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Building teachers expertise in understanding, assessing and developing children s mathematical thinking: the power of task-based, one-to-one assessment interviews Doug Clarke, Barbara Clarke & Anne Roche
Building teachers expertise in understanding, assessing and developing children s mathematical thinking: the power of task-based, one-to-one assessment interviews Doug Clarke, Barbara Clarke & Anne Roche ZDM The International Journal on Mathematics Education ISSN ZDM Mathematics Education DOI / s Your article is protected by copyright and all rights are held exclusively by FIZ Karlsruhe. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your work, please use the accepted author s version for posting to your own website or your institution s repository. You may further deposit the accepted author s version on a funder s repository at a funder s request, provided it is not made publicly available until 12 months after publication. 1 23 ZDM Mathematics Education DOI /s ORIGINAL ARTICLE Building teachers expertise in understanding, assessing and developing children s mathematical thinking: the power of task-based, one-to-one assessment interviews Doug Clarke Barbara Clarke Anne Roche Accepted: 11 June 2011 Ó FIZ Karlsruhe 2011 Abstract In this paper, we outline the benefits to teachers expertise of the use of research-based, one-to-one assessment interviews in mathematics. Drawing upon our research and professional development work with teachers and students in primary and middle years in Australia and the research of others, we argue that the use of the interviews builds teacher expertise through enhancing teachers knowledge of individual and group understanding of mathematics, and also provides an understanding of typical learning paths in various mathematical domains. The use of such interviews also provides a model for teachers interactions and discussions with children, building both their pedagogical content knowledge and their subject matter knowledge. Keywords Teacher expertise Assessment interview Student thinking Teacher knowledge 1 Introduction This paper begins with a brief discussion of research and frameworks of teacher knowledge, drawing largely upon the seminal work of Shulman and the more recent work of Ball and her colleagues. We discuss the relatively recent phenomenon of the use of one-to-one assessment D. Clarke (&) A. Roche Australian Catholic University, 115 Victoria Parade, Fitzroy, VIC 3065, Australia A. Roche B. Clarke Monash University, PO Box 527, Frankston 3199, Australia interviews in building teacher expertise in professional development settings in Australia. We then draw upon two research programs in which the one-to-one interview formed a major component. These projects were the Early Numeracy Research Project (ENRP) and the Australian Catholic University (ACU) Rational Number Project. We provide the reader with considerable detail on the interviews within these projects. We give examples of the kinds of tasks used and the learning framework which underpinned them. We then outline the preparation involved prior to teachers using the interviews, and the steps taken to maintain consistent use of the interviews across many teachers. We share data on student performance on certain tasks. Following this background information on the interviews and their use, we then discuss the benefits of such use to preservice and inservice teachers, in building their expertise. In doing so, we draw upon teacher questionnaire, individual and focus group data as it supports our argument. Both these research projects were large scale, involving hundreds of teachers and thousands of students. With a focus on student learning and growth in such learning over time as evidenced from interview data, the research projects were not originally intended to study the direct contribution of the one-to-one interview to developing teacher expertise. However, the data on this which emerged during the projects were compelling. 2 Teacher knowledge In considering ways to build teacher expertise, an important component is of course teacher knowledge (Fennema and Franke 1992). Shulman (1986) first coined the term pedagogical content knowledge (PCK). He described PCK, D. Clarke et al. the intersection of content knowledge and pedagogical knowledge as the most useful forms of representation of those ideas, the most powerful analogies, illustrations, explanations and demonstrations in a word, the ways of representing and formulating the subject that makes it comprehensible to others (1986, p. 9). Since Shulman s seminal work, researchers in mathematics education have attempted to conceptualise and measure teachers mathematical knowledge for teaching (e.g., Hill, Ball, and Schilling 2008). Such work is important, as research has determined that student outcomes can be improved by enhancing teachers mathematical knowledge for teaching (Hill, Rowan, and Ball 2005). Ball, Thames, and Phelps (2008) many years later proposed the model shown in Fig. 1. Although there is no clear consensus on the components of teachers knowledge and may never be, like Graeber and Tirosh (2008), we regard the ongoing conversations as constructive, and view the attempts to define the construct PCK within mathematics education as part of that larger quest to establish a unique base of professional knowledge, a hallmark of a true profession, for teachers of mathematics (p. 129). In later sections, we will not only provide evidence of the growth in teacher expertise, as it relates to several of the components of the Ball et al. (2008) model, but also argue that such expertise cannot be fitted neatly into these categories. 3 An increased emphasis in Australia on the use of one-to-one assessment In the last 20 years, the inadequacy of a single assessment method administered to students at the end of the teaching of a mathematics topic has been widely acknowledged (Ginsburg 2009). It is increasingly the case that those working at all levels of mathematics education regard the major purpose of assessment as improving instruction and supporting learning (Webb and Romberg 1992), and this has led to a search for appropriate assessment methods to achieve this. The limitations and disadvantages of pen and paper tests in gathering high quality, in-depth data on children s knowledge were well established by Clements and Ellerton (1995). They contrasted the quality of information about Grade 5 and Grade 8 students gained from written tests with that gained through one-to-one interviews. They observed that children may have a strong conceptual knowledge of a topic (revealed in a one-to-one interview) but be unable to demonstrate that during a written assessment. The data suggested that around one-quarter of students responses could be classified as either (a) correct written responses given by students who did not have a sound understanding of the mathematical knowledge, skills, concepts and principles which the questions were intended to cover ; or (b) incorrect written answers given by students who had partial or full understanding. Following the work of Piaget, clinical interviews have been used for many years in mathematics education research (Ginsburg et al. 1998). Typically, such research has been conducted with relatively small numbers of students, and the results not always communicated well to the teaching profession. However, the late 1990s, in Australia and New Zealand, saw the development and use of research-based one-to-one, task-based interviews with large numbers of students, as a professional tool for teachers of mathematics (Bobis, Clarke, Clarke, Gould, Thomas, Wright, and Young-Loveridge 2005). Fig. 1 Framework of mathematical knowledge proposed by Ball et al. (2008, p. 403) Building teachers expertise in children s mathematical thinking 4 Two large-scale research projects involving extensive use of assessment interviews 4.1 The ENRP The ENRP research and professional development program, conducted in Victoria from 1999 to 2001, involved 353 teachers and over 11,000 children aged 5 8 years old (Clarke 2001; Clarke, Cheeseman, Gervasoni, Gronn, Horne, McDonough, Montgomery, Roche, Sullivan, Clarke, and Rowley 2002). There were four key components to this research and professional development project: the development of a research-based framework of growth points in young children s mathematical learning (in Number, Measurement and Geometry); the development of a 40-min, one-on-one, task-based interview, used by all teachers to assess aspects of the mathematical knowledge of all children at the beginning and end of the school year; extensive teacher professional development at central, regional and school levels, for teachers, mathematics coordinators, and principals; and a study of the practices of particularly effective teachers. As part of the ENRP, it was decided to create a framework of key growth points in numeracy learning. Students movement through these growth points in project schools, as revealed in interview data, could then be tracked over time. In creating the growth points, the project team studied available research on key stages or levels in young children s mathematics learning (e.g., Clements et al. 1999; Lehrer and Chazan 1998), as well as frameworks developed by other authors and groups to describe learning. Data relating to growth points and student learning have been reported in a number of publications (see, e.g., Clarke 2004; B. A. Clarke, Clarke, and Cheeseman 2006). There are some parallels in our growth points with the work of Simon (1995), who referred to a hypothetical learning trajectory as the teacher s prediction as to the path by which learning might proceed (p. 135). Our work differs from that of Simon and Clements and Sarama (2004) in that instructional sequences are somewhat implied by our growth points, but not specified (Clarke 2008). The one-to-one interview (assessing Number, Measurement and Geometry) was used with around 11,000 children, taking an average of 45 min, and varying in time according to the interviewer s experience and the responses of the child. The interview followed a very tight script, which indicated precisely which question to ask next, given a particular response to the previous item. The interviews were conducted by the student s regular classroom teacher, following a full day s training on its use, and the opportunity to practise the interview process under the eye of either the school mathematics coordinator (who had received additional training) or a member of the research team. A range of procedures was developed to maximise consistency in the way in which the interview was administered across the schools. The teacher completed a record sheet during each interview, which recorded both students answers and their stated method. In many cases, the method was evident because of the students body language (e.g., use of fingers), but for most tasks the student was asked a question of the kind, please explain your thinking, how did you work that out?, could you do that a different way? ), but always according to the script. There was effectively no time limit on students responses, although when it became clear that the student had little idea on how to attempt to solve a given problem, the teacher would usually move on. The interview provided information about growth points achieved by a child in each of nine mathematical domains: four in Number (counting, place value, addition and subtraction, multiplication and division); three in Measurement (time, length, mass), and two in Geometry (properties of shape and visualisation and orientation). Although the full text of the ENRP interview involved around 60 tasks (with several sub-tasks in many cases), no child moved through all of these. The interviewer made a decision after each task, according to the script. Given success on a particular task, the interviewer continued with the next task in the domain as far as the child could go with success. Given difficulty with the task, the interviewer either abandoned that section of the interview and moved on to the next domain or moved into a detour, designed to elaborate more clearly the difficulty a child might be having with a particular content area. To clarify the notion of a detour, the interview starts with a task where students are asked, using a cup, to take a big scoop of plastic teddy bears from a tub of teddy bears. They estimate the total and then count them, using any method they choose. If they are unsuccessful in correctly counting the collection, they move into a detour which focuses on tasks to do with more and less, oneto-one correspondence, and conservation. Figure 2 includes two questions from the interview (Department of Education and Training 2001). These questions focus on identifying the mental strategies for subtraction that the child draws upon. The strategies used were recorded on the interview record sheet. Since its development, the ENRP interview has been used by teachers and researchers in Australia, New D. Clarke et al. Fig. 2 Two tasks from the ENRP interview 19. Counting Back For this question you need to listen to a story. a) Imagine you have 8 little cookies in your morning snack and you eat 3. How many do you have left?... How did you work that out? If incorrect answer, ask part (b): b) Could you use your fingers to help you to work it out? (It s fine to repeat the question, but no further prompts please). 20. Counting Down To / Counting Up From I have 12 strawberries and I eat 9. How many are left?... Please explain. Zealand, Germany, Sweden, South Africa, Canada and the USA. 4.2 ACU Rational Number Interview Following the ENRP, and requests for a similar interview from teachers of older students, it was decided to develop a one-to-one interview for teachers of 9- to 14-year olds. Given the recognised difficulty with fractions and decimals for many teachers and students (see, e.g., Behr et al. 1983; Steinle and Stacey 2003), it was decided to make rational numbers the focus of the interview. Anne Roche adapted and developed tasks in decimals (see, e.g., Roche, 2005, 2010) and Annie Mitchell in fractions (see, e.g., Mitchell and Horne 2010). An important source of tasks was the Rational Number Project (Behr and Post 1992). Student data on key tasks were reported in D. M. Clarke, Roche, Mitchell, and Sukenik (2006b). As part of this project, this interview was used with 323 students who were completing the last year of primary school. Two sample tasks from the ACU Rational Number interview are given in Fig. 3. These are Construct a Sum from the Rational Number Project (Behr et al. 1985), and Make me a Decimal (Roche and Clarke 2004). These two tasks illustrate the potential of one-to-one interview tasks, as compared to traditional written assessment. The capacity of students to move the cards around in each task has at least two clear benefits. First, the student can place them in particular positions initially, knowing that they are easily changed. Second, the teacher conducting the interview has a window into children s reasoning as they see them move the pieces from place to place. Such rich information would be very difficult to collect from a written assessment. Fig. 3 Sample tasks from the Australian Catholic University Rational Number Interview Construct a sum Place the number cards and the empty fraction sum in front of the student. a) Choose from these numbers to form two fractions that when added together are close to one, but not equal to one. Record the student s final decision. b) Please explain how you know the answer would be close to one. Record any change of solution. Make me a decimal Here are some number cards and some blanks that could be any number a) Could you use some of these cards to show me what two tenths would look like as a decimal? b) 27 thousandths? c) ten tenths? d) 27 tenths? Figure 3. Sample tasks from the Australian Catholic University Rational Number Interview. Building teachers expertise in children s mathematical thinking 4.3 Some information on how the interviews were administered and the data collected Student strategies were recorded in detail on the interview record sheet. For example, for the two ENRP subtraction tasks outlined in Fig. 2, the teacher completes the record sheet, as shown in Fig. 4, recording both the answer given and the strategies used. The emphasis on asking for and recording both answer and strategies is clear recognition that the answer alone is not sufficient, and gives a message to students that their strategies and mathematical thinking are valued (Swan 2002). The act of completing the record sheet requires an understanding of the strategies listed (e.g., modelling all, fact family, count up from, etc.), but of course must be preceded by extensive teacher professional development on its use. This is our first example of the kinds of teacher expertise which are developed prior to and during the use of the interview. Processes used by the ENRP research team to maximise reliability and validity of interview data have been detailed elsewhere (Horne and Rowley 2001). Having data on over 36,000 ENRP interviews for the 11,000 students (with many students being interviewed on several occasions) and around 400 for the ACU Rational Number Interview provided previously unavailable high quality data on student performance. For example, Table 1 shows the percentage of children on arrival at schools (typically 5 year-olds), who were able to match numerals to their corresponding number of dots (B. A. Clarke et al. 2006). Fig. 4 An excerpt from the addition and subtraction tasks on the interview record sheet Table 1 Performance of children on entry to school on selected tasks (%) (n = 1,437) Match numeral to 2 dots 86 Match numeral to 4 dots 77 Match numeral to 0 dots 63 Match numeral to 5 dots 67 Match numeral to 3 dots 79 Match numeral to 9 dots 41 During the ENRP, it became increasingly obvious to the research team that the interview was providing opportunities for development of teachers knowledge. Evidence emerged of teachers greater confidence in the use of mathematical language, and of their growing sense of typical learning paths of their students. 5 Methodology Percent success (%) As mentioned earlier, the research projects were not originally established to study the direct contribution of the one-to-one interview to developing teacher expertise, but the compelling nature of the anecdotal data which emerged early encouraged our team to investigate this more thoroughly. Although there was a range of data which is not reported here (e.g., teachers grouping practices, their planning methods, actual time given to mathematics, and their expectations of student growth), data collection relevant to this article took the following forms: Teacher Entry questionnaire (February 1999), involving 24 items focusing on areas including background information, personal mathematical knowledge, confidence in teaching mathematics, mathematical expectations of their students, and areas of their teaching which they sought to improve (n = 195). Teacher Exit questionnaire (October 2001), involving 21 items focusing on similar areas to the Entry questionnaire, in order to discern changes over time (n = 221). Teachers Highlights and Surprises questionnaire (March 1999), where teachers were simply asked what highlights and surprises were there as a result of conducting the interviews with your students? (n = 198). Changes in Teaching Questionnaire (October 2001), where teachers were asked to nominate the greatest D. Clarke et al. changes in their teaching and in their students as a result of their involvement in the ENRP (n = 220). All of these data are reported in detail in Clarke et al. (2002). Using a two-page questionnaire involving Likert scale items and open response items, 140 preservice teachers at ACU and Monash University were asked to comment on their growing expertise as mathematics teachers through their use of one-to-one assessment inte
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