Download a Landscape of Learning Graphic Landscape of Learning
The rich, open investigations we've developed allow children to engage in mathematizing in a variety of ways.We honor children's initial attempts at mathematizing, at the same time supporting and challenging children to ensure that important big ideas and strategies are being developed progressively. Our approach should not be confused with what is commonly called "developmentally appropriate practice," where teachers assess every child, ascribe a stage, and then match tasks to each child. Our approach emphasizes emergence. Learning, real learning, is messy; it is not linear. We conceive of learning as a developmental journey along a landscape of learning. This landscape is composed of landmarks in three domains: strategies, big ideas, and models.
Strategies, Big Ideas, and Models
Strategies can be observed. They are the organizational schemes children use to solve a problem; for example, they might count by ones, skip-count, or use doubles.
Underlying these strategies are big ideas. Big ideas are "the central, organizing ideas of mathematics—principles that define mathematical order"(Schifter and Fosnot 1993, 35). Big ideas are deeply connected to the structures of mathematics. They are also characteristic of shifts in learners' reasoning—shifts in perspective, in logic, in the mathematical relationships they set up. As such, they are connected to part-whole relations—to the structure of thought in general (Piaget 1977). In fact, that is why they are connected to the structures of mathematics. Through the centuries and across cultures as mathematical big ideas developed, the advances were often characterized by paradigmatic shifts in reasoning. That is because these structural shifts in thought characterize the learning process in general. Thus, these ideas are "big" because they are critical ideas in mathematics itself and because they are big leaps in the development of the structure of children's reasoning. Some of the big ideas you will see children constructing as they work with the materials in this package are unitizing, compensation, and equivalence.
Finally, mathematizing demands the development of mathematical models. In order to mathematize, children must learn to see, organize, and interpret the world through and with mathematical models. This modeling often begins simply as representations of situations, or problems, by learners. For example, learners may initially represent a situation with connecting cubes or a drawing. These models of situations eventually become generalized as learners explore connections across contexts—for example, using a train of connecting cubes that represents a measurement to mark lengths on a paper strip, thus creating a blueprint for cutting strips of certain lengths. Generalizing across contexts allows learners to develop more encompassing mental models to think about situations with—for example, the blueprint becomes a number line model to explore the relationships between addition strategies. At this point, teachers use the emerging model didactically, representing children's invented computation strategies for addition and subtraction as leaps on an open number line. This stage bridges learning from informal solutions specific to a context toward more formal, generalizable solutions—from models of thinking to models for thinking (Beishuizen, Gravemeijer, and van Lieshout 1997; Gravemeijer 1999). Models that are developed well can become powerful tools for thinking.
Learning as Development
Now picture a landscape. See the attached landscape of learning for multiplication and division. On the horizon is a deep understanding of these topics. Along the way are many developmental landmarks—strategies, big ideas, and ways of modeling that as a teacher you will want to notice, support the development of, challenge learners to construct, and celebrate. The units in this package are designed to support children on this journey. Each unit has a different focus and zooms in on a section of the landscape. You will find this information on the first page of the overview in each unit.
Teaching mathematics is about facilitating mathematical development. This means that you cannot get all learners to the same landmarks at the same time, in the same way, any more than you can get all toddlers to walk at the same time, in the same way! All you can do is provide a rich environment, turn your classroom into a mathematical community, and support the development of each child in the journey toward the horizon. 6