As each school year begins, a focus on effective instruction becomes the dominant conversation in most professional development workshops. Building mathematical understanding is possible when instruction results in a solid foundation at each level.
Development of conceptual understanding and facility with mathematical skills are not mutually exclusive. Principles from research and the Singapore math pedagogy agree that students should gain understanding in how and why math works. From here, a scaffolded and coherent set of instructional elements should promote deep math understanding.
“Developing mathematical understanding requires that students have the opportunity to present problem solutions, make conjectures, talk about a variety of mathematical representations, explain their solution processes, prove why solutions work, and make explicit generalizations” ( FRANKE. ET AL ., 20 07) .
As teachers review the instructional pathway, the intentional sequence of problems within a multi-day lesson becomes more apparent. Using multiple approaches with a focus on presenting new concepts in a straightforward concrete or visual format requires that teachers consider pacing which allows students to “go beyond the basics”.
Math in Focus lessons follow an instructional pathway of Learning, Consolidating and Applying.
Learn new concepts and skills through concrete and highly visual examples and teacher instruction
Consolidate concepts and skills through practice, activities and non-routine problems
Apply concepts and skills with extensive problem-solving practice and challenges
Embedded practice is evident in a series of activities that are a part of the scaffolded sequence as shown above and not designed to be optional tasks. These include Hands-On Activities (shown below), Games, Let’s Explore, Math Journal and Put on Your Thinking Cap! problems.
The activity shown above is designed to help students in Grade 3, develop a better understanding of regrouping in division so they can generalize patterns in the base 10 system outside of the procedural computation of long-division. Students have learned in a previous lesson that any odd number divided by 2 will result in a remainder of 1. This allows them to work with familiar patterns and number sense, as well as visualize what that remainder of 1 may look like when it too is shared out. Generalization and visualization are key strategies in understanding mathematics in the elementary grades.