MATHS 1011 Mathematics
Children follow natural developmental progressions in learning and development. As a simple example, children first learn to crawl, which is followed by walking, running, skipping, and jumping with increased speed and dexterity. Similarly, they follow natural developmental progressions in learning math; they learn mathematical ideas and skills in their own way. When educators understand these developmental progressions, and sequence activities based on them, they can build mathematically enriched learning environments that are developmentally appropriate and effective. These developmental paths are a main component of a learning trajectory.
Key Research Questions
1. What objectives should we establish?
2. Where do we start?
3. How do we know where to go next?
4. How do we get there?
Recent Research Results
Recently, researchers have come to a basic agreement on the nature of learning trajectories.1 Learning trajectories have three parts: a) a mathematical goal; b) a developmental path along which children develop to reach that goal; and c) a set of instructional activities, or tasks, matched to each of the levels of thinking in that path that help children develop higher levels of thinking. Let's examine each of these three parts. Goals: The Big Ideas of Mathematics
The first part of a learning trajectory is a mathematical goal. Our goals are the big ideas of mathematics—clusters of concepts and skills that are mathematically central and coherent, consistent with children’s thinking, and generative of future learning. These big ideas come from several large projects, including those from the National Council of Teachers of Mathematics and the National Math Panel.
Development Progressions: The Paths of Learning
The second part of a learning trajectory consists of levels of thinking; each more sophisticated than the last, bwhich lead to achieving the mathematical goal. That is, the developmental progression describes a typical path children follow in developing understanding and skill about that mathematical topic. Development of mathematics abilities begins when life begins. Young children have certain mathematical-like competencies in number, spatial sense, and patterns from birth.5,6 However, young children's ideas and their interpretations of situations are uniquely different from those of adults. For this reason, good early childhood teachers are careful not to assume that children “see” situations, problems, or solutions as adults do. Instead, good teachers interpret what the child is doing and thinking; they attempt to see the situation from the child’s point of view.
Similarly, when these teachers interact with the child, they also consider the instructional tasks and their own actions from the child’s point of view. This makes early childhood teaching both demanding and rewarding. The learning trajectories we created as part of the Building Blocksa and TRIADb projects provide simple labels for each level of thinking in every learning trajectory. illustrates a part of the learning trajectory for counting.
The Developmental Progression column provides both a label and description for each level, along with an example of children's thinking and behavior. It is important to note that the ages in the first column are approximate. Without experience, some children can be years behind this average age. With high-quality education, children can far exceed these averages. As an illustration, 4-year-olds in our Building Blocks curriculum meet or surpass the “5-year-old” level in most learning trajectories, including counting. (For complete learning trajectories for all topics in mathematics,
These works also review the extensive research work on which all the learning trajectories are based.). Instructional Tasks: The Paths of Teaching The third part of a learning trajectory consists of set of instructional tasks, matched to each of the levels of thinking in the developmental progression. These tasks are designed to help children learn the ideas and skills needed to achieve that level of thinking. That is, as teachers, we can use these tasks to promote children's growth from one level to the next. The third column in Figure 1 provides example tasks. (Again, the complete learning trajectory in Clements & Sarama,6,7 includes not only all the developmental levels, but several instructional tasks for each level.)
Although learning trajectories have proven to be effective for early mathematics curricula and professional development,9,10 there have been too few studies that have compared various ways of implementing them. Thus, their exact role remains to be studied. Also, in the early years, several learning trajectories are based on considerable research, such as those for counting and arithmetic. However, others, such as patterning and measurement, have a smaller research base. Further, there are few guidelines for many more sophisticated math topics for teaching older students. These remain challenges to the field.
Learning trajectories hold promise for improving professional development and teaching in the area of early mathematics. For example, the few teachers that actually led in-depth discussions in reform mathematicsclassrooms saw themselves not as moving through a curriculum, but as helping students move through levels of understanding.11 Further, researchers suggest that professional development focused on learning trajectories increases not only teachers’ professional knowledge but also their students’ motivation and achievement Thus, learning trajectories can facilitate developmentally appropriate teaching and learning for all children.