1 Metacognitive teaching approaches Metacognitive teaching strategies and young children's mathematical learning Alison Elliott Faculty of Education University of Western Sydney Nepean A working paper presented at the Australian Association for Research in Education Conference, Fremantle, WA. November 1993 ________________________________________________________________ For further information on this study contact Dr Alison Elliott, Faculty of Education, University of Western Sydney Nepean, PO Box 10, KINGSWOOD 2747, phone 047 360781, or email a.elliott@uws.edu.au Abstract There is growing awareness of the important orchestrating role played by metacognitive activity in skilled mathematical problem solving. Yet, despite their amenability to classroom instruction, metacognitive strategies are seldom explicated in mathematical teaching, and especially in the early years of education. The purpose of this study was to examine the impact of scaffolded mathematics instruction on the learning outcomes of high and low achieving kindergarten children. Specifically, it explored the effectiveness of metacognitively-rich teaching approach within the context of regular classroom teaching. Results indicated that children who participated in the metacognitively-guided mathematics sessions scored significantly higher on tests of mathematics achievement than did children who participated in a "best practice" approach. Of particular interest was the positive effect of the metacognitive approach on children with low scores on a baseline test of mathematics achievement In this paper I will focus on the theoretical background to the research, results of the study, implications for teaching mathematics in early childhood contexts, and future directions for research in early childhood mathematics classrooms. This study was completed with the assistance of University of Sydney (School of Educational Psychology, Measurement, and Technology) Honours students Cathy Neilson and Sarah Wackett. Each student reported on an aspect of the study as part of their 3rd year Honours projects. The study served as a pilot for a larger project that is currently the subject of an ARC funding application. The importance of metacognitive knowledge and strategies to academic success and the need to encourage young children and underachieving learners to self-regulate their learning by planning, evaluating, and regulating their performance on academic tasks has been highlighted by a number of researchers (Paris & Winograd, 1990; Garner, 1987). In particular, metacognitions seem to play an im portant role in mathematical problem solving (Schoenfield, 1985, 1987; Garofalo & Lester, 1985) and indications are that they can be enhanced by specific strategy training within typical classroom environments (Borkowski, Carr, Rellinger & Pressley, 1990; Silver & Marshall, 1990). Few studies, though, have examined ways in which young children can be encouraged to develop and employ metacognitive strategies. If, as suggested by Zimmerman (1990) metacognitive knowledge and self-regulated learning are critical to academic success, play an important role in mathematical problem solving (Jacobson, Lowery & Du Cetter, 1986; Schoenfield, 1985, 1987; Garofalo & Lester, 1985) and are amenable to promotion in classroom settings (Collins & Stevens, 1982; Paris & Winograd, 1990; Zimmerman, 1990), then their development seems as important to effective mathematical learning in the early as the later years of schooling. This may be especially true for many under achieving learners who have dysfunctional metacognitive systems (Silver & Marshall, 1990). The objectives of the study reported here were to examine (a) the feasibility of implementing a metacognitive teaching approach in a naturalistic classroom setting and (b) to investigate its effects on young children's mathematical learning at different achievement levels. Specifically, it was hypothesized that an intervention program that explicated metacognitive knowledge and strategies would result in enhanced mathematical performance, particularly for low achieving children. This study built on earlier research investigating the effects of metacognitive strategy development on young children's mathematical learning in computer supported learning environments (Hall & Elliott, 1992) and served as a pilot for the design of a more comprehensive study of relations between metacognitive teaching approaches, self-system constructs and mathematics competence. Facilitating metacognitive activity Metacognition generally refers to the orchestration of problem- solving activity and knowledge about cognitive states and processes to learning (Kuhl, 1990; Paris & Winograd, 1990; Wellman, 1985; Zimmerman, 1990). It is manifested in the overlapping experiences that engage young children in talking about their problem-solving strategies, and monitoring and regulating their own (or others') thinking and action. In early mathematical activities, metacognitions involve problem specification, forward planning, revising and evaluating strategies, checking outcomes, and developing and testing new strategies to address deficiencies. Relatedly, metacognitive teaching serves to extend new learning by linking it with existing knowledge and facilitating proceduralisation of declarative knowledge. Self-regulation refers to the control of problem-solving activity and the application of knowledge about cognitive states and processes (metacognition) to the learning process. Self-regulated learning strategies are those actions and processes that involve purpose, agency, volition and instrumentality (Wellman, 1985; Zimmerman, 1990). More specifically, self-regulatory mechanisms involve checking outcomes of problem-solving tasks, forward planning, revising strategies, and developing and testing new strategies to address deficiencies. A key function of self-regulatory behaviour is its role in helping children become active participants and controllers of their own learning. But according to Corno (1986) there are specific conditions that can interfere with students' volitional control during learning, such as competing action tendencies, peer pressures and selective attention deficiencies. Similarly, Morine-Dershimer (1983) observed that while most children come to realise that they must take an active role in their own learning, low achievers often remain disengaged from the task at hand. According to Vygotskian perspectives on cognitive development suggest that children's interactions with competent others serve to mediate thinking and problem-solving in the cognitive space between what can be accomplished alone and in collaboration with more capable others, that is, in the zone of proximal development. It is contended that this scaffolding process provides on-going stimulation and motivation for learning, as well as support of a more metacognitive, and particularly, self- regulatory nature. Subsequently, self-regulatory strategies are internalised to become part of the learner's independent repertoire of competencies for application in similar contexts (Rogoff, 1990; Vygotsky, 1978). Simultaneously, the scaffolding process should assist learners develop richly connected cognitive networks of skills and understandings. Over the past few years there has been a considerable research focus in the wider educational literature on strategy development through scaffolded instruction. According to Garner (1987), strategy training studies are undertaken for both theoretical and practical reasons. At the theoretical level they provide the opportunity to explore theoretical propositions about the efficacy of particular cognitive strategies to support learning. On a practical level, explorations of the strategy effectiveness serve as an impetus for classroom based instructional interventions (Brown et al, 1986). The strength of scaffolded context as a setting for strategy development lies in the cognitive support and guidance generated by both teacher and child participants. Gradually, the guidance is internalised to become part of the learner's independent repertoire of competencies. As children gain competence in a particular learning domain the high level of scaffolding can be reduced. It can be rebuilt when children again encounter different or challenging tasks (Day, French, & Hall, 1985). The role of the teacher in this scaffolding process is to promote an awareness of the cognitive demands of the task and to guide activity within a purposeful and goal directed framework focussing on, for example, overall task orientation, and planning, monitoring and evaluating cognitive activity, rather than on the management and organisation of the learning experience itself. He or she concentrates on rectifying discrepancies between the child's action or response and the ideal situation, controlling frustration and risk in problem solving, and demonstrating an idealised version of the problem's solution. Complementary peer interaction, as well as the individual's own structuring of the activity, generate the cognitive and metacognitive dialogues that serve to support and mirror knowledge and thinking. In this sense the interrelatedness of the contextual supports frame the strategies required for of self-regulated learning. One of the difficulties encountered in facilitating the learning of early mathematical ideas and elementary number work is that teachers often misjudge the complexity of the task (Burton, 1990). Baroody (1985), for example, emphasised that learning basic number combinations is not a straightforward, rote memory task. Rather, the process of replacing slow counting procedures with rapid fact retrieval involves considerable counting experience and is dependent on the development of procedures or invented rules. Further, it appears that learners use more than simple associative models. Indeed, the complexity of human information processing and the tasks at hand suggest the existence of a series of rules that differ from child to child. It is in helping establish elaborated cognitive networks and strategies for problem-solving that metacognitions are especially important. Using the theoretical work of Flavell (1978), Corno (1986), Baker and Brown (1984), and Rogoff (1990) as a base, aspects of a metacognitive learning model can be applied to mathematical problem solving in an early childhood context. Specifically, self-regulatory processes can be delineated as (i) Goal and problem identification and clarification; (ii) Planning actions; (iii) Sustaining mindful and purposeful problem-solving; and (iv) Evaluating actions and specifying a rationale for decisions. (i) Goal and problem identification and clarification: Important to self-regulation of learning is children's identification of their goals and determination of what is required in the activity. Scaffolds encourage children to think about the global problem as well as each step of the problem. (ii) Planning actions: The act of planning is important to the ultimate self-regulation of problem-solving. Although supported planning is not always useful in facilitating problem solving (Gauvain & Rogoff, 1989) planning assistance seems to be important to successful problem-solving especially in the initial stages of problem-solving or in new activities. (iii) Sustaining mindful and purposeful problem solving. Strategies for keeping young children "on track" seem critical to problem-solving success. These are operationalised by (a) breaking tasks into small steps and sequencing the steps toward the desired goal; (b) actively monitoring learner progress. For example, questioning the child if she stopped short of the final goal or left out an important step. What happens next? What should you do now? Such questions provide children implicitly with cues to assist their thinking processes; (c) providing immediate and academically oriented feedback. Have another look. What has happened? Does that look right? (Child responds) So what should you do? Think carefully; (d) providing lots of praise for thoughts and ideas as well as correct solutions: Clever boy; that was thoughtful; good thinking. enables children to recognise the importance of the process as well as the product; and (e) encouraging children to use prompts, cues and mnemonic processes within the environment Can you remember that? Think of your birthday cake. How many candles did you have? (iv) Evaluating actions and specifying a rationale for decisions in order to review the process of reaching a conclusion. This process of monitoring and evaluation is considered central to self-regulated learning. Why that way? Tell me why you did that? Think back. Why did you start there? Where else can you start? In an early mathematical contexts activating such processes can involve three main scaffolding techniques: (a) direct guidance, (b) cues, questioning, and discussion, and (c) modelling and demonstration. (a) Direct guidance involves specifying what children must do in a particular situation. What is the first step? OK, count one by one. Point to the first counter.. Ok, now point to the next... (b) A less directive approach asks children to reflect on their actions through the use of cues or hint questions: How will you start this? What do you need to do first? What would happen if Can you remember what you and Jessie did last time? (c) Demonstration and modelling of appropriate problem solving strategies provide children with an idealised representation of a workable solutions: Look I'll show you . (Teacher counts each block pointing as she goes to emphasise the process of matching). Mary, can you do this now? Every one watching.... Within the above categorisations strategies include formulating key questions to stimulate discussion about concepts and skills, pacing of learning (moving from easy to more difficult examples; providing correct answers after repeated errors then moving on before frustration set in), sequencing of actions (you need to do this, then this, then this; a step by step approach), encouraging children to reflect on actions and responses ; re-asking questions, and giving immediate feedback); employing different modes of presenting the same information (both simultaneous and successive, eg. presented a set of dogs- one at a time, or as one array); making multiple representations of the same information (for example a set of 6 objects could be counters, pencils, claps, hands, fingers), building on previous actions by presenting different problems in a similar manner, and relating new ideas to familiar ones. Such scaffolds should serve to reduce the processing burden involved in activating and applying self-regulatory strategies. Importantly, they should encourage the beginnings of planful behaviours, such as goal setting, goal monitoring, self- monitoring and self-evaluating (Zimmerman, 1990). Gradually, we would expect use of the strategies not only to improve mathematical performance during mediated activities, but also to become internalised to enhance performance independent of the support. In the study reported here scaffolding procedures (for example guiding, modelling and cueing) were used to facilitate metacognitive knowledge and strategy development during classroom-based small group mathematics sessions. It was hypothesized that this process of support and explication would result in enhanced mathematical performance, particularly for low achieving children. Method Subjects and setting The source group for the study were 30 children (mean age 5.6 years) representative of a range of family backgrounds typical of suburban Sydney and enrolled in the kindergarten class of a state primary school. Measures of achievement Mathematics achievement (pre and post intervention) was measured using the Test of Early Mathematical Ability (TEMA 2). The TEMA2 test examines both formal and informal components of early mathematical thinking, measuring concepts of relative magnitude, number facts, counting, and calculation skills (Ginsburg & Baroody, 1989). Administered on a individual basis, each TEMA testing is completed in approximately 15 to 20 minutes. Entry level test items are determined according to the age of the child and testing concludes when a child has missed or responded incorrectly to five consecutive items. For the purpose of this study only items relating to numeration were considered when assessing post intervention gains. Teaching approaches Content and activities forming the basis of the teaching sessions were drawn from the NSW Department of School Education's Mathematics K-6 syllabus and adapted for classroom use with the assistance of the class teacher. Content included matching objects by one-to-one correspondence, comparing groups of objects by one-to-one correspondence, ordering groups according to the number of objects in the group, recognising , copying and creating simple number patterns, continuing a pattern and supplying a missing element, reading and writing numerals to 10 (or 20), and counting and ordering objects to 10 (or 20), rote counting backwards and forwards, and elementary arithmetical operations with sets. Most activities required children to handle concrete materials. On some occasions children were required to engage in complementary writing or drawing activities. The metacognitive instructional approach involved attention to the range of strategies described earlier (that is, goal identification, active monitoring, modelling etc). Especially important to this approach was the use of both teacher and child language to scaffold the construction of mathematical ideas and self- regulatory processes. Teachers working with children in the best-practice groups employed the "best practice" strategies they would normally use to teach this content. Teachers assigned to the metacognitive groups were tutored in the use of the strategies and understood the rational behind their employment. The other teachers were instructed to use their "best practice" and to seek guidance from the curriculum and teacher resource books. Typically, "best practice" involved direct guidance with minimal teacher involvement other than that of an encouraging, managerial or confirmatory nature. In contrast to the metacognitive approach, there was little modelling of relevant processes, little if any discussion of "why" or "how", little focus on planning, monitoring and evaluating, and little emphasis on peer interaction. Procedure On the basis of outcomes of the Test of Early Mathematical Ability children were ranked according to performance scores and randomly assigned to two groups that were equally representative of the range of classroom achievement levels. The two groups were further subdivided into high or low achievers based on TEMA2 scores. High achievement group 1 (HA1) and low achievement group 2 (LA2) were assigned to the metacognitive teaching sessions; high achievement group 3 (HA3) and low achievement group 4 (LA4) to the best-practice teaching sessions. Each group (n = 7 or 8) then participated in seven forty minute mathematics teaching sessions over a six week period. Three children were later excluded from the study due to insufficient participation in the intervention phase or non completion of the posttest. The posttest was administered 3 weeks after the conclusion of the teaching sessions. This three week period coincided with a school holiday break. It was expected that this delay would enable us to gain a clearer picture of the more lasting effects of the teaching. Results and discussion Gains scores were computed for each child by subtracting the pretest score from the post test score. As expected after seven sessions of focussed mathematics activity there were improvements in the posttest scores of each group, with the greatest gain score recorded for the low achievement group receiving the metacognitive teaching approach and the lowest score for high achieving children who participated in the "best practice" teaching sessions. Table 1 Mean improvement scores for each group Mean SD High achievement- Metacognitive 6 2.7 High achievement- Best practice 2.7 3.6 Low achievement- Metacognitive 7.8 1.2 Low achievement- Best practice 4.6 1.7 As will be highlighted in the presentation subsequent analyses indicated that both high and low achieving children who participated in the metacognitive groups recorded significantly greater improvements in measures of mathematics achievement than did those who participated in the "best-practice" sessions. __________________ Table 2 about here __________________ From the results outlined in the presentation it is apparent that the metacognitive teaching approach was more effective than the best practice approach in enhancing children's mathematics performance. This finding is consistent with the view that learning emerges from and reflects social discourse. And, it is joint activity between participants, in the form of guided and collaborative activity, that creates the climate for learning. In this intervention the scaffolding of metacognitive knowledge and strategies seemed especially useful in assisting learners to become more conscious and reflective in constructing knowledge. The better performance of children exposed to the metacognitive approach especially the low achievers suggests that the focus on planning, organising, and monitoring had a positive impact children's strategic knowledge and insights into how and when to use a specific strategy. In the scaffolded context children were able to consciously focus on both task content and their understanding of task demands, as well as progress toward achieving gaols. In contrast, children in the best practice group tended to be told what do rather than encouraged to consciously focus on the how and why of the process. Thus there was less opportunity to use and practice planful, systematic or elaborated strategies to deal with problem-solving. Future research directions Borkowski et al's. (1990) recent work on self-system components and metacognitions hint at the importance of motivational and affective states for metacognitive development. They propose that children who feel good about themselves and their ability are more likely to develop strategic behaviours and believe in their utility in problem solving. In this sense the self-system "power(s) metacognition by giving children reasons to learn." (p. 64). Borkowski et al. suggest that children with positive self- systems are likely to activate their metacognitive systems and "increase their metacognitive knowledge because these processes have paid off in the past, elevating performance and enhancing self esteem" (p. 64). Importantly though, they stress the complementary and interactive roles of both self and metacognitive systems. Motivation and good feelings are of little use to a child who does not have the corresponding strategic knowledge and skills. Hence, development of affective and metacognitive systems must go hand-in-hand. Where this does not happen the likely result is poor academic performance coupled with reinforcement of negative self-perceptions and beliefs. If Borkowski et al's (1990) theory about the role of the self- system in metacognitive development is correct, then an intervention aimed developing metacognitive strategies and self- esteem should be especially beneficial to mathematics performance in the short, but especially the longer term. If successful, though, for such a model to to be really useful it must be easily implemented by regular classroom teachers in regular classroom settings. 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