Well-prepared beginning teachers of mathematics have foundational understandings of students’ mathematical knowledge, skills, and dispositions. They also know how these understandings can contribute to effective teaching and are committed to expanding and deepening their knowledge of students as learners of mathematics. |
C.3.1. Anticipate and Attend to Students’ Thinking About Mathematics Content C.3.2. Understand and Recognize Students’ Engagement in Mathematical Practices C.3.3. Anticipate and Attend to Students’ Mathematical Dispositions |

Effective teachers understand how students’ mathematical ideas develop and how to apply such understandings to every aspect of teaching. There is much to learn about students’ mathematical thinking, their engagement in mathematical practices, and their mathematical dispositions. In the following section, we elaborate on the knowledge and pedagogical practices needed by well-prepared beginning teachers of mathematics at the upper elementary level*.*

### UE.5. Students’ Sense Making

Well-prepared beginning teachers of mathematics at the upper elementary level nurture students' proficiency with, and sense making of, mathematical ideas, processes, and practices. [Elaboration of C.3.1 and C.3.2]

To be well prepared for teaching mathematics in upper elementary grades, beginning teachers are ready to support students in developing increasingly sophisticated, and at times increasingly abstract, notions of mathematical ideas. Students in this grade-bands build on their additive thinking to develop the more sophisticated multiplicative thinking. Students explore the nature of fractions and decimals as well as operations involving them. They build on insights from their work with whole numbers and at times must try to avoid overgeneralizing lessons learned. They notice and describe more complex properties of shapes than they recognized previously and can express measurements of shapes in multiple ways, including with algebraic formulas. Fostering these advancements requires that beginning teachers know how mathematical ideas can progress, drawing on knowledge from research on learning progressions. They realize that using concrete, semi-concrete, and abstract representations involves overlap and integration and that their students need opportunities to revisit and try out ideas even while abstractions are developing. They help students understand the logic that makes procedures meaningful and the power and elegance mathematical conventions hold for working collectively on mathematics. Developing this understanding requires that beginning teachers actively monitor the evolution of students’ ideas, be aware of likely misconceptions, and be open to understanding unique ways that students might use to express characteristics and generalizations. Beginning teachers can tailor instruction in ways that build on what students understand and consistently encourage students to stretch their mathematical thinking, such as their willingness to make conjectures or describe mathematics ideas and objects in depth.

Well-prepared beginning teachers help students become more fluent with mathematical ideas, knowing that fluency can free mental space students need to grapple with complex mathematical ideas. One commonly held goal of teachers in upper elementary grades is to support the learning of basic facts. Well-prepared beginners must be sensitive to the negative effects of practices commonly used to enhance skill with such facts, such as timed tests. Skills such as fast retrieval of multiplication facts should not be developed at the expense of sense making or developing productive dispositions toward mathematics. These teachers use methods of supporting students’ need to make sense of ideas while also developing greater fluency and proficiency.

Well-prepared beginners in this grade-band nurture personal and public engagement in mathematical practices. They understand that students’ mathematical identities and perceptions of mathematical status are likely to influence their participation, so they consistently work to engage all students in mathematical discussions involving explanation and critique as well as encourage the belief that each student can make valued contributions. They know that students in upper elementary grades can engage meaningfully in mathematical practices, and they build these practices on the foundations that students bring. For instance, students may believe that mathematical facts are established by the teacher, the textbook, smart classmates, or even popular consensus (i.e., by voting). Beginning teachers help students unpack the limitations of these notions while also engaging students in more robust forms of argument and proof.

Well-prepared beginners attend to the mathematical dispositions of their students. In this grade-band, students can feel greater empowerment to investigate mathematical ideas and independently address mathematical problems. Unfortunately, at this time students may, instead, shift to seeing mathematics as a collection of rules and procedures in which they lack facility and interest. Well-prepared beginners have an emerging repertoire of ways to nurture productive dispositions toward mathematics. For example, when seeing a student’s expression of frustration with a particular idea or practice, a teacher might tell the student that mathematicians often struggle to solve problems or compliment the student for making several attempts to solve the problem, drawing attention to the mathematical practices and processes that are valued in mathematics.

Well-prepared beginning teachers of upper-elementary-grades students know that their students can skillfully and reflectively engage in mathematical work. As a result, they establish routines and provide students with tools to use to assess their mathematical thinking and the mathematical products they produce. This practice goes well beyond directives to “check your work.” Well-prepared beginners help students express their questions and fine-tune their resources for help. They work with students to develop a shared sense of such components of high-quality work as graphs or explanations. They help students understand that accuracy and speed are not the only, and often not the best, measures of quality. They build a classroom culture in which students can provide mathematically useful feedback to their peers and the teacher and the textbook are not viewed as the only sources of mathematical validation.