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The nature and development of middle schoolmathematics teachers knowledge
Kim Beswick Rosemary Callingham Jane Watson
Published online: 25 February 2011 Springer Science+Business Media B.V. 2011
Abstract In this article, we report on the use of a teacher profiling instrument with 62middle school teachers at the start of a 3year professional learning programme. The
instrument was designed to assess the aspects of teachers knowledge identified by
Shulman (1987) refined by Ball et al. (2008) and extended to include teachers confidence
to use and teach various topics in the middle school mathematics curriculum and their
beliefs about mathematics teaching and learning. Based on a hierarchical coding of items,
the application of the partial credit Rasch model revealed that the profile items were
measuring a single underlying construct and suggested that the various facets of teacher
knowledge develop together. We describe the characteristics of four levels of the hierar
chical construct measuring teacher knowledge and understanding for teaching mathematics
in the middle years of schooling, and discuss the unique affordances of a holistic view of
teacher knowledge in contrast to considerations of multiple knowledge categories.
Keywords Mathematics teacher knowledge Middle school teachers Raschmeasurement Teacher beliefs
Introduction
The dilemma of how to measure teacher competence for teaching has been on the edu
cational agenda for more than a century. A major reason for wishing to describe and
measure attributes associated with teaching mathematics is to be able to identify those
where high levels of proficiency are associated with high levels of student achievement.
The apparently multifaceted nature of teachers knowledge for teaching mathematics,
K. Beswick (&) R. Callingham J. WatsonFaculty of Education, University of Tasmania, Locked Bag 1307, Launceston, TAS 7250, Australiaemail: Kim.Beswick@utas.edu.au
R. Callinghamemail: Rosemary.Callingham@utas.edu.au
J. Watsonemail: Jane.Watson@utas.ed.au
123
J Math Teacher Educ (2012) 15:131157DOI 10.1007/s1085701191779
however, has complicated efforts to establish clear links between it and students mathe
matics achievement. In this article, we present evidence that the many aspects of middle
school mathematics teachers knowledge can be conceptualised as contributing to a single
underlying variable that we have called teacher knowledge, thereby laying the groundwork
for future studies in which teacher knowledge and student attainment can be linked. In
addition, we discuss what can be learned about the development of mathematics teaching
expertise from an instrument designed to measure a comprehensive conceptualisation of it.
Conceptualising teacher knowledge
Hill et al. (2007) provide an excellent history of developments in understanding of teacher
competence in the United States with a particular focus on the teaching of mathematics.
The precise nature of the knowledge required for teaching mathematics effectively has
proven difficult to specify, but there is agreement that it comprises more than simply
knowledge of mathematics (Hill et al. 2007; Mewborn 2001). Indeed Shulman (1987)
suggested seven categories of teacher knowledge for teachers across the curriculum:
content knowledge, pedagogical knowledge, pedagogical content knowledge, knowledge
of how students learn, curriculum knowledge, knowledge about the educational context,
and knowledge of the values and purposes of education. Shulmans work provided the
impetus for many subsequent studies focussing on particular categories and at times
enlarging or interrelating them. Zhou et al. (2006) for example, considered the first three,
whereas Kanes and Nisbet (1996) explored content knowledge, pedagogical content
knowledge, and curriculum knowledge. Ball and Bass (2000) examined pedagogical
content knowledge, and Watson (2001) and Watson et al. (2006) evaluated all seven. Chick
et al. (2006) used the concept of pedagogical content knowledge but also expanded it to
consider examples of content knowledge in a pedagogical context and pedagogical
knowledge in a content context. Acknowledging a close link with content knowledge and
knowledge of how students learn, Watson et al. (2008) also considered pedagogical content
knowledge.
Ball et al. (2008) presented an empirically based refinement of Shulmans content and
pedagogical content knowledge types developed from some two decades of systematic
research in the area. Their conception of subject matter knowledge (content knowledge)
comprised common mathematical content knowledge (CCK) that many adults not involved
in teaching might reasonably be expected to have, plus specialised mathematical content
knowledge (SCK) that would not be expected outside the teaching profession and used, for
example, in assessing the mathematical appropriateness of nonstandard solutions to
mathematics problems. In addition, they hypothesised that a further aspect of content
knowledge might be what they termed horizon content knowledge. This involves knowing
how the mathematics being taught at a particular grade level relates to that which is to
come and how current teaching choices may facilitate or obstruct future learning. Building
on Shulmans characterisation of pedagogical content knowledge as involving an amalgam
of content knowledge and pedagogical knowledge, Ball et al. (2008) identified three
knowledge types at the intersections of content knowledge and each of knowledge of
students, knowledge of teaching, and knowledge of the curriculum. They described these
as knowledge of content and students (KCS), knowledge of content and teaching (KCT),
and knowledge of content and curriculum.
Ball et al. (2008) claimed to define knowledge broadly to include skill, habits of mind,
and insight (p. 399) but later stated that their interest was in skills, habits, sensibilities,
132 K. Beswick et al.
123
and judgements as well as knowledge (p. 403) (italics added). Undoubtedly, Ball andcolleagues are interested in more than simple declarative knowledge; their work is based
on what teachers actually do in the full breadth of tasks associated with teaching mathe
matics. Nevertheless, the precise way in which they conceive of knowledge and how
aspects of such a conception beyond facts that are known is incorporated in their model is
not clear. In this article, we attempt to define a broad conception of teacher knowledge that
encompasses all of Shulmans knowledge types, including Ball et al.s refinements, as well
as elements more commonly associated with the affective domain. We suggest that such a
holistic consideration of teacher knowledge reveals important insights that may not be
evident from detailed analytic dissections of the concept.
In particular, we include in our conception of knowledge teachers beliefs and confi
dence. Teachers beliefs have long been recognised as crucial in shaping their practice
(Wilson and Cooney 2002). Beswick (2005, 2007) argued that distinctions between beliefs
and knowledge are contextual in that they vary with time and place; there are many things
that once were known (e.g., that the earth is the centre of the universe) but now are known
not to be true. Essentially whether a statement is considered to be knowledge or a belief is
dependent upon the extent to which there is consensus as to its veracity, and this is usually
a function of the extent to which the claim is useful in making sense of phenomena. In any
case, teachers act upon their beliefs as if they are knowledge.
There is growing evidence that teachers beliefs (whether or not they would be accepted
as knowledge in other contexts) about broad issues or principles concerning the nature of
mathematics, and mathematics teaching and learning, rather than about the use of specific
approaches or tools, are what matter to students learning (e.g., Beswick 2007; Watson and
De Geest 2005). Nevertheless, in the literature promoting studentcentred approaches to
mathematics teaching, certain methods and tools are associated with traditional (and by
implication less effective) teaching. The use of textbooks, for example, has been contrasted
with reform oriented (or studentcentred) teaching (Farmer et al. 2003) and associated with
conventional (as opposed to reform) teaching (Wood et al. 2006).
Confidence is generally positioned in the affective domain and is one of eight dimen
sions of attitude identified from the literature by Beswick et al. (2006). It has been claimed
to be of particular relevance to teachers practice, reflected in enjoyment of mathematics
for its own sake (Beswick 2007). In addition, Watson et al. (2006) reported increased
teacher confidence in relation to topics in the mathematics curriculum that were the focus
of a professional learning programme, suggesting that confidence is associated with
knowledge. It is also possible, however, that increased understanding of the complexities
of teaching mathematics might lead, initially at least, to decreased confidence; hence, the
relationship between competence and confidence is not necessarily straightforward. Nev
ertheless, we were interested to explore whether or not confidence could usefully be
included in a broad conception of knowledge for teaching mathematics.
Measuring teacher knowledge
Mewborn (2001) described how early attempts to identify attributes of mathematics
teaching associated with high levels of student attainment involved crude measures of
teacher knowledge in the form of the numbers of mathematics courses studied or years of
teaching experience. Such efforts failed to establish any clear connections. Comparative
studies such as those of Ma (1999) and Zhou et al. (2006) provided indirect evidence of a
link between teachers knowledge and student achievement by establishing that in
Middle school mathematics teachers knowledge development 133
123
countries such as China, where students mathematical achievement is typically higher
than that of students in the United States, teachers score more highly on measures of
certain aspects of knowledge. Hill et al. (2005) cited studies linking certain classroom
behaviours of teachers, and teachers mathematical proficiency as measured by written
tests, with improved student achievement. They contended that the relevant aspect of
teacher knowledge missing from such studies was how teachers used their mathematics
knowledge in classrooms. With their multiple choice items devised to match the knowl
edge that teachers use in classroom contexts, they reported positive correlations between
teacher knowledge and student achievement.
The seven categories outlined by Shulman (1987) provided the foundation for sub
sequent research, but the categories themselves do not address the method of measuring the
aspects of knowledge and understanding behind the phrases or in fact the comprehensive
meaning of each. Hill et al. (2007) summarised the many methods of measurement used for
this purpose in the United States throughout the 20th century into the 21st, acknowledging
the benefits and limitations of each. Attempts to measure teachers content knowledge have
utilised pen and paper instruments addressing mathematics content knowledge as variously
defined (Hill et al. 2005; Zhou et al. 2006) and general pedagogical knowledge (Zhou et al.
2006). Measuring pedagogical content knowledge has been seen as a greater challenge.
Hence, more intensive methods such as observation and detailed analyses of classroom
interactions (Ball and Bass 2000), interviews in which teachers comment on teaching plans
(Zhou et al. 2006), and workshop assignments completed by prospective teachers (Chick
and Pierce 2008) have been employed.
The study reported here builds on that reported by Watson et al. (2006) by using a
written teacher profile comprising openended tasks and questions as well as Likert format
items. The profile was used as a measure of a comprehensive conception of teacher
knowledge needed for mathematics teaching that includes the types of knowledge iden
tified by Shulman (1987), the additional categories of Ball et al. (2008) and extended to
include teachers confidence and salient aspects of their beliefs. Rather than considering
and attempting to measure each of these different dimensions of teacher knowledge sep
arately, the interest in this study was to consider whether it was legitimate to consider
various types of teacher knowledge as a single construct, and identify how that construct
might develop. The potential of this approach was that it could provide insights into
relationships among the diverse knowledge categories and particularly their development.
The approach is analogous to, for example, considering a students mathematics com
petence as an entity, although it may be composed of several aspects including compe
tence in algebra, geometry and arithmetic that develop at different rates and at different
stages of the students schooling. In addition, if teacher knowledge can be conceived of as
a unidimensional construct, then it could provide a basis upon which it may be possible to
link teacher proficiency, broadly conceived, to student achievement.
Rasch models
Rasch models are a set of measurement models coming under the general umbrella of Item
Response Theory (Stocking 1999). They use the interaction between persons and items to
obtain an estimate of the probabilities of the response of each person on each item, and
conversely of each item to each person. In this way, a set of scores is derived that defines
the position of each person and each item against the underlying construct on the same
measurement scale. This produces a genuine interval scale in units of logits, the logarithm
of the odds of success (Bond and Fox 2007). In this study, the specific model used is the
134 K. Beswick et al.
123
Masters (1982) Partial Credit Model (PCM). The PCM is an appropriate model for use in
this study because it does not assume that every item has the same structure, that is, each
item may have a different number of categories or itemsteps.
Rasch measurement (Rasch 1980) provided a means of examining the extent to which
the multiple aspects of teacher knowledge can be considered to work together to measure a
single underlying variable. Rasch models are underpinned by three assumptions. The first
of which is that the variable under consideration is a unidimensional construct; second,
this construct must be measurable using an additive measure in which a higher value
indicates a greater quantity of the variable, and finally, the items used to operationalise
the construct must be independent of each other (Bond and Fox 2007). The initial step in
using Rasch measurement is to establish the extent to which these assumptions hold for the
data under consideration. In the case of this study, conformity of the teacher profile data
with the model would demonstrate the instrument was indeed measuring an underlying
unidimensional construct, the nature and structure of which could then be considered.
To establish the validity of an instrument, such as the teacher knowledge profile used in
this study, two features must be addressed: a theoretical framework and a measurement
instrument that operationalises this framework. The strongest evidence of validity arises
when the fit of the information obtained through use of the measurement instrument is
closest to the theoretical framework (Messick 1989). In this instance, the theoretical
framework was provided by the conceptualisation of teacher knowledge described earlier,
operationalised through the profile instrument. In addition to the content validity of the
teacher knowledge construct deriving from its theoretical conceptualisation, Rasch mea
surement allows for a consideration of the nature of the construct through the mathematical
description provided by the model (Fisher 1994). Hence, testing the profile instrument
against the Rasch model provided a means of determining the validity of the theorised uni
dimensional construct of teacher knowledge.
Uses of Rasch modelling in mathematics education have included the creation of
measurement scales (e.g., Waugh 2002) and the identification of hierarchies in students
understanding of particular concepts (e.g., Callingham and Watson 2004). Callingham and
Watson were concerned with dichotomously scored items, whereas other studies have used
Masters (1982) Partial Credit Model (PCM) (e.g., Callingham and Watson 2005). The
application of Rasch techniques to Likert items, as in Waugh (2002), requires the use of
partial credit models in which part marks are awarded for various intermediate responses
between complete success or agreement and complete failure or disagreement. Where there
are no missing categories, the Andrich (1978) Rating Scale model is applied. These
models, the PCM and the Rating Scale model, are applicable to items eliciting responses
that reveal increasing amounts of the ability or understanding that they are designed to
measure (e.g., Bond and Fox 2007;Callingham and Watson 2005; Watson and Callingham
2004; Watson et al. 2007).
The study
Participants
The participants in this study were 62 middle school (grades 58, ages 1014 years)
teachers (of whom only 4 claimed to have majors in mathematics) in 10 rural primary
(grades K to 6), district (grades K to 10), and high (grades 710) schools in the Australian
state of Tasmania. Although they catered for different grade ranges all of these school
Middle school mathematics teachers knowledge development 135
123
types followed the same state curriculum. The teachers in each school type were similarly
qualified and employed by the Department of Education (DoE) under the same conditions
of employment. The schools were selected by the DoE to take part in a professional
learning programme designed to assist teachers to facilitate two types of learning for their
students: learning related to the quantitative literacy skills required for active citizenship in
Australia, and learning needed as a prerequisite for studying the higher mathematics in
senior secondary school required for those who would contribute later to the scientific and
technological innovations of Australian society. The profile was administered as a written
survey instrument to teachers when they entered the programme. Due to changes in school
staffing, some teachers entered the programme at its beginning and others at later stages.
Some of the teachers involved in the study reported here had also been participants in an
earlier professional learning programme described by Watson et al. (2006).
Profile description
The profile was designed to cover Shulman (1987) seven knowledge categories including
subsequent refinements such as those by Ball et al. (2008), and to incorporate aspects of
teachers confidence and beliefs. The profile sections and the knowledge types covered in
each are summarised in Table 1. Specifically, it required teachers to: nominate how they
would improve middle school students mathematical understandings and how mathe
matics might be used to enhance students learning more broadly (Section 1); outline a
plan for teaching a mathematics concept that they considered important (Section 2); rate
their confidence on an open scale from low to high in relation to developing their
students understanding of a range of middle school mathematics topics, and their ability
to make connections between mathematics and other curriculum areas, to develop critical
numeracy using the media and to assess their students achievement against the new
Tasmanian curriculum standards (Section 3); respond on open scales from strongly agree
to strongly disagree to 11 items concerning the use of mathematics in everyday life
(Section 4) and 14 belief statements related to mathematics teaching and learning
(Section 5); and suggest both appropriate and inappropriate responses that their students
might give to each of three mathematics problems and describe how they could use each
of the items in their classroom (Section 6). Sections 7 and 8 related to teachers back
ground and perceived professional learning needs. Although not unrelated to teachers
knowledge, the personal details addressed in Section 7 are possible correlates of
knowledge rather than parts of the construct and, in relation to Section 8, it is possible
that teachers with considerable knowledge might nevertheless perceive a need to know
more. These sections, therefore, were not included in the measure of teacher knowledge
and are not shown in Table 1. The complete profile and coding schemes are available
from the authors.
Table 1 illustrates how the various sections of the profile spanned knowledge types and
how the knowledge categories are not necessarily distinct. Ball et al. (2008) acknowledged
a similar problem in that different teachers may bring to bear different aspects of their
knowledge in relation to the same task. In our profile, for example, teachers could
potentially address all of the knowledge types in Table 1 or fewer than we have indicated.
In Table 1, we have indicated those categories of knowledge that were either directly
solicited in the structure of the item or which would seem most difficult to avoid using in
responding to the task. Furthermore, although we viewed teachers confidence and beliefs
as parts of teacher knowledge, they also related more or less directly to other aspects of
knowledge. For example, confidence to teach mathematics topics is related to both
136 K. Beswick et al.
123
Ta
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4
Middle school mathematics teachers knowledge development 137
123
Ta
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Pro
file
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Ty
pes
of
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kn
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ge
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98
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and
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44
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138 K. Beswick et al.
123
common and specialised content knowledge, and also to pedagogical content knowledge in
relation to content and teaching and content and the curriculum. Similarly, many of the
belief statements in Sections 4 and 5 relate to views about mathematical content and the
purposes and priorities of mathematics teaching that constitute part of the context of
teaching the subject.
Data analysis
The data were coded such that the increasing quality of a response, based on progressively
more structurally complex responses, was rated more highly. The decision to focus on the
structure of responses was made prior to the study with the precise form of the coding
emerging from the actual responses of teachers. The application of a coding scheme is
illustrated in relation to Item 3 in Section 6 in which teachers were presented with the
following problem.
Mary and John both receive pocket money. Mary spends of hers, and John spends of his.A. Is it possible for Mary to have spent more than John?
B. Why do you think this? Explain.
The teachers were asked to respond to this item by addressing two questions:
6.1a What responses would you expect from your students? Write down some
appropriate and inappropriate responses (use * to show appropriate responses).
6.1b How would/could you use this item in the classroom? For example, choose one of
the inappropriate responses and explain how you would intervene.
Examples of responses to question 6.1a that attracted each of the possible codes are
shown in Table 2.
Examples of responses to question 6.1b that attracted each of the possible codes are
shown in Table 3.
The data were checked for missing categories in each item, and where necessary
responses were recoded to provide a single continuous rating for each item. Analysis was
carried out using Quest computer software (Adams and Khoo 1996). The results were
Table 2 Examples of responses to question 6.1a
Code Example of suggested student response
0 No response
1 Response not addressing fractions or wholes I dont knowI need helpWhy do I have to do this?
2 Response indicating either a correct fractionrelation to whole or an incorrect relationship towhole
It depends how much pocket money they get. Marycould have spent more than John if she gets morepocket money than him
3 Response containing both appropriate andinappropriate approaches to the problem
Yes, Mary might have bought something moreexpensive
No, is less than *Yes, because Mary might have had lots more than
John
Middle school mathematics teachers knowledge development 139
123
examined in two ways. First, the overall fit to the model was considered, including the fit of
each individual item. Then the relative itemstep difficulties and distribution of itemsteps
were considered.
Fit to the model is the prime quality control process used to evaluate instruments
when using Rasch measurement. If there is misfit, then the assumption of unidimension
ality may be violated (Bond and Fox 2007). Fit is determined by considering the extent to
which the responses to a particular item deviate from those expected by the model (Bond
and Fox 2007; Wright and Masters 1982). The most commonly used measure of fit at test
level is the infit mean square, which has an ideal value of 1. As a rule of thumb for
practical situations, fit values are considered adequate if the infit mean square (IMSQ)
value lies between 0.77 and 1.3 (Adams and Khoo 1996; Keeves and Alagumalai 1999),
and these values are used throughout this study. Fit measures are provided by Quest as part
of the routine output.
The second examination of the data obtained from the analysis involved considering the
item difficulty and the variable map produced by Quest. A variable or Wright map is a
visual representation of the distribution of items and persons along the variable. The unit of
Table 3 Examples of responses to question 6.1b
Code Example of suggested student response
0 No response
1 Response not addressing the mathematicalcontent of the problem
By modelling critical thinking skills
2 A single generic idea for the problem, e.g.,use money, discuss fractions
Compare of 100 and of 20. Which is bigger?
3 Reference to two or more aspects of thesolution without linking them
I would relate the question to their pocket money and askthem how much they receive each week. Then I wouldask what is and what a quarter is and discuss which isgreater.
4 Discussion including reference to partwhole concepts with specific examples
Look at class pocket money and compare and of eachof the amounts. See if of some is more than ofothers
Logit Persons Items X It18
2.0 It8 XXXX It 20 XX It 15
1.0 X It19 61tI31tI01tI
XXX It12 It17 0.0 XXXXX It9 It14
XXXXXXXXXX It6 11tI7tI
1.0 XXXXXXX XXXXXX It4
XXXX It2 It5 2.0 XXX It3
XXXX It1 3.0
Fig. 1 Components of a variablemap
140 K. Beswick et al.
123
measure is the logit, the natural logarithm of the odds of success. Figure 1 shows the
components of a variable map for 50 persons and 20 items. On the left side of the axis, Xs
represent persons, and the position on the scale indicates the ability estimate, increasing
with greater proficiency with respect to the variable being measured. On the right side of
the axis, items are displayed based on their measured difficulty, again being more
demanding with a larger logit value. The item mean is constrained to 0 logits, and this has
the effect of providing a true zero point on the measurement scale. Discontinuities between
items, e.g., between item It4 and the two items It7 and It11, indicate some change in the
demands of the items. Such jumps can be used to identify groups of items that form
clusters having similar demands. These clusters can then be used to describe the devel
opment along the identified variable.
Results and discussion
Overall fit to the model
Overall fit values obtained for the profile instrument were satisfactory for both items
(IMSQI = 1.01) and persons (IMSQP = 1.05), suggesting that the items were working
together satisfactorily to provide a single measurement scale, and that the persons were
also responding to the items in the way anticipated by the profile designers. Hence, the
various aspects of the teachers knowledge addressed by the profile together could legit
imately be considered to work together to measure a single underlying variable that we
have called teacher knowledge. The item separation reliability measure, which indicates
the extent to which the items are spread along the scale, was low at 0.21. This measure is
dependent on the size of the sample of test takers (Linacre 1991) and does not threaten the
validity of the test because it is not related to fit to the model.
Individual item fit
In addition to considering fit at test level, individual items were also considered. This is
important because even when test fit is satisfactory, if particular items do not fit the model,
these may not be measuring the same construct. In this situation, the developer needs to
consider the nature of the misfit and seek an explanation. In some instances, items should be
removed from the scale because they threaten the integrity and validity of the instrument.
Of the total of 51 items, nine showed some misfit with five having infit mean square
values below 0.77 and four having values greater than 1.3. Items with low infit mean
square value (\0.77) exhibit less variability in the data than the Rasch model predicts.These items are measuring the same construct but in an overly predictable manner. This
kind of misfit may suggest dependence among the items. In the profile, these items all
related to confidence with three concerning confidence to develop students understandings
of decimals, percent, and ratio and proportion, and two concerning the newly introduced
Essential Learnings (ELs) curriculum. One of the ELsrelated items concerned confidence
in connecting mathematics to the key elements of the ELs that did not explicitly mention
mathematics, and the other related to confidence to assess the Being Numerate key element
against the ELs standards (teachers were about to undertake this task for the first time). It
is, therefore, not surprising that teachers responded in highly predictable ways (i.e., with
overwhelmingly low confidence) to the latter two items and that there might be dependence
among responses to the three dealing with aspects of rational number. Overfit of this type is
Middle school mathematics teachers knowledge development 141
123
regarded as not providing any threat to the validity of the scale because the items are
measuring the same construct, albeit in ways that are more predictable than the model
expects and with some redundant information because of possible interdependence
(Wright 1991).
The remaining four misfitting items all showed underfit (fit values[1.3) to the modelindicating randomness in the responses. Three of the four underfitting items concerned
belief statements about teaching and learning mathematics. Specifically they were Telling
children the answer is an efficient way of facilitating their mathematics learning, It is
important that mathematics content be presented to children in the correct sequence, and
Effective mathematics teachers enjoy learning and doing mathematics themselves. It is
likely that inconsistencies among teachers interpretations of these items led to responses
being more variable than expected. The other underfitting item concerned the possible
classroom use of a problem that required the comparison of and in relation tounknown wholes. Teachers responses to this item were discussed in detail by Watson et al.
(2006), and it was also included in the professional learning programme described by
Watson et al. (2006). Teachers who had participated in that programme as well as in the
study reported here had been involved in discussions of the ways in which students actually
did respond to the problem and how such problems could be used in teaching. It is possible
that these teachers responded to this item quite differently from their colleagues, leading to
lack of fit to the model. The misfit was, again, relatively small and overall, however, the
items could be considered to work together to provide a scale of teacher knowledge.
The scale of middle school mathematics teachers knowledge
The variable map in Fig. 2 shows the distribution of itemsteps on the righthand side of
the vertical dotted line and persons on the left. The itemsteps are grouped according to the
section of the profile and the knowledge subscale to which they relate. The relationship
between the sections of the profile (see Table 1) and types of knowledge according to
which the itemsteps are grouped in Fig. 2 is shown in Table 4. Itemsteps in Fig. 2 are
denoted by letters (P, PC, NC, CN, and EL) also shown in Table 4, signifying the
knowledge subscale to which they relate. Table 4 also shows how the subscales relate to
the profile sections. In Fig. 2, the numbers after the decimal points indicate the level of
response according to the coding, with .1 denoting the second lowest level at which a
response was obtained.
Itemsteps appearing higher on the scale were more difficult for the teachers to satisfy
or, in the case of Likert scale items, more difficult for them to endorse. Horizontal lines
indicate points at which the authors agreed that there was a shift in the nature and/or
demands of the itemssteps, using the same process as that described by Callingham and
Watson (2005). Identifying levels provided a means of describing a developmental hier
archy along the measurement scale. The levels were named to reflect the distinct differ
ences in the nature of teacher knowledge that characterised each.
Description of the levels
In this section, each of the levels of middle school mathematics teachers knowledge is
described in terms of the demands of itemsteps at each level. Itemstep codes as in Fig. 2
are included throughout. For the first of these, Level 1 (Personal Numeracy), the complete
set of itemsteps is detailed (See Table 5) along with a summary of the distinguishing
142 K. Beswick et al.
123
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Middle school mathematics teachers knowledge development 143
123
characteristics of itemsteps at this level. For other levels, rather than presenting tables
similar to Table 5, the discussion of each begins with a summary of the distinctive features
of that level, followed by paragraphs that highlight in more detail the itemsteps at that
level that distinguish them from those at the previous and subsequent levels. This dis
cussion is designed to assist readers to interpret Fig. 2 although it is possible to gain an
overview of the distinctive aspects of the levels from reading just the introductory para
graphs. The complete set of itemstep codes appearing at each level is shown in Fig. 2.
Level 1: personal numeracy
The itemsteps appearing in the lowest level required teachers to express confidence in
their own capacities to use mathematics in everyday life and moderate confidence in their
ability to develop understandings of most topics in their students. However, itemsteps
requiring recognition of the importance of mathematical topics such as fractions, decimals,
percent, ratio and proportion, and pattern and algebra in everyday tasks did not appear.
Consistent with this, itemsteps concerning beliefs about teaching and learning mathe
matics included agreement that the mathematics taught in their classes was often irrelevant
to students. The most notable feature of this level is the very low levels of responses
demanded by general pedagogical and pedagogical content knowledge itemsteps, all of
which required only single or inappropriate suggestions. This level is, therefore, about
everyday personal functioning without significant links to the mathematics curriculum or
to classroom practice.
Full details of the itemsteps are provided in Table 5, which is essentially an expanded
version of level 1 of Fig. 2. The appearance at the same level of two or more itemsteps for
a single item indicates that more than one response to the item made relatively similar
demands of teachers. For example, both EL11.1 and EL11.2 representing ambivalence and
agreement with the proposition, I can easily extract information from tables, plans and
graphs were both relatively easy for teachers to endorse, and hence both appeared at this
lowest knowledge level. The itemstep representing strong agreement (EL11.3) was suf
ficiently difficult to endorse for it to appear at Level 2.
Level 2: pedagogical awareness
The second level required teachers to express high levels of confidence in relation to their
ability to use mathematics in their everyday lives, and to teach most topics. In terms of
beliefs, it demanded more positive views than the first level of such aspects as the place of
Table 4 Relationship of knowledge subscales in Fig. 2 to profile sections
Profile section (as described in Table 1) Subscale in Fig. 2 Itemsteps
1 Significant factors General pedagogical knowledge(GPK)
P
2 Planning to teach a mathematics or numeracyconcept
3 Confidence Confidence CN
4 Mathematics/numeracy in everyday life Everyday life EL
5 Numeracy in the classroom Numeracy in classroom NC
6 Student survey items Pedagogical content knowledge(PCK)
PC
144 K. Beswick et al.
123
Table 5 Itemsteps at the personal numeracy level (Level 1)
Itemsteps
Knowledgesubscale
Level ofendorsement/response
Code Item
Numeracy ineveryday life
Strong agreement EL1.1 I need to be numerate to be an intelligent consumer
EL2.3 I am confident that I could work out how many tiles I wouldneed to tile my bathroom
Agreement EL8.3 Given the price per square metre, I could estimate how muchcarpet I would need for my lounge room
EL2.2 I am confident that I could work out how many tiles I wouldneed to tile my bathroom
EL11.2 I can easily extract information from tables, plans and graphs
EL3.1 I often perform calculations in my head
EL10.1 I often use mathematics to make decisions and choices ineveryday life
EL6.1 I have difficulty identifying mathematical patterns ineveryday situations
Ambivalence/neutrality
EL7.1 Proportional reasoning is needed to understand claims madein the media
EL4.1 Understanding fractions, decimals, and percents is becomingincreasingly important in our society
EL8.2 Given the price per square metre, I could estimate how muchcarpet I would need for my lounge room
EL2.1 I am confident that I could work out how many tiles I wouldneed to tile my bathroom
EL9.2 Mathematics is not always communicated well innewspapers and the media
EL11.1 I can easily extract information from tables, plans and graphs
Disagreement EL8.1 Given the price per square metre, I could estimate how muchcarpet I would need for my lounge room
EL9.1 Mathematics is not always communicated well innewspapers and the media
Confidence High CN5.1 Measurement
Moderate CN9.1 Mental computation
CN1.2 Fractions
CN2.2 Decimals
CN3.2 Percent
CN4.2 Ratio and proportion
CN7.2 Pattern and algebra
CN8.2 Chance and data
CN10.1 Connecting mathematics to other key learning areas
Low CN1.1 Fractions
CN2.1 Decimals
CN3.1 Percent
CN4.1 Ratio and proportion
CN7.1 Pattern and algebra
Middle school mathematics teachers knowledge development 145
123
Table 5 continued
Itemsteps
Knowledgesubscale
Level of endorsement/response
Code Item
CN8.1 Chance and data
CN12.1 Critical numeracy in the media
CN11.1 Connecting mathematics to key elements of the ELs
CN13.1 Assessment of Being Numerate against the Elsstandards
Numeracy inthe classroom
Agreement NC12.1 Mathematics teaching should assist students todevelop an attitude of inquiry
NC3.1 Teachers of mathematics should be fascinated withhow children think and be intrigued by alternativeideas
NC10.2 Effective mathematics teachers enjoy learning anddoing mathematics themselves
NC9.3 Justifying the mathematical statements that a personmakes is an extremely important part ofmathematics
NC14.2 Often the mathematics work I do in the classroom isnot relevant to the students everyday lives
Ambivalence/neutrality NC3.1 Teachers of mathematics should be fascinated withhow children think and be intrigued by alternativeideas
NC10.1 Effective mathematics teachers enjoy learning anddoing mathematics themselves
NC2.2 I would feel uncomfortable if a child suggested asolution to a problem that I hadnt thought ofpreviously
NC5.2 Allowing a child to struggle with a mathematicalproblem, even a little tension, can be necessary forlearning to occur
NC8.2 Ignoring the mathematical ideas that childrengenerate themselves can seriously limit theirlearning
NC9.2 Justifying the mathematical statements that a personmakes is an extremely important part ofmathematics
NC14.1 Often the mathematics work I do in the classroom isnot relevant to the students everyday lives
Disagreement NC1.1 Mathematics is computation
NC2.1 I would feel uncomfortable if a child suggested asolution to a problem that I hadnt thought ofpreviously
NC5.1 Allowing a child to struggle with a mathematicalproblem, even a little tension, can be necessary forlearning to occur
NC8.1 Ignoring the mathematical ideas that childrengenerate themselves can seriously limit theirlearning
146 K. Beswick et al.
123
struggle in mathematics learning, the importance of the teacher being fascinated with
students thinking, and the relevance of the mathematics they taught. These beliefs mostly
reflect views that are more closely aligned with those presented in the literature as asso
ciated with studentcentred mathematics teaching than those at Level 1. In contrast with the
previous level, itemsteps at this level that related to planning for mathematics teaching
and to pedagogical content knowledge were present although only demanding low levels of
responses. Itemsteps at this level thus required at least some awareness of pedagogical
issues and the beginnings of recognition of the possibility of using specific problems to
reveal students thinking and to facilitate their learning.
Itemsteps at this level indicated agreement or strong agreement with all but one item
related to mathematics and numeracy in everyday life. Ambivalence (EL6.2) or dis
agreement (EL6.3) that mathematical patterns are difficult to identify in everyday situa
tions was the exception and also the only item in this section worded such that
disagreement represented greater mathematical proficiency.
With respect to confidence to teach mathematics, itemsteps at this level demanded
moderate confidence in relation to aspects of the ELs curriculum that were not related to
specific mathematics content, and critical numeracy in the media (CN12.2). High or, in the
case of measurement, very high, levels of confidence in relation to all content areas except
for ratio and proportion, and pattern and algebra appeared at this level.
Itemsteps concerning beliefs about mathematics teaching and learning indicated broad
agreement with a contemporary studentcentred orientation as evidenced by disagreement
with such things as that teaching would be difficult without a text (NC11.1), and that telling
children the answer is an efficient way of facilitating their mathematics learning (NC4.1),
and strong agreement that teachers of mathematics should be fascinated with childrens
thinking (NC3.3). However, itemsteps appearing at this level also indicated ambivalence
Table 5 continued
Itemsteps
Knowledgesubscale
Level of endorsement/response
Code Item
NC9.1 Justifying the mathematical statements that a personmakes is an extremely important part ofmathematics
NC6.1 Mathematical material is best presented in anexpository style: demonstrating, explaining anddescribing concepts and skills
NC7.1 It is important that mathematics content is presentedin the correct sequence
NC13.1 Mathematics in high schools is best taught in mixedgroups of abilities, at least until grade 9
Generalpedagogicalknowledge
Unistructural response P2c.1 Suggested assessment methods and strategies forchosen concept
P1a.1 How would you go about improving studentsnumeracy and mathematical understandings in themiddle years?
Pedagogicalcontentknowledge
Response not addressingthe relevantmathematics
PC1a.1 What responses to, What is 90% of 40? would youexpect from your students
Middle school mathematics teachers knowledge development 147
123
about mathematics being computation (NC1.2), and about a range of practices, such as
using an expository style (NC6.2), and teaching mathematics in mixed ability groups at
least until grade 9 (NC13.2), which are commonly associated with traditional teaching.
Ten itemsteps related to pedagogical content knowledge featured at this level. Sug
gested student responses to each of the three problems that did not address the relevant
mathematics (PC1a.1, PC2a.1, PC3a.1) appeared, along with itemsteps that demanded a
suggestion that addressed a single mathematically relevant aspect (PC1a.2, PC2a.2,
PC3a.2). This level demanded suggestions for classroom uses for the problems that did not
address their mathematical content (PC1b.1, PC2b.1, PC3b.1) or, in the case of the fraction
problem, that suggested the provision of just a single generic idea (PC3b.2).
In relation to general pedagogical knowledge, itemsteps at this level required teachers
to provide multiple suggestions regarding how numeracy in the middle years could be
improved (P1a.2) and either just one (P1b.1) or several (P1b.2) examples of ways in which
they used mathematics to enhance students learning in other key learning areas. In relation
to planning to teach a mathematics concept of the teachers choice, only the lowest item
steps of some aspects appeared at this level. These required teachers to suggest, for
example, only very broad or very narrow understanding goals in relation to the concept
(P2a.1), and to suggest activities related to the concept but which did not link clearly to a
developmental sequence or be likely to provide more than a limited insight into students
understandings (P2b.1).
Level 3: pedagogical content knowledge emergence
The third level was characterised by itemsteps requiring teachers to express high or very
high levels of confidence with respect to both their everyday use of mathematics and the
teaching of mathematics, and increasingly strong beliefs, not necessarily aligned with
studentcentred ideas, about mathematics teaching and learning. The itemsteps demanded
a focus on students understanding as an outcome of teaching and required teachers to
demonstrate some awareness of the likely range of their students responses to mathe
matics problems along with at least some idea of how such problems might be used in
teaching.
Itemsteps at this level demanded strong agreement with statements such as, that
fractions, decimals and percent are becoming increasingly important in our society (EL4.3)
and that quantitative literacy is as necessary to efficient citizenship as reading and writing
(EL5.4), suggesting important links between mathematical understandings and participa
tion in society. At this level, all itemsteps related to confidence to teach mathematical
topics required teachers to express high or very high confidence, including in relation to
ratio and proportion, and pattern and algebra (CN4.3, CN7.3, CN7.4).
Ambivalence concerning the difficulty of teaching mathematics without a text (NC11.2)
was the only itemstep at this level related to beliefs about mathematics teaching and
learning that required other than agreement or strong agreement. Several itemsteps
required endorsement of statements typically aligned with a studentcentred orientation, for
example, strong agreement that ignoring students ideas can limit their learning (NC8.4),
and that justifying mathematical statements is extremely important (NC9.4). Others, such
as those requiring agreement that mathematics is computation (NC1.3) and that mathe
matical material is best presented in an expository style (NC6.3) were consistent with a
traditional view.
148 K. Beswick et al.
123
Itemsteps related to pedagogical content knowledge required teachers to suggest stu
dent responses to all three of the problems that included both appropriate and inappropriate
responses (PC1a.3, PC2a.3, PC3a.3). One or more suggestions for the classroom use of the
piechart problem (PC2b.2, PC2b.3), and two or more disconnected ideas for using each of
90% of 40 (PC1b.3) and the fraction problem (PC3b.3) were demanded by itemsteps at
this level. The itemstep requiring suggestions for the use of the fraction problem that
included reference to fractions and wholes with specific examples (PC3b.4) also appeared
at this level. In terms of general pedagogical knowledge, itemsteps at this level required
teachers to provide appropriate understanding goals for their chosen topic (P2a.2) and to
indicate that their teaching of it generally resulted in their students understanding it
(P2d.1).
Level 4: pedagogical content knowledge consolidation
Itemsteps at this level demanded the highest levels of confidence with respect to teaching
mathematics, and required teachers generally to express mainly very strongly held beliefs
about teaching and learning mathematics. With respect to pedagogical content knowledge,
itemsteps at this level demanded of teachers the ability not only to think of multiple
possible uses of mathematics problems, but also to identify the relevant mathematical
concepts inherent in each, along with relationships among these ideas.
No itemsteps concerning the everyday use of mathematics appeared at this level and
only itemsteps demanding very high levels of confidence with respect to teaching
mathematics topics featured here. These topics related to ratio and proportion (CN4.4),
critical numeracy in the media (CN12.4), assessing Being Numerate (CN13.4), and making
connections between mathematics and key elements of the ELs curriculum (CN11.4).
Itemsteps requiring strong agreement with the value of struggle in mathematics
learning (NC5.4), the desirability of teaching mathematics in the correct sequence (NC7.4),
and using an expository style (NC6.4) appeared at this level along with strong agreement
that mathematics is computation (NC1.4). The item related to using mixed ability groups at
least until grade 9 required either agreement (NC13.3) or strong agreement (NC13.4) at
this level. Agreement that mathematics would be very difficult to teach without a text book
(NC11.3) was also demanded, and itemsteps requiring teachers to express either ambiv
alence (NC4.2), agreement (NC4.3), or strong agreement (NC4.4) that telling students
answers is an efficient means of facilitating their mathematics learning were also included
at this level along with the itemstep requiring teachers to disagree strongly that the
mathematics they do in class is often irrelevant to their students lives (NC14.4).
Itemsteps requiring teachers to discuss a range of possible uses of each of 90% of 40
and the piechart problem, which referred to relevant partwhole ideas and included spe
cific examples, were included at this level (PC1b.4, PC2b.4). This level demanded that
teachers present an integrated rationale for their suggestions about how they would go
about improving middle school students numeracy and mathematical understandings
(P1a.3). In terms of planning, itemsteps at this highest level required appropriate goals
expressed as understandings (P2a.3), teaching and assessment strategies that related to
these goals (P2b.2) and that also included evidence of evaluation of student understanding
(P2b.3). Itemsteps demanded assessment methods including multiple strategies linked to
outcomes (P2c.2), student responses including both engagement and understanding
(P2d.2), and multiple examples of how work across the curriculum could contribute to
understanding of the relevant mathematics (P2e.2).
Middle school mathematics teachers knowledge development 149
123
Knowledge subscales across the levels
The following sections consider each of the subscales of items in the profile instrument.
Two of these, general pedagogical knowledge and pedagogical content knowledge cor
respond to knowledge types identified by Shulman (1987). The other three subscales relate
to teachers confidence and their beliefs about numeracy in everyday life, and numeracy
and mathematics in the classroom. As shown in Table 1, these subscales correspond to
sections of the profile and were designed to access various Shulmans (1987) knowledge
types. They are considered separately in the following discussion because, as shown in
Fig. 2, there were clear differences in the distribution of itemsteps from these subscales
across the levels. Illustrative examples are provided here.
Numeracy in everyday life
Numeracy in everyday life were essentially belief statements, and hence the position on the
scale of itemsteps related to these is a measure of the ease with which the teachers were
able to agree with them (Bond and Fox 2007). For example, strong agreement that
mathematics is best taught in an expository style appeared at Level 4. This means that this
itemstep was difficult for teachers to endorse. In general, itemsteps at the lowest level
(Personal Numeracy) were those easiest to endorse and those at Level 4 (PCK Consoli
dation) were most difficult to agree with.
More than half of all itemsteps related to the everyday use of mathematics appeared at
the Personal Numeracy level (Level 1). These included those demanding disagreement
with the proposition that understanding fractions, decimals and percent is becoming
increasingly important (EL4.1), and agreement that finding mathematical patterns in
everyday life is difficult (EL6.1). Itemsteps related to arguably more common everyday
uses of mathematics demanded responses reflecting more perceived competence. These
included those requiring agreement or strong agreement that being numerate is necessary
for intelligent citizenship (EL1.1, EL1.2), and agreement concerning frequent personal use
of mental computation (EL3.1) and the use of mathematics in everyday decision making
(EL10.1).
Levels 2 and 3 (Pedagogical Awareness and PCK Emergence) included only itemsteps
that expressed positive beliefs about the role of mathematics and numeracy in everyday life
and no itemsteps for these items occurred at Level 4 (PCK Consolidation). The most
difficult itemsteps to endorse included those requiring strong agreement that under
standing fractions, decimals and percents is becoming increasingly important in our society
(EL4.3), and that proportional reasoning is required to understand claims in the media
(EL7.3). These appeared at Level 3 (Pedagogical Awareness) along with strong dis
agreement with the statement, I have difficulty identifying mathematical patterns in
everyday life (EL6.4).
Confidence
Only itemsteps demanding high confidence appeared beyond Level 1 (Personal Numer
acy). The nine itemsteps included at this level that suggested low confidence related to
critical numeracy in the media (CN12.1) and aspects of the ELs curriculum. Itemsteps that
demanded low and moderate confidence in relation to all mathematics topics except for
space and measurement also appeared at this level. High confidence in relation to devel
oping students measurement understandings (CN5.1) appeared at Level 1, whereas high
150 K. Beswick et al.
123
confidence in relation to most topics, including space (CN6.1), as well as making con
nections between mathematics and other curriculum areas (CN10.2) appeared at Level 2
(Pedagogical Awareness) along with very high confidence in relation to measurement
(CN9.2).
At levels 3 and 4 (PCK Emergence and PCK Consolidation) only itemsteps repre
senting high or very high confidence appeared. The most demanding itemsteps concerned
very high confidence in making connections between the mathematics and the ELs
(CN11.4), critical numeracy in the media (CN12.4), assessing numeracy against the ELs
standards (CN13.4), and developing students understandings of percent (CN4.4), all of
which appeared at the PCK Consolidation level (Level 4).
Mathematics and numeracy in the classroom
Of the 11 mathematics and numeracy in the classroom items that fitted the model, eight had
itemsteps ordered across levels such that those at higher levels required beliefs more
consistent with those advocated in contemporary mathematics education literature than
those at lower levels. From Level 1 to Level 4, itemsteps demanded increasing agreement
with such ideas, including that secondary school mathematics is best taught in mixed
ability groups at least until grade 9 (NC13), that a student suggesting an unanticipated
solution would cause discomfort (NC2), that allowing a child to struggle with a mathe
matical problem can be necessary for learning (NC5), that ignoring childrens mathe
matical ideas can seriously limit their learning (NC8), that teachers of mathematics should
be fascinated with how children think and intrigued by new ideas (NC3), and that justifying
ones mathematical statements is an extremely important part of mathematics (NC9).
In contrast to these items, several items behaved in ways that are typically presented in
the literature as contrary to studentcentred thinking. Ambivalence through to strong
agreement was demanded by itemsteps at higher levels in relation to the statements that
mathematics is computation (NC1), that mathematics would be difficult to teach without a
textbook (NC11), and that mathematics is best taught in an expository style (NC6).
Pedagogical content knowledge
Itemsteps related to pedagogical content knowledge occurred mainly at Levels 2 and 3
(Pedagogical Awareness and PCK Emergence). The only itemstep in this group that
appeared at Level 1 (Personal Numeracy) required teachers to suggest a student response to
90% of 40 that did not address appropriate procedures for solving it (PC1a.1). Itemsteps
requiring teachers to suggest student responses that similarly did not address relevant ideas
in relation to the pie chart (PC2a.1), and the fraction problem (PC3a.1) occurred at Level 2
(Pedagogical Awareness).
At the PCK Emergence level (Level 3) itemsteps concerning student responses
demanded suggestions that included both appropriate and inappropriate approaches to 90%
of 40 (PC1a.3) and the fraction problem (PC3a.3), and that mentioned both salient and
irrelevant aspects of the pie graph or its context (PC2a.3). Suggested classroom uses at this
level were reflected in itemsteps that required a single generic idea for solving the pie
chart problem (PC2b.2), reference to two or more relevant but unlinked ideas in relation to
each of the three problems (PC1b.3, PC2b.3, PC3b.3) and, for the fraction problem, a
discussion of fractions and wholes including specific examples (PC3b.4). At the PCK
Consolidation level (Level 4), itemsteps required teachers to provide suggested classroom
uses for 90% of 40 that included a discussion of partwhole concepts with specific
Middle school mathematics teachers knowledge development 151
123
examples (PC1b.4) and similarly for the piechart problem, a discussion of percents and
wholes with specific examples (PC2b.4).
General pedagogical knowledge
In terms of general pedagogical knowledge itemsteps the Personal Numeracy level (Level
1) required only that teachers provide a single example of how they would go about
improving the mathematics/numeracy understandings of their students (P1a.1) and more
than one assessment strategy for their chosen topic (P2c.1). It was not until the level of
PCK Emergence (Level 3) that itemsteps required teachers to suggest understanding goals
that included appropriate understandings and skills (P2a.2) and to consider their students
responses to their teaching in terms of understanding (P2d.1). Finally, at the level of PCK
Consolidation (Level 4), itemsteps demanded ideas for improving students mathematics/
numeracy presented as an integrated rationale (P1a.3). For their chosen topic, teachers
needed to provide appropriate goals expressed as understandings (P2a.3), teaching and
assessment plans linked with the understanding goals within an appropriate time frame and
with attention to the evaluation of students understandings (P2b.3). In evaluating their
students reactions to their teaching, consideration of both engagement and understanding
(P2d.2), and multiple examples of how the broader curriculum could contribute to the
development of the relevant understandings (P2e.2), were demanded.
Discussion
The fact that the data gathered using the teacher profile satisfied the assumptions of the
Rasch model confirmed that for these teachers the profile was indeed measuring a single
underlying construct and thus validated our holistic conception of teacher knowledge for
mathematics teaching. Several important lessons can also be derived from the 4level
structure of teacher knowledge that the model revealed.
The two lowest levels of teacher knowledge identified in this study demanded little in
terms of general pedagogical knowledge or pedagogical content knowledge. In contrast
with this, itemsteps expressing confidence to teach most topics in the middle school
mathematics curriculum were easy to endorse with moderate confidence reflected in item
steps at Level 1 (Personal Numeracy) and high levels of confidence in relation to most
topics featuring at Level 2 (Pedagogic Awareness). This is consistent with the relative ease
with which teachers were able to endorse itemsteps indicating a positive view of their
ability to use mathematics in their own everyday lives. It is apparent that confidence to use
mathematics and even to develop mathematical understandings in students does not imply
levels of general pedagogical knowledge or PCK that could be considered satisfactory for
teachers of middle school mathematics. This finding is timely in view of current shortages
of suitably qualified mathematics teachers in many parts of the world (OECD 2004) that
could tempt education authorities to look to teachers without appropriate qualifications
and/or experience, yet confident of their ability for the role, to fill mathematics teaching
positions. Much is also made about the importance of improving the confidence of pre
service primary teachers in relation to teaching mathematics (e.g., Graven 2004). This
study suggests that although building confidence is desirable, its development appears to
precede that of other aspects of knowledge and hence should not be taken as indicative of
competence.
152 K. Beswick et al.
123
In relation to teachers beliefs about mathematics teaching and learning as described in
the section on mathematics and numeracy in the classroom, the sequencing of itemsteps
across levels suggests increasingly student focussed beliefs at higher levels of knowledge.
Itemsteps at Levels 3 and 4 (PCK Emergence and PCK Consolidation) required beliefs
about the value of struggle with mathematical ideas, the importance of using students
ideas, justifying mathematical claims, fostering inquiry and teaching mixed ability groups.
All of these are aligned with the recommendations in the literature regarding student
centred mathematics education. Consistent with this, the appearance at Level 4 of strong
disagreement that the mathematics work done in class is irrelevant to students lives
suggests that more teacher knowledge is associated with classroom tasks that are more
relevant to students, or at least perceived by teachers to be so.
Itemsteps relating to three items, however, present a different picture. It seems that
teachers with greater knowledge, who have ability measures that fell within PCK Con
solidation (Level 4), would be more likely to equate mathematics with computation, and to
value expository teaching and textbook use. One might not expect that these three items
would be endorsed by highly knowledgeable teachers with a studentcentred approach to
mathematics teaching. However, the teacher beliefs literature suggests that there is nothing
necessarily contradictory in this picture. Rather, it is consistent with Beswicks (2007)
argument that it is teachers beliefs about the nature of mathematics, how mathematics is
learned, students capabilities, and the role of the teacher, rather than about the merits of
differing particular pedagogic approaches and tools, that determine the extent to which
teaching is consistent with constructivist (and hence studentcentred or reform) ideas.
Associations of particular practices with studentcentred mathematics teaching and others
with traditional teaching are not helpful.
The belief that mathematics is computation relates to the nature of mathematics and is
usually associated with an instrumentalist view thereof (Ernest 1989). Beswick (2005)
identified it as a belief relatively unlikely to be held by teachers who could be characterised
as having a problem solving orientation (consistent with studentcentred ideas) to math
ematics teaching. The current study, however, included primary and secondary teachers, of
whom only four (of 62) claimed to have majors in mathematics, whereas Beswicks (2005)
study involved only secondary teachers who taught mathematics to at least two classes. Of
the 25 teachers in that study, three reported having majors in mathematics and a further five
claimed to have studied some mathematics at third year university level. The greater
involvement of these teachers with the discipline of mathematics may explain their broader
views of it although Beswick (2009) reported on one teacher who despite considerable
tertiary study of mathematics had great difficulty in conceiving of the discipline beyond the
school curriculum to the extent that she regarded it as quite different from school math
ematics. It seems that the development of knowledge to teach mathematics may not
influence views of the nature of the discipline and hence the belief that mathematics is
computation may persist in spite of the development of studentcentred beliefs about
teaching and learning.
The wording of these items might also be problematic. They are worded positively, but
the sentiment of the items is in the opposite direction. This finding raises an issue about the
structure of Likert type items when used with Rasch measurement. Bond and Fox (2007)
suggest that items should all be worded in the same direction. In most instances, this
requires the removal of negatively worded items, such as I do not believe Whereitems are considered to be working in the opposite direction to a desired state, even if
worded positively, they might be better reverse coded. This issue needs further exploration
and discussion.
Middle school mathematics teachers knowledge development 153
123
From a measurement perspective, there is also scope to refine the teacher knowledge
scale described here. This study represents an initial attempt to consider various aspects of
teacher knowledge on a single scale, and it was decided to keep as much as possible in the
scale. Future considerations will focus on whether it is necessary to retain all items in each
group and the merits of collapsing categories to obtain better discrimination.
Conclusion
The application of Rasch modelling to an instrument designed to measure multiple facets
of mathematics teacher knowledge has shown that teacher knowledge can be conceived of
as a unidimensional construct that underpins many separate aspects. This study has shown
that a holistic conception of teacher knowledge made up of multiple aspects but that is,
nevertheless, a single construct is possible and offers some unique insights into the nature
and development of mathematics teachers knowledge. The scale that resulted shows that
teacher knowledge develops from competence with everyday personal numeracy, unin
formed confidence in ones ability to teach most of the mathematical concepts that con
stitute the middle school mathematics curriculum, and disagreement or ambivalence in
relation to most of the tenets of studentcentred mathematics teaching. Progress is by way
of initial ideas about general pedagogy, increasingly strongly held beliefs largely aligned
with the contemporary studentcentred literature, and confidence to teach and to use
mathematics that appears to be in advance of the general pedagogical knowledge and PCK.
It is evident that personal mathematical competence, confidence to teach mathematics, and
beliefs that are largely aligned with studentcentred agendas do not imply more than the
most basic and inadequate general and content specific pedagogical knowledge. Further
more, the significant demands of Level 4 (PCK Consolidation) itemsteps in terms of both
general pedagogical knowledge and PCK suggest that these aspects of teacher knowledge
develop together.
Previous work that has built upon Shulmans seminal categorisation of knowledge types
has added much to our understanding of what constitutes teacher knowledge and how it
relates to specific content areas such as mathematics. A crucial contribution of Ball and
colleagues (e.g., Ball et al. 2008) has been to identify aspects of mathematical knowledge
uniquely required for teaching mathematics. Ball et al. (2008) pointed to the potential of
their approach to identify aspects of mathematics knowledge for teaching that are espe
cially relevant to improving students learning and for designing more effective mathe
matics curricula for preservice teachers. They acknowledge, however, that further work
and refinement are necessary for these ends to be achieved. Of particular importance will
be clarification and delineation of the various categories. In this study, we were mindful of
the complexity of teaching mathematics, also acknowledged by Ball et al. (2008). Teachers
do not always employ the same sort of knowledge in apparently equivalent situations, and
they draw upon a range of types of knowledge in relation to many of their everyday tasks,
moving among them seamlessly and flexibly. Analysing and categorising their knowledge,
although useful in many respects, risks losing an appreciation of the complexity of the
work of teaching mathematics and may never be possible with complete clarity.
A striking feature of the variable map shown in Fig. 2 is the differing levels of teacher
knowledge that are tapped by the various knowledge subscales. The lowest level (Personal
Numeracy, Level 1) featured beliefs about everyday life, confidence and beliefs about
mathematics teaching and learning that were easy for all of the teachers involved in the
study to endorse. Itemsteps belonging to all five groups are relevant at the middle levels
154 K. Beswick et al.
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(Pedagogical Awareness, Level 2, and PCK Emergence, Level 3), whereas the highest
knowledge level (PCK Consolidation, Level 4) is characterised by itemsteps demanding
the highest levels of confidence in relation to the most difficult aspects of the middle school
mathematics curriculum, beliefs that are most difficult to endorse, and quite sophisticated
understandings of PCK for mathematics teaching and general pedagogy.
The inclusion of teacher belief statements in the measure of teacher knowledge is
consistent with the definition of beliefs as any propositions that an individual regards as
true and distinguished from knowledge only by the degree of consensus that they attract
(Beswick 2005, 2007). In showing beliefs to be contributors to a unidimensional teacher
knowledge construct, this study lends weight to what was previously a theoretically based
contention.
The use of Rasch measurement in relation to these items has raised some interesting
questions for future research. In particular, it suggests that knowledgeable teachers of middle
school mathematics hold sets of beliefs that include aspects that are consistent with the
mathematics education literature promoting studentcentred teaching, and others that may
not be. Precisely how teachers perceive the role of exposition and textbooks and their views
of computation in relation to the discipline of mathematics warrant further exploration. At
the very least, the findings present a warning against simplistic assumptions about what
teachers believe and how readily observable practices relate to underlying beliefs.
It is important to note that the findings reported here are based on a particular group of
middle school teachers, the nature and structure of whose knowledge may not be the same as
that of other teachers. Indeed it is possible that the unidimensionality found in this instance
may not hold in other contexts. Nevertheless, the outcomes of this study suggest that there is
value in pursuing holistic conceptions and measures of mathematics teachers knowledge as
well as ongoing detailed analyses of categories. Such approaches may be a better basis for
future comparison of teachers knowledge with their students achievement in mathematics
as they avoid the delineation problems inherent in attempting to distinguish categories while
still acknowledging the inherent complexity of the construct. In addition, we believe that the
profile instrument used in this study has the potential to be adapted, refined and applied in a
range of settings including the evaluation of professional learning programmes.
Acknowledgments The authors would like to thank Natalie Brown for her valuable contribution to initialdiscussions of the delineation and naming of levels on the teacher knowledge scale, and Suzie Wright for herassistance with the coding and analysis of the data. This research was funded by Australian ResearchCouncil Linkage Grant No. LP0560543.
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The nature and development of middle school mathematics teachers knowledgeAbstractIntroductionConceptualising teacher knowledgeMeasuring teacher knowledgeRasch models
The studyParticipantsProfile descriptionData analysis
Results and discussionOverall fit to the modelIndividual item fitThe scale of middle school mathematics teachers knowledge
Description of the levelsLevel 1: personal numeracyLevel 2: pedagogical awarenessLevel 3: pedagogical content knowledge emergenceLevel 4: pedagogical content knowledge consolidation
Knowledge subscales across the levelsNumeracy in everyday lifeConfidenceMathematics and numeracy in the classroomPedagogical content knowledgeGeneral pedagogical knowledge
DiscussionConclusionAcknowledgmentsReferences