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


    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:

    R. Callinghame-mail:

    J. Watsone-mail:


    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


    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.


  • 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


  • 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.


  • 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 u...


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