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IMPORTANCE OF UNDERSTANDING THE VERTICAL LEARNING PRoGRESSIoN To Deepen STuDENTS' mathematical understanding of conceptual modelsOne way to deepen one’s conceptual understanding is to make connections among important conceptual models within and between mathematics topics by unpacking the vertical learning progressions. Learning progression describes students’ reasoning as it becomes more sophisticated, and as “...hypothesized descriptions of the successively more sophisticated ways student thinking about an important domain of knowledge or practice develops as children learn about and investigate that domain over an appropriate span of time” (Corcoran, Mosher & Rogat, 2009, p. 37). In mathematics education, the notion of learning trajectories has been important in trying to understand the progression of mathematical concepts and how students’ learning progresses and becomes more advanced and sophisticated. To better understand how student come to understand concepts, mathematics education researchers have focused on hypothetical learning trajectories, which include “the learning goal, the learning activities, and the thinking and learning in which the students might engage” (Simon, 1995, p. 133). These goals provide direction for teachers as they plan learning activities and are critical for teachers to predict the potential reasoning, misconceptions, and learning of students. Clements and Samara’s work (2004, 2009) takes on a curriculum developers’ stance as they use learning trajectories to develop a specific set of sequenced instructional activities focused on early geometric reasoning in young children. These researchbased tasks are designed to promote the child’s construction of the skills and concepts of a particular level with a set of sequenced instructional activities hypothesized to be a productive route (Clements & Samara, 2009). Confrey and her colleagues (2009) define a learning trajectory as “a researcher conjectured, empirically supported description of the ordered network of constructs a student encounters through instruction (i.e., activities, tasks, tools, forms of interaction and methods of evaluation), in order to move from informal ideas, through successive refinements of representation, articulation, and reflection, toward increasingly complex concepts over time (Confrey et al., 2009, p. 347).” Understanding the learning trajectory requires an understanding of the components of the learning progression that are important for teachers as they plan instruction, anticipate students’ responses, differentiate for diverse learners, and assess students’ learning. The work with learning trajectories also supports vertical teaming by teachers, for it “allows an exciting chance for teachers to discuss and plan their instruction based on how student learning progresses. An added strength of a learning trajectories approach is that it emphasizes why each teacher, at each grade level along the way, has a critical role to play in each student’s mathematical development” (Confrey, 2012, p. 3). For example, there may be many ways to implement and interpret the CCSS or even the districtcreated teaching and assessment standards. Teachers may understand them, initially, at a surface level, as the big ideas or list of objectives on which to base their lessons. However, deeper understanding of how these standards are implemented and interpreted in different grade levels is quite a complex task. According to the Confrey (2012, pp. 78), there are five elements that can help unpack a learning trajectory. Teachers need to understand:
A great resource for teachers is the website TurnonCC https://turnonccmath.net/. The website describes and offers learning trajectories of how concepts, and student understanding, develop over time by connecting the Common Core Standards across the vertical grades and providing examples and activities for educators. In our work Table 1.2 Mapping the learning progression for the handshake problem Learning progression for the handshake problem Benchmark Problem: At a neighborhood party, there were a total of 25 guests. All 25 guests shook hands to introduce themselves. How many distinct handshakes happened at this party? Show the different ways you can represent your thinking.
Strategies, representations, and misconceptions To introduce the Handshake Problem across the vertical grades, teachers in grades 34 might begin with the related problem of the Valentine Exchange. This is a realworld scenario that most students can relate to regardless of the holiday. It is a problem about each student (30) giving something/item to each person (29) in the class. The problem begs for a multiplication problem structure of 30 x 29 because you do not have to give yourself a card or something. The simpler problem is of course to start with a smaller number of guests at the party to see if there is a pattern. As shown in the various examples above from Figures 1.4 to 1.12, there are a variety of strategies and approaches to this rich problem. The common misconception for the handshake problem is sometimes counting oneself or double counting which is allowed in the Valentine Exchange because there is a give and take of the cards, whereas in the handshake, the give and take of the shake is counted as one. with lesson study with vertical teams, we used the tools on TurnonCC to help our teachers look across the learning trajectory to describe a “typical student conceptual growth, from prior knowledge and informal ideas, through intermediate understandings, to increasingly complex understandings, over time.” In addition to this great resource, we have used the progressions documents for the Common Core Math Standards from University of Arizona. The progression documents explain why “standards are sequenced the way they are, point out cognitive difficulties and pedagogical solutions, and give more detail on particularly knotty areas of the mathematics” (http://math.arizona.edu/~ime/progressions). The deep understanding of the mathematical learning progressions involves important aspects of mathematical knowledge for teaching. Mathematics knowledge for teaching (Hill, Sleep, Lewis & Ball, 2007) includes understanding of general content but also having domainspecific knowledge of students. More specifically, mathematical knowledge for teaching includes practicebased knowledge of “being able to pose meaningful problems, represent ideas carefully with multiple representations, interpret and make mathematical and pedagogical judgments about students’ questions, solutions, problems, and insights (both predictable and unusual)” (Ball, 2003, p. 6). Understanding students’ learning trajectories in different content domains can help teachers develop the specific knowledge of students and the pedagogical content knowledge necessary for highleverage teaching. Each chapter will zoom in on a learning progression and share a lesson study vignette to illustrate how teachers can design instruction of important mathematical concepts with strategies and tasks for modeling the mathematical ideas (see Table 1.2). Think about it! Pick an important concept you teach at your grade level. Unpack the learning trajectory in terms of development of the ideas underlying a concept; strategies, representations, and misconceptions; and the underlying concepts are in between to bridge the gaps between the standards. 
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