert & Wearne, 1993; Walsh & Sattes, 2005; Reys et al., 2012).
Lesson Study and professional development support teachers to adapt their pedagogy (Ong,
Lim, & Ghazali, 2010) and become more cognisant of their practice to develop a robust
understanding of how to develop students’ mathematical thinking (Watson & De Geest,
2005). This resonates in their ability to question and improve their own practice (Jaworksi,
1998), and enables the monitoring of progress in mathematics outcomes for students (Askew
et al., 1997).
Teachers will implement the lesson whilst being publicly observed. All observers are
provided with the lesson plan so that they know what they are looking for, and they measure
whether the objectives of the lesson are achieved (Fernandez & Yoshida, 2004), and whether
students achieved these goals (Tachibana, 2007).
Once observed, teachers will discuss how the students performed and what difficulties they
had before making suggestions to improve the lesson prior to re-implementing it (Fernandez
& Yoshida, 2004; Tachibana, 2007). This enables teachers to take part in deeper reflection
about practice and content, and formulate questions, engaging teachers in professional
conversation (Isoda et al., 2007; Fernandez &Yoshida, 2004) to improve the Instructional
Core (City et al., 2009) of the lesson.
The difference here lies within the ideas of problem solving rather than review and practice,
and developing understanding of place value with improved practice to implement
meaningful problem solving (Reys et al., 2012).
The desired outcome of lesson study for this action research was to exemplify the benefits of
collaboration; by improving teachers’ conceptions of their students’ understanding of place
value, and adapting their practice to address students’ learning needs. Through this process,
classrooms would become ‘deprivatised’ for the purpose of strengthening the Instructional
Core to improve students’ understanding of place value.
7
3 Methodology
3.1
Outline and Justification
In order to answer the key research questions, participants took part in a Lesson Study
professional learning cycle. The Lesson Study cycle had a multifaceted intention. Whilst the
aim was to improve teacher capacity by developing their understanding of their own practice
and their students’ conceptions of place value, it was also vital to ascertain whether the
process of Lesson Study impacted students’ learning.
As discussed in the Literature Review, students who do not develop sound knowledge of
whole number place value have difficulties when interpreting decimal fractions (Steinle &
Stacey, 2004), and may have misconceptions which impact their understanding of the number
system. The method and implementation of the Lesson Study was designed to address these
misconceptions at a foundational level.
At Sundale Primary School, Year 3 students were identified as ideal candidates for the
research lesson. Past National Assessment Program – Literacy and Numeracy (NAPLAN)
data indicates that added value from years three to five was declining. This may have been a
result of misconceptions in students’ early years numeracy knowledge. Therefore, it was
appropriate to choose a matched cohort to determine whether targeted interventions in a
single Lesson Study lesson could improve students’ understanding of place value before
transitioning into senior school.
The Lesson Study model was not only utilised as a tool to improve student learning, but was
also used with the intention of expediting opportunities for teachers to collaboratively analyse
and make reference to assessment data in order to develop greater conceptions of students’
understanding of place value.
The development of teacher conceptions regarding their students’ misconceptions was used
to form the basis of a single lesson focus and provide structured collaboration to discuss ways
of addressing said misconceptions through the design of a well-structured lesson.
Figure 1. Lesson Study Implementation. From Lesson study: A Japanese approach to
improving mathematics teaching and learning (p. 32), by C. Fernandez & M. Yoshida., 2004,
Mahwah, NJ: Lawrence Erlbaum.
8
Ongoing planning, observations, feedback and review, as seen in Figure 1, were used to
improve teacher practice, not only through the use of assessment and planning, but the ways
in which teachers question to scaffold.
3.2
Participants
Teacher participants were invited verbally and in written form, to take part in the research on
a voluntary basis. They were selected to ensure that all participants were available to teach
students from the same year level (Year 3).
Teacher participants were selected to provide different levels of experience to the team and
ensure that the Lesson Study would be purposeful for all participants. The team consisted of
two Year 3 teachers, including a first year graduate, a second year graduate and an Assistant
Principal who was acting in a Year 3 teaching role. Additionally, there was the researcher
who also taught at the school as an accomplished Year 5 teacher.
Sixty four Year 3 students were invited to participate in the research via a letter explaining
the nature of the research to parents. The parents of 38 students responded, giving consent for
their children to be part of the research. All student participants were between 8 to 10 years
old, 24 student participants were female and 14 were male.
3.3
Data Collection Tools
The research was conducted under a quasi-experimental method whereby a variety of
qualitative and quantitative data were used to collect evidence in an attempt to address the
research questions.
Table 1
Data Collection Tools
Data Collection
Time of collection
Tool
Purpose
Paper Based Place
Value Assessment
Prior to the commencement of
Lesson Study
Establish a base line of students’ current
level of place value understanding
Lesson Observation
Prior to and during Lesson
Study
Establish a base line of teacher practice
and view implementation of the
collaboratively planned lesson
Audio Recording
and Photographs
All observed lessons
Collect data and evidence of teacher
practice and student responses
Lesson Plans
After each planning cycle
Collect evidence of lesson changes
Student Interviews
Prior to and post Lesson study
Collect formative (prior) and summative
(post) assessment data
Exit Task
At the completion of each
observed Lesson Study lesson
Evidence of student responses
(formative)
Teacher Interview
Post Lesson Study
Gauge teacher perception about the
impacts of Lesson Study on practice
9
A paper based place value assessment based on the Mathematics Online Interview (DEECD,
2013) was collected from teachers to form a better understanding of students’ understanding
of place value after a sequence of lessons had been taught in class. These data were used to
develop an understanding of students’ understandings in place value and were used to inform
the direction of the research.
Prior to the Lesson Study, Year 3 teachers’ mathematics classes were observed to form a base
line understanding of teachers’ ‘normal’ practice. Audio recordings were used to analyse
exchanges between each teacher and their students without interrupting the normal
procedures of the class.
Ten minutes of dialogue was taken from the introduction of each teacher’s lesson to ensure
that the exchanges were uninterrupted. This information was coded into a table (see
Appendix H) and was used as a tool to provide teacher feedback. This was done to make
teacher participants more aware of their questioning practice before the Lesson Study. It is
important to consider that responses were coded by the researcher, based on questioning
levels of Bloom’s Taxonomy (1956, as cited in Walsh & Sattes, 2005). Some additional
categories were added. The analysis may therefore present some accuracy issues, resulting
from the limitations of only one person categorising these questions.
Teachers conducted one-on-one interviews (see Appendix E) with their students to collect
quantitative and qualitative data to probe deeper into students’ understanding of place value.
The ‘Assessment for Common Misunderstandings’ interviews were sourced from the
Department of Education and Early Childhood website. These interviews are based on the work of
Di Siemon et al. (2011) and include a question related to research conducted by Ross (1989).
Upon the completion of the Lesson Study implementation, students from the sample were reinterviewed one week after the lesson. They were however, only reinterviewed on three
questions, which related to the lesson aims. Only two of the three questions are discussed in
this report (see Appendix E). Achievement gains were measured after each lesson cycle and
used to inform the next cycle of planning.
An Exit Task (ET) was carefully constructed by the team (see Appendix A). The ET was
given to students at the end of each lesson and was used to gauge whether the aims of the
lesson were met, and to learn more about students’ understanding of place value after each
lesson cycle.
Whilst observing colleagues’ lessons, each teacher was allocated a role during the lesson (see
Appendix G). These roles alternated with each lesson implementation. They included;
observing student’s verbal responses, observing student’s written responses and recording
responses to teachers’ questions and taking photographs. This evidence formed the basis of
the post lesson debriefs.
Lesson Study plans (see Appendix D) were collected at each stage of the planning to
document evidence of any adaptations that were made to the lesson. Evaluations were also
included in the lesson plans.
These plans documented adaptations to the lesson after each implementation cycle, with the
intention of demonstrating whether teachers’ conceptions of students’ understanding and their
questioning changed throughout the entirety of the Lesson Study. It was however limited
because teacher discussion was not adequately recorded in the evaluation section of the
lesson plans.
10
A teacher interview (see Appendix I) was administered one-on-one with each teacher
participant upon the completion of the Lesson Study cycle. This was used to promote
reflective and evaluative responses that focused on the impact of the Lesson Study on
teachers’ conceptions of their students’ understanding of place value. Questions were
designed to probe for any perceived changes in teacher practice as a result of being involved
in the Lesson Study action research.
As the implementation of Lesson Study progressed, it became evident that student data,
including pre and post Assessment Interviews (AIs) and ETs, teacher interviews, and
observation proformas were the most significant data collection tools when monitoring the
impact of the Lesson Study in relation to the research questions.
3.4
Lesson Study Stages
Twice weekly meetings over the duration of three weeks were scheduled to discuss the
planning and pedagogy which would support students’ understanding of misconception areas
identified from the analysis of the paper based place value test and student interviews.
During the first of the meetings, teachers met to analyse students’ responses from the paper
based place value test, which were inconclusive, therefore led to uncertainties about students’
understanding of place value. As a result, one-on-one student interviews were implemented.
Evidence of what students already knew and their misconceptions, identified from the
interviews, informed the discussion and basis for selecting an area of focus for the place
value Lesson. The next meeting was based on the process of Lesson Study and looking at
lesson plan (see Appendix D) pro-forma exemplars to familiarise participants with the
planning process.
During the second week of meetings, preliminary ideas about the lesson were discussed in
conjunction with ways to target interventions towards addressing misconceptions that were
highlighted in the one-on-one interviews. Professional Development was provided to the
team in response to data gathered from lesson observations about quality questioning,
providing opportunities for direct feedback to individual teachers.
The last week was dedicated to collaboratively planning the lesson, pre-empting student
responses and making provisions for these through the development of questions which were
appropriate for scaffolding student learning. Choice of appropriate materials to target their
misconceptions was discussed and inserted into the lesson plans as responses to student
needs.
Each participating teacher implemented the lesson. The first year graduate teacher was the
first to implement the lesson, followed by the second year graduate and then the Assistant
Principal. At the end of each lesson, students were given the ET to determine whether the
single lesson had had an impact on students’ understanding of place value and to detect the
effectiveness of the lesson after each cycle.
Student ETs were collaboratively analysed in search of evidence of students’ understanding
which related to the learning intention. These data revealed information that was then used to
adapt the lesson to make changes to the questions and the learning intention, in order to
reinforce the importance of various aspects within the lesson. This process was repeated with
each subsequent lesson implementation, with small and varied alterations being made at each
stage.
11
4 Results
In this section you will find a summary of results which highlight changes in student
responses to AIs (see Appendix E & F) and ETs (see Appendix A & B).
Pre and Post AI results are discussed broadly, showing the progress that each cohort of
students made with each lesson implementation. Analyses of student ET responses are
discussed to demonstrate a correlation between student ‘response types’ and the impact these
had on student gains in the Post AI.
It is an important consideration when reading this section to acknowledge that four students
across all the cohorts had no room for improvement because they had already demonstrated
strong place value knowledge on their pre AI and therefore did not make progress. Their
results are however discussed in the results in terms of the quality of their ET responses.
4.1
Kylie’s Class
Data shows that in the sample of twelve students from the first cohort, the largest area of
growth in the post assessment is the shift in which students assign knowledge of tens and
ones to 26 in Question One of the post AI (see Appendix E & F).
Table 2
Students’ Pre and Post Interview Results from Kylie’s Class
Question Criteria
Question One
 No response
 Additive Place Value Knowledge
 Multiplicative Place Value Knowledge
Question Two
 No response
 Additive Place Value knowledge
 Multiplicative Place Value Knowledge
Student
Results
Pre AI
Student
Results
Post AI
4
7
1
3
4
5
10
2
0
8
2
2
Results from the first lesson implementation are displayed in Table 2. These data show
changes in students’ responses to pre and post assessment data. In the Post AI, results show
an increase from 8% to 42% of the first cohort giving multiplicative responses to Question
One; recognising tens as countable units.
This result indicates the most significant shift in student understanding of place value
following the first lesson. These data show that students who either responded with additive
responses, such as 20 + 6, or incorrectly assigned meaning to the digits within a number,
decreased from 58% to 33%.
Only two students were able to correctly answer the post AI relating to the targeted
misconception in Question Two (see Appendix F). There was no change in 67% of students’
responses to Question Two. Only one student, who had previously been unresponsive to
Question Two, was able to make progress.
12
Table 3
Exit Task Response Types, Lesson One
Last Question
No Response
Students’ responses
6
Misconception
1
Additive
-
Multiplicative
5
Results in Table 3, show student response ‘types’ to the last question of the ET (see
Appendix A) for the first lesson. Students who responded with a yes or no response were
classified as ‘No Response’ because they did not justify their response or associate their
response with place value.
4.1.1.1 No Response
Results indicate a correlation between students who were unresponsive on the last question of
the ET and those who made little or no improvement on the post AI. This was particularly
evident in relation to Question Two. Only one student was able to respond to Question Two
(additively) and the rest made no gains. One student regressed in this question.
4.1.1.2 Misconception Response
Unsurprisingly, the student who answered by naming the 2 in 26 as the two circles used to
enclose the blocks in the first question of the ET (see Appendix A) made no progress on the
post AI on either question.
4.1.1.3 Multiplicative Responses
Analysis of the ET, show that the remaining five students answered multiplicatively,
identifying that 2 in 26 represented two tens and the 6 was six ones. Only two students were
able to respond to Question Two of the post AI multiplicatively. The remaining students,
however, reverted back to additive responses or made no movement on either question. These
results show that students who were able to respond multiplicatively on the ET, were slightly
more inclined to demonstrate strong, multiplicative place value knowledge.
4.2
Miranda’s Class
Table 4
Students’ Pre and Post Interview Results from Miranda’s Class
Question Criteria
Student
Results
Pre AI
Student
Results
Post AI
Question One
 No response
 Additive Place Value knowledge
 Multiplicative Place Value Knowledge
5
5
4
1
3
10
Question Two
 No response
 Additive Place Value knowledge
 Multiplicative Place Value Knowledge
10
2
2
6
0
8
Data synonymous with lesson two shows more significant shifts in students’ understanding of
place value as a direct comparison before and after the lesson took place. As shown in Table
4, this class had shown a relatively even spread of students’ responses for Question One,
prior to the lesson implementation. Student results showed that they either; did not respond,
13
responded additively and/or assigned false meaning to the numbers, or assigned strong place
value knowledge in terms of tens and ones.
Post results indicate substantial improvement for the first assessment question, with 71% of
students able to describe place value in terms of tens and ones compared with 29% prior to
the lesson, marking a decrease in the number of students who viewed the 2 in 26 as 20
individual objects. Only one student was non-responsive in the post interview to Question
One, and only 21% of students answered by renaming numbers (additively), compared with
36% in the pre AI.
Furthermore post assessment data shows an increase in the number of students who were no
longer distracted by the different visual image of 26 in Question Two (see Appendix F). In
the post AI, 57% of students could assign knowledge of tens and ones to describe how the
number 26 was recorded, compared to only 14% in the pre AI. However 43% of students still
remained unresponsive, showing no signs of having overcome their misconception about how
26 was represented in Question Two.
Table 5
Exit Task Response Types, Lesson Two
Last Question
Number of responses
No
Misconception
Response
-
Reference
to fives
5
Reference
to tens
9
Response ‘types’ and classifications varied from the first lesson due to modifications made to
the ET (see Appendix B) before the second implementation lesson. ET responses shown in
Table 5 indicate that nine students were able to explain Julie’s error (see Appendix B) by
relating it back to countable units of ten.
4.2.1.1 Reference to Ten
Six students answered the post AI demonstrating strong place value knowledge on both
questions of the post AI. Most students made some significant gains, four of which had been
previously unresponsive to Question Two on the pre AI. Whilst this result shows increased
sophistication of students’ understanding of place value, three students were unable to retain
this understanding for the post AI.
4.2.1.2 Reference to Five
Five participants only made reference to groups of five on the last question of the ET (see
Appendix B), one of which had been unable to answer the pre AI questions but responded to
the post AI, demonstrating strong place value knowledge for both questions. This was an
anomoly for this cohort. The remaining four students either made no movement or regressed
in question two.
These data indicate that students who made reference to ten made more significant gains in
the post AI for lesson two.
14
4.3
Pablo’s Class
Table 6
Students’ Pre and Post Interview Results from Pablo’s Class
Question Criteria
Student
Results
Pre AI
Question One
 No response
 Additive Place Value knowledge
 Multiplicative Place Value Knowledge
Question Two
 No response
 Additive Place Value knowledge
 Multiplicative Place Value Knowledge
Students
Results
Post AI
6
2
4
0
2
10
6
3
3
2
0
10
Results following the third lesson implementation show the greatest shift in student responses
due to the larger proportion of movement for the relatively smaller sample size of twelve
students.
As a direct comparison to Lesson Two, Table 6 suggests that students continued to make
greater gains in Question One, compared with Miranda’s implementation, despite the smaller
sample size. Additionally, where 50% of students were unresponsive to this task in the preassessment, all students made some progress on this question.
83% of students in Pablo’s class answered in the preferred way, using tens and ones to
display strong place value knowledge on Question Two. This shows a large shift in students’
understanding of place value.
Although 50% of the class had exhibited the misconception in Question Two, compared with
71% of Miranda’s class, before implementation, only 17% of Pablo’s class maintained the
misconception compared with 43% in Miranda’s class after the lesson.
Table 7
Exit Task Response Types, Lesson Three
Last Question
Number of responses
No
Misconception
Response
-
Reference
to fives
10
Reference
to tens
2
The last ET presented very different results from Miranda’s class, despite the task remaining
the same. Table 7 shows that the majority of students referred to groups of five rather than
ten, but seemed to strengthen their place value knowledge according to the post AI results.
4.3.1.1 Reference to Ten
Only two of the students had made reference to groups of ten. Interestingly, one succeeded in
achieving the preferred response which demonstrated strong place value knowledge on both
questions of the post AI. The other made minimal gains on Question One only by renaming
20+6.
15
4.3.1.2 Reference to Five
All students who made reference to the 4 fives, except the student mentioned above,
responded with strong place value knowledge on both questions of the post AI. Results
indicated that whilst students did not necessarily refer to countable units of ten on the ET,
they were generally able to relate their learning of ‘groups’ to any countable unit, including
ten.
16
5 Discussion
5.1
Teacher Practice
5.1.1 First Implementation
The problem of practice was centred on teacher questioning. Questions were pre-formulated
into the lesson scripts but teachers were seldom able to memorise and therefore implement
them. However as the lessons progressed, more of the higher order questions became evident
in the lessons, resulting in richer responses from students.
The first implementation was by Kylie and the introduction to the lesson adhered to the
lesson plan. In the beginning, Kylie was able to question students effectively to allow them to
verbalise that when two groups of 5 were added together it would make ten; and because
there were four groups of five, the total would make two tens and there would be 7 left over
(see Appendix D). Therefore the link to place value was clear and the questioning used was
effective to highlight that the 2 in 27 were represented by the 2 tens and the 7 were the
remaining buttons.
Whilst explaining the task to students, Kylie demonstrated how to draw an enclosure to make
groups (see Appendix D), resulting in different arrangements of the blocks. The task required
students to record the number of blocks used to make the enclosure, and write a number
sentence that would reflect the way the number was recorded, using place value (see
Appendix C). However when Kylie modelled the task on the board, the number sentence
reflected the number of blocks on each side 8+8+4+4= 24. In the modelled number sentence
there was no link to tens and ones and how this representation related to place value.
Essentially, this resulted in students not recognising the relationship between the different
representation of numbers and place value, as was the intention of the lesson.
The team noticed that the place value connection was lost on account of student ET responses
whereby, students’ number sentences were not based on place value, and few students chose
to elaborate on a yes or no response in the last question of the ET (see Appendix A). The
majority of students had difficulty explaining the connection between the first diagram and
the written notation of 26 blocks. This meant that the purpose of the lesson was not actualised
and it became evident that whilst the lesson was collaboratively planned, the team was unsure
about the purpose that the lesson aimed to serve; therefore students couldn’t have performed
as expected.
Whilst the lesson was not altered dramatically by the team after the first implementation, time
limits and discussion about the important features (scaffolding) of the lesson became a focus.
Members of the team had had different understandings of the lesson aims, particularly in
terms of making the connection between place value and different representations of
numbers.
17
5.1.2 Second Implementation
Figure 2. Gus’ response prior to returning to change his drawing and number to 27.
In Miranda’s lesson, Gus, a student who was considered to be high achieving, was invited to
the whiteboard to draw a representation of the model enclosure he had made with blocks
during the lesson. He was asked to show the groups and record how many blocks he had
used. The total number was 24 blocks, however before completing his drawing, as seen in
Figure 2, Gus became ‘confused’ and stated that he had used 27 blocks.
His model enclosure had shown, 2 sevens and 2 fives to make 24, however, as he explained
this representation to the class and began to draw, he was influenced by his group of seven,
which he interpreted as ones. This showed that he was distracted by the representation of his
grouping and provided evidence of his misconception. However Miranda did not correct him
because her aim was to guide students using questioning. Ultimately, his post assessment data
showed that this was not effective to address his misconceptions.
Generally, however, there was a greater emphasis on the tens and ones language, but post
lesson discussion highlighted the need to balance questions with moments of explicit
teaching. Miranda had done little to explicitly intervene, as she was reluctant to guide the
student responses, on account of believing she should question the students and not direct
them.
Despite having discussed the importance of using the materials to make the link between the
different representations to tens and ones explicit, Miranda felt rushed and skipped the
modelling. Additionally there was no modelling of a number sentence, which removed
another opportunity for students to verbalise the relationship between the different
representations of the blocks to place value, tens and ones.
Results indicate that the use of tens and ones language made a difference to students’
understanding; however 6 of the 14 students from this sample class made no progress in
overcoming their misconceptions. This highlights the importance of using materials such as
ten frames to model number sentences.
Explicit instruction was needed to show students how to relate their representations back to
place value. The team also discussed how important it was to create opportunities for the
students to verbalise their understanding, by framing questions which would enable them to
verbalise the connections between different representations and how the number of blocks
was recorded.
18
5.1.3 Third Implementation
By the third lesson more explicit links were made using materials, by dragging each block
from the enclosure into a tens frame. This explicitly highlighted the tens as countable units,
with ‘some more’ left over. The language that accompanied this explicit instruction became
more fluent and more familiar to the teacher. Additionally, questioning moved from closed,
fact finding language to probing questions which encouraged students to prove, elaborate and
explain the relationships that could be seen. This shifted the learning from teacher directed
conversations to students building consensus.
Pre-formulating questions and including them in the lesson plan scripts, actively observing,
and discussing teacher questioning after each lesson was accountable for highlighting a
questioning practice amongst the Lesson Study team.
Whilst the implementation of the lessons did not necessarily provide ongoing opportunities to
improve this practice in the short term, post Lesson Study teacher interviews revealed that all
teachers felt that they were more conscious of their questioning style. Two participating
teachers have reported pre-formulating questions and including them in their lessons since
being involved in Lesson Study. Pablo, the Assistant Principal, worked with other learning
teams across the school to implement a miniature lesson study and referred to literature to
develop high order probing questions for the lesson.
The Lesson Study also suggests a shift in the use of language and teachers’ developing a
better understanding of scaffolding practice.
5.2
Teachers’ Conceptions of Students’ Understanding in Place Value
Whilst analysing responses to a place value task from the early assessment data, it was
evident that teachers’ preconceived ideas about what their students could achieve lead to
assumptions about their students’ understanding of place value based on their perceived
ability.
Figure 3. Exemplifies some students’ representations of 36 on a place value assessment, two
of which were marked correct.
The task required students to draw 36 using tens and ones. Many students drew
representations of ten such as long strokes, without demonstrating whether they understood
that the representation should show a countable unit of ten. Teachers marked these
representations as correct, assuming that their higher attaining students had not taken care, or
hadn’t been concerned with drawing each ten. However it was unclear to the researcher
whether this was in fact a misunderstanding or a result of haphazard responses.
19
Discussion revealed that the initial task was insufficient in probing students’ understanding of
place value. This prompted teachers to interview students using the misconceptions
interviews from DEECD (2012) to learn more (see Appendix E). These findings became a
key piece of evidence about teachers’ conceptions of their students’ understanding in place
value. It also highlighted how teacher assumptions can conceal students’ true level of
understanding in relation to place value, based on choice of task and the question techniques
used to probe students’ knowledge.
In a post interview with Miranda she said that the diagnostic assessment results “blew my
mind away, in terms of the misconceptions, that I had no idea that my students had.”
Kylie said, “I was so surprised by the information that, you know, came out of that initial
testing.”
Miranda felt that by interviewing students, it became evident that those students who had
previously succeeded on typical place value tasks were able to say the correct answers, but
when probed, demonstrated a lack of understanding compared to the level of understanding
the team had initially assumed they were capable of.
Kylie also discussed how she had perceived that some of her students were “advanced in their
conceptual understanding of place value” but they “were actually, just good at speaking the
lingo and following processes without really understanding why they were doing what they
were doing”.
The teachers had been unaware of the difficulties that their students had been having with
their understanding of place value prior to the Lesson Study. This was particularly apparent
when the representation of a number changed as per Question Two of the AIs (see Appendix
E).
The teacher interviews also revealed that discussing student responses from the lesson, and
their assessment data, had improved their conceptions of students’ inability to understand the
value of ten, and its relationship to base ten and how that impacts the way numbers are
recorded.
Kylie emphasised that “You really have to know what you’re looking for, to really uncover
those misconceptions”. The team affirmed this comment by acknowledging that these
difficulties were probably a result of students never being explicitly taught, but were easily
overcome when they were effectively challenged.
Kylie and Pablo acknowledged the danger of making assumptions about students’
understanding of place value based on expectations that students already have the knowledge
as specified at Level Three, AusVELS (VCAA, 2012). They also acknowledged the
importance of ensuring that students have a robust understanding of place value concepts at
“this crucial time”, because the development of mathematics depends on these foundations,
and “concepts continue to get harder and nothing else makes sense”.
Kylie and Miranda reported that the use of diagnostic assessment had altered the way that
they regarded students’ development of place value, and perceived this as important and
something that they would definitely do again.
Teachers articulated how important diagnostic and formative assessment was to assist them
understand what students did and did not understand. This is integral to teaching and
learning, and teachers explained how assessment was used to help them teach at students’
20
point of need, and how important this was for teachers to resist making assumptions about
what students know.
All teachers changed their conceptions of students’ understanding of place value as a result of
their participation in the Lesson Study.
5.3
Student Understanding of Place Value
5.3.1 Kylie’s Class
Post AI data indicated that students in Kylie’s class who had demonstrated some
understanding of the final question on the original ET (see Appendix A) were more likely to
make some improvement on post AI questions. In contrast, students that gave ‘no response’
were almost all linked to making no progress.
This finding was not surprising. Discussion in the teaching team revealed that the majority of
students’ number sentences related to the number of blocks in the first question (see
Appendix A), and were almost all indicative of additive thinking, showing no evidence of
students’ understanding of multiplicative structures associated with the task.
Typically, responses showed number sentences such as 20 + 6 =, 10 + 10 + 6 = or similar.
These responses provided evidence that students were noting groups of ten but not exploring
multiplicative countable units. They were not relating the 2 in 26 to 2 X 10 and the six as 6
ones, which could cause further misconceptions, whereby students were more likely to view
the 2 as 20. This could indicate to students that the number is 206 rather than 26, due to the
placement of an unnecessary zero, or students believing there were 20 tens.
Marcus was another student who did not respond to the last question of the ET but showed a
multiplicative response of 2 tens and 6 ones instead of writing a number sentence in the
question before (see Appendix A). He was able to answer the first question of the post AI test
multiplicatively. His misconception about the different representations on the second
question remained. This indicated that he was able to succeed on the first question by having
learnt the language associated with 26 at face value. However, he did not necessarily
understand the connection between multiplicative and additive structures of place value in
terms of, tens as a countable unit and how numbers can be represented differently.
Like Marcus, Reja was an exception to the finding and did not respond to the last question on
the ET, but, in contrast to Marcus, was able to make some progress on the post delayed test.
Her number sentence was written as 10+10+1+1+1+1+1+1+1=26; her results showing she
was only able to progress to as far as understanding additive structures of place value.
The number of students that had not responded indicated that the ET was too ambiguous and
unclear for students and therefore prompted the team to change the ET. Additionally, the
responses that students gave on the ETs made their thinking much more evident to teachers.
21
Figure 4. Exit Task responses from the first lesson.
Student ET results highlight that students were able to perform relatively well on some place
value tasks without having a robust conceptual knowledge. The ET, as shown in Figure 4,
exemplifies students’ ability to count the blocks, and draw them using longs and minis (tens
and ones). In Question One of the post interview, 75% of students were able to count the 26
counters and name the 2 as either 20 or 2 tens, but this number significantly dropped when a
distractor/ different representation was presented (see Appendix E & F). This result
emphasises that when students are asked relevant questions that delve deeper into that
conceptual knowledge, students are unable to relate their knowledge of the number to place
value. Results generally showed face value understanding.
5.3.2 Miranda’s Class
After the second lesson implementation there was a correlation between students that made explicit
reference to tens and ones in the last question of the ET, and those who were able to succeed in the
post delayed test to the highest degree by providing multiplicative responses.
Figure 5. Exit Task response from the second lesson.
The students who made the greatest progress were able to write a number sentence in additive
terms in the ET and match it to the number of blocks, and then refer to units of ten in the last
question of the ET, as shown in Figure 5. Typically these responses discussed how Julie had
been ‘confused’ because each group only had five buttons. Therefore two groups were
needed to make ten. Responses such as these showed an understanding of the place value in
both additive and multiplicative terms, and an understanding of the significance of ten.
22
However there were some exceptions to this finding.
Interestingly, Gus, the student we met earlier, did not write a number sentence to match the
number of blocks shown in the ET (see Appendix B) but was able to respond with a
multiplicative response to the last question of the post AI. It is apparent that he and one other
student may not have made a cognitive connection between 20 objects as 2 tens to bridge his
multiplicative interpretation of place value to an understanding of its additive, renamed
structure.
Additionally, Gus did answer Question One of the post AI to the highest level. Whilst it was
one which required a multiplicative response, the lesson had emphasised the language of tens
and ones. This result appeared to be a result of face value understanding associated with the
learned language around written numbers, similar to Marcus in the first lesson.
Interestingly, there was a significant increase in the number of correct responses in relation to
how students represented tens and ones to match how many blocks there were. This finding
would suggest that there was improvement in students’ understanding about ten as a unit and
therefore more attention to detail was given by students when drawing their tens and ones in
the ET.
5.3.3
Pablo’s Class
Figure 6. Exit Task response from the third lesson.
After the third lesson was implemented, there were some counterintuitive results, which upon
further analysis provided some interesting findings. The ET results showed that students were
not responding as expected to the last question; rather, they were rarely making reference to
units of ten to explain Julie’s error. Students were mostly referring to how she had grouped
by fives. There were rarely explicit links made to the idea that 2 groups of five would make
ten and there were two tens.
This was an unexpected result. Furthermore, it was expected that students that made
reference to ten would have made progress, but of the two students that did make reference to
ten, one of them was unable to maintain or understand the significance of this unit in the post
AI, and the other had already achieved a perfect score in the pre AI so there was little we
could conclude from his data.
Whilst the comparison between 5 and ten was not explicit in most responses, it became
evident that most students had given multiplicative responses. Therefore the similarity
between responses from Miranda’s students and Pablo’s class was that; students who
provided a multiplicative response irrespective of the unit value, such as 4 fives are 20 and
then there is one more group of five and three left to make 23, showed that they could assign
23
a value to one group of objects, and tended to perform best in the post delayed test, with some
exceptions.
Student data and responses provided some unexpected results, however data and student ET
responses would suggest that the process of the Lesson Study lead to improvement in
students’ understanding of place value.
5.4
Conclusions
Prior to the Lesson Study, teachers made assumptions that their students had understood the
foundations of place value. They were unaware of their students’ misconceptions in place
value. However, teacher conceptions changed as a result of collaborative discussions and
evidence of student understanding from ETs and AIs.
Teachers’ use of language, scaffolding practices and intervention strategies showed
improvement by the last Lesson Study cycle. Typically, this was associated with
collaborating to make instruction more explicit. Opportunities to observe colleagues
developed teacher’s understanding of what instruction does and does not work in practice.
Additionally, team discussion continued to develop teachers’ PCK and understanding of the
lesson aims with each lesson implementation.
Using models such as ten frames became a vital tool to highlight the correct language of
place value. When implemented into the lesson, students’ were provided with more
opportunities to demonstrate, explain and justify their learning.
The research highlights that Lesson Study, as a tool for professional learning, improved
teacher practice, particularly when targeting misconceptions. Teachers must have robust PCK
to know when and how to intervene and when to facilitate conversation to enable students to
build consensus.
As teacher PCK improved, students showed increased understanding of place value by
developing their understandings of countable units. These results were shown in post AI data
from the last cohort of students.
Through collaboration of planning, observing and reviewing through Lesson Study, the
hypothesis made at the beginning of the research was actualised. This research demonstrated
that a single lesson can improve students’ understanding of place value, develop teachers’
conceptions of students’ understanding and alter their practice by targeting the Instructional
Core.
5.5
Implications
The systematic enquiry associated with Lesson Study and its collaborative teacher practices
has the potential to change school cultures by ‘deprivatising’ classrooms.
Lesson Study can influence professional learning practice and has the potential to improve
teacher PCK. Additionally, Lesson Study has the potential to influence the way teachers’ use
scaffolding when informed by evidence of students’ learning, inherent in student assessment.
If a whole school culture of professional learning is present, powerful intervention strategies
for teaching areas of Mathematics that are prone to misconceptions, such as place value, can
improve student learning.
24
Lesson Study also has the potential to promote shared practice whereby scaffolding and tasks
are calculated, trialled and effective. This could ensure students are more likely to overcome
misconceptions, or avoid misconceptions entirely. Furthermore this can support students to
have deeply rooted foundational knowledge which provides students with avenues to develop
abstract thinking.
Ultimately, if students are given opportunities to move from concrete to abstract thinking
they are more likely to succeed with Mathematics and build greater confidence to pursue
Mathematics in Secondary school.
25
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28
7 Appendices
Appendix A- Exit Task One
Appendix B- Exit Task Two
Appendix C- Investigation Sheet
Appendix D- Lesson Plans
Appendix E- Pre and Post Assessment Interview Questions
Appendix F- Pre and Post Assessment Rubrics
Appendix G- Observation Proformas
Appendix H- Coded Questions
Appendix I- Teacher Interview Questions and Audio
Appendix J- Teacher Interview Transcript
29
Appendix A
Exit Task One
Name:
Noel wants to use these fences to enclose his paddock. How Using longs and minis, how would Noel correctly represent
many blocks are used for his paddock all together?
this number?
Write a number sentence that shows what your picture
represents?
Draw here if needed.
Does the way we write the number of blocks, have anything
do with how they are grouped in the first diagram? Explain.
Response:
30
Appendix B
Exit Task Two
Name:
Noel wants to use these fences to enclose his paddock. How Write a number sentence that shows what your picture
many blocks are used for his paddock all together?
represents?
Julie looked at this picture and said, “I think that shows the
number 43”, is she correct?
If you do not think Julie is correct, explain why not.
Using longs and minis, how would Noel correctly represent
this number?
31
Appendix C
Investigation Sheet
Show your groups
Number of blocks
Draw or write a number sentence
using tens and ones to show how
many blocks you have
32
Appendix D
Lesson Plans
Lesson One
Learning Activities
and Questions
-different representation
doesn’t structure of
number
1. Grasping the
Problem
Setting
Expected Student
Responses
Teacher response
to students
reaction/things to
remember/
Questions
List some responses How will we use
higher order
questioning?
You have some counters
in front of you, how
would group them to
help you count?
27Five groups of 5 with 2
left over
Choose a student to
write the number up on
the board.
Hmmm? How did…. so
and so count this
collection so quickly?
That’s interesting, I
wonder if 5 has
anything to do with how
I write the number of
counters there?
I wonder if 2 has
anything to do with the
number of counters
there are in my picture?
I wonder about seven
(there is no distractor
here)
grouped the fives
together to make ten
counted by ones
Evaluation
This section
worked well but
was a little bit
long, we should
condense this.
Apply
What strategy would
you use to count
them?
pushed the counters
together and counted
by two
counted by fives
There are five in
each group
There are five
groups
No, because there is
no five in the
number
Create
How could you
prove that your
groups are related to
the number of
blocks?
Because 2 is twenty
and it has 4 fives
and 7 has a five in it
too.
Analyse- comparing
How could we work
out how many tens
that is?
There are two left
over.
UnderstandingWhat are we trying
to understand?
The language of
two tens and some
more, was very
clear and students
begun to use this
language also.
Students were able
to verbalise that
two groups could
be added to make
groups of tenresulting in two
tens and seven
more
33
There are twenty?
There are two tens.
There are 2 tens
Point out that the
two is two tens and
we use the fives to
make ten.
2. Presentation of
the problem
format
What’s this?- show a
picture of a farm or zoo
enclosure.
-make this section
more explicitly
related to place
value.
Ask students to
separate their fence
sides.
What do we need to
keep the animals in?
Give each pair 24 mab
ones.
Students may have
Can you make a four
sided enclosure to keep difficulty with the
the tigers in using these? MAB, They may
double count each
Okay, so can anyone see corner block.
-some may count 28
any groups that can be
made from the length of -tens and ones
-repeated addition of
their enclosure fence?
the length of each
-ask students to share
side eg. 6+6+6+6
their responses on the
-multiplicative
board.
responses 2x10+
2x2
- 8+8 +4+4 and
So how many did we
other combinations
need all together?
How do we write the
28,
number of blocks,
24
needed to make this
enclosure?
Point out the
separate groups by
emphasising the
separation of each
fence
Children were
relatively quiet
here, the focus
needs to remain on
the tens and ones.
Ensure students
verbalise the
connection
between the
different
representations
and how the
number is
recorded.
Ensure students
use the materials
to prove that each
representation is
equal to 24 as two
tens and 4 more
Who can show me
what groups they
used to make this
number?
Do the groups tell us
how to write this
number?
How we could use
34
one of these (tens
frame, unifix, paddle
pop sticks, MAB)
Can someone show
me using something
else?
Everyone has used
something different
to show me the same
answer, but what is
the same about all
these solutions?
3. Solving the
Main Problem
1. Draw your
enclosure.
2. Show your
groups
3. Number of
blocks
4. Draw or write a
number sentence
to show how
many blocks you
have.
Differentiation
was evident in this
section.
prepare a hand out
and have them write
on it.
There are 2 groups
Show them how to
of … and …group the sides of
answers which result each enclosure.
in a double count.
provide unifix, tens
frames and pop
sticks to enable
them to regroup into
tens and ones.
Are we expecting
students to show a
different number
sentence for each
enclosure? This
could be good as
an extension task.
When modelling
the number
sentence, we need
to ensure it is
based on tens and
ones.
The task requires
students to draw
enclosures. This
was too time
consuming, in
retrospect the
drawing of the
enclosures is not
important
therefore this may
be best removed.
35
3.
Polishing and
Reporting Individual
Solution Methods
encourage the use of
symbols.
Have a variety of
students blue tac their
solutions on the board;
both their written
number sentence to
represent tens and ones
and their groups
Have students explain
how they recorded.
group the double
digit responses and
group the three digit
responses on the
board
for example 46
40+6
4 tens 6 ones
groups of ten and
ones
10+10+10+10+6
regrouping
responses2 tens +26 ones
Analyse
What stays the
same, what
changes?
CreateWhat would happen
if we had 231
cubes?
This section was
quickly paced and
students were able
to see the pattern
on the record
sheets- the
representations
change but the
number of blocks
remains the same.
Many number
sentences were not
indicative of place
value knowledge.
This needs to be
modelled or the
wording on the
recording sheet
needs to be
changed.
4x10 +6
(4x10) +6
or 146
10² +(4 x10) + 6
14 tens 6 ones etc.
Evaluating
Have students build
consensus of the
most efficient ways
of writing ‘tens and
some more’
1.Summary and
Announcement of Next
Lesson
point out that even
though the size of the
groups change and the
representation of that
number can change it is
This was also very
clear. Our
learning intention
needs to be
reworded so that
students
understand the
purpose of the
lesson from the
beginning.
36
still written in terms of
tens and some more
Last question on
exit task elicited
different
responses, this
needs to be
reworded to
ensure all students
are clear on what
it is asking.
Each teacher/class
may need a
different follow up
lesson depending
on the students’
level of
understanding.
37
Lesson Two
Learning intention: To investigate different groupings of numbers and how we write them
Learning Activities
and Questions
different
representation doesn’t
structure of number
4. Grasping the
Problem
Setting (5mins)
Expected Student
Responses
Teacher response
to students
reaction/things to
remember/
Questions
List some responses How will we use
higher order
questioning?
You have some counters
in front of you, how
would group them to
help you count?
27Five groups of 5 with 2
left over
Choose a student to
write the number up on
the board.
How did…. so and so
count this collection so
quickly?
That’s interesting, I
wonder if 5 has
anything to do with how
I write the number of
counters there?
I wonder if 2 has
anything to do with the
number of counters
there are in my picture?
I wonder about seven
(there is no distractor
here)
Evaluation
worked well, range
of strategies used,
articulated well.
link was made to
tens and ones
(place value)
grouped the fives
together to make ten
counted by ones
Apply
What strategy would
you use to count
them?
pushed the counters
together and counted
by two
counted by fives
There are five in
each group
There are five
groups
No, because there is
no five in the
number
Because 2 is twenty
and it has 4 fives
and 7 has a five in it
too.
There are two left
over.
Emphasise correct
tens using materials
Create
How could you
prove that your
number is always
24?
Some students
verbalised the
misconception
Analyse- comparing
How could we work
out how many tens
that is?
UnderstandingWhat are we trying
to understand?
Point out that the
38
There are twenty?
two is two tens and
we use the fives to
make ten.
There are two tens.
There are 2 tens
5. Presentation of
the problem
format (20mins)
What’s this?- show a
picture of a farm or zoo
enclosure.
Ask students to
separate their fence
sides.
What do we need to
keep the animals in?
Give each pair 24 mab
ones.
Students may have
difficulty with the
MAB, They may
double count each
corner block.
Okay, so can anyone
see any groups that can -some may count 28
be made from the length -tens and ones
of their enclosure fence? -repeated addition of
the length of each
-ask students to share
side eg. 6+6+6+6
their responses on the
-multiplicative
board.
responses 2x10+
2x2
- 8+8 +4+4 and
So how many did we
other combinations
need all together?
How do we write the
28,
number of blocks,
24
needed to make this
enclosure?
Can you make a four
sided enclosure to keep
the cows in using these?
Point out the
separate groups by
emphasising the
separation of each
fence
Emphasis seems to
be on the
groupings rather
than the place
value. We need to
spend more time
emphasising the
place value by
taking the
groupings and
demonstrating
how these can be
grouped as tens
and ones using a
tens frame.
(tens and ones
reflects the way we
write our number)
Who can show me
what groups they
used to make this
number?
Three solutions
drawn on the
whiteboard
simultaneously
39
Does the way that we
group the number,
change the way we
write it? Why? Why is it
24 when e.g. it’s two
groups of 5 and two
groups of 7?
Do the groups tell us
how to write this
number? (no it’s
written using place
value – tens and
ones)
How we could use
one of these (tens
frame, unifix, paddle
pop sticks, MAB)
Can someone show
me using something
else?
Everyone has used
something different
to show me the same
answer, but what is
the same about all
these solutions?
6. Solving the
Main Problem
(20mins)

Make your
enclosure.

Show your
groups

Number of
blocks

Draw or write a
number sentence
– emphasis on
place value, to
show how many
blocks you have.
prepare a hand out
and have them write
on it.
There are 2 groups
Show them how to
of … and …group the sides of
answers which result each enclosure.
in a double count.
Students were
confused about the
template and some
students came up
with one solution
but spread it
across the
template.
Students spend a
long time drawing
their groups
provide unifix, tens
frames and pop
sticks to enable
them to regroup into
tens and ones.
40
4.
Polishing and
Reporting Individual
Solution Methods
(10mins)
encourage the use of
symbols.
different approach
was taken by some
students so the
comparison was
difficult to make
Have a variety of
students blue tac their
solutions on the board;
both their written
number sentence to
represent tens and ones
and their groups
group the double
digit responses and
group the three digit
responses on the
board
responses were not
necessarily
indicative of place
value (the number
sentences)
Have students explain
how they recorded.
Analyse
What stays the
same, what
changes?
for example 46
40+6
4 tens 6 ones
Does the way that we
groups of ten and
group the number,
ones
change the way we
10+10+10+10+6
write it? Why? Why is it regrouping
24 when e.g. it’s two
responsesgroups of 5 and two
2 tens +26 ones
groups of 7?
4x10 +6
(4x10) +6
or 146
10² +(4 x10) + 6
CreateWhat would happen
if we had 231
cubes?
14 tens 6 ones etc.
Evaluating
Have students build
consensus of the
most efficient ways
of writing ‘tens and
some more’
1.Summary and
Announcement of Next
Lesson (5mins)
point out that even
though the size of the
groups change and the
representation of that
41
number can change it is
still written in terms of
tens and some more
42
Lesson Three
Learning intention: To investigate different groupings of numbers and how we write them
Learning Activities
and Questions
different
representation doesn’t
structure of number
7. Grasping the
Problem
Setting (5mins)
Expected Student
Responses
Teacher response
to students
reaction/things to
remember/
Questions
List some responses How will we use
higher order
questioning?
You have some counters
in front of you, how
would group them to
help you count?
27
Five groups of 5 with 2
left over
Choose a student to
write the number up on
the board.
How did…. so and so
count this collection so
quickly?
That’s interesting, I
wonder if 5 has
anything to do with how
I write the number of
counters there?
I wonder if 2 has
anything to do with the
number of counters
there are in my picture?
I wonder about seven
(there is no distractor
here)
Evaluation
Language clear,
link made to place
value.
Enabled students
explain in their
own way.
grouped the fives
together to make ten
counted by ones
Apply
What strategy would
you use to count
them?
pushed the counters
together and counted
by two
counted by fives
There are five in
each group
There are five
groups
No, because there is
no five in the
number
Because 2 is twenty
and it has 4 fives
and 7 has a five in it
too.
There are two left
over.
Emphasise correct
tens using materials
Analyse- comparing
How could we work
out how many tens
that is?
UnderstandingWhat are we trying
to understand?
Point out that the
two is two tens and
we use the fives to
make ten.
43
There are twenty?
There are two tens.
There are 2 tens
8. Presentation of
the problem
format (20mins)
Recorded a variety
of different
number sentencesreinforced
different
groupings of ten.
5mins
What’s this?- show a
picture of a farm or zoo
enclosure.
Questions were
open endedStudents guided
the conversationJustifying,
explaining.
What do we need to
keep the animals in?
Give each pair 24 mab
ones.
Can you make a four
sided enclosure to keep
the cows in using these?
Students may have
difficulty with the
MAB, They may
double count each
corner block.
-some may count 28
-tens and ones
-repeated addition of
the length of each
side eg. 6+6+6+6
-multiplicative
responses 2x10+
2x2
- 8+8 +4+4 and
other combinations
10 mins
Have three students
with different solutions
draw their
representations on the
whiteboard. These
should be drawn up
during task time.
So how many did we
have all together?
28,
24
Point out the
separate groups by
emphasising the
separation of each
fence
Elaborated on
more sophisticated
answers,
multiplication
Do the groups tell us
how to write this
number? (no it’s
written using place
value – tens and
ones)
Model how a tens
frame can be used to
write/explain a
corresponding
number sentence.
44
Create
How could you
prove that your
number is always
24?
What do you notice
about these
solutions? (direct
them to 2 tens and 4
ones)
Does the way that
we group the
number, change the
way we write it?
Why? Why is it 24
when e.g. it’s two
groups of 5 and two
groups of 7?
9. Solving the
Main Problem
(20mins)
Use 28 to model the
expectations for filling
out the template, on
the IWB ie. each line
requires a different
grouping- this will
require modelling using
lines to represent how
many or dots in circles
 Show your
groups

Number of
There are 2 groups
hand out a recording
of … and …sheet for the
answers which result students to fill out
in a double count.
Intervene as
Different
responses,
students explored
with renaming and
different
representations.
45
blocks

Draw or write a
number sentence
– emphasis on
place value, to
show how many
blocks you have.
Some students may
miscount blocks due
to one-to-one
correspondence
Student may not
know how to share
blocks into four
sides.
necessary
provide tens frames
enable them to
regroup into tens
and ones
Some students
filled out the no. of
blocks first (they
saw the pattern)
select two samples
from students for
each way that the
numbers are
grouped
differently.
Display one under
the other, in three
columns.
Very effective
discussion.
Discuss the number of
blocks-tell them that
each bag will have an
even number of blocks
3.
Polishing and
Reporting Individual
Solution Methods
(10mins)
Have a variety of
students blue tac their
solutions on the board;
both their written
number sentence to
represent tens and ones
and their groups
Have students explain
how they recorded.
Does the way that we
group the number,
change the way we
write it? Why? Why is it
24 when e.g. it’s two
groups of 5 and two
groups of 7?
for example 46
40+6
4 tens 6 ones
groups of ten and
ones
10+10+10+10+6
regrouping
responses2 tens +26 ones
4x10 +6
(4x10) +6
or 146
10² +(4 x10) + 6
14 tens 6 ones etc.
Students described
the pattern. They
were using place
value language.
encourage the use of
symbols.
Analyse
What stays the
same, what
changes?
CreateWhat would happen
if we had 231
cubes?
Evaluating
Have students build
consensus of the
most efficient ways
of writing ‘tens and
some more’
46
1.Summary and
Announcement of Next
Lesson (5mins)
come back to the
learning intention
point out that even
though the size of the
groups change and the
representation of that
number can change it is
still written in terms of
tens and some more
Lesson Plans templates taken from: Fernandez, C., & Yoshida, M. (2004). Lesson Study: A
Japanese Approach to Improving Mathematics Teaching and Learning. Mahwah, NJ:
Lawrence Erlbaum.
47
Appendix E
Pre and Post Assessment Interview Questions
Materials


26 counters in a suitable jar or container
7 bundles of ten icy pole sticks or straws and 22 single sticks or straws
Instructions
Bold type indicates what should be said.
Question One
Empty container of counters in front of student and ask: “Can you count these as quickly as
possible and write down the number please?” Note how the count is organised and what is
recorded.
If not 26, ask, “Are you sure about that? How could you check?”
Once student has recorded 26, circle the 6 in 26 and ask, “Does this (point to the 6) have
anything to do with how many counters you have there?” Indicate the collection. Note
student’s response.
Circle the 2 in 26 and repeat the question. Note student’s response. Place counters back in the
container.
Distractor question.
Place bundles and sticks in front of the student and ask, “Can you make 34 using these
materials please?” Note student’s response. If student asks or moves to unbundle a ten, say,
“Before you do that, is there any way you could use these (pointing to the bundles of ten)
to make 34?” Note student’s response. Remove sticks.
Question Two
Tip out the container of 26 counters and ask student to count these again and record the
number. Note response, then ask, “Can you put these into groups of four please?” Once
this is completed, point to the 26 that has been recorded and circle the 6. Ask: “Does this
have anything to do with how many counters you have?” Circle the 2 in 26 and repeat the
question. Note student responses.
Taken from: Department of Education and Early Childhood Development. (2012).
Assessment for Common misunderstandings- level 2 Place-Value. Retrieved from
http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/a
ssessment/Pages/lvl2place.aspx.
48
Appendix F
Pre and Post Assessment Rubrics
Question One
Student responses to this task indicate the meanings they attach to 2-digit numerals. A
version of this task was originally employed by Ross (1989) who identified five stages in the
development of a sound understanding of place-value, each of which appears in some form in
the advice below.
Response
Observed
response
Suggestions
Interpretation/Suggested teaching
response
Little/no
response
May not understand task

Response given
but not
indicative of
strong placevalue
knowledge, eg,
refers to 6 ones
or physical
arrangement
such as “2
groups of 3” for
circled 6, and
“twenty” for
circled 2.
Says 6 ones and
2 tens fairly
quickly
Names
Repeat at a later date
Suggests 26 is understood in terms of
ones, or 20 (ones) and 6 ones, may not
trust the count of 10 or see 2 as a count
of tens


Check extent to which child trusts
the count for 10 by counting large
collections (see Tool 2.2)
Practice making, naming and
recording tens and ones,
emphasising the count of tens in
the tens place and the count of
ones in the ones place
Appears to understand the basis on which
2-digit numbers are recorded

Consolidate 2-digit place-value by
comparing 2 numbers (materials,
words and symbols),
ordering/sequencing (by ordering
5 or more 2-digit numbers or
placing in sequence on a rope
from 0 to 100), counting forwards
and backwards in place-value
parts starting anywhere (eg, 27,
37, 47 (clap), 46, 45, 44, 43, …),
and by renaming (eg, 45 is 4 tens
and 5 ones or 45 ones)
49

Consider introducing 3-digit
place-value
Question Two
Student responses to this task indicate the strength of their understanding of place-value by
exploring the extent to which they can be distracted by the regrouping and the perceptual
image it presents (6 groups of 4 and 2 ones remaining). Interestingly, some students who
referred to the 2 in 26 as “twenty” in the first instance are prompted to refer to the 2 in 26 as
“2 tens” after the grouping exercise.
Response
Observed
response
Suggestion
Interpretation/Suggested teaching
response
Little/no
response or
refers to 6 as
the number of
groups of 4 and
2 as the 2
remaining ones
Distracted by the visual arrangement to
override whatever else they may know
about what ‘26’ means, suggests little/no
place-value knowledge. May not
understand task, does not trust the count
of 10


Names
Check extent to which child trusts
the count for 10 by counting large
collections (see Tool 2.2) and
review subitising and part-partwhole ideas for 10 (see Level 1)
Practice making, naming and
recording tens and ones,
emphasising the count of tens in
the tens place and the count of
ones in the ones place
Is not distracted Suggests place-value ideas not well
by visual image established, may not trust the count of 10
or regrouping,
but refers to 2
 Check trust the count, review
as “twenty”
subitising and part-part-whole
ideas for 10 and making, naming
and recording tens and ones (see
above)
Consolidate 2-digit place-value by
comparing 2 numbers (materials, words
and symbols), ordering/sequencing (by
50
ordering 5 or more 2-digit numbers or
placing in sequence on a rope from 0 to
100), counting forwards and backwards
in place-value parts starting anywhere
(eg, 27, 37, 47 (clap), 46, 45, 44, 43, …),
and by renaming (eg, 45 is 4 tens and 5
ones or 45 ones)
Says 6 ones and Appears to understand the basis on which
2 tens fairly
2-digit numbers are recorded
quickly
 Consolidate 2-digit place-value by
comparing 2 numbers (materials,
words and symbols),
ordering/sequencing (by ordering
5 or more 2-digit numbers or
placing in sequence on a rope
from 0 to 100), counting forwards
and backwards in place-value
parts starting anywhere (eg, 27,
37, 47 (clap), 46, 45, 44, 43, …),
and by renaming (eg, 45 is 4 tens
and 5 ones or 45 ones)
 Consider introducing 3-digit
place-value
Taken from: Department of Education and Early Childhood Development. (2012).
Assessment for Common misunderstandings- level 2 Place-Value. Retrieved from
http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/a
ssessment/Pages/lvl2place.aspx.
51
Appendix G
Observation Proformas
Student
Verbal and Physical Evidence
Student response and strategy
Evidence of student understanding
Question or suggestion for
the team about the student
response
52
Teacher Questions
Situation/what was
asked
Did students respond
as expected?
What could the teacher
action be, to support or
extend students?
Suggestions for
debrief:
53
Student Evidence
Written
What evidence was collected/
photographed?
What does it show in relation to
the lesson goal?
Suggestion or question for
the team.
54
Facilitator
Lesson study
teacher: What
worked well?
Each team member
report on findings
Exit task review
What changes need to
made to the lesson
and/or exit task?
10min
10 mins per member
20 min
30min
55
Appendix H
Coded Questions
The Taxonomy Table
Total
Prompt
Probe
Create
Evaluate
Analyse
Apply
Clarify
Remember
Understand
Bloom’s Taxonomy Categories
Kylie
7
12
2
1
3
-
-
6
-
31
Miranda
Pablo
Total
4
6
17
6
2
20
3
4
9
1
3
5
5
4
12
1
1
2
0
3
4
4
23
24
9
Adapted from Bloom’s Taxonomy, taken from:
Walsh, J.A. & Sattes, B.D. (2005). Quality Questioning: Research-Based Practice to Engage Every Learner. Washington, D.C: Corwin Press.
56
Appendix I
Teacher Interview Questions
Q1. How did the lesson study process impact your conceptions of students’ understanding in place
value?
Q2. What were the most significant findings for you about the student’s understanding about place
value? How will these findings affect your future planning of place value lessons?
Q3. Tell me about any changes in your planning and teaching practice that are evident to you, if any,
as a result of your involvement in the Lesson Study.
Q4. Can you describe your learning as a result of the Lesson Study process, if any?
Q5. How important do you think the use of student data is to inform your planning?
Q.6 How important do you think collaboration when planning?
Q.7 Can you describe what has had the greatest impact on your teaching as a result of your
involvement in the Lesson Study process?
57
Appendix J
Transcript
Kylie
Q1. How did the lesson study process impact your conceptions of students’ understanding in
place value?
This really opened my eyes to the fact that, [pause], um you really have to know what you’re
looking for to really uncover those misconceptions because I didn’t realise that some of my
kids who I thought were probably a lot more advanced in their conceptual understanding of
place value were actually just good at speaking the lingo and following processes without,
um, really understanding why they were doing what they were doing so, um. Yeah, I think
it’s just made me more aware, you know, about the importance of actively looking for those
misconceptions instead of waiting for them to pop, cause they might not.
Q.7 Can you describe what was the most valuable part of the Lesson Study process for you?
I was so surprised by the information that, you know, came out of that initial testing.
Miranda
Q1. How did the lesson study process impact your conceptions of students’ understanding in
place value?
Yep, the Lesson Study absolutely blew my mind away, in terms of the misconceptions that I
had no idea that my students had. I think a lot of, it showed a lot of the testing we do, ahh
both the formal testing and informal testing, was still testing, [pause] testing conceptions that,
we knew that they would be able to tell us the right answers, but when we delved deeper they
actually didn’t understand as much as we thought. We didn’t , so it’s really made us think
about our teaching and our, and our.. yeah, assessments.
58