## Tuesday, August 29, 2017

### Three Teachers, One Lesson on Teaching Trigonometry through Problem Solving in a Lesson Study

by Allan M. Canonigo amcanonigo@up.edu.ph

This article discusses the different ways students solved a given problem involving trigonometry and how the teacher made use of the students’ solutions in introducing and developing conceptual understanding of sine, cosine, and tangent. In this study, the teacher introduced a problem to the class and then allowed the students to solve the problem in groups using their prior knowledge and understanding of some mathematics concepts. There were five teachers who were involved in the lesson study, three of whom implemented the same lesson in their respected classes. Results show that in all three classes, students used graphical representation to understand the problem and to present the solution. The diagrams or graphical representations were essential tools for students’ mathematical thinking. This is consistent with the study of Greeno and Hall (1997), particularly regarding the algebraic, numerical, and graphical representations. In particular, most of them used the unit circle to arrive at their solutions.

In all these classes, the students were not able to provide much reasons to justify or explain their solutions. However, the problem has already provided opportunity for students to make connections, justify their solutions, and make sense of sine, cosine, and tangent. Two of the teachers emphasized the unit circle method in introducing sine, cosine, and tangent. Two other teachers utilized the students’ solutions in introducing the concept of sine, cosine, and tangent. Although these teachers vary in their approaches to utilizing students’ answers and solutions, two of them attempted to ask probing questions to elicit students’ justifications to their solutions. This helped the students to make a clear connection of previous mathematical concepts which were needed to solve the problem.

In planning a lesson, the teachers involved in the lesson study team realized that in order to be effective in teaching, students’ current knowledge and interests must be placed at the center of their instructional decision making. Although they wrote all their intentions in the plan prior to the implementation of the lesson, they learned to adjust their instruction to meet the students’ learning needs. They also realized that instead of trying to fix weaknesses and fill gaps, they can make use of students’ existing proficiencies – by making use of the students’ solution to the problem in order to help them understand the concept of trigonometric functions.

As shown in this study, the students could solve a problem in different ways when they were given the opportunity to do so. The students were able to work in groups effectively and came up with a solution and the reasoning behind that solution. On the other hand, it is very important that the teachers are able to process these solutions to develop conceptual understanding of sine, cosine, and tangent. For the teachers involved in this study, it was a challenging task for them to introduce the lesson and develop students’ conceptual understanding through problem solving by utilizing students’ solutions and answers.

The teachers found the lesson study a rich learning experience. Through planning the lessons collaboratively, they were able to deepen their subject matter knowledge as well as their understanding of how to teach sine, cosine, and tangent. It provided them with the opportunity to actually see and be sensitive to how students processed their thinking, how students’ misconceptions and difficulties could arise, and how it was an eye-opener to observe how the students struggled with the problem, and how teachers used students’ solutions to develop conceptual understanding in different ways. They were able to see that a good lesson is one that meets the learning needs of the students. Such teachers are responsive both to their students and to the discipline of mathematics. It is therefore recommended that, whenever mathematics teachers use “real-world” contexts for teaching mathematics, they maintain a focus on mathematical ideas.