Teaching Math to ELL Students: A "Lesson Study" Structure Helps Students Think Independently and Then Work with Partners to Arrive at a Solution
Anhalt, Cynthia, Farias, Laura, Farias, Salvador, Olivas, Josie, Ulliman, Melanie, Leadership
Arizona middle school mathematics teachers of predominantly Latino students teamed up with a researcher at the Center for the Mathematics Education of Latinos/as in the fall of 2006 to examine effective strategies in teaching mathematics. We followed a traditional "lesson study" structure--which is rooted in professional development practices in Japan--that allowed all of us to design a lesson. Then, each participating teacher taught the lesson as it evolved, with modifications based on the team's observations and debriefings.
Our discussions, and the negotiations that took place on what constitutes effective teaching, were based on and influenced by our individual teaching experiences, as well as the book, "The Teaching Gap," by James Stigler and James Hiebert. Our overall goal for the evolving lesson was for students to initially think independently and then to collaborate with a partner to arrive at a solution.
The lesson involved addition of fractions with unlike denominators, which the middle school students had not worked on yet this year. First, students were given the following word problem:
"Cecilia uses 2/6 pound of cheddar cheese and 1/4 pound of mozzarella cheese to make nachos. How much cheese does she use in all?"
Adding hands-on tools
Students were initially given time to think independently. Most of them readily added the fractions by adding the numerators and adding the denominators, and wrote down a solution of 3/10. After discussions took place among partners, only a few pairs of students changed their solutions to 7/12, recognizing that it was necessary to find a common denominator.
Not until additional hands-on tools were provided to students, namely fraction bars, did they begin to think critically about their approach, since they had to "justify" their solutions using the fraction bars. It was at this point that most of the students were able to make connections between the fractions, the fraction bars, and why they needed to find a common denominator to solve the problem.
By allowing students to make mistakes initially (adding the numerators and the denominators) we were able to observe how students would later correct their own thinking after identifying their errors. The impact of students realizing their own mistakes and understanding why a common denominator is necessary for adding fractions will have a lasting effect on their continued learning of fractions.
Making sense of mathematics
We realize that all students need to make these connections; however, this approach proved to be especially crucial for English Language Learners, as they may not always fully understand everything the teacher is saying during English instruction. Teaching-by-telling often has a negative impact, because some students whose academic English may not be developed appropriately for grade-level expectations will never have an opportunity to learn particular concepts.
In our research lesson we deliberately planned for students to fully engage during a problem-solving and concrete experience with the concept of the need for a common denominator. We did not focus on telling them initially how to add fractions with unlike denominators by a procedure.
While it was difficult to observe exactly when they were making a mistake in the process of adding fractions, it was powerful to observe the "discovery" students later made on their own. Teachers did intervene, but only to ask questions and to continue engaging students in the discourse of the mathematics that was involved in the lesson. We found that, by designing the lesson as we did, the ELL students were able to actively engage in academic discourse regarding fractions in a meaningful way.
Collaboration is key
During the teaching and critiquing of the lesson, lesson study allowed us to work collaboratively in planning, teaching and observing each other. …