# The Landscape of Learning and Learning Trajectories

### From MathWiki

*The following is a short summary I wrote on some work of Catherine Fosnot (an Education Professor in New York, who is involved in pre- and inservice teacher training), regarding how students are thought to learn mathematics. The reference information is included below in case anyone is interested in looking into it any further.*

**The Landscape of Learning and Learning Trajectories**

Fosnot & Dolk (2002) address the "how" of mathematics learning in a broad sense - something that is relevant to our project of improving student experiences with geometry in the elementary years. They state that "traditionally," mathematics learning was viewed as a linear model of development: progressive units were expected to be an extension of previously learned information, which was viewed as being consistent with the cumulative nature of mathematics curricula.

They assert that curricula and lessons were designed presupposing that students will progress at the same pace through an identical linear path of learning. These presupposed learning trajectories determined expectations of student development, and students who did not progress along this linear path were deemed to be "falling behind." However, Fosnot adopts the idea of a "hypothetical learning trajectory" from Simon (1995). Learning trajectories are claimed to be "hypothetical" since until students actually attempt a problem, "we can never be sure what they will do or whether and how they will construct new interpretations, ideas and strategies" (Fosnot & Dolk, 2002, p.22).

In this way, Fosnot & Dolk (2002) do not reject the idea of a learning trajectory, but rather emphasize that a linear model does not account for "real" learning since real learning "is messy" (p.23). Instead, they suggest the metaphor of a landscape. In the landscape of learning, "knowledge of models, strategies, and big ideas," are essential to organizing a coherent hypothetical learning trajectory. In other words, teachers and curriculum developers still need to have a clear and organized idea of what they want their students to know. However, the landscape of learning also comes with an appreciation for divergence from a linear progression - students may have to move "backwards" in a hypothetical learning trajectory to a previously covered idea in order to understand what is currently being taught. Landmarks (models, strategies, connections, big ideas) are central to a learning trajectory, but the path through which students acheive these landmarks does not have to progress linearly.

In several books on this topic, Fosnot suggests learning landscapes for the topics of Number Sense, Fractions, Decimals, & Percents, and Multiplication and Division. For example, for the fractions landscape, landmarks include: "doubles numerator to multiply by two", "uses a common whole to add and subtract", "clock model", and "using landmark percents (e.g. 50%, 10%)" (Fosnot & Dolk, 2002, p.136-137)

If we are trying to piece together a coherent view of visual and spatial reasoning, the idea of a hypothetical learning trajectory along a learning landscape for related conceptual/procedural skills may be worth considering.

`Reference:` Fosnot, C.T., & Dolk, M. (2002). *Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents*. Portsmouth: Heinemann.

Available in the ERC Library - Call Number 372.72 FOS

Back to Mordy's Links

Back to Spatial Reasoning Home