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ZDMThe International Journal onMathematics Education ISSN 1863-9690 ZDM Mathematics EducationDOI 10.1007/s11858-011-0345-2

Building teachers’ expertise inunderstanding, assessing and developingchildren’s mathematical thinking:the power of task-based, one-to-oneassessment interviewsDoug Clarke, Barbara Clarke & AnneRoche

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ORIGINAL ARTICLE

Building teachers’ expertise in understanding, assessingand developing children’s mathematical thinking: the powerof task-based, one-to-one assessment interviews

Doug Clarke • Barbara Clarke • Anne Roche

Accepted: 11 June 2011

� FIZ Karlsruhe 2011

Abstract In this paper, we outline the benefits to teachers’

expertise of the use of research-based, one-to-one assessment

interviews in mathematics. Drawing upon our research and

professional development work with teachers and students in

primary and middle years in Australia and the research of

others, we argue that the use of the interviews builds teacher

expertise through enhancing teachers’ knowledge of indi-

vidual and group understanding of mathematics, and also

provides an understanding of typical learning paths in vari-

ous mathematical domains. The use of such interviews also

provides a model for teachers’ interactions and discussions

with children, building both their pedagogical content

knowledge and their subject matter knowledge.

Keywords Teacher expertise � Assessment interview �Student thinking � Teacher knowledge

1 Introduction

This paper begins with a brief discussion of research and

frameworks of teacher knowledge, drawing largely upon

the seminal work of Shulman and the more recent work of

Ball and her colleagues. We discuss the relatively recent

phenomenon of the use of one-to-one assessment

interviews in building teacher expertise in professional

development settings in Australia.

We then draw upon two research programs in which the

one-to-one interview formed a major component. These

projects were the Early Numeracy Research Project

(ENRP) and the Australian Catholic University (ACU)

Rational Number Project. We provide the reader with

considerable detail on the interviews within these projects.

We give examples of the kinds of tasks used and the

learning framework which underpinned them. We then

outline the preparation involved prior to teachers using the

interviews, and the steps taken to maintain consistent use

of the interviews across many teachers. We share data on

student performance on certain tasks.

Following this background information on the interviews

and their use, we then discuss the benefits of such use to

preservice and inservice teachers, in building their exper-

tise. In doing so, we draw upon teacher questionnaire,

individual and focus group data as it supports our argument.

Both these research projects were large scale, involving

hundreds of teachers and thousands of students. With a

focus on student learning and growth in such learning over

time as evidenced from interview data, the research pro-

jects were not originally intended to study the direct con-

tribution of the one-to-one interview to developing teacher

expertise. However, the data on this which emerged during

the projects were compelling.

2 Teacher knowledge

In considering ways to build teacher expertise, an impor-

tant component is of course teacher knowledge (Fennema

and Franke 1992). Shulman (1986) first coined the term

pedagogical content knowledge (PCK). He described PCK,

D. Clarke (&) � A. Roche

Australian Catholic University, 115 Victoria Parade, Fitzroy,

VIC 3065, Australia

e-mail: [email protected]

A. Roche


B. Clarke

Monash University, PO Box 527, Frankston 3199, Australia


123

ZDM Mathematics Education

DOI 10.1007/s11858-011-0345-2

Author's personal copy

the intersection of content knowledge and pedagogical

knowledge as ‘‘the most useful forms of representation of

those ideas, the most powerful analogies, illustrations,

explanations and demonstrations—in a word, the ways of

representing and formulating the subject that makes it

comprehensible to others’’ (1986, p. 9). Since Shulman’s

seminal work, researchers in mathematics education have

attempted to conceptualise and measure teachers’ mathe-

matical knowledge for teaching (e.g., Hill, Ball, and

Schilling 2008). Such work is important, as research has

determined that student outcomes can be improved by

enhancing teachers’ mathematical knowledge for teaching

(Hill, Rowan, and Ball 2005).

Ball, Thames, and Phelps (2008) many years later pro-

posed the model shown in Fig. 1.

Although there is no clear consensus on the components

of teachers’ knowledge and may never be, like Graeber and

Tirosh (2008), we regard the ongoing conversations as

constructive, and ‘‘view the attempts to define the construct

PCK within mathematics education as part of that larger

quest to establish a unique base of professional knowledge,

a hallmark of a true profession, for teachers of mathe-

matics’’ (p. 129). In later sections, we will not only provide

evidence of the growth in teacher expertise, as it relates to

several of the components of the Ball et al. (2008) model,

but also argue that such expertise cannot be fitted neatly

into these categories.

3 An increased emphasis in Australia on the use

of one-to-one assessment

In the last 20 years, the inadequacy of a single assessment

method administered to students at the end of the teaching

of a mathematics topic has been widely acknowledged

(Ginsburg 2009). It is increasingly the case that those

working at all levels of mathematics education regard the

major purpose of assessment as improving instruction and

supporting learning (Webb and Romberg 1992), and this

has led to a search for appropriate assessment methods to

achieve this. The limitations and disadvantages of pen and

paper tests in gathering high quality, in-depth data on

children’s knowledge were well established by Clements

and Ellerton (1995). They contrasted the quality of

information about Grade 5 and Grade 8 students gained

from written tests with that gained through one-to-one

interviews. They observed that children may have a

strong conceptual knowledge of a topic (revealed in a

one-to-one interview) but be unable to demonstrate that

during a written assessment. The data suggested that

around one-quarter of students’ responses could be clas-

sified as either (a) correct written responses given by

students who did not have a sound understanding of the

mathematical knowledge, skills, concepts and principles

which the questions were intended to ‘‘cover’’; or

(b) incorrect written answers given by students who had

partial or full understanding.

Following the work of Piaget, clinical interviews have

been used for many years in mathematics education

research (Ginsburg et al. 1998). Typically, such research

has been conducted with relatively small numbers of stu-

dents, and the results not always communicated well to the

teaching profession. However, the late 1990s, in Australia

and New Zealand, saw the development and use of

research-based one-to-one, task-based interviews with

large numbers of students, as a professional tool for

teachers of mathematics (Bobis, Clarke, Clarke, Gould,

Thomas, Wright, and Young-Loveridge 2005).

Fig. 1 Framework of

mathematical knowledge

proposed by Ball et al.

(2008, p. 403)

D. Clarke et al.

123


4 Two large-scale research projects involving extensive

use of assessment interviews

4.1 The ENRP

The ENRP research and professional development pro-

gram, conducted in Victoria from 1999 to 2001, involved

353 teachers and over 11,000 children aged 5–8 years old

(Clarke 2001; Clarke, Cheeseman, Gervasoni, Gronn,

Horne, McDonough, Montgomery, Roche, Sullivan,

Clarke, and Rowley 2002).

There were four key components to this research and

professional development project:

• the development of a research-based framework of

‘‘growth points’’ in young children’s mathematical

learning (in Number, Measurement and Geometry);

• the development of a 40-min, one-on-one, task-based

interview, used by all teachers to assess aspects of the

mathematical knowledge of all children at the begin-

ning and end of the school year;

• extensive teacher professional development at central,

regional and school levels, for teachers, mathematics

coordinators, and principals; and

• a study of the practices of particularly effective

teachers.

As part of the ENRP, it was decided to create a

framework of key ‘‘growth points’’ in numeracy learning.

Students’ movement through these growth points in project

schools, as revealed in interview data, could then be

tracked over time. In creating the growth points, the project

team studied available research on key ‘‘stages’’ or ‘‘lev-

els’’ in young children’s mathematics learning (e.g., Cle-

ments et al. 1999; Lehrer and Chazan 1998), as well as

frameworks developed by other authors and groups to

describe learning. Data relating to growth points and stu-

dent learning have been reported in a number of publica-

tions (see, e.g., Clarke 2004; B. A. Clarke, Clarke, and

Cheeseman 2006).

There are some parallels in our growth points with the

work of Simon (1995), who referred to a hypothetical

learning trajectory as ‘‘the teacher’s prediction as to the

path by which learning might proceed’’ (p. 135). Our work

differs from that of Simon and Clements and Sarama

(2004) in that instructional sequences are somewhat

implied by our growth points, but not specified (Clarke

2008).

The one-to-one interview (assessing Number, Mea-

surement and Geometry) was used with around 11,000

children, taking an average of 45 min, and varying in time

according to the interviewer’s experience and the responses

of the child. The interview followed a very tight ‘‘script’’,

which indicated precisely which question to ask next, given

a particular response to the previous item. The interviews

were conducted by the student’s regular classroom teacher,

following a full day’s training on its use, and the oppor-

tunity to practise the interview process under the eye of

either the school mathematics coordinator (who had

received additional training) or a member of the research

team.

A range of procedures was developed to maximise

consistency in the way in which the interview was

administered across the schools. The teacher completed a

record sheet during each interview, which recorded both

students’ answers and their stated method. In many cases,

the method was evident because of the students’ body

language (e.g., use of fingers), but for most tasks the stu-

dent was asked a question of the kind, ‘‘please explain your

thinking’’, ‘‘how did you work that out?’’, ‘‘could you do

that a different way?’’), but always according to the script.

There was effectively no time limit on students’ responses,

although when it became clear that the student had little

idea on how to attempt to solve a given problem, the tea-

cher would usually move on.

The interview provided information about growth points

achieved by a child in each of nine mathematical domains:

four in Number (counting, place value, addition and sub-

traction, multiplication and division); three in Measure-

ment (time, length, mass), and two in Geometry (properties

of shape and visualisation and orientation). Although the

full text of the ENRP interview involved around 60 tasks

(with several sub-tasks in many cases), no child moved

through all of these. The interviewer made a decision after

each task, according to the script. Given success on a

particular task, the interviewer continued with the next task

in the domain as far as the child could go with success.

Given difficulty with the task, the interviewer either

abandoned that section of the interview and moved on to

the next domain or moved into a ‘‘detour’’, designed to

elaborate more clearly the difficulty a child might be

having with a particular content area.

To clarify the notion of a detour, the interview starts

with a task where students are asked, using a cup, to take a

‘‘big scoop’’ of plastic teddy bears from a tub of teddy

bears. They estimate the total and then count them, using

any method they choose. If they are unsuccessful in cor-

rectly counting the collection, they move into a ‘‘detour’’

which focuses on tasks to do with ‘‘more’’ and ‘‘less’’, one-

to-one correspondence, and conservation.

Figure 2 includes two questions from the interview

(Department of Education and Training 2001). These

questions focus on identifying the mental strategies for

subtraction that the child draws upon. The strategies used

were recorded on the interview record sheet.

Since its development, the ENRP interview has been

used by teachers and researchers in Australia, New

Building teachers’ expertise in children’s mathematical thinking

123


Zealand, Germany, Sweden, South Africa, Canada and the

USA.

4.2 ACU Rational Number Interview

Following the ENRP, and requests for a similar interview

from teachers of older students, it was decided to develop a

one-to-one interview for teachers of 9- to 14-year olds.

Given the recognised difficulty with fractions and decimals

for many teachers and students (see, e.g., Behr et al. 1983;

Steinle and Stacey 2003), it was decided to make rational

numbers the focus of the interview. Anne Roche adapted

and developed tasks in decimals (see, e.g., Roche, 2005,

2010) and Annie Mitchell in fractions (see, e.g., Mitchell

and Horne 2010). An important source of tasks was the

Rational Number Project (Behr and Post 1992). Student

data on key tasks were reported in D. M. Clarke, Roche,

Mitchell, and Sukenik (2006b). As part of this project, this

interview was used with 323 students who were completing

the last year of primary school.

Two sample tasks from the ACU Rational Number

interview are given in Fig. 3. These are Construct a Sum

from the Rational Number Project (Behr et al. 1985), and

Make me a Decimal (Roche and Clarke 2004).

These two tasks illustrate the potential of one-to-one

interview tasks, as compared to traditional written assess-

ment. The capacity of students to move the cards around in

each task has at least two clear benefits. First, the student

can place them in particular positions initially, knowing

that they are easily changed. Second, the teacher con-

ducting the interview has a window into children’s rea-

soning as they see them move the pieces from place to

place. Such rich information would be very difficult to

collect from a written assessment.

19. Counting Back

For this question you need to listen to a story.

a) Imagine you have 8 little cookies in your morning snack and you eat 3.

How many do you have left? ... How did you work that out?

If incorrect answer, ask part (b):

b) Could you use your fingers to help you to work it out? (It’s fine to repeat the question,

but no further prompts please).

20. Counting Down To / Counting Up From

I have 12 strawberries and I eat 9. How many are left? ... Please explain.

Fig. 2 Two tasks from the

ENRP interview

Construct a sum

Place the number cards and the empty fraction sum in front of

the student.

a) Choose from these numbers to form two fractions that when

added together are close to one, but not equal to one. Record

the student’s final decision.

b) Please explain how you know the answer would be close to

one.

Record any change of solution.

Make me a decimal

Here are some number cards and some blanks that could be any number.

a) Could you use some of these cards to show me what “two tenths” would look like as a decimal?

b) 27 thousandths?

c) ten tenths?

d) 27 tenths?

Figure 3. Sample tasks from the Australian Catholic University Rational Number Interview.

0 2 7 1

Fig. 3 Sample tasks from the

Australian Catholic University

Rational Number Interview

D. Clarke et al.

123


4.3 Some information on how the interviews were

administered and the data collected

Student strategies were recorded in detail on the interview

record sheet. For example, for the two ENRP subtraction

tasks outlined in Fig. 2, the teacher completes the record

sheet, as shown in Fig. 4, recording both the answer given

and the strategies used. The emphasis on asking for and

recording both answer and strategies is clear recognition

that the answer alone is not sufficient, and gives a message

to students that their strategies and mathematical thinking

are valued (Swan 2002).

The act of completing the record sheet requires an

understanding of the strategies listed (e.g., modelling all,

fact family, count up from, etc.), but of course must be

preceded by extensive teacher professional development on

its use. This is our first example of the kinds of teacher

expertise which are developed prior to and during the use

of the interview.

Processes used by the ENRP research team to maximise

reliability and validity of interview data have been detailed

elsewhere (Horne and Rowley 2001). Having data on over

36,000 ENRP interviews for the 11,000 students (with

many students being interviewed on several occasions) and

around 400 for the ACU Rational Number Interview pro-

vided previously unavailable high quality data on student

performance. For example, Table 1 shows the percentage

of children on arrival at schools (typically 5 year-olds),

who were able to match numerals to their corresponding

number of dots (B. A. Clarke et al. 2006).

During the ENRP, it became increasingly obvious to the

research team that the interview was providing opportuni-

ties for development of teachers’ knowledge. Evidence

emerged of teachers’ greater confidence in the use of

mathematical language, and of their growing sense of

typical learning paths of their students.

5 Methodology

As mentioned earlier, the research projects were not orig-

inally established to study the direct contribution of the

one-to-one interview to developing teacher expertise, but

the compelling nature of the anecdotal data which emerged

early encouraged our team to investigate this more thor-

oughly. Although there was a range of data which is not

reported here (e.g., teachers’ grouping practices, their

planning methods, actual time given to mathematics, and

their expectations of student growth), data collection rele-

vant to this article took the following forms:

• Teacher Entry questionnaire (February 1999), involv-

ing 24 items focusing on areas including background

information, personal mathematical knowledge, confi-

dence in teaching mathematics, mathematical expecta-

tions of their students, and areas of their teaching which

they sought to improve (n = 195).

• Teacher Exit questionnaire (October 2001), involving

21 items focusing on similar areas to the Entry

questionnaire, in order to discern changes over time

(n = 221).

• Teachers’ Highlights and Surprises questionnaire

(March 1999), where teachers were simply asked

‘‘what highlights and surprises were there as a result

of conducting the interviews with your students?’’

(n = 198).

• Changes in Teaching Questionnaire (October 2001),

where teachers were asked to nominate the greatestFig. 4 An excerpt from the addition and subtraction tasks on the

interview record sheet

Table 1 Performance of children on entry to school on selected tasks

(%) (n = 1,437)

Percent

success

(%)

Match numeral to 2 dots 86







123


changes in their teaching and in their students as a

result of their involvement in the ENRP (n = 220).

All of these data are reported in detail in Clarke et al.

(2002).

Using a two-page questionnaire involving Likert scale

items and open response items, 140 preservice teachers at

ACU and Monash University were asked to comment on

their growing expertise as mathematics teachers through

their use of one-to-one assessment interviews and impli-

cations for their future teaching (McDonough, Clarke, and

Clarke 2002). In addition, five ACU students were inter-

viewed individually with a particular emphasis on what

they had learned in relation to individuals’ understanding

of mathematics and strategy use. A focus group discussion

was also conducted with six other students, focusing upon

implications for teaching of what they had learned from the

interviews. Audiotapes of all interviews and focus groups

were transcribed. Two researchers both coded all data, and

identified themes emerging from the data.

‘‘All research is a search for patterns, for consistencies’’

(Stake, 1995, p. 44). An interpretative perspective (Erick-

son, 1986) was taken in identifying themes from the

unstructured open response items. The experience of

working with the teachers within the project and the

framework which underpinned the professional develop-

ment program in which the use of the interview was

embedded (see Clarke et al., 2002) gave the researchers

some advance warning of their likely responses to open

items. However, it was decided not to impose an initial

structure on the data, but rather to see what emerged. The

process of categorising the data and developing these

themes was as follows. Phrases that represented particular

pedagogical ideas were used as the data units.

All of the responses were read and the main themes

identified by two of the researchers, working independently.

These were then debated, the terminology clarified, and a

set of themes was determined by consensus. One researcher

then categorised all data units according to the agreed

themes to allow unique categorisation within the one theme.

This is a form of data reduction in that it groups information

into ‘‘a smaller number of sets, themes or constructs’’

(Miles & Huberman, 1994, p. 69). As most teachers had

more than one response to the given item, different parts of

their response were often coded to different themes. This

was generally not a complicated process, as the different

responses by a teacher were usually on different themes,

often listed as separate dot points. For example, the fol-

lowing statement was given three different codes as shown.

• I have a greater understanding of how children learn.

[KHCL: knowledge of how children learn]

• Working to children’s own ability and needs. [ALN:

addressing learning needs]

• Knowledge of the growth points helps make my

planning more detailed. [GPIP: growth points inform

planning]

A second researcher then examined the decisions of the

first and challenged the categorisation if necessary.

Essentially, the data categorised within a theme were

examined to assure that each theme did indeed represent a

coherent construct, clearly distinct from others. The two

researchers then negotiated on the meanings and categori-

sation. While the main benefit from these steps was that the

researchers came to some agreed understandings of the

scope of particular terms, the process ensured a systematic

and thorough approach.

In the following section, particular tasks, data from

teachers, and insights from other researchers are used to

build the argument of the power of the interview as an

important tool in building teachers’ knowledge and

expertise in understanding, assessing and developing chil-

dren’s mathematical thinking.

6 The role of the interview in developing teachers’

expertise in mathematics instruction

Sowder (2007), in discussing the goals of professional

development for mathematics teachers, emphasised the

goal of developing an understanding of how students think

about and learn mathematics. She claimed that student

thinking could be thought of as an interpretive lens that

‘‘helps teachers to think about their students, the mathe-

matics they are learning, the tasks that are appropriate for

the learning of that mathematics, and the questions that

need to be asked to lead them to better understanding’’

(p. 164).

At a professional development day, towards the end of

the third and final year of the ENRP, 220 teachers were

asked in an open response question to identify changes in

their teaching practice (if any), which they believed had

come as a result of their 3-year involvement in the project.

There were several common themes (Clarke 2008), many

of which related directly to the professional learning

experienced through the use of the interview. They were, in

decreasing order of frequency:

• using growth points to inform planning (63 responses);

• using knowledge of individual understanding to address

learning needs (49);

• challenging and extending children and having higher

and/or more realistic explanations;

• having more confidence in teaching mathematics (28);

D. Clarke et al.

123


• enjoying mathematics more and making mathematics

more interesting (27); and

• having a greater knowledge of how children learn (24).

Several of these themes are evident in the following

response from a teacher:

The assessment interview has given focus to my

teaching. Constantly at the back of my mind I have

the growth points there and I have a clear idea of

where I’m heading and can match activities to the

needs of the children. But I also try to make it

challenging enough to make them stretch.

This highlights the direct impact of the understanding and

use of the interview on this teacher’s expertise. This theme

was repeated many times in the responses. In arguing the

value to teachers of the interview, it is important to note the

strong impact of the ENRP on student learning. The research

design included teachers and students in 35 ‘‘trial’’ schools

who were involved fully in the program and 35 statistically

matched ‘‘reference’’ (or control) schools, where students

were also interviewed (see Bobis et al. 2005, for more

detail). In every mathematical domain at every year level,

there were significant differences in favour of students in the

ENRP trial schools (Horne and Rowley 2001).

We would claim that following appropriate professional

development preparation, the regular use of a research-

based, one-to-one interview by teachers with their students

has contributed to teacher knowledge and expertise in a

range of areas, particularly in relation to teachers’ knowl-

edge of students’ mathematical thinking.

Our experience is that the claims made here for prac-

tising teachers apply also to a large extent to preservice

teachers (McDonough et al. 2002). On a questionnaire

which used a Likert scale (McDonough et al. 2002, p. 219),

the number of preservice teachers out of 140 agreeing or

strongly agreeing with a given statement is shown in

parentheses:

The interview …

(a) gave me new insights into how young children think

when doing maths (135);

(b) gave information that would help me to plan for and

teach that child (129);

(c) gave insights that would help me to plan for and teach

all children (92);

(d) gave me insights into the types of questions to ask

young children to assess their understandings and

strategy use (120).

McDonough et al. (2002) summarised the data from the

preservice teachers’ study as showing that the use of the

interview had enhanced the knowledge and skills of pre-

service teachers in the following ways:

• Preservice teachers are more aware of the kinds of

strategies that children use, including their variety and

level of sophistication.

• Preservice teachers have seen the power of giving

children one-to-one attention and time, without the

distraction and influence of their peers.

• The interview provides a model of the kinds of

questions and tasks that are powerful in eliciting

children’s understandings.

• The interview and subsequent discussion stimulate

preservice teachers to reflect on appropriate classroom

experiences for young mathematics learners. (p. 223)

In the ENRP, the evidence of impact of the knowledge

and use of the interview on teacher expertise was a ‘‘by-

product’’ of a larger study. In the study of preservice

teachers, the impact of the interview was the explicit focus.

We focus now on the characteristics of the interview and

related growth points or big ideas in enhancing teacher

knowledge and expertise. Our evidence, both formal and

anecdotal, is that the preparation for and use of the inter-

view offers the benefits discussed in the following sub-

sections to teachers, thereby building their expertise. We

will now consider each of these points in detail, including

our justification for their inclusion.

6.1 A clearer, evidence-based understanding of student

thinking in mathematics and what students know

and can do

Through participating in a project where many teachers are

using interviews to collect data on their students, the tea-

cher gains insights into what individuals, their class group

and, in our case, a broader cohort across the state are able

to do. One of the advantages of administering the assess-

ment interview at both the beginning and end of the school

year is that teachers are provided, face-to-face, with

exciting evidence of growth in student understanding over

time. For example, for the matching dots tasks in Table 1,

the percentage success on each item increased to 99, 98,

97, 94, 99 and 82%, respectively, by the end of the school

year. By considering a single class’ data and the state data,

a teacher gains a sense of what typical performance looks

like over a year, thus informing reasonable expectations of

students in particular mathematical domains at particular

year levels, sometimes in contrast to published curriculum

expectations.

There was evidence in the ENRP that through extensive

use of the interview, teachers developed more realistic

expectations of what children knew and could do. Early in

the project, teachers made comments such as, ‘‘my greatest

surprise was that most children performed significantly

better than I anticipated. Their thinking skills and strategies


123


were more sophisticated than I expected’’ (Clarke et al.

2002, p. 260). In contrast, teachers were surprised with the

difficulty that many children appeared to have with tasks

relating to abstracting multiplication (Sullivan, Clarke,

Cheeseman, and Mulligan 2001), ordering whole numbers,

reading clocks, and identifying the triangles on a page of

triangles and non-triangles. An overall change in teachers’

expertise was in their awareness of the considerable range

of levels of mathematical understanding in their classes.

This was quantified in the ENRP, when teachers were

asked, at the beginning and the end of the project, to

indicate whether none, some, most or all of their children

could do certain tasks. For example, teachers of Preps

(5 year-olds in the first year of school) were asked how

many of their children by the end of the year would know

that four hundred and two is written 402 and knows why

neither 42 or 4002 is correct. At the beginning of the

project, 61% of the teachers said that none of their children

would know that, while at the end of the project, the per-

centage had dropped to 30% (Clarke et al. 2002). This was

a consistent pattern in the data, where teachers, through the

use of the interview, were far more likely to indicate that

some or most of their children would know a particular

mathematical idea, and far less responded none or all,

evidence that they were far more aware of the diversity of

understanding in their classrooms.

We know that from even as early as age five, there is

considerable variation in the kinds of understanding children

bring with them to school (Ginsburg, et al. 1998), and this

variation persists, although there is evidence that the relative

performance of individuals can change considerably (Clarke

et al. 2002). The use of interviews such as the ENRP and

ACU Rational Number Interview can help us to quantify this.

An important additional feature of both interviews is

that they have a ‘‘high ceiling’’. If students continue to

have success in a particular mathematical domain, they are

presented with more and more difficult tasks, well beyond

the normal expectations for their grade level. This has at

least two benefits. First, all students are provided with a

challenge. Second, teachers’ horizon content knowledge

(Ball et al. 2008) is enhanced, as they are posing tasks and

hearing student responses to tasks which are well beyond

their usual content focus for the grade they teach, and they

gain an enhanced sense of where the mathematics on which

they normally focus is heading.

6.2 An understood framework/growth points/typical

learning trajectory for students in a given domain

The growth points in the ENRP informed the creation of

interview tasks and the recording, scoring and subsequent

data analysis, although the process of development of

interview and growth points was very much a cyclical one.

In discussions with teachers, we came to describe growth

points as key ‘‘stepping stones’’ along paths to mathemat-

ical understanding. They provide a kind of mapping of the

conceptual landscape (Fosnot and Dolk 2002). However,

we do not claim that all growth points are passed by every

student along the way.

In developing the ENRP framework of growth points, it

was intended that the framework would

• reflect the findings of relevant research in mathematics

education from Australia and overseas;

• emphasise important ideas in early mathematics under-

standing in a form and language readily understood

and, in time, retained by teachers;

• reflect, where possible, the structure of mathematics;

• allow the description of the mathematical knowledge

and understanding of individuals and groups;

• form the basis of planning and teaching;

• provide a basis for task construction for interviews, and

the recording and coding process that would follow;

• allow the identification and description of improvement

where it exists;

• enable a consideration of those students who may

benefit from additional assistance; and

• have sufficient ‘‘ceiling’’ to describe the knowledge and

understanding of all children in the first three years of

school. (Clarke 2001)

To clarify further what is meant by growth points, the

six growth points for the ENRP domain of Addition and

subtraction strategies are shown in Fig. 5.

We do not claim that all growth points are passed by

every student along the way. As Van den Heuvel-Panhui-

zen (2001) emphasised, ‘‘a teaching-learning trajectory

should not be seen as a strictly linear, step-by-step regime

in which each step is necessarily and inexorably followed

by the next’’ (p. 13). For example, one of our growth points

in Addition and Subtraction involves ‘‘count-back’’,

‘‘count-down-to’’ and ‘‘count-up-from’’ in subtraction sit-

uations, as appropriate. But there appears to be a number of

children who view a subtraction situation (say, 12 - 9) as

‘‘what do I need to add to 9 to give 12?’’ and do not appear

to use one of those three strategies in such contexts.

The key here is that teachers’ expertise was enhanced by a

clear understanding of typical trajectories in young children’s

mathematical understanding, through the combination of the

growth points and their experience in using the interview.

6.3 Revelations about ‘‘quiet achievers’’

in the classroom

In response to a written question on highlights and sur-

prises following their first substantial use of the Early

Numeracy Interview, one teacher commented:

D. Clarke et al.

123


In every class there is that quiet child you feel that

you never really ‘know’—the one that some days

you’re never really sure that you have spoken to. To

interact one-to-one and really ‘talk’ to them showed

great insight into what kind of child they are and how

they think (ENRP teacher March 1999).

A number of teachers noted that the one-to-one inter-

view enabled some ‘‘quiet achievers’’ to emerge, and

several noted that many were girls. There appeared to be

some children who did not involve themselves publicly in

debate and discussion during whole-class or small-group

work, but given the individual time with an interested

adult, were able to show what they knew and could do.

Another teacher noted:

The greatest highlight was that no matter at what

level the children were operating mathematically, all

children displayed a huge amount of confidence in

what they were doing. They absolutely relished the

individual time they had with you; the personal feel,

and the chance to have you to themselves. They loved

to show what they can do (ENRP teacher March

1999).

The experience of the interview meant that many

teachers became more sensitive to quiet achievers, and

realised that a child not offering much in whole class dis-

cussions did not necessarily mean that they did not have

full understanding of the strategies and concepts being

addressed.

6.4 Enhanced subject matter knowledge and PCK

The evidence from the ENRP demonstrates that the use of

the interviews contributes to enhanced teacher knowledge

(Clarke et al. 2002; Clarke 2008). In the middle years,

many teachers acknowledge their lack of a connected

understanding of rational number (Lamon 2007), often

using limited subconstructs (sometimes only part-whole),

and limited models (such as the ubiquitous ‘‘pie’’). Many

teachers using the rational number interview have reported

that their own understanding of rational number (e.g., an

awareness of subconstructs of rational number such as

measure and division and the distinction between discrete

and continuous models) has been enhanced as they observe

the variety of strategies their students draw upon in

working on the various tasks and complete the record sheet.

In professional learning settings, quite a few middle school

teachers have difficulty in solving the rational number tasks

shown in Fig. 3.

Some might presume that teacher PCK in the first

3 years of school would not be an issue. However, many

teachers reported that terms such as ‘‘counting on,’’ ‘‘near

doubles’’, and ‘‘dynamic imagery’’ were unfamiliar to

them, prior to their involvement in the ENRP. It is inter-

esting to consider whether this is specialised content

knowledge or PCK (see, e.g., Ball and Hill 2002; Hill and

Ball 2004; Hill, Ball and Schilling 2008). As mentioned

earlier, it is difficult to categorise exactly the kinds of

knowledge which are evident in teachers’ practice (Graeber

and Tirosh 2008), but we would argue there is little doubt

that both subject matter knowledge and PCK are enhanced

by the use of such interviews.

6.5 An awareness of common strategies used

by students

In questionnaire data (Clarke 2008), teachers reported that

the training for, and use of the interviews gave them an

awareness of strategies in solving problems with which

they were not previously familiar. The ACU Rational

Number Interview provided examples of this. In the frac-

tion comparison task, students were asked to decide which

of two fractions was the larger, for eight pairs, giving

1. Count-all (two collections) Counts all to find the total of two collections. 2. Count-on Counts on from one number to find the total of two collections. 3. Count-back/count-down-to/count-up-from Given a subtraction situation, chooses appropriately from strategies including count-back, count-down-to and count-up-from. 4. Basic strategies (doubles, commutativity, adding 10, tens facts, other known facts) Given an addition or subtraction problem, strategies such as doubles, commutativity, adding 10, tens facts, and other known facts are evident. 5. Derived strategies (near doubles, adding 9, build to next ten, fact families, intuitive strategies) Given an addition or subtraction problem, strategies such as near doubles, adding 9, build to next ten, fact families and intuitive strategies are evident. 6. Extending and applying addition and subtraction using basic, derived and intuitive strategies

Given a range of tasks (including multi-digit numbers), can solve them mentally, using the appropriate strategies and a clear understanding of key concepts.

Fig. 5 ENRP growth points for

the domain of addition and

subtraction strategies


123


reasons for their decisions. These data are discussed in

considerable detail in Clarke and Roche (2009). The frac-

tion pairs presented to the student are shown in Fig. 6.

Each pair, typed on a card, was placed in front of the

student one pair at a time, and the student was asked to

point to the larger fraction of the pair, explaining their

reasoning. There was no time limit involved.

Researchers report frequently that students use strategies

in solving fraction comparison tasks which they are unli-

kely to have been specifically taught. The use of residual

thinking (Post and Cramer 2002) and benchmarking (or

transitive, Post, Behr and Lesh 1986) are likely to be evi-

dence of conceptual understanding and lead to a successful

choice. The term residual refers to the amount which is

required to build up to the whole. So, in comparing 5/6 and

7/8, students may conclude that the first fraction requires

1/6 more to make the whole (‘‘the residual’’), while the

second requires only 1/8 to make the whole, so 7/8 is lar-

ger. The use of benchmarks involves the student comparing

two fractions of interest to a third fraction, often 1/2 and

sometimes 1. A student using this strategy appropriately

would say that 5/8 is larger than 3/7 because the first

fraction is greater than one half, while the second is less

than one half. Post et al. (1986) referred to benchmarking

as a transitive strategy, where the transitive property is

used in relation to an external value, the benchmark

fraction.

Since using the ACU Rational Number Interview with

their students, and given the opportunity in professional

learning settings to discuss student strategies, many

teachers have indicated to the authors that they now ensure

that all their students are exposed to strategies such as

benchmarking and residual thinking, strategies of which

many were not previously aware. We would argue that this

is specialised content knowledge (Ball et al. 2008).

6.6 An awareness of common difficulties

and misconceptions present in students

As teachers have the opportunity to observe and listen to

students’ responses, they become aware of common diffi-

culties and misconceptions. For example, many children in

the first 5 years of school (Grades Prep to 4) were unable to

give a name to the shape on the left in Fig. 7. It was not

expected that they would name it ‘‘right-angled triangle,’’

but simply ‘‘triangle’’. Because it did not correspond to

many students’ ‘‘prototypical view’’ (Lehrer and Chazan

1998) of what a triangle was (i.e., a triangle has a hori-

zontal base and ‘‘looks like the roof of a house’’—either an

isosceles or equilateral triangle), some called it a ‘‘half-

triangle, because if you put two of them together you get a

real triangle.’’ Many students also nominated the two

shapes on the right in Fig. 7 as triangles. In fact, in a later

task in the interview, 20% of students at the end of Grade 4

were unable to select correctly the triangles from a page of

nine shapes (Clarke 2004).

Following the use of the interview, it was clear from a

teaching perspective that it was important to focus on the

properties of shapes, and to present students with both

examples and non-examples of shapes, as they were com-

ing to terms with definitions.

A common, incorrect strategy in fraction comparison

tasks is the use of ‘‘gap thinking’’ (Pearn and Stephens

2004), often evident in students’ responses to task (g) in

Fig. 6. Some students claim that 5/6 and 7/8 are equivalent,

because they both require one ‘‘bit’’ to make a whole. In

this case, the students are focusing on the gap between 5

and 6 and the gap between 7 and 8, but not considering the

actual size of the pieces. This gap thinking is really a form

of additive rather than proportional thinking, where the

student is not considering the size of the denominator and

therefore the size of the relevant parts (or the ratio of

numerator to denominator), but merely the absolute dif-

ference between numerator and denominator.

If a teacher has an understanding of these kinds of

common misconceptions through observing them first hand

in an interview setting and discussing them in professional

development settings, they are well prepared to recognise

them when students demonstrate them in a classroom sit-

uation, taking the opportunities to confront them in con-

versations with students, or by careful choices of examples.

Some may argue that it is sufficient for teachers to be

presented with these misconceptions as part of professional

learning programs. On the contrary, we argue how much

more powerful it is for a teacher to observe such miscon-

ceptions among their own students during interviews,

a) 3/8 7/8 e) 2/4 4/2

b) 1/2 5/8 f) 3/7 5/8

c) 4/7 4/5 g) 5/6 7/8

d) 2/4 4/8 h) 3/4 7/9

Fig. 6 The eight fraction pairs used in the interviewFig. 7 Triangle and non-triangle shapes in the interview

D. Clarke et al.

123


either before or after extensive teaching of the relevant

content.

6.7 Improved questioning techniques, including

the opportunity to see the benefits of increased

wait time

Researchers studying particularly effective teachers’ prac-

tice within the ENRP, noted that the interview appeared to

provide a model for classroom questioning (Clarke and

Clarke 2004). In interviews with the research team,

teachers indicated that they found themselves making

increasing use of questions of the following kind:

• How did you work that out?

• Is there a quicker way to do that?

• How are these two problems the same and how are they

different?

• Would that method always work?

• Is there a pattern in your results? (Clarke et al. 2002)

Wait time has been an important topic in the literature

for many years (Tobin 1987). Tobin found that teachers

who had been trained to extend their wait time reduced

their number of utterances per unit time, interrupted stu-

dent discourse less frequently, and reduced students’ fail-

ure to respond to teacher solicitations. At the same time,

there was evidence that students used the extra time for

thinking, and that the average length of students’ utterances

increased.

Teachers in the ENRP observed the power of waiting for

children’s responses during the interview, noting on many

occasions the way in which children who initially appeared

to have no idea of a solution or strategy, thought long and

hard and then provided a very rich response. Such insights

then transferred to classroom situations, with teachers

claiming that they were working on allowing greater wait

time (Clarke 2001).

6.8 The opportunity to use tasks from the interview

as models or inspirations for developing

classroom tasks

The capacity of the teacher to take the information on the

record sheet and ‘‘map’’ student performance in relation to

the growth points or ‘‘big ideas’’ is a key step in the process

of using the interview to inform teaching practice. After

conducting the interview, teachers are likely to ask the

reasonable question in relation to planning, ‘‘so now

what?’’ If they have a clear picture of individual and group

performance in particular mathematical domains, they are

then in a position, hopefully with the support of colleagues,

to plan appropriate classroom experiences for individuals

and groups.

The tasks in the interview did provide a model for the

development of different but related classroom activities.

For example, in the Place Value section of the Early

Numeracy Interview, students are asked to type numbers

on the calculator as they are read by the teacher or read

numbers that emerge as they randomly pick digits and

extend the number of places (ones, tens, hundreds, etc.) of

the number on the screen. Seeing the potential of the cal-

culator as a tool for exploring and extending place value

understanding, teachers tried tasks such as ‘‘type the largest

number on the calculator which you can read.’’ Such a task

provides an opportunity for the teacher to challenge their

students to make the number even larger. This task, re-

visited regularly, provides a helpful measure of growth in

student understanding over time, and therefore can be used

as an ongoing assessment tool.

Construct a Sum (Fig. 3) is an example of where a task

used in an assessment interview can be adapted for use as

an instructional activity. Teachers have used the same

materials, with students working in pairs, and invited them

to make the largest sum they can with two fractions, the

smallest sum, the sum closest to 3, and so on.

In this way, classroom tasks modelled on or inspired by

those from the interviews, used together with the kinds of

appropriate probing of students’ thinking discussed earlier,

provided powerful responses to what had been learned

from the interview, and led to the kinds of improved

understanding which teachers were seeking.

7 Conclusion

There is a strong, demonstrated link between teacher

knowledge and student performance (Hill et al. 2008).

Throughout this article, we have argued that knowledge

gained from the understanding and use of one-to-one

interviews is multi-faceted. We have given examples of

how both subject matter knowledge (particularly specia-

lised content knowledge and horizon content knowledge)

and PCK can be enhanced.

In relation to PCK, the major benefits to teachers do not

fall neatly into the categories of Ball et al. (2008). Rather, it

is the knowledge of students’ understanding, thinking and

reasoning that is most evident. Even and Tirosh (1995)

emphasised the importance of knowledge of students in

teachers’ responses to students’ questions, ideas and

hypotheses. The discussion of fraction pairs earlier pro-

vides examples of how students think about the subject

matter, and this knowledge can enhance greatly a teacher’s

response to what students offer in the classroom.

Ginsburg (2009) noted that good teaching involves

‘‘understanding the mathematics, the trajectories, the

child’s mind, the obstacles, and using general principles of


123


instruction to inform the teaching of a child or group of

children’’ (p. 126). Not surprisingly, we would argue that

this article provides compelling evidence that the task-

based, one-to-one assessment interview can make a major

contribution to such understanding, thereby greatly

increasing teacher expertise.

Acknowledgments We are grateful to our colleagues in the Early

Numeracy Research Project team (Jill Cheeseman, Ann Gervasoni,

Donna Gronn, Pam Hammond, Marj Horne, and Andrea McDonough,

from Australian Catholic University, and Glenn Rowley and Peter

Sullivan from Monash University), and our collaborator in our

rational number work: Annie Mitchell (Australian Catholic

University).

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