5th Grade Lesson Study – Introductory Lesson on Division with Decimals
December 7, 2016
What are the essential understandings that students need when dividing decimals, and what can we do to ensure all students learn something new towards 5NBT.7?
By providing students with opportunities to start the problem-solving process on their own without a teacher-prescribed strategy, represent their thinking, share their thinking with their peers, and listen to their peers’ way of thinking, students’ current understandings are illuminated in a way that drives decision-making throughout the lesson. In this lesson, 19 out of 23 students were able to provide some representation of their thinking and/or problem-solving strategy on a concept that they’ve had no previous experience with.
Developing expertise in estimating with decimals, paired with students’ reasoning behind their estimates, prior to experiences using computation with decimals strongly supports understanding. Estimation supports reasoning, elicits evidence of students’ place value understanding and relative size of decimals, which were found to be 2 of 3 key ingredients in developing understanding of decimal computations (per van de Walle). Given significant evidence that students rarely refer back to their estimate after determining a more precise answer (in this lesson, and based on the experiences of all teachers in the group), there is a need to support students in increasing their buy-in of estimation as a tool for problem-solving. Also important to initially developing understanding of division with decimals, understanding the relationship between multiplication and division can serve as an entry point for students, and simultaneously supports understanding of both models of division (partitive and quotitive).
A third essential understanding of dividing with decimals lies in the concept of equivalence. It seemed that in working towards developing generalizable methods for dividing with decimals, equivalence will become increasingly important. Understanding that 1.1 could also be called 11 tenths will support the generalized method of re-writing the expression that involves decimals (3.6 divided by 0.4) to work with an equivalent expression that does not (36 divided by 4).
Therefore, there seems to be a potential instructional sequence that involves significant work with estimation of decimal values to determine relative size of decimals (individually and with computation), place value understanding and equivalence, with equivalence having the closest connection to the more common generalizable methods for dividing with decimals. What is more certain is that introducing procedures or algorithms, such as the long-division algorithm, before these understandings are in place can inhibit students’ reasoning, intuitive thinking and repertoire of problem-solving strategies.
Ideas for further study…
Why are students not transferring use of strategies/models to new situations?
How do we know when students are making meaning of models in a way that they will choose to use it in another situation?