self-monitoring by lesson reports from teachers in problem-solving ...

SELF-MONITORING BY LESSON REPORTS FROM TEACHERS IN PROBLEM-SOLVING MATHS LESSONS Christina Collet, Regina Bruder and Evelyn Komorek Technical University Darmstadt The following article presents chosen results from a teacher training on the learning of mathematical problem-solving in connection with self-regulation. The lessons documented during several weeks and the work products submitted by the teachers show specific further training effects. INTRODUCTION AND THEORETICAL BACKGROUND In Germany education standards (KMK, 2003) have been developed which are implemented at the moment. Problem-solving plays a central role in these standards, and special competencies are required from the teachers to enhance problem-solving in maths lessons. Within the scope of a project supported by the DFG (German Research Foundation) the Technical University Darmstadt pursues the aim to enhance this teaching competency on the basis of an approved, material-based and daily-life adapted teaching concept for the learning of problem-solving in connection with self-regulation (Komorek et al., 2006). The results of a study with student training measures on specific problem-solving strategies and interdisciplinary self-regulation stategies reveal specific effects regarding the mathematical performance of the students (Perels et al., 2005). On the basis of these training results a teaching concept for the enhancement of problem-solving in connection with self-regulation was established for teacher further training courses (Collet et al., 2006). The teacher further training of the school year 2004/2005 was focussed on the following aspects: • A teaching concept for the learning of problem-solving • The enhancement of the self-regulation of students by homework Teachers who participated in the study were trained on special research contents like problem-solving (PS), problem-solving and self-regulation (PSR), self-regulation (SR) and the safeguarding of mathematical basis competencies (CG: “Quasi-control group”1). In a further training at the beginning of the school year and supported by supervision (curricular-based (CB), webbased (WB), no supervision (NO)) during the school year the teachers had to go through defined fields of competency, depending on the further training content. Following a (moderate) constructivist approach and in order to reach corresponding effects with the teachers, practical exercises were part of the further training at the beginning of the school year and of the supervision during the

1

CG-group is no control group in the proper meaning of the word as the teachers of the group underwent further

education for mathematical basic competencies and internal differentiation.

2007. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 169-176. Seoul: PME.

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Collet, Bruder & Komorek school year, allowing to experience problem-solving strategies or to construct own problems in the sense of the teaching concept. Theoretical background and classification of the study Shulman (1986) describes six knowledge categories of teacher competencies. The present study analyses the PCK (pedagogical content knowledge) in connection with action elements for problem-solving in combination with self-regulation. The problem with this kind of studies is that there is no clear horizon of expectation (standards) in Germany so far. In the American standards for maths teachers (NBPTS, 2001) those competencies are already included, however not operationalized. The purpose of the present study is to contribute to the development and the use of suitable instruments to describe further training effects. Qualitative instruments and conventional quantitative methods as well as cross-section and process surveys for the analysis of efficiency are applied in a field work with both teachers and students, especially lesson reports and work products. The work products are particular problems, long-term homework, learning controls and teaching drafts. On the basis of submitted work products it is possible to assess if a teacher is able to implement developed knowledge of subjects treated in his further training. The lesson reports of the teachers allow to evaluate their self-perceived ability with regard to the integration of the further training concept in regular maths lessons. Questions and hypothesis concerning the chosen tools The present study wants to analyse the following research questions: To what extent are self-developed tasks reflecting further training effects? (work products) • Which effects on the self-perceived implementation of the further training content by the participating teachers has the intervention with the further training concept? (lesson report) Positive trends concerning the implementation of the further training content are expected from the lesson reports for those further training groups which were supervised throughout the school year (CB, WB). The tasks developed by the teachers were expected to show the individual implementation of the concept, following Galperin (1974) in three grades of professionality levels depending on different action orientations (try-and-error-, pattern-, field orientation). The aim of the further training is to acquire at least one pattern orientation. This means that only examples without variation are adopted. However, in the long run the aim is to achieve field orientation, allowing the teacher to generate own examples which are in line with the concept. The quality of the work products alone does not guarantee the successful implementation of the concept in maths lessons but is considered as an essential prerequisite. •

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Collet, Bruder & Komorek STUDY DESIGN 48 teachers (Gymnasium teachers and teachers from other school types) from 9 schools with classes of levels 7 and 8 took part in the study. At the beginning of the school year 2004/2005 the teachers participated in a further training with four training modules according to the aforementioned competencies (PS, PSR, SR, CG). Two groups were supervised throughout the school year, either by curricular-based training courses (CB) or by webbased coaching (WB). Table 1 shows the design of the study with different variations of the further training content and coaching. PS CB WB NO

11

PSR 4 8 6

SR 8 4

CG

8

Table 1: Design of further training method and content: number of teachers Instruments of data collection Four developed instruments were applied with the teachers to analyse the further training effects from different points of view (www.math-learning.com). A repertory grid survey (qualitative) and a teacher questionnaire (quantitative with qualitative elements) served as basis for a pre- and post comparison. The repertory grid survey allows to record the ideas teachers may have on maths problems by asking them to specify characteristic features of problem pairs (Collet et al., 2006; Lengnink et al., 2003). The teacher questionnaire deals with teaching aims in connection with the enhancement of subject-related and multidisciplinary competencies, attitude and experience with respect to the integration of individual learning possibilities in the lessons and cognitive requirements in the homework. Moreover the teachers’ ideas about good maths lessons are collected in concept maps. For the internalization of aspects of the teaching concept and the documentation of the concept implementation a self-monitoring instrument, the lesson report, was used as processual investigation tool. The teachers were asked to document their maths lessons continuously over a period of 10 weeks. The lesson report consists of 34 items established against the background of the teaching concept on the learning of problem-solving and self-regulation in connection with basic structural elements of “good” maths lessons. In order to gain access to their knowledge regarding the successful concept implementation and to maintain the teacher further training effects by individual experience, the teachers are asked to submit at least one own work product at the end of the further training. Moreover a student performance test with problem-solving tasks was run to show possible effects of the further training (Collet et al., 2006). A standardized student questionnaire was employed to determine, among others, the self-regulative competency and the perception of the lessons. Both instruments were adopted in the pre- and post-survey.

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Collet, Bruder & Komorek RESULTS OF THE STUDY Selected general results The repertory grid survey revealed quantitative and qualitative growth in the description of problem features, allowing to draw corresponding conclusions from the further training effects (Komorek et al., 2006). The teacher questionnaire demonstrates the stability of the values allocated to subject-related and multidisciplinary teaching aims as well as to the commitment and self-perception of teachers. There is a significant increase in the enhancement of individual learning and cognitive homework requirements. The students manifested in their student performance test significant increases in performance and a more frequent use of heuristics (Komorek et al., 2006). Results of the work products The work products submitted by the teachers at the end of the school year were evaluated by means of a category system, the results were reported to the teachers. 17 teachers submitted 38 work products in total (8 working sheets, 5 particular problems, 4 learning controls, 8 long-term homeworks, 13 lesson designs). The following aspects were taken into account for the evaluation of the work products: subject-related criteria, aspects of goal orientation (transparency of the goal for the students, clearness of the goal by the teacher), motivation potential, internal differentiation, cognitive activation (especially variation of degree of difficulty and task type), student activities and self-regulation. The results of the submitted work products prove the successful concept implementation by the teachers (Komorek et al. 2006). An exemplary work product (shortened version) of a teacher from the problem-solving group (PS) shows a specific implementation of the concept (in the sense of field orientation). The task, following an upgrading of requirements, starts with low requirements and becomes more complex with every subtask. Source of the task is a so-called “two-minute task”, taken from a Hungarian TV-show in the sixties. Original task from a TV-show: “The semicircular disc glides along two legs of a right angle. Which line describes point P on the perimeter of the half circle?” (Engel, 1998). The problem for students modified by the teacher is:

P A

Make a construction of beer coasters or similar materials to visualize the problem. • Which line describes point P? 0 • Explain the form of the curve. A student who was working with a dynamic geometry software explained her solution of the two latter subtasks as follows: •

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Collet, Bruder & Komorek Eva:

I didn’t produce a curve but a line. All points of P are on this line. When we move the segment AB and A'B' on the coordinates, line G is always following the movement in a way that it goes through point P and P'. I do not have an explanation for line G, but I believe that whenever a point is attached to G and AB or A'B' is moved, it describes a curve.

The teacher reflects upon the use of the task as follows: Teacher:

Experimental homework is useful and arouses the cusiosity of many students. The goal was reached to make a maximum of students deal with this most demanding geometric problem. The written documentation of the problem-solving process revealed a multitude of individual perceptions, approaches and procedures of the students and also showed the causes of possible problems.

Results of the lesson reports 1296 lesson reports were presented by 38 teachers who had documented their lessons in 15 to 41 periods. From the items collected in a four-stage answer format (3: “very true” to 0: “not true”) the following five scales were established: • Inner and outer mathematical problem-solving (Cronbachs α=.89; 6 Items) • Use of strategies – reflection on action (Cronbachs α=86; 4 Items) • Internal differentiation – individual learning (Cronbachs α=.80; 4 Items) • Formation of exercise processes (Cronbachs α=.73; 6 Items) • Accomplishment of lesson target (Cronbachs α=.86; 4 Items) The first three scales were subsumed in a superordinate scale “problem-solving and self-regulation“(Cronbachs α=.94; 14 Items). This scale includes elements for the enhancement of partial actions of problem-solving, the integration of heuristic and self-regulative elements in the lessons and the heterogenity in class. The following is focussed on this scale. Analyses of trends in connection with the further training method (CB, WB, NO) show that curricular coaching or webbased coaching are considered to be a cause for significant positive trends towards the implementation of the concept. On the other hand the input in the form of a single compact course does not create a positive trend, a certain starting level in the context of the enhancement of basic competencies notwithstanding. Figure 1 illustrates the trends of the curricular further training group (CB) and of the further training group which was not supervised during the school year (NO).

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Fig 1: Trend analysis (scale: “use of strategies – reflection on action“) As a processual instrument the lesson report reveals the effect of a concrete intervention, resulting of an additional further training course. Coaching throughout the school year should influence the variables of the lesson report in a way that the participation in this further training produces a significant difference between self-perceived teacher action after the further training course at the beginning of the school year (first intervention phase) and after the intervention (second intervention phase). It was not possible to run a classical baseline phase without placing excessive burden on the teachers. The effect of the additional training on the curricular-based group (CB) was analysed with an intervention analysis (ARIMA (1st order autoregressive model)). On the 18th day of the study a significant training effect can be observed (figure 2). As for the integration of heuristic elements it is stated that in the Gymnasiums heuristic tools as well as strategies and principles are adopted. The teachers of other school types are rather focussing on heuristic tools. Most frequently used heuristic tools were the informative figure (122), table (86), graph/diagram (64) and strategies like forward and backward working (121). The frequency of mentioned heuristics by the teacher is put in brackets.

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Fig. 2: Intervention analysis (scale: “problem-solving and self-regulation“) Conclusion and outlook The results of all instruments prove the successful implementation of the concept. The work products of teachers are suitable to show competencies acquired in the further training. Due to the positive results of this study the collection of work products and their evaluation became part of the e-learning training courses (www.prolehre.de), organized by the Technical University Darmstadt since 2005. The lesson report as processual instrument allows to draw conclusions from the self-perceived action competence with respect to the implementation of the concept. The results show that in addition to a single compact course further coaching during the school year is necessary to enhance the integration of the concept ideas. The multi-perspectivity of the applied tools allows a connection of teacher knowledge and practical action with regard to the learning of problem-solving and self-regulation according to the proposal of da Ponte et al. (2006). The results of a follow-up study (one year after the further training) with 11 project classes are currently evaluated. They are expected to reveal the long-term effects of the further training courses on the students.

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Collet, Bruder & Komorek Literature Collet, C. & Bruder, R. (2006). Evaluation of a teaching concept for the development of problem solving competencies in connection with self-regulation. In: Novotná, J., Moraová, H., Krátká, M., Stehlíková, N. (Eds.): Proceedings of the 30th PME International Conference, 2, p. 345-352. Engel, A. (1998). Problem-Solving Strategies. Springer-Verlag: New York; Berlin; Heidelberg. Galperin, P. (1974). Die geistige Handlung als Grundlage für die Bildung von Gedanken und Vorstellungen. In: Galperin, P., Leontjew, A. N. (Eds.): Probleme der Lerntheorie, p. 33-49. Kultusministerkonferenz (KMK) (2003). Beschlüsse der Kultusministerkonferenz. Bildungsstandards im Fach Mathematik für den Mittleren Schulabschluss. www.kmk.org/schul/Bildungsstandards/Mathematik_MSA_BS_04-12-2003.pdf (last call: 30.8.2006). Komorek, E., Bruder, R., Collet, C., Schmitz, B. (2006). Inhalte und Ergebnisse einer Intervention im Mathematikunterricht der Sekundarstufe I mit einem Unterrichtskonzept zur Förderung mathematischen Problemlösens und von Selbstregulationskompetenzen. In: Prenzel, M. & Allolio-Näcke, L. (Eds.): Untersuchungen zur Bildungsqualität von Schule. Abschlussbericht des DFG-Schwerpunktprogramms. Münster: Waxmann, S. 240-267. (English version in print – published: April 2007) Lengnink, K. & Prediger, S. (2003). Development of personal constructs about mathematical tasks – a qualitative study using the repertory gird methodology. In: Pateman, N. A., Dougherty, B. J., Zilliox, J. T. (Eds.): Proceedings of the 27th PME International Conference, 4, p. 39-46. National Board for Professional Teaching Standards (NBPTS) (2001). NBPTS Adolescence and Young Adulthood Mathematics STANDARDS (for teachers of students ages 14-18+). Second Edition. Perels, F., Gürtler, T. & Schmitz, B. (2005). Training of self-regulation and problem-solving competence. Learning and Instruction, 15, p. 123-139. Ponte da, J. P., Chapman, O. (2006). Mathematics teachers´ knowledge and practices. In: Gutiérrez, A., Boero, P. (Eds.): Handbook of Research on the Psychology of Mathematics Education. Sense Publishers, Rotterdam, p. 461-494. Schoenfeld, A.H. (1985). Mathematical problem solving. Orlando, Florida: Academic Press, Inc. Shulman, L. S. (1986). Those who unterstand: Knowledge growth in teaching. Educational Researcher, 15(2), p. 4-14. Zimmerman, B. J. (2000). Attaining self-regulation: A social cognitive perspective. In: Boekaerts, M., Pintrich, P. R., Zeidner, M.: Handbook of self-regulation. San Diego, CA: Academic Press, p. 13-41.

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