The nature and development of middle school mathematics teachers’ knowledge

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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 3-year 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 multi-faceted nature of teachers knowledge for teaching mathematics,

    K. Beswick (&) R. Callingham J. WatsonFaculty of Education, University of Tasmania, Locked Bag 1307, Launceston, TAS 7250, Australiae-mail: Kim.Beswick@utas.edu.au

    R. Callinghame-mail: Rosemary.Callingham@utas.edu.au

    J. Watsone-mail: Jane.Watson@utas.ed.au

    123

    J Math Teacher Educ (2012) 15:131157DOI 10.1007/s10857-011-9177-9

  • 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 inter-relating 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 non-standard 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 student-centred 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 student-centred) 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 open-ended 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 uni-dimensional 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 item-steps.

    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 uni-dimensional 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

    uni-dimensional 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

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  • 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 item-step difficulties and distribution of item-steps

    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 part-whole 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 ELs-related 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 inter-dependence

    (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 item-steps on the right-hand side of

    the vertical dotted line and persons on the left. The item-steps 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 item-steps are grouped in Fig. 2 is shown in Table 4. Item-steps 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.

    Item-steps 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 items-steps, 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 item-steps at each level. Item-step codes as in Fig. 2

    are included throughout. For the first of these, Level 1 (Personal Numeracy), the complete

    set of item-steps 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 item-steps 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 item-steps 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 item-step codes appearing at each level is shown in Fig. 2.

    Level 1: personal numeracy

    The item-steps 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, item-steps

    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, item-steps 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 item-steps, 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 item-steps 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 item-steps 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 item-step 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 Item-steps

    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 Item-steps at the personal numeracy level (Level 1)

    Item-steps

    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

    Item-steps

    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 student-centred mathematics teaching than those at Level 1. In contrast with the

    previous level, item-steps at this level that related to planning for mathematics teaching

    and to pedagogical content knowledge were present although only demanding low levels of

    responses. Item-steps 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.

    Item-steps 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, item-steps 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.

    Item-steps concerning beliefs about mathematics teaching and learning indicated broad

    agreement with a contemporary student-centred 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, item-steps appearing at this level also indicated ambivalence

    Table 5 continued

    Item-steps

    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

    Uni-structural 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 item-steps 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 item-steps 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, item-steps 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 item-steps 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

    student-centred ideas, about mathematics teaching and learning. The item-steps 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.

    Item-steps 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 item-steps 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 item-step at this level related to beliefs about mathematics teaching and

    learning that required other than agreement or strong agreement. Several item-steps

    required endorsement of statements typically aligned with a student-centred 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

  • Item-steps 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

    pie-chart 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 item-steps at

    this level. The item-step 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, item-steps 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

    Item-steps 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,

    item-steps 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 item-steps concerning the everyday use of mathematics appeared at this level and

    only item-steps 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).

    Item-steps 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 item-steps 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 item-step requiring teachers to disagree strongly that the

    mathematics they do in class is often irrelevant to their students lives (NC14.4).

    Item-steps requiring teachers to discuss a range of possible uses of each of 90% of 40

    and the pie-chart problem, which referred to relevant part-whole 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, item-steps 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). Item-steps 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 item-steps 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 item-steps 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

    item-step was difficult for teachers to endorse. In general, item-steps 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 item-steps 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). Item-steps 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 item-steps

    that expressed positive beliefs about the role of mathematics and numeracy in everyday life

    and no item-steps for these items occurred at Level 4 (PCK Consolidation). The most

    difficult item-steps 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 item-steps demanding high confidence appeared beyond Level 1 (Personal Numer-

    acy). The nine item-steps included at this level that suggested low confidence related to

    critical numeracy in the media (CN12.1) and aspects of the ELs curriculum. Item-steps 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 item-steps repre-

    senting high or very high confidence appeared. The most demanding item-steps 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

    item-steps 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, item-steps 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 student-centred thinking. Ambivalence through to strong

    agreement was demanded by item-steps 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

    Item-steps related to pedagogical content knowledge occurred mainly at Levels 2 and 3

    (Pedagogical Awareness and PCK Emergence). The only item-step 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). Item-steps

    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) item-steps 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 item-steps 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), item-steps required teachers to provide suggested classroom

    uses for 90% of 40 that included a discussion of part-whole concepts with specific

    Middle school mathematics teachers knowledge development 151

    123

  • examples (PC1b.4) and similarly for the pie-chart problem, a discussion of percents and

    wholes with specific examples (PC2b.4).

    General pedagogical knowledge

    In terms of general pedagogical knowledge item-steps 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 item-steps 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), item-steps 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 4-level

    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, item-steps 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 item-steps 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 item-steps

    across levels suggests increasingly student focussed beliefs at higher levels of knowledge.

    Item-steps 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.

    Item-steps 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 student-centred 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 student-centred or reform) ideas.

    Associations of particular practices with student-centred 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 student-centred 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 student-centred 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 uni-dimensional 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 student-centred mathematics teaching. Progress is by way

    of initial ideas about general pedagogy, increasingly strongly held beliefs largely aligned

    with the contemporary student-centred 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 student-centred 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) item-steps 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. Item-steps belonging to all five groups are relevant at the middle levels

    154 K. Beswick et al.

    123

  • (Pedagogical Awareness, Level 2, and PCK Emergence, Level 3), whereas the highest

    knowledge level (PCK Consolidation, Level 4) is characterised by item-steps 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 uni-dimensional 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 student-centred 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 uni-dimensionality 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

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