Each two-page lesson of Knowing Mathematics is written for one instructional session (40 to 50 minutes in length) and contains the following parts:
- Warm-Up Exercises: These exercises prepare students for the topic of the lesson or act as a review of previous lessons.
- Getting Ready for the Mathematical Conversation: After the Warm-Up exercises, students briefly discuss the lesson title. The title is designed to focus on the main topic of the lesson and is often a question so students are encouraged to predict what they will learn.
- Mathematical Conversation: The main part of the first page, the instruction page, shows a mathematical conversation between the teacher (Ms. Park) and her students. Students read and act out the conversation, stopping for discussions at strategically placed locations. The instruction page is designed for students to think about the mathematics they are learning.
- Guided Practice: As a group, with teacher guidance, students ask questions as they try and discuss exercises based on the topic of the lesson. Teaching support for these exercises is scaffolded.
- Independent Exercises: For the second part of the lesson, students work independently on exercises that are monitored by the teacher. Acting as a coach, the teacher asks students to discuss their strategies or to notice aspects of the exercises at strategically placed pauses. The exercises are carefully designed to support the learning of the mathematical topic just discussed. Review sections are always included to maintain coherence in learning. The teaching support fades for some topics (such as word problems) over the course of several lessons and for some topics over a single lesson.
- Reflect and Discuss: Students share and discuss strategies and opinions about the lesson topic:
- Reflection and Discussion: A problem or question is posed to students that is designed to provide insight into students' understandings of the lesson topic.
- Return to the Lesson Title: Students return to the lesson title where they are encouraged to summarize what the lesson was about and monitor their progress.
You will notice that throughout the instructional model, students are asked for their strategies and opinions.
This approach is supported by research in psychology by Ann Brown and colleagues paraphrased here:5
- Verbal reportseven without feedback from another personcan benefit learning and problem solving. Most commonly this occurs when the type of report required is a reason for an action or a statement of a rulepossibly a rule created by the learner.
- Good problem solvers identify and evaluate their problem-solving efforts more than poor learners, stating rules and evaluating their efficiency.
- Instructions to state a rule accelerate learning.
- Instructions to state a rule facilitate transferstudents are more likely to use their knowledge in different contexts.
- Learning an explanation for a procedure and the procedure makes transfer more likely.6
Aspects of the instructional format are similar to that of reciprocal teaching (used in the reading interventions Early Success and Soar to Success):
||Students work in a small group together with a teacher.
||The whole group focuses on reading and understanding the instruction page.
||Instead of the routines of predict, clarify, question, and summarize used in reciprocal teaching, some features of mathematics learning are made visible by the cast of characters: asking questions about the content, predicting future topics, and connecting past learning with current topics.
Teaching Guide Format