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School pupil change associated with acontinuing professional developmentprogramme for teachersJane Watson a & Kim Beswick aa Faculty of Education , University of Tasmania , AustraliaPublished online: 20 Jan 2011.
To cite this article: Jane Watson & Kim Beswick (2011) School pupil change associated with acontinuing professional development programme for teachers, Journal of Education for Teaching:International research and pedagogy, 37:1, 63-75
To link to this article: http://dx.doi.org/10.1080/02607476.2011.538273
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Journal of Education for TeachingVol. 37, No. 1, February 2011, 63–75
ISSN 0260-7476 print/ISSN 1360-0540 online© 2011 Taylor & FrancisDOI: 10.1080/02607476.2011.538273http://www.informaworld.com
School pupil change associated with a continuing professional development programme for teachers
Jane Watson* and Kim Beswick
Faculty of Education, University of Tasmania, Australia
Taylor and FrancisCJET_A_538273.sgm(Received 22 May 2010; final version received 14 September 2010)10.1080/02607476.2011.538273Journal of Education for Teaching0260-7476 (print)/1360-0540 (online)Original Article2011Taylor & Francis371000000February [email protected]
This paper reports on the evaluation of a six-day programme that providedprofessional learning to middle school teachers with the aim of equipping them toassist their pupils to achieve improved numeracy outcomes. A teacher profilinginstrument designed to measure varied aspects of teachers’ knowledge forteaching mathematics was administered to 29 teachers at the beginning and end ofthe programme. As well, over 670 of their pupils were surveyed at the beginningof the programme and the end of the school year in relation to the mathematicalexperiences they had in their classrooms and their performance on basic numeracytasks. Evidence of changed classroom practice and improved pupil performancewas observed.
Keywords: mathematics; teacher professional learning; pupil change
Introduction
Measuring the effectiveness of professional development programmes for teachers isfraught with technical and ethical issues. Asking teachers for feedback and self-reported change is the easiest and least controversial way to assess a programme but,as Mewborn (2003) acknowledged, this method has the weakness of a lack of hardevidence of improvement of teachers’ knowledge or pedagogical performance, or ofimprovement of outcomes for their pupils (e.g., Karagiorgi and Charalambous 2006).Reports in the literature of the ineffectiveness of much educational research (e.g.,Burkhardt and Schoenfeld 2003) and the absence of rigorous evidence in evaluationsof educational interventions led the Australian Councils of Deans of Education (2004,6) to call for ‘Greater and higher quality provision [of professional learning],evidenced through quantitative and qualitative evaluation, including studentoutcomes’. Professional development aims to effect change in teachers’ beliefs andknowledge (Beswick 2007) and hence their classroom practices (Wilson and Cooney2002) in anticipation of positive impacts on pupil learning. Given the acknowledgeddifficulty of changing teachers’ beliefs (Cooney 2001), a complex mixture of learningprogramme and evaluation processes is needed to effect and measure change. Ethicalissues also impact upon the rigour of evaluations. Teachers involved in researchprojects have to be given the freedom to withdraw or not complete instruments, or torefuse to have their pupils tested. If teachers for whom a programme has not beensuccessful choose to withdraw, this can bias results.
This paper reports on the effectiveness of a programme designed to enhance aspectsof teacher knowledge believed to contribute to successful teaching for numeracy in the
*Corresponding author. Email: [email protected]
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64 J. Watson and K. Beswick
middle years of schooling. In the third iteration of the programme, reported here,teacher profiling instruments and pupil surveys of their classes were administered atthe beginning and end of the programme. The analysis of outcomes from theseinstruments is used to answer the following research questions:
(1) What changes in teacher pedagogical knowledge for teaching numeracy wereevident following a six-day continuing professional development programmein middle school numeracy?
(2) What changes were evident in the beliefs and numeracy outcomes of pupilsin the classes of the teachers involved in the programme?
(3) How did teachers perceive the effectiveness of the programme?
Context
The context within which this study is set reflects the interest in knowledge for teach-ing that has evolved in recent years in the mathematics education research community(see Chick 2007; Hill, Schilling, and Ball 2004). Generally it has grown from the workof Shulman (1987) who suggested seven types of teacher knowledge required forsuccessful classroom teaching: content knowledge; general pedagogical knowledge;pedagogical content knowledge; knowledge of students as learners; curriculumknowledge; knowledge of education contexts; and knowledge of education ends,purposes, and values. Although Shulman’s work was generic across subject matter,others (see Hill, Schilling, and Ball 2004) have linked, combined, and elaboratedShulman’s ideas in an effort to describe adequately the knowledge needed to teachmathematics. The significance of Shulman’s approach is that it focuses both onaspects of knowledge (e.g., of students) from educational psychology as well as on thecrucial elements of content knowledge and pedagogical content knowledge. Theexpectation from following Shulman’s lead is that improving teachers’ knowledge inthese areas will improve their pupils’ learning outcomes. Although such an expecta-tion has been abroad for a very long time (e.g., Stones 1978), the actual collecting ofdata is very difficult to organise and there may be potential lurking variables that arenot measured. Although some work has linked teacher mathematical knowledge forteaching to pupil outcomes (e.g., Hill, Rowan, and Ball 2005), there appears to be nodirect evidence of teacher change associated with changed pupil performance.
In their review of research on the effects of mathematics teaching on schoolpupils’ learning, Hiebert and Grouws (2007) acknowledged the complexity of thelearning environment and the difficulty of isolating and measuring the actions ofteachers and consequent change in pupil learning. They concluded that the opportu-nity to learn is the key piece in the puzzle. By this they meant more than being taughtcontent or spending time on task. They included ‘consideration of students’ entryknowledge, the nature and purpose of the tasks and activities, and the likelihood ofengagement’ (379). Provision of these aspects of opportunity to learn require most, ifnot all, of the knowledge types identified by Shulman, and particularly his notions ofcontent knowledge, pedagogical content knowledge, and knowledge of students aslearners.
The question of how to provide teachers with the professional learning to enablethem to develop strategies for the classroom that would or could result in opportunitiesfor pupils to learn mathematics can be considered in terms of developing teachers’
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knowledge as conceptualised by Shulman. It is helpful also to consider the ends towhich teachers use that knowledge in their classrooms. Hiebert and Grouws (2007)suggested two such ends, namely of teaching for skill efficiency and teaching forconceptual understanding. Teaching for skill efficiency is teacher-directed involvingmodelling followed by practice. Teaching for conceptual understanding is morecomplex and has two key features: teachers and pupils attend explicitly to conceptsand pupils struggle with important mathematics. The focus is more on pupils and whatthey require to be able to engage and struggle with important concepts.
Sowder (2007) addressed the principles that should guide professional develop-ment for teachers of mathematics. Her goals also reflect Shulman’s (1987) types ofknowledge as well as Hiebert and Grouws’ (2007) key features. In particular, shereferred to the principles developed by Hawley and Valli (1999), which were influen-tial in the design of the programme reported here. These included analysis of pupilperformance against benchmarks, consultation with teachers on their perceived needs,basing expectations in the school environment, provision for collaborative problem-solving, on-going commitment by the school system, evaluating multiple sources ofinformation, providing opportunity for theoretical development, and the expectationand support for genuine change. Sowder also suggested teachers investigate their ownpractice and develop habits of reflection, and noted various ways of describing teacherchange in relation to cognitive, psychological, and sociological factors. She acknowl-edged, however, that little has been documented on the association of teacher changewith pupil change.
One way to consider teacher change is to develop a profiling instrument, as doneby Watson (2001), reflecting Shulman’s (1987) types of teacher knowledge for teach-ing. Such a profiling instrument has been adapted for various content areas in themathematics curriculum and shortened to be used as a survey instrument (seeBeswick, Watson, and Brown 2006; Watson, Beswick, and Brown 2006).
Method
Professional development programme
The continuing professional development programme (CPD), run in three consecutiveyears, was an initiative of Tasmania’s Department of Education (DoE). In the thirdyear, the focus of this research, the 12 schools in the programme were single campusdistrict schools (K–10) with a middle school programme; most of these schools hadsignificant percentages (but not necessarily large numbers) of their Grade 7 studentsstruggling with numeracy. Each school selected three or four teachers to take part inthe programme, which consisted of three two-day professional development sessionsat a site away from the schools during the first two (of three) teaching terms of theschool year. Thirty-seven teachers began the programme with 35 completing it. Thethird year was the only year in which pupils were surveyed at both the beginning (n =970) and end (n = 677) of the programme. Relief and travel were provided for allteachers by the DoE. Due to staff changes in the participating schools, some teacherswho completed the initial teacher profile were replaced by others who completed thefinal profile.
The six-day programme was based on the principles of Hawley and Valli (1999)and was consistent with the more recent reviews by Sowder (2007) and Hiebert andGrouws (2007). Initial planning was informed by feedback sought from middle school
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66 J. Watson and K. Beswick
teachers on issues they felt were important and topics with which they wanted assis-tance. These included mental computation, proportional reasoning (fractions, deci-mals, and percents), and assessment. The programme provided the opportunity forteachers to develop their own skill efficiency and conceptual understanding (Hiebertand Grouws 2007) and to consider how they could provide similar opportunities fortheir pupils. The content focus for the first two days (end of March) was mentalcomputation, including a presentation on student difficulties based on pupil surveysfrom the beginning of the previous year and other available research. Materials wereprovided, including a mental computation teaching package (Dole and McIntosh2005), printed readings, and $AUD300 worth of concrete materials. Activitiesincluded using 1–100 boards, experiencing ‘Today’s number is…’ (McIntosh, DeNardi, and Swan 1994) with fractions, viewing a DVD of children’s multiplicationstrategies, and touring online resources.
The second two-day session focussed on proportional reasoning after an initialtime for sharing and reflecting on activities undertaken since the first session, with anemphasis on the changes teachers had implemented and how their pupils hadresponded. Groups of teachers completed the proportional reasoning tasks and usedscoring rubrics emphasising the communication of both answers and mathematicalunderstanding to mark each others’ work. Links were made to Tasmanian curriculumdocuments and some formal ideas for teaching proportional reasoning were intro-duced, as well as modelling with counters. Activities included work with number linesand ordering large and small numbers, paper folding to create fractions, as well astime to explore websites recommended by the DoE. Teachers were asked to trial someof the activities with their classes and bring work samples to the next two-day session.
The final two-day session was held in mid-August, with a major focus on deci-mals. It began with a sharing time on trialled activities. The results of the initialsurveys of pupils in the teachers’ classes were presented and the misconceptionsdisplayed with respect to some of the problems were discussed in detail; in particularthe difficulty pupils had in explaining the reasoning for their answers was highlighted.Work with decimals included concrete materials, decimal squares, links to fractions,numeral expanders, number lines as linear models, ‘Today’s number is…’ with deci-mals, and games involving decimals and calculators. Time was allocated for collabo-rative planning in school groups. Three times during the programme, $AUD250 wasprovided to each school to spend on resources, including further concrete materials,books or software.
Instruments
The initial teacher profiling instrument (Watson et al. 2006) used with the teachers atthe beginning of the programme was developed from that of Watson (2001) andreflected Shulman’s seven types of teacher knowledge.
Six items asked about teacher confidence in relation to particular topics empha-sised in the programme. Another five asked about confidence in relation to mentalcomputation, for example with respect to basic number facts and operations with frac-tions. Beliefs about numeracy in the classroom were measured by the 14 items usedin Beswick, Watson, and Brown (2006). Examples of items included the following:mathematics is computation; telling children the answer is an efficient way of facili-tating their mathematics learning; it is important that mathematics content be
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presented to children in the correct sequence; effective mathematics teachers enjoylearning and ‘doing’ mathematics themselves; mathematics would be very difficult toteach without a textbook.
Teachers’ knowledge of students as learners (KSL) and pedagogical content knowl-edge (PCK) were assessed in items asking for suggested pupil responses (correct andincorrect) to questions and how they might use the questions in the classroom. An itemused on both profiles is shown in Figure 1 (with the space to answer removed). Therubric for assessing responses, shown in Table 1, was adapted from that used byWatson, Beswick, and Brown (2006) and reflects increasingly complex understandingof the mathematics, the pupils, and potential teaching strategies in the classroom.Figure 1. Example of a question used to assess teachers’ KSL and PCK.Teachers were asked at the start of the programme to describe the areas of personalprofessional development they would like addressed by the project, for examplerelated to personal understanding of mathematics, resources and concrete materials,and teaching for understanding. At the end of the programme, teachers were askedabout their perceptions of the extent to which these needs had been met.
The pupil survey had two parts, asking pupils what they thought of mathematicsand of mathematics in their classrooms (Part A) and assessing their mathematicalthinking for a series of problems involving basic number skills (Part B). The 25 itemsin Part A were made up of 16 relating to pupils’ attitudes to mathematics, as used byBeswick et al. (in press), and nine relating to their perceptions of their mathematicsclassroom environments. The attitude items comprised two representing each of eight
Figure 1. Example of a question used to assess teachers’ KSL and PCK.
Table 1. Rubric for teacher responses to 25% of 80.
Level Profile Question 1 Profile Question 2
0 No response No response1 Responses not addressing percent or part-
whole or answers without any explanation
Response not addressing the mathematical content of the problem
2 Response indicating either the correct percent relation to the whole or an incorrect relation
One or more generic ideas for the problem, e.g., use discount, discuss percent
3 Response containing both appropriate and inappropriate approaches to the problem
Reference to relevant idea/s without linking them
4 NA Discussion including reference to percents and fractions/wholes with specific examples
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68 J. Watson and K. Beswick
aspects of attitude to mathematics identified in the literature. These were a like ordislike of mathematics, an inclination to engage in or to avoid mathematics, beliefsabout whether one is good or bad at mathematics, beliefs that mathematics is impor-tant or unimportant, useful or useless, easy or difficult (Ma and Kishor 1997), and thatmathematics is interesting or uninteresting (McLeod 1992), along with confidence oranxiety in relation to mathematics (Ernest 1988). Seven of the nine classroom envi-ronment items were adapted from the Constructivist Learning Environment Survey(Taylor, Fisher, and Fraser 1993) and two were added to ascertain pupils’ perceptionsof the frequency with which they used concrete materials and were asked to explaintheir thinking in their mathematics lessons. Each of the 25 items required a responseon a five-point Likert scale ranging from ‘strongly agree’ to ‘strongly disagree’ for theattitude items, and from ‘never’ to ‘very often’ for the classroom environment items.Part A questions were the same on both student surveys.
In Part B pupils were asked to explain their understanding in relation to basicnumeracy concepts. The layout of the questions is shown for two problems inFigure 2 (with the space for writing removed).Figure 2. Examples of questions from Part B of the student survey.To test the concepts involved while avoiding the possibility of memorising theanswers to the first survey, which teachers had access to during the CPD programme,the format of questions in Part B remained the same but the numbers used in all excepttwo questions were changed. In both versions Item 4 presented 16 decimal numbersbetween ‘0’ and ‘1.0’, represented with up to three decimal places, with the request topick pairs that summed to ‘1.0’. Item 6 presented eight fractions less than ‘1’ with therequest to circle the two closest to ‘1’. The modifications from the first to the secondsurvey for the other items are shown in Table 5 in the Results. Item 8, involving a piechart, was used in both versions with just the context and hence segment labelschanged. The items were presented in formats similar to that shown in Figure 2, withspace for explaining answers.
Figure 2. Examples of questions from Part B of the student survey.
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Analysis
The attitudes and belief questions were analysed using paired t-tests for the teachersand pupils who answered both surveys. The pupil responses to Part B of the surveywere mainly to questions with the same structure but different numbers. The analysisof difference in performance for each question was therefore based on Wilcoxon SignedRank tests. In this analysis pupils’ improvement is recorded as a positive change ofcode, and diminished performance is recorded as a reduction in code. No change fromearlier performance is recorded as zero for all pupils (whatever code, e.g., ‘0’, ‘1’, or‘2’, was received on both surveys). This is a conservative test since pupils receivingthe highest score on an item for each survey could not improve for that item.
Results
Research question 1: teacher change
Beliefs
Teachers were asked to rate their confidence in teaching various topics in the middleschool mathematics curriculum on a continuous scale that was translated to ‘1’ forleast confident to ‘5’ for most confident. Average confidence increased on all topics.Changes significant at the 0.05-level based on the paired t-tests were observed forfractions, percent, ratio and proportion, numeracy across the curriculum, criticalnumeracy in the media, basic facts, operations with fractions, and operations withdecimals.
Teachers’ beliefs related to the mathematics classroom changed very little over therelatively brief time of the programme (March to August). Some of the statementspresented were not directly discussed or debated during the programme, rather, theemphasis of the programme was on promoting practices such as the use of rich tasks,allowing pupils to struggle with challenging ideas, and valuing mathematicalprocesses such as explaining thinking and justifying solutions.
KSL and PCK
The item shown in Figure 1, based on anticipating students’ responses to ‘25% of80’, is used to illustrate teacher change in respect to KSL and PCK. Three categoriesof response were identified for teachers’ suggestions of the appropriate and inappro-priate responses their pupils would give to the question (Question 1 in Figure 1).These categories were: (1) suggestions not addressing percent; (2) suggestions ofincomplete or incorrect pupil responses; and (3) suggestions of pupils’ explainedprocedures. Table 2 contains a summary of these groupings and the number of eachtype on the initial and final profiles. The total number of suggestions increased from92 to 111 over the programme with a doubling of suggestions of incomplete orincorrect pupil responses, and fewer suggestions not explicitly addressingpercentages.
Table 3 contains a summary of the strategies for using 25% of 80 (or 60) in theclassroom for pupils experiencing difficulties with this type of problem. Few teachersin either the initial or final profile suggested just ‘demonstrating’ the procedure.Relating the problem to number concepts and suggesting the use of various aids bothincreased between the initial and final profiles. The number of responses increased
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70 J. Watson and K. Beswick
from 49, for the 28 teachers in the initial profile, to 87 in the second profile; thisrepresented an average increase of 1.36 suggestions per teacher during theprogramme, which is encouraging given the programme’s focus on the concepts andimportant mathematics.
Table 2. Summary of suggested student responses for 25% of 80 (or 25% of 60 in finalsurvey).
Suggestions from 28 teachersNo. of initial
responsesNo. of final responses
Not addressing percent:Use calculator; Don’t know; No response; Random answer;
Answer only (20); Estimate
33 20
Incomplete or incorrect responses:Confuses operation: 25*80, 80/25, 80–25, 80+25; Confuses operation, role of decimal place or percentage:2500*80, 2.5, 25
17 37
Explained procedures:Percent as a fraction: 25% is a quarter so divide by 4;Repetitive division: 50% is 1/2, and 1/2 of 80=40, 25% is 1/2 of
1/2 so 1/2 of 40=20;Addition using familiar percentages: 10% of 80=8, 5% of 80=4,
25%=10%+10%+5% so 8+8+4=20;Complex calculation: 25% of 100=25, 80=100–20 and 25% of
20=5 so 25% of 80=25–5=20;Calculation using an easier number: 25% of 8=2 so 25% of
80=20;Geometric approach: using 1/4 of pie-chart or square, or position
on a number line
42 54
Total 92 111
Table 3. Summary of approaches suggested for use in the classroom for 25% of 80 (or 25%of 60 in final survey).
Strategy suggestions (n = 28)No. of initial
responsesNo. of final responses
Not addressing content (percent):Think mathematically/collaborate; Have students explain;
Question students; Use calculator; Check answer; Teacher-led whole class discussion; Formative or preliminary assessment
15 13
Demonstrate/explain procedure 1 4Using aids:Concrete materials; Counters; Grids; Decimal squares; Number
lines; Money/sales
11 16
Based on related number concepts:10% related to 90% – build; compare percents; fractions/
decimals; What is the whole?
22 55
Total 49 87
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It was also possible using the rubric in Table 1 to measure change in levels ofresponse. Two teachers did not respond on the initial profile. Other teachers couldsuggest at least an appropriate or inappropriate response for the percent questions butthe responses for how the problem could be used in the classroom were more varied.Of the 16 teachers who could improve their responses on Question 1 (Figure 1), 11did so (69%), whereas five stayed the same. Of the 13 teachers at the highest level onthe initial profile, nine remained at that level (69%) and four declined one level on thefinal profile. On Question 2 (Figure 1), only two teachers responded at a lower levelon the final profile than the first, whereas eight stayed at the same level; 18 improved(64%), some by several levels. Only two teachers responded to the second question atthe highest level on both surveys.
Research question 2: pupil beliefs and performance
Beliefs and classroom experience
A total of 674 pupils answered 25 questions about their attitudes to mathematics andthe frequencies of activities taking place in their classrooms on both surveys. For allitems the most positive response was scored ‘5’. Significant change at the 0.05-level(based on paired t-tests) occurred on only six items, five of which related to thefrequency of particular activities in their mathematics classrooms. The six items arelisted in Table 4 with their relevant statistics. These items reflect some of the princi-ples emphasised in the professional learning programme and point to the pupils’changed perceptions of their mathematics classroom environments. Responses aboutfinding problems too easy reflect the view of Hiebert and Grouws (2007) about strug-gling with important mathematics, which was included in the programme, but tryingdifficult problems still remains a challenge. Class discussion and listening werestressed throughout the CPD and the increased use of equipment may have reflectedthe provision of resources that was part of the programme. The largest change, for ‘Iam asked to explain my maths thinking’, is consistent with teachers’ feedback at thesecond and third two-day sessions.
Table 4. Paired t-tests for change in beliefs and what happens in the classroom.
Item Nov–Mar Std dev Effect size t sig
6. Maths helps to develop my mind and teaches me to think
.098 1.043 0.094 2.43 .015
12. I usually keep trying with a difficult problem until I have solved it
−.098 1.142 0.086 −2.224 .026
15. Most of the time I find maths problems too easy and unchallenging
−.114 1.138 0.100 −2.605 .009
17. I try to make sense of other students’ ideas about maths
.105 1.158 0.091 2.13 .033
20. I use equipment in my classroom to help me with my maths work (e.g. counters, charts)
.135 1.330 0.101 2.374 .018
21. I am asked to explain my maths thinking .194 1.292 0.150 3.524 .000
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72 J. Watson and K. Beswick
Performance
The pupils’ mathematical thinking questions were presented with the request to ‘showor explain how you worked out the answer’. The rubrics for coding included the expla-nation as well as the correctness of the answer. Although it could be argued that somechanges to numerical values made problems more difficult in the second survey, it isalso likely that some items became slightly easier. The actual numerical content of theitems on the pupil surveys is shown in Table 5. A total of 677 pupils completed bothPart B surveys. The Wilcoxen Signed Rank test was used for each item to judge studentimprovement, diminished performance, or no change over the two surveys. Theoutcomes for each question are shown in Table 5. Although it would be hoped thateven better results would be achieved, these are all in the overall positive direction. Apaired t-test was performed for the 677 pupils who completed both surveys. The t-valuefor the entire scale was 17.1, indicating not only a highly significant p-value but alsoan observable change in performance for some classrooms.
Questions 1A and 1B focused on the meaning of the equal sign as indicatingbalance in an equation, rather than as a signal to perform an operation in order to puta number in a box. Many of the teachers in the programme had not recognised this asan important issue for focus in the classroom and expressed surprise when reportingtheir pupils’ responses to questions about the meaning of the equal sign. As seen inTable 5, pupil improvement was significant; for example, a pupil justified an answerof 14 for the initial survey with ‘I added 6 + 5 and that equals 11 and then you plusthe 3 and that equals 14’ whereas in the final survey, the answer of nine was justifiedwith ‘7 + 6 = 13 and to make that true 4 + 9 = 13 because they’re equal to each other’.
Research question 3: teacher satisfaction
The satisfaction of teachers with the CPD programme was judged by a comparison ofresponses to their needs in specific areas as indicated on the initial teacher profile and
Table 5. Test statistics by question.
Question March NovemberWilcoxon statistic
Two-tailed p value
1A 6+5= [square]+3 7+6=4+ [square] 13.259 .0001B 42+38= [square]+39 27+ [square]=26+58 9.410 .0002 25% of 80 25% of 40 5.596 .0003 6×7 8×7 2.477 .0134 Decimal sums Same item 8.184 .0005 3÷0.5 4÷0.5 3.640 .0006 Fractions Same item 6.163 .0007 87–58=31 96–67=31 1.743 .0818 Pie chart of foods Pie chart of sports 6.413 .0009 Mary, 1/4; John, 1/2 Paul, 1/3; Helen, 2/3 4.845 .00010A 360 divided by 9 250 divided by 5 7.281 .00010B 360 divided by 90 250 divided by 50 10.800 .00011 4×3/4 3×2/3 4.534 .00012 3×24 4×16 3.773 .000
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responses to the changes they had experienced in these areas as a result of theprogramme. In relation to teachers’ personal understanding of mathematics oneteacher said in March that he ‘need[ed] help teaching key elements of maths andapplying them to everyday situations’. In August he said, ‘My understanding of howto approach teaching numeracy has greatly improved throughout the project. Beingtrained as a materials, design and technology teacher, it has opened a lot of opportu-nity for new tasks within my maths class’. In reply to the prompt about understandingstudents as learners in the initial profile, one teacher indicated she was ‘alwayslearning’. In August, her response was, ‘Excellent – far more able to deal with verylow ability and extension students’. In terms of teaching for understanding one teacherrelated in the following experience in August: ‘The funny thing about this one is thatnow that I try to have this as the essential focus, the kids’ numeracy books are not sofull and the room is full of work’.
When asked to make suggestions for the future or any other feedback, teacherssuggested follow-up sessions, including involving more teachers from their schools,extending the programme around the state, and balancing theory and practice.
Discussion and conclusion
The intervention reported in this paper was based on best suggested practice (seeHawley and Valli 1999). Evidence from the teacher profiles and student surveyssuggests that it was associated not only with teacher change but also with changes tostudents’ experiences of their mathematics classrooms and numeracy understanding.These changes were significant overall and highly significant in some schools andclasses. That it was not even greater is always a concern to educators but thisprogramme shows that progress can be made on pupils’ numeracy levels with adedicated programme and the outcomes suggest several recommendations for thefuture. These were echoed by the teachers themselves and include extension to thelength of the programme, follow up, and coverage of more schools, grades andregions of the state.
Besides the improvement in student numeracy performance, the changes in pupils’perceptions of their classroom environments were encouraging and confirmed theclaims made by teachers that they had changed their practices, particularly in askingpupils to explain their thinking. This, combined with pupils being more likely to tryto make sense of other pupils’ ideas, supports the implication that classroom practiceswere changed over the months of the project and the rest of the school year. There isalso evidence that pupils were using concrete materials more in the classroom, mostlikely those supplied by the project or bought with money from the project. All of thisis consistent with the comments of teachers regarding their perceptions of theprogramme’s effectiveness but the more rigorous evaluations of both teacher andpupil change have provided a sounder basis for planning future initiatives than wouldteacher self-reports alone.
Watson, Beswick, and Brown (2006) reported the value of a teacher profile similarto that used in this study, in measuring various aspects of teacher knowledge. Theobserved improvements in pupil performance reported here that accompanied theteacher changes suggest that the profile was indeed accessing knowledge that wasrelevant to teachers’ abilities to impact their pupils’ learning. In particular, teachers’abilities to suggest appropriate and inappropriate pupil responses to particular numeracy
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74 J. Watson and K. Beswick
tasks and to suggest possible classroom uses of such tasks appears to be a promisingapproach to measuring teachers’ knowledge.
Further research with similar objectives will be able to employ more sophisticateddesigns and techniques. In this programme it was not possible to identify teachers withtheir pupils and an overall numerical scale of teacher knowledge for teaching mathe-matics was not devised. Future research may be able to provide closer links betweenchange in teacher performance and change in their pupils’ performances. Hill, Rowanand Ball (2005) have begun to link teacher ability with pupils’ improved performancein Grades 1 and 3, but their study does not consider teacher change as well. Thecurrent study has provided some first steps in this direction, which is important forthose planning, funding, and delivering professional development in mathematicseducation and potentially in other discipline areas as well.
The need to improve school pupils’ numeracy performance is often brought to thepublic’s attention, for example in reports of national testing and in the light of the imple-mentation of a National Curriculum in Australia. The outcomes of this programme indi-cate that improvement is possible but that there is no ‘quick fix’ and that it will not bean inexpensive exercise. Dedicated time must be provided for teachers to be challengedand assisted to extend their own mathematical thinking, to appreciate the typical think-ing of pupils, to allow for planning to revise their teaching practices, and to reflect onchange. Teachers and their pupils must change together. This programme has provideda model for potential success in initiating change and evaluating its impacts.
AcknowledgementsThe authors thank the Department of Education Tasmania for permission to report on thisprogramme as part of a consultancy undertaken by them. The ‘Being Numerate in the MiddleYears’ programme was coordinated by Denise Neal, assisted by Umesh Pratap and TichFerencz, satisfying the ethical requirements of the Department.
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