# Clip 1/11: Pre-Lesson

## Overview

This video describes how the Problem of the Month "First Rate" was used in the school for the past two weeks and how this end of lesson was developed. It gives insight into how the unit developed over time. Condon describes how students worked on the Problem of the Month over the last two weeks. The teacher and coach discuss the mathematical goals for the lesson and goal for the school around flexible thinking. They pose the question: How are students identifying the need for units or terms for numerator and denominator? What strategies do students have for comparing ratios?

## Teacher Commentary

COACH LINDA FISHER: I am impressed with the courage of the team to drop formal instruction and allow students to construct their own understanding. I like how the Problems of the Month allow students to explore and develop an understanding of this rich mathematical idea.

A school-wide goal is to develop flexible thinking in students. Hopefully the lesson will allow ideas about making one measure constant or the same to make a comparison. Questions will helpfully draw out this generalization. Then students are given "naked numbers," numbers without units, to see if they can come up with the idea of unit rate.

Often U.S. teachers are criticized for having lesson goals around procedural steps or techniques. I am impressed that the lesson study team is focusing on important mathematical generalizations: In order for the rates to be compared, the quantity in the numerator or the quantity in the denominator needs to be the same. Unit rate is a convenient way to compare quantities. The team is also very curious about how students learn, noting that a strategy or way of representing a rate that is most comfortable to me as a teacher may not be the mot natural or comfortable way for a student to make sense of the same situation. The teacher and coach want to explore which representations make most sense to students.

I am curious to see how students learning and thinking have progressed from the introductory lesson on rate experiments to the current lesson. How much will students learn without direct instruction?