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

18) or set goals to work toward without a clear vision of the intended learning. When they do try to assess their own achievement without understanding the learning targets they have been working toward, their conclusions can’t help them move forward.

      The following three actions will help teachers communicate learning goals effectively so that students can connect to mathematics.

      1. Eliminating jargon: Eliminate jargon that is intended for the teacher and instead incorporate empowering language that provides focus and motivation.

      2. Making goals concrete: Communicate learning goals with vivid and concrete language.

      3. Using imagery and multiple representations: Promote mathematics concepts as visually connected to numerical values and symbols.

      Table 1.1 provides some examples of these three actions.

Table 1.1: Actions for Communicating Learning Goals

Strategy	Description
Eliminating jargon	Instead of using the language from the standard to create the learning target, use vocabulary and terminology that make sense and are motivating, and then explicitly teach new vocabulary words.
Making goals concrete	Use language that clarifies what the student is doing and how.
Using imagery and multiple representations	Encourage students to represent their mathematics learning goals in different forms, such as with words, a picture, a graph, an equation, or a concrete object, and encourage students to link the different forms.

      Eliminating Jargon

      Learning goals can be difficult for students to grasp when they contain pedagogical jargon and seem to be crafted more for education experts than for students. We don’t mean to discredit the use of academic language; however, when academic language becomes a barrier because it prevents students from connecting with the material, teachers have to re-evaluate how they’re communicating about mathematics. When learning goals are ambiguous, they don’t provide the focus, motivation, or inspiration students need to reach targets. Mathematics teachers must align learning goal language to desired learning outcomes for students using everyday language and connect it to academic language by showing the students the goal written in various ways. Judit Moschkovich (2012) states that instruction needs to move away from a monolithic view of mathematical discourse and consider everyday and academic discourses as interdependent, dialectical, and related rather than assume they are mutually exclusive. Additionally, learning goals should make appropriate connections to academic language when scaffolding is present. Figure 1.3 shows a standard followed by the rewritten student-friendly, jargon-free statement for two grade levels and algebra II.

Figure 1.3: Transforming a learning goal by eliminating jargon.

      Obtaining student feedback is the best way to determine if learning goals make sense to students. Creating a focus group of students (a committee to eliminate jargon) that vet learning goals is a strategy to ensure your learning goals are student friendly. In this process, the teacher asks students in the focus group to circle nouns and verbs that seem ambiguous or don’t seem very connected to everyday language.

      Figure 1.4 is an example of how a student focus group would provide feedback on the first draft of a learning goal.

Figure 1.4: Transforming a learning goal with student feedback.

      The students from the group in figure 1.4 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