week 3_Taking Action Chapter 2_Establishing Mathematics Goals to Focus Learning.pdf

CHAPTER 2 Establish Mathematics Goals to Focus Learning The Analyzing Teaching and Learning activities in this chapter engage you in exploring the effective teaching practice, Establish mathematics goals to focus learning. According to Principles to Actions: Ensuring Mathematical Success for All...

CHAPTER 2 Establish Mathematics Goals to Focus Learning The Analyzing Teaching and Learning activities in this chapter engage you in exploring the effective teaching practice, Establish mathematics goals to focus learning. According to Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014, p. 12): Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions. Goals should set the course for a lesson and provide support and direction for teachers' instructional decisions. For example, the selection of instructional tasks should follow from the stated goals, hence providing a road map for the lesson. Goals can help guide teachers' decision making during a lesson, such as determining which questions to ask or identifYing which student-generated strategies and ideas to pursue. Additionally, goals are part of a progression oflearning. Goals are a key part in determining what tasks are relevant to the planned learning progression, what representations might be highlighted during a lesson or sequence oflessons, and what will be the focus of mathematical discourse in a lesson. In this chapter, you will- explore and compare different goal statements created for a lesson on exponential functions; consider the ways in which lesson goals can support teaching and learning by connecting goals to specific teaching moves in both narrative and video cases; review key research findings related to the importance of establishing mathematics goals to focus learning; and  analyze the relationships among your classroom goals, your teaching practices, and possible student learning outcomes. For each Analyzing Teaching and Learning (ATL) activity, make note of your responses to the questions and any other ideas that seem important to you regarding the focal teaching practice in this chapter, establish mathematics goals to focus learning. If possible, share and discuss your responses and ideas with colleagues. Once you have written down or shared your ideas, then read the analysis where we offer ideas relating the Analyzing Teaching and Learning activity to the focal teaching practice. Exploring Lesson Goals We begin the chapter by asking you to engage in Analyzing Teaching and Learning 2.1. Here you will compare two different goal statements that might be written for a lesson on exponential functions. Analyzing Teaching and Learning 2.1 Comparing Goal Statements 1. Review goal statements A and Band consider these questions: How are they the same and how are they different? How might the differences matter? Goal A: Students will identify a function ofthe form y = b"' as an exponential function where x is the exponent and b is the base. Students will be able to substitute values for x and b to evaluate exponential functions. Goal B: Students will understand that exponential functions grow by equal factors over equal intervals and that, in the general equation y = b", the exponent (x) tells you how many times to use the base (b) as a factor. 2. If needed, read (or reread) the Case of Vanessa Culver in chapter 1. In what ways does goal B (lines 3-5 in the case) align with Ms. Culver's teaching practice? 16 Taking Action Grades 9-12 Analysis of ATL 2.1: Comparing Goal Statements ATL 2.1 asks you to consider how goals A and Bare similar and different. While both goals address the same mathematical content, goals A and B expect much different types of mathematical work and thinking from students. To meet goal A, students need to identifY y = bx as an exponential function, substitute values into the exponential function, and evaluate the function. Notice that the underlined verbs imply memorization (e.g., identifying a function of a given form) and executing procedures (e.g., substituting and evaluating). Tasks aligned with goal A might provide students with values for x and b (perhaps embedded in a word problem) and ask students to create and evaluate exponential functions. Prior to completing such tasks, students are often provided with the definition and form of an exponential function. While germane to students' mathematical learning, these skills do not invoke conceptual understanding, thinking, and reasoning around exponential growth and the behavior of exponential functions. In goal B, students are expected to understand exponential growth and what it means for x to be the exponent and b to be the base in an exponential function y = bx. Such an understanding is essential for recognizing when real-world or mathematical relationships can be modeled with exponential functions. Tasks aligned with goal B, such as the Pay It Forward task in Ms. Culver's lesson, provide scenarios where students can model and understand exponential growth in a variety of ways on the basis of their prior knowledge of the growth of linear functions and ways of representing linear relationships. The differences between goals A and B matter because they require very different mathematical activity from students, which in turn generates differences in the nature of students' mathematical learning. Exploring How Lesson Goals Support Teaching and Learning in the Case of Vanessa Culver Ms. Culver identified goal B as her intention for students' learning in the lesson featured in the case. She used the mathematical goal to focus learning in several ways. Ms. Culver selected a task that would help students meet her goals for students' learning (lines 10-18).1he Pay It Forward task provided a context that supported students in making sense of exponential functions, could be modeled in several ways (that would make exponential growth apparent), and promoted thinking and reasoning. Ms. Culver made tools available to help students explore exponential growth, such as graph paper and graphing calculators (lines 21-22). She asked questions to help students attend to how the pattern was growing, such as, "How do the number of good deeds increase at each stage? How do you know?" (lines 24-25). Ms. Culver sequenced the presentations to build up students' understanding of exponential growth (lines 32-39), particularly in selecting different representations of exponential growth and progressing Establish Mathematics Goals to Focus Learning 17 from diagrams (group 4) to tables (group 3) to equations (groups 1 and 2) to graphs (group 5). In the whole-group discussion, Ms. Culver provided opportunities for a number of students to explain exponential growth using the context of the problem and representations shared by different groups (lines 44-89). Students used their developing understanding of exponential growth to determine which function (y = 3x or y = 3x) correctly modeled the Pay It Forward situation. Hence, Ms. Culver's mathematical goals for the lesson provided direction for determining what task to use, what questions to ask throughout the lesson, and how to structure the whole- group discussion in order to focus students' learning on understanding exponential growth. Having goals (and a task) that focused her instructional decisions on promoting students' understanding of mathematics, rather than rote procedures or facts without understanding, was an essential first step. Considering How Lesson Goals Support Teaching and Learning With ATL 2.2, we will go into the classroom of Shalunda Shackelford, where students are examining graphs that model the speed of a bike and truck over a given time period (see the next page). 18 Taking Action Grades 9-12  The Bicycle and Truck Task A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle's distance and K(t) represent the truck's distance. ~ ~ -,.5 ctl e / 0 / -- ·---- t: ~ E.g Cl) (,) c ~ c Time (in seconds) 1. Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. 2. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. 3. Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words. 4. Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack. Taken from Institute for Learning (2015a). Lesson guides and student workbooks are available at ifl.pitt.edu. Establish Mathematics Goals to Focus Learning 19  Note that some inconsistencies exist in how the graphs model the real-life movement of a bike and truck (e.g., a vehicle would not come to an immediate stop at 9 seconds). Consistencies and inconsistencies in how the graphs model the real-life movement of a bike and truck can foster productive mathematical discussion among students. A teacher might ask students to identifY ways in which the graphs are not realistic and discuss why, or he or she might ask them to consider how they would change the graph to better model the real-life movement of a bike and a truck. A graph that more realistically depicts the movement of a bike and truck is available at http://www.nctm.org/PtAToolkit. Ms. Shackelford has three content goals for her students. She wants them to understand the following: 1. The language of change and rate of change (increasing, decreasing, constant, relative maximum or minimum) can be used to describe how two quantities vary together over a range of possible values. 2. Context is important for interpreting key features of a graph portraying the relationship between time and distance. 3. The average rate of change is the ratio of the change in the dependent variable to the change in the independent variable for a specified interval in the domain. According to Principles to Actions (NCTM 2014), "Teachers need to be clear about how the learning goals relate to and build toward rigorous standards" (p. 12). It is important to note that the Bike and Truck task fits within a sequence oflessons on creating and interpreting functions. Ms. Shackelford incorporated the lessons into her curriculum to engage students with mathematical ideas aligned with the mathematics standards adopted by her state. Specifically, the Bike and Truck task provides students with opportunities to explore rigorous standards related to functions and modeling, as identified in figure 2.1. 20 Taking Action Grades 9-12  Examples of Rigorous State Connection to the and National Standards Bike and Truck Task For a function that models a relationship Questions 1-3 ask students to interpret between two quantities, interpret key key features of the graph portraying the features of graphs and tables in terms of relationship between time and distance the quantities, and sketch graphs showing traveled for a bike and a truck. key features given a verbal description of the relationship. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* Relate the domain of a function to its Question 3 asks students to use the graph and, where applicable, to the domain of the function to determine quantitative relationship it describes. which vehicle was the first to reach For example, if the function h(n) gives 300 feet. the number of person-hours it takes to Question 4 also provides opportunities assemble n engines in a factory, then the to relate the domain of the function to positive integers would be an appropriate its graph as students consider changes in domain for the function.* time when determining average rate of change. Calculate and interpret the average Question 4 asks students to consider the rate of change of a function (presented average rate of change for the bike and symbolically or as a table) over a specified truck on the basis of the graph. interval. Estimate the rate of change from Question 1 may prompt intuitive a graph.* discussions of rate of change as students identify which graph represents the bike (steady rate) and which graph represents the truck. Question 3 may also encourage intuitive discussions of average rate of change as students determine whether (and why) the bike or the truck arrived at 300 feet first. *Indicates that the standard provides opportunities for mathematical modeling Fig. 2.1. Aligning the Bike and Truck task to rigorous standards Establish Mathematics Goals to Focus Learning 21  Prior to the Bike and Truck lesson, Ms. Shackelford has been working on facilitating mathematical discussions and targeting mathematical practices in very deliberate ways. Hence, in addition to content goals, Ms. Shackelford also has process goals for her students to engage in mathematical discourse, problem solving, mathematical argumentation, and mathematical modeling. The video clip from Ms. Shackelford's classroom has three segments of discussion (see More4U at nctm.org/more4u for the video clip). First, Ms. Shackelford introduces a common misconception, framing it as a question from her "imaginary friend Chris." She draws students' attention to the "flat" horizontal portion on the graph of the truck K(t) between 9 and 12 seconds. Ms. Shackelford explains that "Chris" thinks that the truck was traveling on a "straight path" during that interval, and she asks for students to come to the front of the room and explain why they agree or disagree with Chris. Second, a student presents an explanation for question 3 (page 19). Third, while the class verbally agreed with the student's explanation for question 3, Ms. Shackelford checks for additional questions and misunderstandings. Analyzing Teaching and Learning 2.2 The Case of Shalunda Shackelford Watch the video clip of the discussion of the Bike and Truck task in Ms. Shackelford's classroom. Consider the extent to which the content and process goals she has established for the lesson are evident in the discussion and what she did to keep students focused on the main points of the lesson. Identify specific instances in which Ms. Shackelford makes an instructional decision that is directly related to her goals and what students say or do as a result of that move. You can access and download the video and its transcript by visiting NCTM's More4U website (nctm.org/more4u). The access code can be found on the title page of this book. Analysis of ATL 2.2 Ms. Shackelford used the mathematical content and process goal to focus learning. She selected a task and asked additional questions that would address content goals 1, 2, and 3 and maintain students' perseverance in solving and making sense of problems (process goal). (Note that evidence of content goal 3 is not present in this video clip. When you view clip 2 in chapter 71 watch for Ms. Shackelford's students to discuss average rate of change.) Ms. Shackelford introduced a misconception (e.g., interpreting the graph as the path of the truck; lines 1-6) so that students must consider how time and distance vary together (content goal1). Ms. Shackelford also pressed students for their misconceptions regarding how the graphs model ----- ---------------------- 22 Taking Action Grades 9-12 which vehicle "got there first (lines 68-90)."These instructional moves provided opportunities for students to use the context to interpret key features of the graph (content goal2) and to consider how the path of the bike and truck are modeled with mathematics. Modeling with mathematics (process goal) also occurred as students used mathematical representations (graph) and concepts/ideas (rate of change, domain, range) to make sense of the path of the bike and truck, the associated changes in time and distance, and how this related to speed. (Note that some features of the graph do not model the real-life movement of a bike and truck, and asking students to identify and explain these inconsistencies could serve these goals as well.) Ms. Shackelford supported students as they persisted in their problem-solving and sense- making efforts (process goal) by pressing them to clarifY their own thinking and understanding and to ask questions if they disagreed or did not understand. For example, even after students expressed verbal agreement with one student's use of the graph to explain question 3, Ms. Shackelford asked students to share what still wasn't making sense to them about the situation (lines 66-68). Next, Ms. Shackelford created an opportunity for students to present and defend opposing opinions (process goal) when Jacobi and Charles came to the front of the room (lines 7-52) and when students were asked to explain what ideas they agreed or disagreed with at the end of the clip (lines 73-90). She positioned students to construct viable arguments, explain and defend their ideas, and critique the reasoning of their classmates. Finally, Ms. Shackelford provided opportunities for students to engage in mathematical discourse (process goal), including defending their position; she asked questions and pressured students for explanations and meaning (e.g., "You agree, why?"; lines 23, 88). In the video clip, we see Ms. Shackelford make several purposeful moves aligned with her goals for the lesson. According to Principles to Actions (NCTM 2014), "The establishment of clear goals not only guides teachers' decision making during a lesson but also focuses students' attention on monitoring their own progress toward the intended learning outcomes" (emphasis added; p. 12). Ms. Shackelford expects students to monitor their own learning, and she communicates this by pressing students to express whether they agree or disagree with "Chris" (e.g., content goal1, understanding how two quantities vary together) and what they do not understand following the explanation of question 3 (e.g., content goal2, using the context to interpret key features of the time/distance graph). In these ways, the goals also support students' monitoring of their own learning and understanding. Establish Mathematics Goals to Focus Learning: What Research Has to Say The cases in this chapter provide examples of teachers using goals to inform instructional decisions and focus students' learning. Teachers' goals addressed important aspects of students' understanding of mathematics and aligned with state and national standards. It is important Establish Mathematics Goals to Focus Learning 23 to note that teachers' goals (and the tasks selected to accomplish those goals) did not exist in isolation (e.g., a fun or interesting task; an activity for a Friday or the day before winter break). Rather, the goals supported teachers' decision making because they were embedded within sequences oflearning progressions (Daro, Mosher, and Corcoran 2011) and intended to develop students' understanding of important mathematical ideas (Charles 2005). According to Principles to Actions (NCTM 2014), goals connected to learning progressions and big mathematical ideas help teachers consider how to support students as they make transitions from prior knowledge to more sophisticated mathematical understandings (Clements and Sarama 2004; Sztajn et al. 2012). Stein (2017) indicates that goals impact teaching and learning (1) by guiding teachers' instructional decisions and (2) by impacting the nature and focus of students' work. First, as illustrated by the cases in this chapter, goals for students' mathematical learning should support teachers' decisions in selecting tasks, asking questions, and framing the direction of whole-group discussions. In student-centered lessons, students often suggest or develop a wide array of mathematical ideas and strategies. Mathematical goals can help teachers determine which ideas and strategies to pursue and serve as "reference points" for guiding mathematical discussions (Ball1993; Stein 2017). In fact, goals are identified as an important first step for teachers in considering how to select and sequence students' mathematical work and ideas when orchestrating mathematics discussions (Stein et al. 2008). Hence, teachers with a sound understanding of instructional goals and the multiple pathways that students can (and cannot) take to reach them are better equipped to support students' learning of mathematics (Leinhardt and Steele 2005). Second, research indicates that teachers' use of goals to guide instruction supports students' ability to monitor their own mathematical learning (Clarke, Timperley, and Hattie 2004; NCTM 2014; Zimmerman 2001). When teachers explicitly refer to goals during a lesson, students are better able to self-assess and focus (or refocus) their learning, which is an important factor in student achievement (e.g., Ames and Archer 1988; Engle and Conant 2002; Fuchs et al. 2003; Henningsen and Stein 1997). Promoting Equity by Establishing Mathematics Goals to Focus Learning Teachers' use of goals to focus instructional decisions supports students' learning of mathematics in general, and specific types of goals can enhance opportunities to learn mathematics for traditionally marginalized students. Principles to Actions identifies "high expectations" as one of several required supports for promoting access and equity in learning meaningful mathematics. Research indicates significant gains in students' learning and reductions in achievement gaps when teachers communicate clear expectations, express 24 Taking Action Grades 9-12 challenging but attainable goals, and create an environment in which students feel supported to attain high goals (Boaler and Staples, 2008; Marzano 2003; McTighe and Wiggins 2013). "High expectations" do not imply difficult or complex mathematical procedures and concepts beyond students' reach. Rather, goals and expectations should establish learning progressions that build up students' mathematical understanding, increase students' confidence in their own ability to do mathematics, and, in doing so, support students' identities as mathematical learners. Too often, instructional tracking and deficit beliefs regarding the mathematical abilities of students of color, students who are poor, or students for whom English is not their first language lead to different opportunities to engage with interesting and rigorous mathematical content (Jackson et al. 2013; Phelps et al. 2012; Walker 2003). When students' mathematical abilities are underestimated, students [receive] fewer opportunities to learn challenging mathematics. Low-track students encounter a vicious cycle oflow expectations: Because litde is expected of them, they exert little effort, their halfhearted efforts reinforce low expectations, and the result is low achievement (Gamoran 2011). (NCTM 2014, p. 61) Furthermore, if mathematical goals and expectations focus primarily on rote skills and procedures, without attention to meaningful mathematics learning, low-track and marginalized students will not develop a deep understanding of mathematics (Ellis 2008; Ellis and Berry 2005). Instead, instructional goals (and the tasks aligned with those goals) should promote students' reasoning and problem solving (e.g., goal Bin ATL 2.1; Ms. Shackelford's content and process goals). Such goals communicate the belief and expectation that all students are capable of participating and achieving in mathematics; in other words, such goals communicate a growth mindset (Boaler 2015; Dweck 2006). Hence, goals can support equitable instruction by setting clear and high expectations, promoting students' mathematical reasoning and problem solving, and communicating the growth mindset that all students are capable of engaging in meaningful mathematical activity. Relating Establish Mathematics Goals to Focus II Learning// to Other Effective Teaching Practices Establish mathematics goals to focus learning is closely connected to several other effective teaching practices. Since this is the first chapter in the book that is focused on an effective teaching practice, in this section we connect establish mathematics goals to focus learning to other effective teaching practices. Specifically, we discuss the synergy between goals and tasks, questions, and facilitating discourse. In subsequent chapters, the connections between the focal effective teaching practice and other practices are woven throughout the chapter. Establish Mathematics Goals to Focus Learning 25 Implement Tasks that Promote Reasoning and Problem Solving If goals represent the destination for students' mathematical learning from a given lesson, then tasks are the vehicles that move students from their current understanding toward those goals. Tasks provide opportunities for students to learn and understand the mathematical content and processes necessary to achieve learning goals. Specifically, goals for students' reasoning and problem solving require tasks that promote reasoning and problem solving. (Such tasks are discussed in chapter 3.) If the tasks students encounter in mathematics class only provide procedural practice, students are not going to attain goals for thinking, reasoning, and understanding mathematics (Stein et al. 2009). Finally, if tasks and goals align and focus on promoting students' reasoning and problem solving, using goals to inform instructional decisions could also support implementing tasks in ways that provide and maintain students' opportunities for reasoning and problem solving throughout a mathematics lesson. Pose Purposeful Questions With clear goals in mind, teachers can ask questions that prompt students to engage with the mathematical ideas aligned with those goals. (Posing purposeful questions is discussed in chapter 5.) Knowing the goals for a lesson can help teachers craft questions before and during a lesson by considering (or responding to) students' specific ideas, strategies, or misconceptions. In this way, goals can support lesson planning just as they support instructional decisions during a lesson. Teachers' questions can help focus students' work and thinking on important aspects of the task or mathematics, thus supporting students' attainment of the lesson goals. Students' responses allow teachers to assess students' progress toward the intended goals and to determine next instructional steps. Teachers' questions can also support students' self- assessment of their own progress. Facilitate Meaningful Mathematical Discourse Goals can inform instructional planning and decisions around facilitating mathematical discourse. A clear focus on goals provides a clear frame for the mathematical ideas to be elicited during the whole-group discussion and can help teachers determine what strategies, ideas, representations, and so forth to select for presentation and discussion. (Facilitating meaningful mathematical discourse is discussed in chapter 7.) Having goals in mind also supports teachers' assessment of students' learning by enabling them to know what to look and listen for as evidence of students' progress toward the goals. 26 Taking Action Grades 9-12 Key Messages Establish clear goals to focus students' learning, and explicitly communicate these goals to students. Establish goals that promote mathematical understanding, reasoning, and problem solving. Create goals within learning progressions that build students' understanding of important mathematical ideas. Use goals to guide instructional decisions and focus students' learning. Support students' use of goals to monitor and assess their own progress. Taking Action in Your Classroom: Establishing Mathematics Goals To Focus Learning The Taking Action in Your Classroom activity provides an opportunity to apply some of these key findings in your classroom. Taking Action in Your Classroom 2.1 Consider a lesson that you have recently taught, in which the learning goal was not explicit. Rewrite the learning goal so that the mathematical idea you wanted students to learn is explicit. How might your more explicit goal statement guide your decision making before and during the lesson? Consider a lesson you will teach in the near future and the current learning goals for this lesson. What type of mathematical work and thinking does the goal expect of students: memorization and procedures or thinking, reasoning, and sense making? If needed, rewrite the goal to require thinking, reasoning, and sense making in students' mathematical activity. Consider how to use your goal statement to guide your instructional decisions before and during the lesson. Consider whether the new goal aligns with the task you planned to use or whether a new task is needed. Establish Mathematics Goals to Focus Learning 27