Learning Trajectories

Wood, Bruner, and Ross (1976) and Bruner (1986) developed the concept of leading children’s learning forward through “scaffolding”. This involves the teacher providing a pedagogical trajectory to support children’s movement into new territories. In articulating Bruner’s notion of guided participation, Rogoff (1991) argued that the teacher’s main role is to “build bridges from children’s current understanding to reach new understanding through processes inherent in communication” (p. 351). Later, Bruner (1996) drew on Vygotsky’s notion of the “zone of proximal development” (Vygotsky, 1978, p. 86) when he further defined scaffolding as a logical structuring of ideas to be understood in an order that leads children to develop further and faster than they would on their own. A variety of images for teachers’ roles in scaffolding learning have been presented. 

In describing quality teaching, Wood (1991, p. 109) used the term “leading by following”, noting that the most effective scaffolding draws on the interests and understandings of the child. Cobb and McClain (1999) described an instructional sequence that follows a conjectured learning trajectory that “culminates with the mathematical ideas that constitute our overall instructional intent” (p. 24). Hiebert et al. (1997) used the term “residue” to describe the knowledge that children gain from teaching that may be used as a basis for further planning of sequences of tasks aimed at the development of further particular residues over time. Scardamalia, Bereiter, McLean, Swallow, and Woodruff (1989) portrayed learning trajectories as social phenomena, with teachers employing scaffolding to create more general pathways of potential development of mathematical concepts and procedures. Lerman (1998) discussed the teachers’ roles in setting up loci of development—social interactions with mutual appropriation by teachers and students. Simon (1995) demonstrated how the continually changing knowledge of the teacher creates change in expectations of how students might learn a specific idea. 

A hypothetical learning trajectory provides the teacher with a rationale for choosing a particular instructional design; thus, I (as a teacher) make my design decisions based on my best guess of how learning might proceed. This can be seen in the thinking and planning that preceded my instructional interventions … as well as the spontaneous decisions that I make in response to students’ thinking. (pp. 135-136) Simon used the word “hypothetical” to suggest that all three parts of the trajectory are likely to be somewhat flexible, with teachers changing the learning goals and adapting aspects of planned activities in response to (a) their perceptions of students’ levels of understanding and (b) their on-going evaluations of students’ performance of classroom tasks. Thus actual learning trajectories cannot be known in advance. Further, Simon noted that such a trajectory is made up of three components: the learning goal that determines the desired direction of teaching and learning, the activities to be undertaken by the teacher and students, and a hypothetical cognitive process, “a prediction of how the students’ thinking and understanding will evolve in the context of the learning activities” (p. 136). 

In discussion of Simon’s paper, Steffe and Ambrosio (1995) described teachers’ working hypotheses of what students could learn as being determined by the teacher as she interprets the schemes and operations available to the student’s actions in solving different tasks in the context of interactive mathematical communication. The anticipation is based on the teacher’s knowledge of other students’ ways of operating, on the teacher’s knowledge of the particular mathematics of that student, and on results of the teacher’s interactions with that student. (p. 154) In this discussion, Steffe and Ambrosio raised an important question—one that is at the heart of this paper: “How does a teacher modify a task that fails to activate certain schemes?” (p. 155). This is a question that we return to below. Throughout these varied discussions about scaffolding of learning via the creation of specific learning trajectories, the general picture is one of a teacher planning to create a context in which the class will follow one learning trajectory.

The concepts of “remedial work” and “extension work”, as well as teachers’ everyday experience of some students being more successful than others, suggest that actual learning trajectories are likely to take a shape. “Ability” grouping, setting or streaming present a further model, with teachers aiming to lead groups or whole classes of students to different learning goals (see Figure 3). However, the literature on the negative effects of such grouping in primary and lower secondary schools is extensive (see for example Boaler, 1997; Gamoran, 1992; Ireson, Hallam, Hack, Clark, & Plewis, 2002; Mousley, 1998; Zevenbergen, 2003).

Negative effects of differentiated learning expectations can include lowering for some students of teachers’ expectations, self- and peer-expectations, self concepts, opportunities for positive modelling and mentoring, and student motivation. In fact, this solution to diversity has the potential to exacerbate disadvantage due to self-fulfilling prophesy effects (Brophy, 1983). In our experience, teachers who use groups in this way are aware of these potential effects but also want to set achievable goals for all students. They realise that most classes have pupils with sufficiently divergent needs that any one task may not be appropriate for all. Clearly there is a need to research forms of pedagogy that may help teachers to adapt classroom tasks to the needs of the range of individual pupils in their classes. Thus it is important to research how teacher may modify tasks that fail to enable some students to meet specific learning goals.

 “Clearly … the trajectories followed by those who learn will be extremely diverse and may not be predictable” (Lave & Wenger, 1991) 

In choosing to focus on learning trajectories, we embrace a metaphor that, for all its appeal, implies that learning unfolds following a predictable, sequenced path. Everyone knows it is not that simple; researchers and educators alike acknowledge the complexity of learning. As Simon (1995) emphasized, learning trajectories are essentially provisional. We can think of them as the provisional creation of teachers who are deliberating about how to support students’ learning and we can think of them as the provisional creation of researchers attempting to understand students’ learning and to represent it in a way that is useful for teachers, curriculum designers, and test makers.

I firmly believe that a critical part of our mission as researchers is to produce something that is of use to the field and serves as a resource for teachers and curriculum designers to optimize student learning. No doubt this includes creating, testing, and refining empirically based representations of students’ learning for teachers to use in professional decision-making and, further, investigating ways to support teachers’ decision-making without stripping teachers of the agency needed to hypothesize learning trajectories for individual children as they teach. This focus would add a layer of complexity to our research on learning and invite us to think seriously about how to support teachers to incorporate knowledge of children’s learning into their purposeful decision-making about instruction.

Further, I suggest we consider, in the end, “Whose responsibility is it to construct learning trajectories?” (Steffe, 2004, p. 130). If we researchers can figure out how to supply teachers with knowledge frameworks and formative assessment tools to facilitate their work, teachers will be able to exercise this responsibility with increasing skill, professionalism, and effectiveness. Because of the growing popularity of learning trajectories in education circles, it is worth thinking hard about the role of learning trajectory representations in teaching, and in particular, whether a learning trajectory can exist meaningfully apart from the relationship between a teacher and a student at a specific time and place. Simon’s (1995) perspective on teaching and learning suggests not. As the field moves forward with research on learning trajectories and strive for coherence in learning across the grades, I would like to remain mindful of both the affordances and constraints this particular type of representation offers for teachers and students alike.