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Teaching for transfer: Transfer of whose learning?

Jennie Bickmore-Brand
Murdoch University

Kim Pitts-Hill
Swan Education District

The research findings from each of the classrooms in this study raise important issues when applied to two current popular practices in education namely the employment of group work and the use of heuristics to enhance student learning. In regard to employing group work to enhance student learning, these research findings revealed that classroom dynamics and teacher-student relationships can actually work against achieving positive effects from group-work activities and problem-solving. The second trend, particularly employed in the mathematics, language and science areas is that of teaching heuristics to enhance student learning. These research findings revealed that the problem-solving approaches of the students were so idiosyncratic as to question the validity or at the very least challenge the notion of transfer of heuristic frameworks to settings outside the taught context.

Background

This paper presents the research findings of three classrooms investigating mathematics learning. All three classes were at upper-primary level located in the metropolitan area of Perth, Western Australia. Class A consisted of 26 girls from a private school and a female teacher. Students were of middle to high socio-economic status backgrounds. Class B had 6 girls and 19 boys with a female teacher. The students came from a middle socio-economic background. Class C was a year 7 group of 26 students, approximately equal numbers of boys and girls and from a low socio-economic background. Class C had a male teacher. Although this provides background, socio-economic status was not factored into the discussion, nor was the sex of the participants.

Methodology and data analysis

In Class A and B the data collection centred on (a) weekly classroom observations over a nine-month period by Bickmore-Brand as a participant observer in the classroom; and (b) regular tape and video-recorded interviews with the key participants in the study. Data collected in this ethnographic framework included keeping field notes, and collecting samples of children's work as well as copies of classroom mathematics materials. A NUD*IST analysis was made of the classroom observations. Video- and audio-tapes were transcribed, a personal diary was kept and a range of norm referenced pre-test and post-tests were given. Observations were triangulated between the participants and by three external "informed" critics.

In Class C, data were collected over a three-month period using written, video and audio recordings by Pitts-Hill as a participant observer during the weekly lessons. Student and teacher talk in the lessons was transcribed and analysed. Stimulated recall interviews with the students in post lesson sessions was also transcribed. Discussions with the teacher clarified various points not clear from the transcribed data. Careful readings, supported by a MAKITAB learning interaction analysis system, and a NUD*IST analyses formed the basis of the data interrogation. The mathematics lessons in all three classes in the study required problem solving. The students either employed small group co-operative learning techniques to solve problems, or sat at tables of 4-6 participants that worked independently.

Literature review

Popular practice 1: Working in small groups

The idea of creating a community atmosphere in a classroom where learners are grouped to share problems and successes with other members of the group has become increasingly popular (Slavin, R.E. 1992; Boomer, 1988; Cairney, 1987). The importance of creating a context for sharing meanings should not be underestimated (Wells, 1981). A shared context is important, for example, in making "public" what Barrett (1985) talks about as "private" or personal knowledge. Pimm (1987) refers to this as "talking for others" as separate from "talking for oneself." Burnes and Page (1985) suggest that a teacher who creates and encourages a shared context in the classroom is going to be in a better position to select appropriate materials and instructional procedures. A shared context helps to reduce the distance between what learners bring with them to the classroom and the content of what is being taught.

Habermas (1970, 1972, 1978, 1984) supports this emphasis away from each individual constructing meaning in an isolated environment, to a recognition of the individual as part of the social milieux. The classroom environment provides varied and complex opportunities for the social construction of meaning. As Bishop (1985) describes the individual-group relationships

...each individual person in the classroom group creates her own unique construction of the rest of the participants, of their goals, of their interactions between herself and the others and of all the events, tasks, mathematical contents which occur in the classroom. (p. 26)
Consequently the classroom can be described as a culture in its own right. The "knowers" are bound with their culture, suggesting that all knowledge is mediated by social interactions. Social influences are a major determinant in what counts as knowledge in classrooms (see discussions from German interactionist literature, for example, Bauersfeld, 1988; 1992; 1995; Voigt, 1985; 1994; 1995). Knowledge is constructed "intersubjectively" and is socially negotiated between all participants in the classroom. The classroom climate that the teacher sets up for children's learning will not only reflect what the teacher knows (for example, about mathematics), but what he/she believes about how children learn (Lubinski, Thornton, Heyl & Klass, 1994).

Teacher operations in the classroom are also influenced by the construct the teacher holds about a particular discipline and what is important to be learned from that discipline. The teacher also has understandings about how people learn this discipline and relationship of the subject within it and will therefore select learning materials which suit how they perceive students learn in that discipline. The teacher will assess the aspects of the subject they regard to be fundamental, thus providing cues to students about what is valued. The teacher's processes have in turn been shaped by the student's successes and responses within the learning environment she has created. This is shown in Figure 1. The figure is not intended to be linear but rather represent influences which filter back and forward in classroom interactions continually shaping the discourse.

Figure 1: How beliefs affect instruction

Thus learning is shaped through a negotiation and sharing of knowledge. Te acher's beliefs and practices are continually shaped by the experiences in the classroom. Teachers attempting to adopt constructivist ideas have aimed at establishing learning environments which nurture interest and understanding through cooperation and through social interaction which tolerates dissonance (Pateman & Johnson, 1990).

Cooperative learning

Investigation into small-groups and learning has been undertaken by a number of researchers over the last 20 years (Johnson & Johnson, 1975, 1994; Schmuck & Schmuck, 1983; Sharan & Sharan, 1984; Slavin, 1983). Concern in the areas of student discussion and teacher behaviour is highlighted by the current world wide interest in small group cooperative learning and the claims that suggest students do as well, if not better, academically, in cooperative groups than when they are taught by more traditional methods (Slavin, 1983; Sharan & Sharan, 1984; Johnson & Johnson, 1994; King, Barry, Maloney, & Tayler, 1993; Meloth, 1990). Davidson (1990) conducted a major review of some 70 studies comparing cooperative learning with traditional whole class teaching. The findings came out strongly in favour of cooperative and small group approaches (see also Davidson & Kroll, 1991). Slavin (1992) also found a higher achievement score for students working in cooperative learning situations when compared with that for students working in traditional classrooms. Students' efforts were attributed to a combination of cooperative and competitive incentives that motivated them to encourage one another in the learning environment. On the other hand, a number of research studies have produced results which reject the idea that cooperative learning leads to greater individual gains (see Gooding, 1990; Healy, Hoyles & Sutherland, 1990; Pirie & Schwarzenberger, 1988).

Much of the research into cooperative learning using small groups has concentrated on the nature of the task, the reward structures and student achievement but little insight has been provided into the role of the teacher's instruction or the content or the form of the student interactions. In a review of research Meloth, Deering and Sanders (1993) found that it was rare that teachers were reported as providing "information that would help students attend to and communicate, important lesson content effectively" (p.5). Furthermore, Meloth, Deering and Sanders (1993) found that fewer than 5% of studies in a review by Johnson, Johnson and Maruyama (1983) "examined the content of peer-group discussions, making it unclear whether the quality of students discussions contributed to learning gains (p.5)".

Traditionally, mathematics classrooms are very competitive places (Mousley & Clements, 1990), often characteristically different from other learning environments (Clarke & Clarke, 1990; Stigler & Baranes, 1988). Current educational reforms have influenced State curricula to now include notions of cooperative group learning in their materials. (See The Mathematics Framework document issued by the Victorian Ministry of Education (1988), the Western Australian Curriculum Framework, the Education Department of Western Australia's Student Outcome Statements (1995), and A National Statement on Mathematics for Australian Schools (AEC, 1990)).

Often cooperative learning is accompanied by the methodology of problem-based learning. Wheatley (1991) argued, as have Gooding and Stacey (1991), that the strength of problem-centered learning is that students can operate at their own cognitive levels and use their preferred learning styles. A major stumbling block for the implementation of cooperative learning in upper schools remains in the assessment area. Even when "who" and "what," is being assessed, has been made explicit at the outset, workload inequities continue to exist (Clarke, 1995).

Clearly, more research is needed on the role of cooperative group learning in the mathematics classroom. "How To ..." documents such as Dalton's (1985) "Adventures in Thinking," although very practical for group problem-solving across the curriculum, lack a strong research rationale. Ellerton and Clements (1991) raise the question that may need to be asked when applying cooperative learning in mathematics classroomsæunder what circumstances is peer discussion likely to be valuable? (p. 103). The research reported in this paper sheds light on the nature of peer discussion when classroom interaction patterns are analysed.

Popular practice 2: Teaching heuristics

An heuristic is a metacognitive device which explicates the use of thinking strategies and thinking aloud discourse in the classroom. Whilst cognition refers to thinking and knowing, metacognition is defined by Metcalf and Shimanura (1994) as "our knowledge about how we perceive, remember, think and act - that is, what we know about what we know" (p. Xi). Metacognitive thinking is accomplished by the thinker.

Probably the most explicit work concerning metacognitive classroom practices in mathematics education has come from Cobb (1990b). He listed four points to describe metacognitive approaches:

  1. Explaining how an instructional activity that a small group has completed was interpreted and solved;

  2. Listening and trying to make sense of explanations given by others;

  3. Indicating agreement, disagreement, or failure to understand the interpretations and solutions of others;

  4. Attempting to justify a solution and questioning alternatives in situations where a conflict between interpretations or solutions has become apparent. (p. 208)
The role of the teacher in developing metacognition can be divided into three areas of assistanceæscaffolding, (Bickmore-Brand, 1998) modelling, and direct instruction in metacognitive strategies (Cazden, 1983). Direct instruction is when "the adult not only models a particular utterance but directs the child to say or tell or ask" (Cazden, 1983, p. 14). Cazden argues that, when adults use this form of dialogue it may be construed that the content is probably especially valued. Greeno (1980) suggests that strategic principles can be made explicit during instruction in problem-solving of, for example, geometry, but need not necessarily be interpreted as the teacher's imposition of prescribed steps for students.

A teacher's explicit and often ritualistic rephrasing and refining of the learner's expressions at times forms part of what Furniss and Green (1991) term a "joint construction" (Furniss and Green also refer to the dialogue as "aided instruction" or "collaborative construction" (p. 48). Leinhardt and Putnam (1987) use the descriptionæ"lesson parser" which signals to the students recognisable parts of the lesson which they can expect to experience. Pateman and Johnson (1990) also note that this dialogue is based on responses from learners, who in turn are being given feedback about their processing and expression. Chapman (1992) refers to a teacher signaling ways of speaking that are appropriate to a particular subject-area, and that these are developed as part of the social practices of the classroom (p. 41).

Some typical routine heuristics used in Class A were: Guess and Check, Draw a Picture, make an Organised List, Make a Table, Work Backwards, Look for a Pattern, Use Logical Reasoning. A problem solving strategy regularly used in both Class A and C had been modelled after Polya's (1981) four phase framework (1. Understand the problem, 2. Devise a plan, 3. Carry out the plan, and 4. Look back and check). The problem solving heuristic was presented as a list of key words or phrases at times supported by diagrams that guided the student in solving the problem. Class C of this study involved students responding to a problem by generating or proposing an idea, insight, explanation or answer; then negotiating, arguing and reacting to that suggestion before coming to a final recommendation about whether it should be included in the group's response to the task. Following this, students wrote or recorded the answer, and looked back and checked their work. In Class B the teacher was very concerned that the students not b e compelled to follow any one strategy and encouraged a range of problem-solving approaches, which were developed by the class as they constructed their understandings.

Students discover for themselves

Where a shared community exists, there is opportunity for free exchanges of opinion and the expertise and contributions of individual members can be maximised. This may be achieved by a change of roles rather than the traditional role where the learner is dependent on the teacher. There should be multiple roles, rather than the traditional stance between teacher and student. Rather than an abandonment of instruction, however, there is a suggestion of instructional diversity (Holdaway, 1979) and the application of a variety of professional techniques, which cater for a range of learning styles and backgrounds.

Most children come to school with well-established thinking skills which have enabled them to survive in a real and meaningful world (Young-Loveridge, 1989). Davies (1993) investigated factors which inhibited the idiosyncratic interpretations of experience and found, that although children began school confident in their own ability to perceive and reason independently, this independence increasingly deferred to the teacher's knowledge the further students moved through the school system. Observations of children performing real-life problem-solving tasks showed that children preferred to use their own routes or have their own strategies even when school routines have been taught and rehearsed (Ginsburg, 1982; Saxe & Posner 1983; Scribner, 1984). Carraher and Schlieman (1985) found that most students displayed a wide range of self-generated strategies and competencies in performing mathematical computations in real world settings (for example, as candy sellers on the streets of Brazil). These strategies and competencies are rarely evident in the classroom when the same students perform algorithms involving the same properties of the numeration system.

Students can be assisted by their teachers to become aware of what they already know (Novak, 1986) and to evaluate their thinking. In this way teachers can assist students to become empowered by their capacity to learn how to learn, and to recognise what works best for themselves. Noddings (1993) has concerns that "turning students loose 'to construct' will not in itself ensure progress toward genuinely mathematical results" (p. 38), hence the role of the teacher can be quite crucial.

The socio-constructivist model (Yackel, Cobb, Wood, Wheatley, & Merkel, 1990) encourages effective communication between teacher and student. Teachers that adopt this model assume that the student contributions will be personally meaningful and that teachers should assist the students to state their ideas in mathematically meaningful ways. The teacher's intention is to work with the students as they try to express the concepts with which they are grappling. The teacher's interactions overtly model dialogue about mathematical issues for the students.

In order to be successful, students need to turn their language and their thinking upon themselves, so that they are not only able to direct their own thought processes, but they are also able to distinguish between possible interpretations of what they hear (Wilson & Wing Jan, 1993). Students may also find it helpful if they are able to represent their ideas in other forms which are less dependent on the context from which the information was originally drawn (Dalton, 1985; Dalton & Boyd, 1992).

The teachers' intent

While the teachers in Classes A, B and C stated that they valued the development of metacognitive processing in each of their students, their classroom practice reflected different methodologies. The Class A teacher was explicit about the ways in which she encouraged the students to use thinking strategies. The teacher asked students to record problem-solving steps in their workbooks. The teacher taught and practised of each of the following metacognitive strategies into her mathematics classes: Guess and Check, Draw a Picture, Make an Organised List, Make a Table, Work Backwards, Look for a Pattern, Use Logical Reasoning.

The Class B teacher attempted to develop her students' metacognitive approaches through the integration of the mathematical ideas with real life situations in a problem-based approach. The teacher's intent was to raise the students' awareness of being critically literate about their mathematical processing (critically numerate).

In Class C the teacher aimed to teach a general problem solving heuristic or set of steps for solving problems. This is shown by Figure 2.

Figure 2: Diagram of a general problem solving heuristic.

In Class C students were expected to use the heuristic while working in small groups of four on a common task. Before beginning the task the students were taught how the diagram showed the general nature of solving a problem. The teacher explained that in discussing the problem early ideas of how the problem could be solved would arise. These ideas would be discussed further by the group and be either accepted or rejected as part of the solution to the problem. If the idea being discussed was accepted as being part of the solution, it was recorded (generally on paper) then checked for group agreement that it was still part of the solution. A review of the heuristic was conducted briefly at the start of each lesson prior to students setting to work on the mathematical tasks.

Evidence of taught metacognitive practices

Close analysis of lesson transcripts of students solving problems in these three classes showed interesting results. Across all three classes there was little apparent use of any heuristic structures that had been taught by any of the teachers. Where students were using individual forms of metacognitive thinking they did not seem to be doing so with any conscious strategic approach.

Both Classes A and B were given a novel task to complete ("On what day and on what year will your 21st birthday fall?"). In Class A, where major heuristics had been taught, there was little demonstration of any predisposition to use the heuristic over that of Class B. In Class A there were only three (3) students that demonstrated any use of a system to solve the problem. In Class B, where the heuristic was not taught, eight (8) students appeared to use a system starting on the day on which they were born.

Regardless of whether the children knew what happened in a leap year or calculated the answer correctly, all children in Classes A and B demonstrated an almost idiosyncratic way of solving the problem. There were only two students from each of Class B and Class A that solved the problem without scaffolding assistance from Bickmore-Brand during the interview. There was no evidence to suggest that children from Class A, students that had been given explicit teaching about problem-solving strategies, actually applied them to this situation. Nor was there any evidence to suggest that Class A were more systematic than Class B who had experienced more real life opportunities to apply metacognitive strategies. Approximately one third of the students in Class B approached the task in a systematic way, whereas only three (3) students in Class A seemed to follow a strategy. In general Class A students were more willing to accept a prompt.

Clearly students failed to demonstrate any transfer of content previously taught in each of the classrooms. When analysing the post test (Placement Test J concept items- standardised test) for example, one third of Class B answered the questions on time and money incorrectly even though these spatial concepts had been explored within major problem-based topics throughout the year. One-third of Class A answered the question on decimals incorrectly, which also had been a major concept dealt with at regular intervals throughout the year. Ove r half of Class A made mistakes with the questions relating to number lines, even though they had had a considerable experience with number lines during the year.

The problem solving heuristic taught to students in Class C did not occur till later in the series of lessons observed. During the observation of lessons prior to the teaching of the problem solving heuristic, students participated in discussions and exhibited as much, if not more, evidence of using the elements of the problem solving heuristic than the lessons that followed the actual teaching of the heuristic. The evidence indicated that in conditions that did not explicate a problem solving heuristic or framework, students demonstrated effective problem solving strategies through verbal discussions within the small group learning context.

Following the introduction of a particular problem solving heuristic in Class C, it appeared that there was confusion in the student attempts to guide the problem solving process by following the model. In later situations where use of the heuristic may have aided the problem solving process, students did not attempt to utilise the model. In situations where the solution to a problem was reached early in the lesson, the students did not appear to have a clear recollection of the process by which the solution was obtained or in fact if they had made use of the heuristic framework at all. These findings are supported by Ross, Rolheiser and Hogaboam-Gray (1996) who found that students preferred to use their own procedures and instruments, rather than those developed by exemplary teachers. Given a choice, students may prefer to follow their instincts in solving problems.

One view (D.J. Clarke, personal communication, April, 1996) posits that students that are good problem solvers may naturally make use of problem solving heuristics. That is, good problem solvers have developed, either intentionally or unintentionally, a method or framework for solving problems. Following the teaching of the problem solving heuristic, students engaged in further problem solving tasks and were encouraged by the teacher to use the heuristic to guide their processes. As was apparent in the post lesson interviews conducted in Class C, most students indicated that there was nothing more they could see to learn about the problem solving heuristic. If students understand the essential elements of an heuristic (in this case, "idea", "discussion", "acceptance", "rejection", "writing" and "recording") and the interaction among those elements, then there may be little else that can be highlighted that will help students to solve problems. As one student stated "we always do that anyway".

Had the study of Class C sought to develop a problem solving heuristic, one source of data may have been the analysis of untrained student problem solving. In like manner, if our understanding of problem solving heuristics, frameworks and guiding questions has developed from the study of effective learners and problem solvers, it is not surprising then, to find these processes already occurring.

Findings

In Classes A, B, and C the students showed evidence that they were aware of the emphasis their teacher was taking toward thinking strategies. In many ways students in Class B had greater opportunity for idiosyncratic problem solving. This might be expected to give them an advantage over those in Class A when applying such strategies to an unfamiliar task, or those in Class C that were required to use a general form of problem solving heuristic. The results however show no real distinction between the three classes. This raises questions about the practice of instructing students in problem solving protocols. Should students be expected to rote learn problem solving strategies and develop patterns of metacognitive thinking that are imposed or should they be encouraged to develop their own metacognitive practices?

Giving clues or possible strategies

On a few occasions the teacher in Class C refocussed the students on the problem solving heuristic and asked about the particular step at which the students were operating. The teacher did this through questioning and seeking student explanations of progress. This had the effect of guiding the students in the solving of the problem. For example, if students suggested that the group was satisfied they had completed the discussion of an aspect, then the teacher would suggest that they should seek agreement to accept or reject the idea, in accordance with the heuristic framework. However, it was apparent that the level of teacher modelling was insufficient. As discussed earlier, approximately one-third of the students in Class B approached the task in a systematic way, and only three students in Class A appeared to follow any strategy. Of these two classes, Class A students were more willing to accept a prompt.

Giving clues reduces student ownership of a problem

Lesson data from all three classrooms suggests that by not frequently refocussing student metacognitive thinking back to the framework of an heuristic or strategy, and instead, opting to give students clues to the solving of the problem, the teacher may inadvertently regain control of the problem solving process. As a result student ownership and student investment in seeking a justifiable solution may be weakened through the absence of understanding and the increased frequency of attempts at guessing the solution. In effect the skills required to solve a problem, including use of heuristic frameworks, may be undermined by teacher hints and clues.

In this segment of lesson transcript from Class C, students were designing a bridge to hold a three kilogram weight (representing cars). Up until this point the students had been engaged in ideas, negotiating and arguing, agreeing and rejecting, recording and checking. This had occurred in no particular order, for 32 minutes. From this time on the teacher's input shifts the students' concentration from the seeking of a solution that will satisfy the problem to trying to guess the answer they think the teacher has in mind, as shown by the following passage. This is evident in Bryce's comment "hey wait, wait, he said it might be the other way round,"

TeacherYeah, I think you need to have another think.
TeacherYou may have found a bit of a clue in that.
StudentSorry.
StudentDon't look.
TeacherBut it may just be around the wrong way. See if you can find...
NickLook here.
Teacher...and try not to pinch ideas from other people (teacher leaves).
BryceHey wait, wait, wait, he said it might be the other way round.
Amber(Giggles) cars go like that (indicates on diagram).
BryceNo other way round.
NickStraight up... Na. Not that thing, it's not, you know, it's not that flexible.

Student discussions do not neatly fit into rote taught strategy use and metacognitive frameworks

Clearly evident in the transcript above is the rapid flow of discussion points, ideas, acceptance and rejections of ideas. In another lesson Class C students were seeking to solve the problem of moving people across a river using a canoe that has limitations on how many children and adults it may carry at any one time. By the 13th minute in the lesson, the students are beginning to use scrap paper to draw the movement of the canoe back and forth across the river. Two minutes later there is a clearly stated agreement as to the answer of how many trips it takes to convey all the people to the other side. The students decide to demonstrate th e answer to the teacher using drawing pins to represent people. The group dialogue has been rapid and focussed as shown by this excerpt.

Bryce...then one guy goes across, then little boy comes back. Nick how'd you do it?
Nick and
Bryce
(talking over the top of each other) the boys come back...you got it,
BryceYou got it, nah.
NickI was right Bryce.
BryceShow us how you got it then.
Amber(interjects and starts explaining) 2 boys go across, 1 boy comes back,
Nick(interjects) One man comes across ...
BryceThat's one (man)
Amber(carries on no break) one man goes across
NickOne boy comes back
AmberAnd one boy comes back (echoes Nick)
NickTwo boys across
AmberTwo boys go across (echo)
NickThen two boys go across
AmberAnd one boy comes back (states this step independently).
NickOne comes back (echoes Amber)
AmberOne man goes across (initiates new statement)
BryceThat's two (men)
NickOne boy back.
AmberThen one boy goes back and (echo)
NickTwo across

The dialogue shows only the audible, literal, words uttered by the students. Of course many other sounds, gestures and non-verbal movements are made that contribute to the communication between and among the members of the group. The speed at which the dialogue occurs is astounding. Many thoughts and contributions are made often with split second timing, the difference between heading in one direction and another is often only effected by a short "burst" of talk, a word or part of a sentence. Much like a "willy willy" (an Australian term meaning miniature hurricane) the group revolves at incredible speed but travels forward at a much lesser speed. Sometimes it seems as though students are tuned into the thoughts of others in their group and know where the discussion is heading and when their contribution is accepted as part of the solution and when it is not. The problem solving heuristic seems to be in use but not in an overt and explicit manner. The social construction of meaning takes precedence.

The role of the teacher

Teacher monitoring

In some situations the teacher's input to the operation of group discussions changed the nature of the thinking students were using. By interrupting the very complex dialogue of the group problem solving process as shown above, with questions about student thinking and the prescribed problem solving heuristic or strategy, the teacher appeared to shift student thinking from cognition to metacognition. However this shift was made explicitly. That is, the natural state of student metacognitive thinking, or the awareness of thinking processes, has shifted from one of background awareness to one of foreground awareness. The students have stopped thinking about seeking a solution to the problem in order to check (with the teacher's prompting) at what stage they are in relation to the problem solving heuristic.

Furthermore by suggesting potential clues as to a solution the teacher has shifted the nature of the problem to one that is more closed, having a definite solution, a solution that other groups may have already attained, and a solution to which this group are very close. Up until this point the high number of student proposals of ideas, negotiations and arguments, reviews and recordings had caused a high quality of discussion.

It is clear that the nature of the teacher's role must be carefully understood so that the quality of student discussions is allowed to continue at appropriate times. Secondly the time for debriefing or explicitly discussing metacognitive thinking may be better placed at the conclusion of lesson task time rather than midway through a task. Students may make significant gains if a more structured approach to teacher input allowed clear times or parts of the lesson, when metacognitive processing would be explicitly discussed. This may be shown by Figure 3.

Figure 3: Reflection time as a specific part of lesson time.

Teacher as scaffolder

The ways in which each teacher has chosen to use scaffolding are qualitatively different. All three teachers have relied on constructing situations in which they can provide a framework for mathematical skill and concept development. In Classes A and B teachers used the students' understanding in the construction of the scaffolding, although this had qualitative differences. Both teachers started with a problem or concept targeted specifically for a student for the class.

The differences for the Class A and Class B teachers lie in how each scaffolding dialogue was constructed. Class A teacher signalled her preferred pathway for the development of a concept, and although clearly aware of a student's misunderstanding continued to drive on with a predetermined method for approaching that skill or concept as can be seen in this transcript excerpt.

Teacher (Class A):Right you realised that whatever the denominator was multiplied by, the numerator was then multiplied by the same number. One times two is two. What else did we say to prove that one third does actually does equal two sixths? We came up with a way to prove that two sixths was equal to a third. Do you remember what we said yesterday? We said that we know if I had two sixths of a cake I would have just the same amount as someone that had one third... and what was that way Georgia?
Georgia:Because in the top number goes into the bottom number three times and in the second fraction the top number goes into the bottom number three times as well.
Teacher (Class A):That's in this instance but it doesn't always happen. I was actually talking about this little thing here that we talked about yesterday, what did we say about that number Angela?
Angela: That you times it by ... you can ... it's a whole number and you can say two ... it's a whole number ...
Teacher (Class A):Right you're saying two halves of two is a whole number right so you're saying two halves is one. One third multiplied by two halves equals two sixths. And all we are doing is multiplying a third by?
Whole Class:One.

The teacher in Class B however shaped and developed a skill or concept by moving down a jointly constructed pathway between her and the student or class. She worked with student ideas as they came up, for example in the following transcript, Joe wanted the sandwiches cut diagonally, and Ben suggested quarters. Her explanation of a skill or concept continued to be reworked in an effort to refine the communication of the idea rather than presenting a system for approaching the task.

Teacher stops at one group who are working on sandwiches.
TeacherSo you'll have to write something about bread. How many loaves of bread you will need.
JoeWe're going to have sandwiches and cut them like this (makes 2 diagonal cuts with his hands). How much bread will we need?
TeacherRight. How many slices in a loaf of bread?
Ben C24.
TeacherI was going to say 24 as well. Right so if you get 24 and you put one on top of the other (demonstrates with her hands) so that's ...?
JoeTwelve.
TeacherTwelve Sandwiches. If you do them into ...? (makes a diagonal cut with her hands)
JoeHalves.
Ben CNo quarters (makes 2 diagonal cuts)
TeacherQuarters. So there'll be 4 lots of 12...
William48.
Teacher48 sandwiches.
Ben CNot everybody would eat it so you'd be able to have 24 quarters.
TeacherIs that for 2 loaves of bread?
JoeMrs?
TeacherRemember we said there were 12 slices (demonstrates by drawing square with her fingers), and we cut those 12 into 4.
William24.
TeacherYou've got 24 slices but when you make them into double sides (demonstrates with hands) that reduces it to how many stacks (demonstrates with hands) of bread?
William12.
TeacherRight. Now if you cut that 12 into lots of 4 how many is that?
William48.
TeacherAnd you've got 2 loaves of bread. (pause) What's 48 plus ...?
Joe48.
Ben C90, 92
TeacherNot 92, 48 and 48. What's 8 and 8?
William16.
Teacher16. So it's ...?
William96.
TeacherAnd if there's 32 of us approximately, how many will each child have? Approximately?
WilliamThey'll have about half.
TeacherHow many 30's in 96?
Ben C3.
TeacherAbout 3. Would it be 3 whole sandwiches or just three quarters of a sandwich? So each can have about one round of sandwiches.
Later, the group reported back to the class
TeacherSo you decided what to do about your sandwiches?
Ben CSome people don't like sandwiches, but we decided that you could get 3/4 of a sandwich each and we needed 2 loaves of bread.

The teacher in Class B has attempted to develop the students' fraction knowledge in a context which is purposeful for the students. In addition, the concept has been tempered with a critical numeracy understanding of the skills they are using ("Some people don't like sandwiches").

Over the longer time period, Class A teacher continued to raise the difficulty of the concept development. Increased demands were placed on the students to apply their conceptual understanding of fraction concepts whereas, Class B teacher tended to focus on the immediate conceptual difficulty the children were facing as they tried to solve the problem they were currently tackling.

In Class C the teacher asked questions of the target group in relation to the heuristic as discussed above. The students were required to indicate on the heuristic model where the group was operating, however generally this did not result in any significant discussion and did not provoke any further reflection from the teacher or the students. Conversely, students were aware that if they did not achieve success in a task the teacher was quite likely to help them with clues or possible strategies. On a few occasions the teacher refocussed the students on the problem solving heuristic and the stage at which the students were operating. This had the effect of guiding the students in the solving of the problem, for example, by suggesting that if the group was satisfied they had completed the discussion of an aspect then they should seek agreement to accept or reject the idea. However, it was apparent that the level of teacher scaffolding was insufficient.

The findings from this study indicate that it is important not only for teachers to scaffold clear structures for thinking, but to also scaffold clear times and opportunities for thinking development. Student and teacher construction of an agreed process for thinking development appears to be an important step in the learning process.

Teacher as model

Brown and Palincsar (1987) argue that teacher modelling and the definition of specific roles and strategic tasks, provide a firm foundation for student learners. Results of studies found that long term maintenance, transfer and generalisation of learning was accomplished for students that had received the teacher modelling. Personal experience of Pitts-Hill in the use of the reciprocal model (Palincsar & Brown, 1989) found that students easily undertake the specific roles following teacher modelling.

Had the teachers in this study demonstrated using the heuristic to solve a problem they needed solving, thus modelling the implied processes and strategies of the heuristics and metacognitive strategies as genuine users, students may have achieved even higher levels of problem solving success. For example, if the teacher in Class C had encouraged the students to monitor similarly their progress according to the heuristic on a regular basis, higher levels of problem solving success may have been achieved by students. Similarly, if students had been encouraged to take an active leadership role in solving the problem through directing the problem solving strategies of the group, solutions to problems may have been achieved more effectively. Class C teacher modelling of regular reflection on the group's position on the model may have also encouraged students to take on a group monitoring role. Regular reflection may also have encouraged higher levels of recording and questioning. Teachers should also develop an understanding through observation of the variety of discussions that students create. This understanding impacts upon teacher planning of time for activities, the content of lesson problems, the need for student expertise and experience and the general culture that needs to be created in order to foster rich problem solving discussions.

Teacher beliefs concerning the teaching of metacognitive processing

While each teacher stated that they valued the development of metacognitive processing in each of their students, their classroom practice reflected different beliefs about how this was best accomplished. In Class A, the teacher was explicit about the ways in which she encouraged the students to use thinking strategies. Not only did she ask students to record problem-solving steps in their workbooks, but included the teaching and practice of several metacognitive strategies into her mathematics classes including; Guess and Check, Draw a Picture, Make an Organised List, Make a Table, Work Backwards, Look for a Pattern, Use Logical Reaso ning.

Alternatively, the Class B teacher attempted to develop her students' metacognitive approaches through the integration of the mathematical ideas with real life situations and tried to raise the students' awareness of being critically literate about their mathematical processing (critically numerate).

In Class C, it was apparent that the strategies used by students were not clearly linked back to the framework of problem solving that was taught. Further it was evident that the teacher did not clearly articulate these links to students. Had the general problem solving heuristic been taught in relation to specific problem solving strategies students may have exhibited far greater levels of achievement. This is shown by Figure 4.

Heuristic stepStrategies

Ideas generationdraw a diagramwork backwardslook for patternstry a simpler problem
Group discussionasking questionstaking turnsgood/bad pointslistening skills
Decision makingseek consensushands up voteadd up good and bad pointslook for justifications
Recordingwrite reportbuild modeldevelop simulationdraw a diagram
audio recordvideo recordoral reporttake photograph
Checkingtry a simpler problemdraw a diagramwork backwardstry under other conditions

Figure 4: Strategies grouped by the problem solving heuristic steps used in Class C.

Heuristics

Student heuristics

It is feasible then to suggest that students may already hold a framework for solving problems or at the very least that students have ways of developing strategies and frameworks that help them solve problems. If a problem-solving framework is specified by the teacher, students may have difficulty in reconciling or assimilating two models, that is the teachers model and their own model, however, there was no evidence in the literature reviewed that focussed on student difficulty reconciling similar or competing heuristic frameworks. Consequently, problem solving heuristics and frameworks developed by students could provide rich information for the teacher on how students perceive the problem solving process.

It would appear that the efforts by each teacher in this study to develop thinking strategies among their students was not as powerful a factor as allowing the students to use their own idiosyncratic ways when processing a problem. However, there was little evidence to suggest that the teachers in this study had recognised the power of student developed metacognition. In Class A, for example, students had experienced mainly textbook-oriented activities which had involved the application of a protocol or strategy. In Class B, students had a variety of lesson content but none of which expressly aimed to validate the students' own metacognitive practices. In Class C the teacher had pre-planned the series of lessons focussing on the teaching of the heuristic framework. It may be concluded that each teacher had pre-empted the needs of students to the extent that he or she was unable to see the opportunity of utilising student developed metacognition. The teacher was in effect preparing their students for "more of the same" - more of what they had been doing in the classroom. The implication here is that teachers need to spend specific time examining the nature of the student learning that is occurring in classrooms. In the current Western Australian educational climate this need for careful analysis of student achievement is supported by the move towards an outcome focussed learning environment and the promotion of developmental learning.

The place of explicit instruction in problem solving strategies and metacognitive thinking

The development of clear understandings relating to metacognitive thinking and problem solving strategies is an important factor in developing effective learning in students. The teachers' understandings about thinking should be developed with the students in a relationship that promotes wide understandings. The relationship of the strategy to the broader heuristic framework should be made explicit. One way to accomplish this is to teach problem solving strategies within the contextual framework of the problem solving heuristic. Using specific content would promote: It is suggested that a staged learning process would facilitate this strategy development. In this graduated learning process, an early stage would involve the teaching and learning of strategies in a suitable problem context and within the framework of a problem solving heuristic. A joint construction of meaning around the heuristic would be encouraged. In later stages these strategies are applied in an environment characterised by reduced teacher intervention and involving broader problem solving situations with less emphasis being placed on the overt use of the heuristic. In essence the majority of the teachers' cognitive intent is undertaken in the first stage of the process, but fully realised in the process overall. This is shown in Figure 5.

Figure 5: A graduated learning process (after Furness and Green, 1991).

In this way learning has both a clear teacher input as to how problems may be solved but is supported and developed in the second stage where students undertake the solution of problems in more idiosyncratic ways.

Shared metacognition - students and teachers reflect

From the results of this study, it is clear that students do not necessarily prefer teacher taught metacognitive structures and frameworks. Conversely, students may well benefit from a co-construction of learning about metacognition, discovering and using naturally occurring metacognitive practices and practising and reflecting on effective problem solving in the light of metacognitive learning. By revisiting Figure 1 in the light of the previous discussions, the interactions of the teacher and the student in the learning process ("teacher's choices" and "student's success") are shown in relation to the use of metacognitive frameworks and heuristics ("instruction"). The implication is that greater effective learning is experienced by the student, and greater understanding of the student's perceptions of the learning are gained by the teacher. Where there is a greater overlap of these perceptions there is a better understanding of the learning process from both the teacher and the student.

Summary and conclusion statements

Teachers A and C tended to favour a more teacher centred approach, however when all three classes were analysed the students' problem solving behaviours tended towards a more idiosyncratic approach. The study raised the question of transfer and whose learning was taking place in the classroom. The findings from studying these three classrooms would suggest support for the current educational drive toward more student outcome focussed learning environment. The following recommendations have been made to support teachers in developing a more student-centred focus with their students.

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Please cite as: Bickmore-Brand, J. and Pitts-Hill, K. (1999). Teaching for transfer: Transfer of whose learning? Proceedings Western Australian Institute for Educational Research Forum 1999. http://www.waier.org.au/forums/1999/bickmore-brand.html


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