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CHAPTER 1

Providing and Communicating Clear Learning Goals

The New Art and Science of Teaching framework begins by addressing how teachers will communicate with students about what they need to learn. It addresses the teacher question, How will I communicate clear learning goals that help students understand the progression of knowledge in mathematics they are expected to master and where they are along that progression?

This design area includes three elements related to tracking students’ progress and celebrating their success. Together, these three elements—(1) providing scales and rubrics, (2) tracking student progress, and (3) celebrating success—create a foundation for effective feedback. In this chapter, we describe specific strategies for implementing these elements in a mathematics classroom.

Scales and rubrics are essential for tracking student progress, and tracking progress is necessary for celebrating success. The desired joint effect of the strategies associated with these three elements is that students understand the progression of knowledge they are expected to master and where they currently are along that progression. When learning goals are designed well and communicated well, students not only have clear direction, but they can take the reins of their own learning. As Robert J. Marzano (2017) articulates in The New Art and Science of Teaching, students must grasp the scaffolding of knowledge and skills they are expected to master and understand where they are in the learning, and this happens as a result of the teacher providing and communicating clear learning goals.

Element 1: Providing Scales and Rubrics

Scales and rubrics articulate what students should know and be able to do as a result of instruction. The content in a scale or rubric should come from a school or district’s standards. As an example of how teachers might do this, we include the learning progression for mathematics from Achieve the Core (n.d.) in figure 1.1 (page 12) and in figure 1.2 (page 13) for secondary-level mathematics.

For element 1 of the model, we address the following two specific strategies in this chapter.

1. Clearly articulating learning goals

2. Creating scales or rubrics for learning goals

Source: Achieve the Core, (n.d.).

Figure 1.1: Learning progression for mathematics, grades K–8.

Source: National Governors Association Center for Best Practices & Council of Chief State School Officers, 2013.

Figure 1.2: Learning progression for mathematics, secondary level.

Clearly Articulating Learning Goals

Mathematics learning goals are most effective when teachers communicate them in a way students can clearly understand; however, students must also feel as though they “own” the goals. Student ownership is the process of allowing students the freedom to choose their goals and take responsibility for measuring their progress toward meeting them. Student ownership occurs most effectively when students are able to connect to mathematics using natural, everyday language. Stephen Chappuis and Richard J. Stiggins (2002) explain that sharing learning goals in student-friendly language at the outset of a lesson is the critical first step in helping students know where they are going. They also point out that students cannot assess their own learning (see element 2, tracking student progress, page 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 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 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 mathematics problems students can answer correctly in three minutes, the winner of math drills, or how well students perform on a standardized test. While there may be benefits to these types of celebrations, they are not as conducive to reliable measurement as progress on a proficiency scale, which allows the teacher to celebrate knowledge gain—the difference between a student’s initial and final scores for a learning goal. To celebrate knowledge gain, the teacher recognizes the growth each student has made over the course of a unit. Mathematics teachers can also use the strategies of status celebration (celebrating students’ status at any point in time) and verbal feedback (emphasizing achievement and growth by verbally explaining what a student has done well) throughout the unit.

Figure 1.14 presents the self-rating scale for element 3, celebrating success.

Figure 1.14: Self-rating scale for element 3—Celebrating success.

GUIDING QUESTIONS FOR CURRICULUM DESIGN

When teachers engage in curriculum design, they consider this overarching question for communicating clear goals and objectives: How will I communicate clear learning goals that help students understand the progression of knowledge they are expected to master and where they are along that progression? Consider the following questions aligned to the elements in this chapter to guide your planning.

• Element 1: How will I design scales and rubrics?

• Element 2: How will I track student progress?

• Element 3: How will I celebrate success?

Summary

Providing and communicating clear learning goals involves three elements: (1) providing scales and rubrics, (2) tracking student progress, and (3) celebrating success. In the mathematics classroom, how teachers state these learning goals can make the difference between students reaching proficiency or not. They can support students in thinking in complex ways about mathematics, but only if they are communicated in a way that students understand and that inspires them to solve problems. Tracking student progress and celebrating success is not only important in the classroom but crucial in mathematics, as students are continually pursuing perseverance in problem solving and need support and affirmation to help them along the way.