Reference Work Entry

Encyclopedia of Mathematics Education

pp 272-275


Hypothetical Learning Trajectories in Mathematics Education

  • Martin SimonAffiliated withSteinhardt School of Culture, Education, and Human Development, New York University Email author 


Learning Teaching Constructivism Teacher thinking Learning progressions


Learning Teaching Constructivism Teacher thinking Learning progressions


Hypothetical learning trajectory is a theoretical model for the design of mathematics instruction. It consists of three components, a learning goal, a set of learning tasks, and a hypothesized learning process. The construct can be applied to instructional units of various lengths (e.g., one lesson, a series of lessons, the learning of a concept over an extended period of time).

Explanation of the Construct

Simon (1995) postulated the construct hypothetical learning trajectory. Simon’s goal in this heavily cited article was to provide an empirically based model of pedagogical thinking based on constructivist ideas. (Pedagogical refers to all contributions to an instructional intervention including those made by the curriculum developers, the materials developers, and the teacher.) The construct has provided a theoretical frame for researchers, teachers, and curriculum developers as they plan instruction for conceptual learning.

Simon (1995. P. 136) explained the components of the hypothetical learning trajectory:

The hypothetical learning trajectory is made up of three components: the learning goal that defines the direction, the learning activities, and the hypothetical learning process – a prediction of how the students’ thinking and understanding will evolve in the context of the learning activities.

There are a number of implications of this definition including the following:
  • Good pedagogy begins with a clearly articulated conceptual goal.

  • Although students learn in idiosyncratic ways, there is commonality in their ways of learning that can be the basis for instruction. Therefore, useful predictions about student learning can be made.

  • Instructional planning involves informed prediction as to possible student learning processes.

  • Based on prediction of students’ learning processes, instruction is designed to foster learning.

  • The trajectory of students’ learning is not independent of the instructional intervention used. Students’ learning is significantly affected by the opportunities and constraints that are provided by the structure and content of the mathematics lessons.

To elaborate the last point, the second and third components of the hypothetical learning trajectory, the learning activities and the hypothetical learning process, are interdependent and co-emergent. The learning activities are based on anticipated learning processes; however, the learning processes are dependent on the nature of the planned learning activities. Clement and Sarama (2004a, p. 83) reaffirmed this point.

Although studying either psychological developmental progressions or instructional sequences separately can be valid research goals, and studies of each can and should inform mathematics education, the power and uniqueness of the learning trajectories construct stems from the inextricable interconnections between these two aspects.

They went on to define learning trajectories as follows.

We conceptualize learning trajectories as descriptions of children’s thinking and learning in a specific mathematical domain and a related, conjectured route through a set of instructional tasks designed to engender those mental processes or actions hypothesized to move children through a developmental progression of levels of thinking, created with the intent of supporting children’s achievement of specific goals in that mathematical domain (c.f. Clements 2002; Gravemeijer 1999; Simon 1995) (p. 83).

According to Simon (1995), a hypothetical learning trajectory was part of a mathematics teaching cycle that connects the assessment of student knowledge, the teacher’s knowledge, and the hypothetical learning trajectory. The cycle is meant to capture a progression in which an instructional intervention is made based on the hypothetical learning trajectory. Student knowledge/thinking is monitored throughout. This monitoring leads to new understandings of student thinking and learning, which, in turn, leads to modifications in the hypothetical learning trajectory. The mathematics teaching cycle also stresses that, in the context of teaching, teachers develop additional knowledge of mathematics and mathematical representations and tasks. All modifications in teacher knowledge contribute to changes in the revised hypothetical learning trajectory. Thus, an implication of the mathematics teaching cycle is that a big part of good teaching is the ability to analyze student learning in order to revise the instructional approach.

The mathematics education research community picked up the hypothetical learning trajectory construct, and 9 years after the original article, Clements and Sarama (2004b) edited a special issue of Mathematics Thinking and Learning on hypothetical learning trajectories. Although the hypothetical learning trajectory construct grew out of constructivist ideas, it has been adapted for use with social learning theories (e.g., McGatha et al. 2002).

Two lines of research grew out of the original work on hypothetical learning trajectories. The first, conducted by Simon and his colleagues, is an attempt to explicate the mechanisms of conceptual learning, that is, to provide a framework for generating hypothetical learning processes in conjunction with learning activities. (See Tzur and Lambert 2011; Simon et al. 2010; Tzur 2007; Simon et al. 2004; Simon and Tzur 2004; Tzur and Simon 2004). Whereas research grounded in constructivist ideas has a tradition of modeling students’ thinking at various points in their conceptual learning, postulation of the hypothetical learning trajectory construct called for modeling the learning process itself, the means by which the students’ thinking changes as they interact with the instructional tasks and setting.

The second line of research, which grew out of the original hypothetical learning trajectory work, is research on learning trajectories in mathematics (also referred to as “learning progressions”; see discussion of learning progressions in this volume). Learning progressions research is an attempt to provide an empirical basis for instructional planning.

Trajectories involve hypotheses both about the order and nature of the steps in the growth of students’ mathematical understandings and about the nature of the instructional experience that might support them in moving step-by-step toward their goals of school mathematics (Daro et al. 2011, p 12).

Not only have a significant number of researchers gotten involved in this line of research, but the Common Core Standards (CCSSO/NGA 2010) in the United States has leaned heavily on the learning progressions work to date. A key issue as research on learning progressions develops is whether a central idea in Simon’s hypothetical learning trajectory will be maintained. That is, will the learning process continue to be seen as interrelated with the instructional approach or will various stakeholders in mathematics education seize on particular learning progressions as the way that students learn. The quote above from Daro et al. seems to imply that there is a set of learning steps, and then instruction is built to foster that sequence of steps. This stands in contrast to a view that any particular sequence of steps is in part a product of the instructional experiences provided to the students. Clements and Sarama pointed to an important implication of the perspective based on Simon’s original definition:

Thus, a complete hypothetical learning trajectory includes all three aspects. … Less obvious is that their integration can produce novel results. … The enactment of an effective, complete learning trajectory can actually alter developmental progressions or expectations previously established by psychological studies, because it opens up new paths for learning and development.


Constructivism in Mathematics Education

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