Assessment and Accountability: Strategies for Inquiry-Style Discussions
Beto, Rachel A., Teaching Children Mathematics
Teachers' actions are what encourage students to think, question, solve problems, and discuss their ideas, strategies, and solutions. The teacher is responsible for creating an intellectual environment where serious mathematical thinking is the norm.
--Principles and Standards for School Mathematics (NCTM 2000, p. 18)
Teachers at my school recently watched a video of a model inquiry lesson, in which the instructor gathered her fifteen students on a rug to discuss fractions and share drawings and ideas. My neighbor whispered, "Sure, it's easy when you have just fifteen kids. Forget it! I've got thirty!" How does a teacher with a large class facilitate a mathematical discussion that produces important ideas, engages all students, and includes assessment? It may sound impossible, but many strategies can make inquiry-style discussions accessible to all teachers.
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As mathematics instruction shifts from direct instruction in which students memorize algorithms to inquiry-style instruction in which students discover properties of numbers, teachers must modify the classroom environment and assessments. Susan Jo Russell defines computational fluency as "efficiency, accuracy, and flexibility" (p. 154). A strong understanding of the base-ten number system helps children choose strategies that demonstrate computational fluency, which requires both basic skills (to assist problem solving) and conceptual awareness (to justify answers). When students work on problems alone, share strategies, then practice the new strategies, they build flexibility from seeing one problem solved in multiple ways; accuracy arises from using these strategies to verify answers and justify solutions. In order to create efficiency and flexibility, I alternate mathematical discussions with practice using a variety of problems and hands-on investigations.
One way to create rich discussions, both in small and large groups, is to post a problem on large paper and gather the students in front of the problem with papers, pencils, and something to write on. Without teaching an algorithm, the teacher asks students to solve the problem using their own strategies and prior knowledge. Then students share their work and confirm or debate one another's reasoning. The emphasis is on how each student arrived at the answer. The teacher's main question and comment are the following:
1. How do you know?
2. Please label your pictures with numbers.
Asking this question when students are correct as well as incorrect is important because mistakes are an integral aspect of inquiry-based discussions. Students gain number sense when they make, discover, and analyze mistakes. Figures 1 and 2 illustrate how a student's mistake can initiate discussion.
To make these mathematics discussions powerful opportunities for all students, teachers must foster a student-centered environment, make students accountable for participating, and take notes about student understanding. This article describes specific strategies that teachers can implement to create powerful participation and discusses how to assess students during mathematical discourse.
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Setting the Scene
In inquiry-based instruction, students play the lead role while the teacher makes sure that students are listening to one another and building meaning from one another's work. Creating the discussion-centered classroom begins on the first day of school. Students must learn to pay attention to one another and be accountable for what others share with the class in all subjects.
The following three discussion "rules" provide a good base for the strategies discussed in this section:
1. Keep your eyes on the speaker.
2. Think about what others say.
3. Always be ready to explain another's ideas and/or offer your own.
Teachers should establish clear expectations that all students are aware of what has just been spoken and that students will ask questions when they do not hear or understand the speaker. …