Chapter 8: Comments On Learning And Teaching Early And Elementary Mathematics

Clements and Sarama’s informative review centers on three key questions for improving mathematics education: How do students learn mathematics? What pedagogical strategies are supported by research? What approaches to professional development are supported by research? Their review of the first and third issues is brief. Similarly, our comments will focus on the second issue.

This section fittingly focused on the importance of learning trajectories as a guide to understanding children’s mathematical learning. Such guides are invaluable for formative assessment and instructional planning by helping to answer such questions as, What prior knowledge does a student need to know to achieve a particular goal? What should a child be taught next? A learning trajectory is particularly important when students run into difficulty and can help teachers think backwards to remedy the source(s) of a learning problem. For example, if a child cannot identify the larger of two number neighbors (e.g., “Which is more 8 or 7?”), a key issue is whether the child is ready for instruction in this skill or needs to master more basic prerequisites. That is, does the child understand the counting-based rule for comparing numbers and accurately apply it to small numbers (e.g., “4” is more than “3” because “4” comes after “3” when we count)? Does the child know the counting sequence well enough to readily determine the number that comes after another (e.g., after 7 comes 8)? Does the child even understand the comparative term “more” or the meaning of number words such as “three” and “four?”