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MAKING SENSE OF MATHEMATICS FOR TEACHING THE SMALL GROUP

When the students struggle to determine how to name thousandths, the teacher’s questions turn to helping students find patterns in how decimals are named. She asks the students what she would have to do to find hundredths with a whole divided into tenths. Relying on her mathematics knowledge for teaching, she strategically decides to use tools to move from a number line diagram to a decimal square. She does this because she observes the students struggling to apply the pattern to partition length. She recognizes that switching to a different representation—an area model—supports students in determining the pattern. To push the students’ thinking, she asks them to extend their understanding of the relationship between tenths and hundredths to determine the next decimal place. The students make sense of the task by responding to the teacher’s questions and engaging in Mathematical Practice 8, “Look for and express regularity in repeated reasoning.” The teacher uses questions to collect evidence that the students have made sense of thousandths using decimals. Not only can questions serve as scaffolding of student learning, but targeted questions are also a helpful technique for collecting useful evidence of student learning. The following section will discuss the final step in the TQE process: gathering and making use of evidence to enhance student learning.

Evidence By providing rigorous tasks and using questions that engage learners in the Mathematical Practices, teachers are able to collect evidence of students’ understanding of the learning goal during small-group instruction. To gather evidence of student learning, the teacher will best use small-group time if he or she knows the common errors to look for during instruction. In this way, the teacher can make sure to ask questions that will elicit those common errors if they exist with students. The teacher can look for both evidence of common errors and evidence that students have met the learning goal during small-group instruction and subsequently use this evidence within a formative assessment process. When used appropriately in a formative assessment process, the evidence guides both task implementation and questioning and serves as part of a cyclical process in planning future learning opportunities. Preparing opportunities to collect this evidence should occur prior to pulling the small group. However, as important as evidence is in guiding instruction, according to Dylan Wiliam (2018), teachers “rarely plan in detail how they are going to find out where the students are in their learning” (p. 83). For this reason, the TQE process requires that teachers make these needed preparations; it incorporates planning to collect evidence of student understanding and preparing tasks and questions. The following list provides examples of sources of evidence that can inform the TQE process.