had previously learned the break-apart strategy and were able to relate to it. Because the break-apart strategy is indeed a commutative strategy, the teacher incorporated student feedback and instead used the terminology students were familiar with in the updated learning goal. It was not as important for students to know the word commutative as it was for them to be able to connect a problem (multiplying two numbers) to how they would solve it (the break-apart strategy). A great time for seeking feedback for eliminating jargon is before collaborative planning sessions. Teachers can provide a list of upcoming learning goals to students and develop an interactive game where students suggest alternate nouns or verbs. The teachers then bring the feedback to collaborative planning time for discussion and implementation.
Making Goals Concrete
According to researchers Sean M. McCrea, Nira Liberman, Yaacov Trope, and Steven J. Sherman (2008), people who think about the future in concrete rather than abstract terms are less likely to procrastinate. This is because a vivid picture of the future makes it seem more real and thus easier to prioritize. Learning goals are pictures of the future; they must appear in concrete language so students feel motivated to meet them. Figure 1.5 shows a learning goal stated in student-friendly language revised to be more concrete.
Figure 1.5: Transforming a learning goal by using concrete language.
In the examples in figure 1.5, the original learning goals don’t specify what kind of problem the student is solving, and they don’t identify a particular strategy to determine the unknown. The intention of “I can” learning targets is to increase clarity by homing in on intended learning.
Additionally, in mathematics instruction, teachers should explicitly communicate technology tools within the learning goals that can enhance the learning. Will students have the option of using a collaborative digital tool to reason through a problem or will they be solving the problem on a sheet of paper? An example of a learning goal with the use of technology is, “I can solve word problems using fractions and show my thinking by creating a video representation.”
Using Imagery and Multiple Representations
Using imagery and representations in mathematics means presenting information in the form of a diagram or chart, for example, or representing information as a mental picture with a concrete image. Visual representation strategies are important for students as they help to support student learning in mathematics for different types of problems. Researchers note that the ways we posture, gaze, gesture, point, and use tools when expressing mathematical ideas are evidence that we hold mathematical ideas in the motor and perceptual areas of the brain—which is now supported by brain evidence (Nemirovsky, Rasmussen, Sweeney, & Wawro, 2012). The researchers point out that when we explain ideas, even when we don’t have the words we need, we tend to draw shapes in the air (Nemirovsky et al., 2012). According to Boaler (2016), we use visual pathways when we work on mathematics, and we all need to develop the visual areas of our brains. One problem with mathematics in schools is that teachers present it as a subject of numbers and symbols, ignoring the potential of visual mathematics for transforming students’ mathematical experiences and developing important neural pathways. The National Council of Teachers of Mathematics (NCTM) has long advocated the use of multiple representations in students’ learning of mathematics (see Kirwan & Tobias, 2014; Tripathi, 2014). But in many classrooms, teachers still employ the traditional approach of mathematics instruction focused on numbers and symbols. To ensure students develop understanding of mathematics through multiple representations, teachers must ensure that learning goals address 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
