The New Art and Science of Teaching Mathematics. Robert J. Marzano

this strategy.

      The example in figure 1.6, derived from Boaler’s (2016) research, shows how to transform a learning goal using visualization through imagery.

Figure 1.6: Transforming a learning goal using imagery.

      In the first example in figure 1.6, using imagery to transform a learning goal allows students to visualize how they will represent a multiplication problem with an array. Many learning goals call for the use of arrays or a visual representation, but this isn’t always meaningful for students unless they see an example right from the beginning of, and throughout, the lesson until they have built understanding.

      When teachers communicate learning goals, it’s important that the communication extends beyond a written statement visible in the classroom or on a device. Carla Jensen, Tamara Whitehouse, and Rachael Coulehan (2000) find that teachers can support students in connecting to mathematical terminology and symbolic notation through verbal communication. The dialogic nature of communicating about mathematics supports students in accessing new mathematical terms and processes.

       Creating Scales or Rubrics for Learning Goals

      An effective tool for creating rubrics and accessing standards-based rubrics is the free online tool ThemeSpark Rubric Maker (www.themespark.net). To measure mathematical thinking, you might want to create a scale for a specific skill like reasoning, problem-solving, or perseverance. Figure 1.7, adapted from Engage NY (2013), shows a rating scale for the skill of reasoning.

      Source: Adapted from Engage NY, 2013.

Figure 1.7: Rating scale for reasoning.

      We recommend that teachers use the scale in figure 1.8 (page 18) to rate their current level of effectiveness with providing scales and rubrics.

Figure 1.8: Self-rating scale for element 1—Providing scales and rubrics.

      Element 2: Tracking Student Progress

      Tracking student progress in the mathematics classroom is similar to tracking student progress in any content area: the student receives a score based on a proficiency scale, and the teacher uses the student’s pattern of scores to “provide each student with a clear sense of where he or she started relative to a topic and where he or she is currently” (Marzano, 2017, p. 14). For each topic at each applicable grade level, teachers should construct a proficiency scale (or learning progression). Such a scale allows teachers to pinpoint where a student falls on a continuum of knowledge, using information from assessments. A generic proficiency scale format appears in figure 1.9.

Figure 1.9: Generic format for a proficiency scale.

      The proficiency scale format in figure 1.9 is designed so that the only descriptors that change from one scale to the next are those at the 2.0, 3.0, and 4.0 levels. Those levels articulate target content, simpler content, and more complex content. Teachers draw target content from standards documents; simpler content and more complex content elaborate on the target content. For example, figure 1.10 shows a proficiency scale for a grade 8 mathematics standard.

Figure 1.10: Proficiency scale for graphing functions at grade 8.

      The elements at the 3.0 level describe what the student does essentially as the learning standard states. The 2.0 level articulates simpler content for each of these elements, and the 4.0 level articulates beyond what the teacher taught.

      Figure 1.11 shows an individual student’s progress on one topic for which there is a proficiency scale. The student began with a score of 1.5 but increased his or her score to 3.5 over five assessments. The strategy of using formative scores throughout a unit of instruction helps teachers and students monitor progress and adjust if necessary. This is different from summative scores, which represent a student’s status at the end of a particular point in time. To collect formative scores over time that pertain to a specific proficiency scale, the mathematics teacher uses the strategy of utilizing different types of assessments, including obtrusive assessments (which interrupt the flow of classroom activity), unobtrusive assessments (which do not interrupt classroom activities), or student-generated assessments.

Figure 1.11: Student growth across five assessments on the same topic.

      For further guidance regarding the construction and use of proficiency scales, see Formative Assessment and Standards-Based Grading (Marzano, 2010a) and Making Classroom Assessment Reliable and Valid (Marzano, 2018). By clearly articulating different levels of performance relative to the target content, both teachers and the students themselves can describe and track students’ progress. They can use a line graph or bar graph of the data to show students’ growth over time.

      Figure 1.12 (page 20) shows a student proficiency scale with a self-reflection component for planning. This can help with the strategy of charting student progress as a student sets a goal relative to a specific scale at the beginning of a unit or grading period and then tracks his or her scores on that scale. At the end of the unit or grading period, the teacher assigns a final, or summative, score to the student for the scale.

Figure 1.12: Student proficiency scale for self-rating and planning.

      Visit go.SolutionTree.com/instruction for a free reproducible version of this figure.

      We recommend that teachers use the scale in figure 1.13 to rate their current level of effectiveness with element 2, tracking student progress.

Figure 1.13: Self-rating scale for element 2—Tracking student progress.

      Element 3: Celebrating Success

      Celebrating success in the mathematics classroom should focus on students’ progress on proficiency scales. That is, teachers should celebrate students for their growth. This may differ from what teachers traditionally celebrate in the classroom. For instance, a teacher might be used to celebrating how many