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Lesson Study: Improving Teachers’

Conceptions of Students’

Understanding in Place Value

Emilie Dawson

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Contents 1. 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

2 Literature Review ............................................................................................................................ 4

2.1 Background of Place Value...................................................................................................... 4

2.2 Base Ten Materials .................................................................................................................. 5

2.3 Lesson Study ........................................................................................................................... 7

3 Methodology ................................................................................................................................... 8

3.1 Outline and Justification ......................................................................................................... 8

3.2 Participants ............................................................................................................................. 9

3.3 Data Collection Tools .............................................................................................................. 9

3.4 Lesson Study Stages .............................................................................................................. 11

4 Results ........................................................................................................................................... 12

4.1 Kylie’s Class ........................................................................................................................... 12

4.2 Miranda’s Class ..................................................................................................................... 13

4.3 Pablo’s Class .......................................................................................................................... 15

5 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

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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).

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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.

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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.

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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).

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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 pre-

grouped 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).

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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

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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.

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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.

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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

Tool

Time of collection 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

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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 re-

interviewed 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.

http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/misunderstandings.aspx

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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.

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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 Student

Results

Pre AI

Student

Results

Post AI

Question One

No response 4 3

Additive Place Value Knowledge 7 4

Multiplicative Place Value Knowledge 1 5

Question Two

No response 10 8

Additive Place Value knowledge 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.

13

Table 3

Exit Task Response Types, Lesson One

Last Question No Response Misconception Additive Multiplicative

Students’ responses 6 1 - 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


Results

Pre AI

Student

Results

Post AI

Question One

No response 5 1



Question Two

No response 10 6



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,

14

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

No

Response

Misconception

Reference

to fives

Reference

to tens

Number of responses - - 5 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.

15

4.3 Pablo’s Class

Table 6

Students’ Pre and Post Interview Results from Pablo’s Class


Results

Pre AI

Students

Results

Post AI

Question One

No response 6 0



Question Two

No response 6 2



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 pre-

assessment, 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

No

Response

Misconception

Reference

to fives

Reference

to tens

Number of responses - - 10 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.

16

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.

17

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.

18

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.

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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.

20

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’

21

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.

22

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.

23

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

24

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.

25

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.

26

6 References

Askew, M., Brown, M., Rhodes, V., Wiliam, D., & Johnson, D. (1997). The contribution of

professional development to the effectiveness in the teaching of numeracy. Teacher

Development: An International Journal of Teachers’ Professional Development,1(3),

335-356.

Bartolini Bussi, M., G. (2011). Artefacts and utilization schemes in mathematics teacher

education: Place value in early childhood. Journal of Mathematics Teacher

Education, 14(2), 93-112.

Broadbent, A. (2004). Understanding place value: A case study of the base ten game.

Australian Primary Mathematics Classroom, 9(4), 45-46.

City, E. A., Elmore, R. F., Fiarman, S. E., & Teitel, L. (2009). Instructional rounds in

education: A network approach to improving teaching and learning. Cambridge, MA:

Harvard Education Press.

Collins, D, E., & Wright, R.. (2009). Developing conceptual place value: Instructional design

for intensive intervention. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing

divides: (Proceedings of the 32nd annual conference of the Mathematics Education

Research Group of Australasia, pp. 169-653). Palmerston North, NZ: MERGA.

Council of Australian Governments. (2008). National numeracy review report. Retrieved

from http://www.coag.gov.au/sites/default/files/national_numeracy_review.pdf.

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.

Department of Education and Early Childhood Development. (2013). Mathematics Online

Interview. Retrieved from

http://www.eduweb.vic.gov.au/edulibrary/public/teachlearn/student/mathscontinuum/

onlineinterviewbklet.pdf.

Fernandez, C., & Yoshida, M. (2004). Lesson study: A Japanese approach to improving

mathematics teaching and learning. Mahwah, N.J: Lawrence Erlbaum Associates.

Fuson, K. (1990). Issues in place-value and multidigit addition and subtraction learning and

teaching. Journal for Research in Mathematics Education, 21(4), 273- 279.

Goos, M., Dole, S., & Makar, K. (2007). Designing professional development to support

teachers’ learning in complex environments. Mathematics Teacher Education and

Development, 8(1), 23-47.

Hiebert, J., &Wearne, D. (1992). Links between teaching and learning place value with

understanding in first grade. Journal for Research in Mathematics Education, 23(2),

98-122.

http://www.coag.gov.au/sites/default/files/national_numeracy_review.pdf

http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/lvl2place.aspx


http://www.eduweb.vic.gov.au/edulibrary/public/teachlearn/student/mathscontinuum/onlineinterviewbklet.pdf

http://www.eduweb.vic.gov.au/edulibrary/public/teachlearn/student/mathscontinuum/onlineinterviewbklet.pdf

27

Hiebert, J., & Wearne, D. (1993). Instruction tasks, classroom discourse, and students’

leaning in second-grade arithmetic. American Educational Research Journal, 30 (2),

393-425.

Isoda M., Stephens, M., Ohara, Y., & Miyakawa, T (Eds.). (2007). Japanese Lesson Study in

Mathematics: Its impact, diversity and potential for educational improvement (1st

Ed.). Singapore: World Scientific.

Jaworski, B. (1998). Mathematics teacher research: Process, practice and the development of

teaching. Journal of Mathematics Teacher Education, 1(1), 3-31.

Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications

for primary grades. Journal of Research for Childhood Education, 1(2), 75-86.

Marshman, M., Pendergast, D., & Brimmer, F. (2011). Engaging the middle years in

mathematics. In Julie Clark, Barry Kissane, Judith Mousley, Toby Spencer & Steve

Thornton (Eds.), Mathematics: Traditions and [New] Practices (Proceedings of the

34th annual conference of the Mathematics Education Research Group of Australasia

and the Australian Association of Mathematics Teachers, pp. 500-507). Adelaide:

AAMT and MERGA.

Ong, E.W., Lim, C.S., & Ghazali, M. (2010). Examining the changes in novice and

experienced mathematics teachers’ questioning techniques through the lesson study

process. Journal of Science and Mathematics, 33(1), 86-109.

Resnick, L, B. (1983). A developmental theory of number understanding. In Ginsburg (Ed.),

The development of mathematical thinking (pp. 109-151). New York: Academic

Press.

Reys, R.E., Lindquist, M.M., Lambdin, D.V., Smith, N.L., Rogers, A., Falle, J., Frid, S., &

Bennet, S. (2012). Helping children learn mathematics (1st Ed.). Milton, Queensland;

JohnWiley & Sons.

Rogers, A. (2012). Steps in developing a quality whole number place value assessment for

Years 3-6: Unmasking the “experts”. In J. Dindyal, L. P. Cheng & S. F. Ng (Eds.),

Mathematics education: Expanding horizons (Proceedings of the 35th annual

conference of the Mathematics Education Research Group of Australasia, pp. 648-

653). Singapore: MERGA.

Ross, S. (1989). Parts, wholes, and place value: A developmental view. National Council of

Teachers of Mathematics, 36(5), 47-51.

Sagor, R. (1991, March). What project LEARN reveals about collaborative action research.

Educational Leadership, 48(6), 6-10.

Saxton, M., & Cakir, K. (2006). Counting-On, trading and partitioning: Effects of training

and prior knowledge on performance on base-10 tasks. Child Development, 77(3),

767-785.

28

Shulman, L.S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard

Educational Review, 57(1), 1-20.

Siemon, D., Beswick, K., Brady, K., Clarke, J., Faragher., R & Warren, E. (2011). Teaching

mathematics foundations to middle years. Victoria, Australia: Oxford.

Steinle, V., & Stacey, K. (2004). A longitudinal study of students understanding of decimal

notation: An overview and refined results. In, I. Putt, R. Faragher, & M. McLean

(Eds.), Mathematics education for the third millennium: Towards 2010 (Proceedings

of the 27th annual conference of the Mathematics Education Research Group of

Australasia pp. 541-548). Sydney: MERGA.

Stephens, M. (2009). Numeracy in practice: teaching, learning and using mathematics.

Retrieved from

http://www.coag.gov.au/sites/default/files/national_numeracy_review.pdf.

Tachibana, M. (2007). Teaching and assessment based on teaching guides. In M. Isoda, M.

Stephens, Y. Ohara, & T. Miyakawa (Eds.), Japanese Lesson Study in Mathematics:

Its impact, diversity and potential for educational improvement (pp. 42-47).

Singapore: World Scientific.

Thomas, D. J. Jr. (2011). Becoming a teacher leader through action research. Kappa Delta Pi,

47(4), 170-173.

Thomas, N. (2004). The Development of structure in the number system. In M.J. Hoines &

A. Fuglestad (Eds.), (Proceedings of the 28th

annual conference of the International

Group for the Psychology of Mathematics Education, pp. 305-312). Bergen, Norway:

Bergen University College.

Thomas, N. D., & Mulligan, J. (1999). Children's Understanding of the Number System.

Retrieved from

http://www.merga.net.au/documents/RP_Thomas_Mulligan_1999.pdf.

Victorian Curriculum Assessment Authority. (2012). AusVELS. Retrieved from

http://ausvels.vcaa.vic.edu.au/.

Walsh, J.A. & Sattes, B.D. (2005). Quality Questioning: Research-Based Practice to Engage

Every Learner. Washington, D.C: Corwin Press.

Watson, A., & De Geest, E. (2005). Principled teaching for deep progress: Improving

mathematical learning beyond methods and materials. Educational Studies in

Mathematics, 8(2), 209-234.

http://www.coag.gov.au/sites/default/files/national_numeracy_review.pdf

http://www.merga.net.au/documents/RP_Thomas_Mulligan_1999.pdf

http://ausvels.vcaa.vic.edu.au/

29

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

30

Appendix A

Exit Task One

Name:

Noel wants to use these fences to enclose his paddock. How many blocks are used for his paddock all together?

Draw here if needed. Response:

Using longs and minis, how would Noel correctly represent this number? Write a number sentence that shows what your picture represents? Does the way we write the number of blocks, have anything do with how they are grouped in the first diagram? Explain.

31

Appendix B

Exit Task Two

Name:

Noel wants to use these fences to enclose his paddock. How many blocks are used for his paddock all together?

Using longs and minis, how would Noel correctly represent this number?

Write a number sentence that shows what your picture 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.

32

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

33

Appendix D

Lesson Plans

Lesson One

Learning Activities

and Questions

-different representation

doesn’t structure of

number

Expected Student

Responses

Teacher response

to students

reaction/things to

remember/

Questions

Evaluation

1. Grasping the

Problem

Setting

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.

Hmmm? How did…. so

and so count this

collection so quickly?

List some responses

grouped the fives

together to make ten

counted by ones

pushed the counters

together and counted

by two

counted by fives

How will we use

higher order

questioning?

Apply

What strategy would

you use to count

them?

This section

worked well but

was a little bit

long, we should

condense this.

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)

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.

Create

How could you

prove that your

groups are related to

the number of

blocks?

Analyse- comparing

How could we work

out how many tens

that is?

Understanding-

What 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 ten-

resulting in two

tens and seven

more

34

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.

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 tigers in using these?

Okay, so can anyone see

any groups that can be

made from the length of

their enclosure fence?

-ask students to share

their responses on the

board.

So how many did we

need all together?

How do we write the

number of blocks,

needed to make this

enclosure?

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

28,

24

Ask students to

separate their fence

sides.

Point out the

separate groups by

emphasising the

separation of each

fence

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

-make this section

more explicitly

related to place

value.

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

35

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.

There are 2 groups

of … and …-

answers which result

in a double count.

prepare a hand out

and have them write

on it.

Show them how to

group the sides of

each enclosure.

provide unifix, tens

frames and pop

sticks to enable

them to regroup into

tens and ones.

Differentiation

was evident in this

section.

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.

36

3.

Polishing and

Reporting Individual

Solution Methods

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.

for example 46

40+6

4 tens 6 ones

groups of ten and

ones

10+10+10+10+6

regrouping

responses-

2 tens +26 ones

4x10 +6

(4x10) +6

or 146

10² +(4 x10) + 6

14 tens 6 ones etc.

encourage the use of

symbols.

group the double

digit responses and

group the three digit

responses on the

board

Analyse

What stays the

same, what

changes?

Create-

What would happen

if we had 231

cubes?

Evaluating

Have students build

consensus of the

most efficient ways

of writing ‘tens and

some more’

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.

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.

37

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.

38

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

Expected Student

Responses

Teacher response

to students

reaction/things to

remember/

Questions

Evaluation

4. Grasping the

Problem

Setting (5mins)



would group them to

help you count?

27-


left over

Choose a student to


the board.

How did…. so and so

count this collection so

quickly?

List some responses

grouped the fives


counted by ones

pushed the counters


by two

counted by fives

How will we use

higher order

questioning?

Apply

What strategy would

you use to count

them?

Emphasise correct

tens using materials

worked well, range

of strategies used,

articulated well.

link was made to

tens and ones

(place value)


wonder if 5 has



counters there?

I wonder if 2 has


number of counters




here)

There are five in

each group

There are five

groups


no five in the

number

Because 2 is twenty

and it has 4 fives


too.

There are two left

over.

Create

How could you

prove that your

number is always

24?

Analyse- comparing

How could we work

out how many tens

that is?

Understanding-

What are we trying

to understand?

Point out that the

Some students

verbalised the

misconception

39

There are twenty?

There are two tens.

There are 2 tens

two is two tens and

we use the fives to

make ten.

5. Presentation of

the problem

format (20mins)



enclosure.

What do we need to



ones.

Can you make a four


the cows in using these?

Okay, so can anyone

see any groups that can

be made from the length

of their enclosure fence?

-ask students to share

their responses on the

board.

So how many did we

need all together?

How do we write the

number of blocks,

needed to make this

enclosure?

Students may have

difficulty with the

MAB, They may

double count each

corner block.

-some may count 28

-tens and ones


the length of each

side eg. 6+6+6+6

-multiplicative

responses 2x10+

2x2

- 8+8 +4+4 and

other combinations

28,

24

Ask students to

separate their fence

sides.

Point out the

separate groups by

emphasising the

separation of each

fence

Who can show me

what groups they

used to make this

number?

Three solutions

drawn on the

whiteboard

simultaneously

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)

40

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?


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.

There are 2 groups

of … and …-


in a double count.

prepare a hand out

and have them write

on it.

Show them how to

group the sides of

each enclosure.

provide unifix, tens

frames and pop

sticks to enable

them to regroup into

tens and ones.

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

41

4.

Polishing and


Solution Methods

(10mins)

Have a variety of



both their written

number sentence to


and their groups


how they recorded.


group the number,

change the way we



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

responses-

2 tens +26 ones

4x10 +6

(4x10) +6

or 146

10² +(4 x10) + 6

14 tens 6 ones etc.


symbols.

group the double

digit responses and

group the three digit

responses on the

board

Analyse

What stays the

same, what

changes?

Create-

What would happen

if we had 231

cubes?

Evaluating

Have students build

consensus of the

most efficient ways


some more’

different approach

was taken by some

students so the

comparison was

difficult to make

responses were not

necessarily

indicative of place

value (the number

sentences)

1.Summary and


Lesson (5mins)

point out that even




42



tens and some more

43

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

Expected Student

Responses

Teacher response

to students

reaction/things to

remember/

Questions

Evaluation

7. Grasping the

Problem

Setting (5mins)



would group them to

help you count?

27


left over

Choose a student to


the board.

How did…. so and so

count this collection so

quickly?

List some responses

grouped the fives


counted by ones

pushed the counters


by two

counted by fives

How will we use

higher order

questioning?

Apply

What strategy would

you use to count

them?

Emphasise correct

tens using materials

Language clear,

link made to place

value.

Enabled students

explain in their

own way.


wonder if 5 has



counters there?

I wonder if 2 has


number of counters




here)

There are five in

each group

There are five

groups


no five in the

number

Because 2 is twenty

and it has 4 fives


too.

There are two left

over.

Analyse- comparing

How could we work

out how many tens

that is?

Understanding-

What are we trying

to understand?

Point out that the

two is two tens and

we use the fives to

make ten.

44

There are twenty?

There are two tens.

There are 2 tens

8. Presentation of

the problem

format (20mins)

5mins



enclosure.

What do we need to



ones.

Can you make a four


the cows in using these?

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?

Students may have

difficulty with the

MAB, They may

double count each

corner block.

-some may count 28

-tens and ones


the length of each

side eg. 6+6+6+6

-multiplicative

responses 2x10+

2x2

- 8+8 +4+4 and

other combinations

28,

24

Point out the

separate groups by

emphasising the

separation of each

fence


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.

Recorded a variety

of different

number sentences-

reinforced

different

groupings of ten.

Questions were

open ended-

Students guided

the conversation-

Justifying,

explaining.

Elaborated on

more sophisticated

answers,

multiplication

45

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

of … and …-


in a double count.

hand out a recording

sheet for the

students to fill out

Intervene as

Different

responses,

students explored

with renaming and

different

representations.

46

blocks

Draw or write a

number sentence

– emphasis on

place value, to

show how many

blocks you have.

Discuss the number of

blocks-tell them that

each bag will have an

even number of blocks

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)

3.

Polishing and


Solution Methods

(10mins)

Have a variety of



both their written

number sentence to


and their groups


how they recorded.


group the number,

change the way we



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

responses-

2 tens +26 ones

4x10 +6

(4x10) +6

or 146

10² +(4 x10) + 6

14 tens 6 ones etc.

select two samples

from students for

each way that the

numbers are

grouped

differently.

Display one under

the other, in three

columns.


symbols.

Analyse

What stays the

same, what

changes?

Create-

What would happen

if we had 231

cubes?

Evaluating

Have students build

consensus of the

most efficient ways


some more’

Very effective

discussion.

Students described

the pattern. They

were using place

value language.

47

1.Summary and


Lesson (5mins)

point out that even






tens and some more

come back to the

learning intention

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.

48

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





49

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 Suggestions Names

Observed

response

Interpretation/Suggested teaching

response

Little/no

response

May not understand task

Repeat at a later date

Response given

but not

indicative of

strong place-

value

knowledge, eg,

refers to 6 ones

or physical

arrangement

such as “2

groups of 3” for

circled 6, and

“twenty” for

circled 2.

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

Says 6 ones and

2 tens fairly

quickly

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)

http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/efficient.aspx

50

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 Suggestion Names

Observed

response

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

Check extent to which child trusts

the count for 10 by counting large

collections (see Tool 2.2) and

review subitising and part-part-

whole 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

by visual image

or regrouping,

but refers to 2

as “twenty”

Suggests place-value ideas not well

established, may not trust the count of 10

Check trust the count, review

subitising and part-part-whole

ideas for 10 and making, naming

and recording tens and ones (see

above)


comparing 2 numbers (materials, words

and symbols), ordering/sequencing (by

http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/efficient.aspx

http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/lvl1trust.aspx

51

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

2 tens fairly

quickly

Appears to understand the basis on which

2-digit numbers are recorded


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





52

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

53

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:

54

Student Evidence

Written

What evidence was collected/ photographed?

What does it show in relation to the lesson goal?

Suggestion or question for the team.

55

Facilitator

Lesson study teacher: What worked well? 10min

Each team member report on findings 10 mins per member

Exit task review 20 min

What changes need to made to the lesson and/or exit task? 30min

56

Appendix H

Coded Questions

The Taxonomy Table

C

lari

fy

Bloom’s Taxonomy Categories

Pro

be

Pro

mp

t

Tota

l

Re

me

mb

er

Un

de

rsta

nd

Ap

ply

An

alys

e

Eval

uat

e

Cre

ate

Kylie 7

12 2 1 3 - - 6 - 31

Miranda 4 6 3 1 5 1 - 3 - 23

Pablo 6 2 4 3 4 1 - 4 24

Total 17 20 9 5 12 2 0 9 4

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.

57

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?

58

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.

Lesson Study: Improving Teachers’ Conceptions of … · Lesson Study: Improving Teachers’ Conceptions of ... education in secondary ... Can Lesson Study improve teachers’ conceptions

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