NUMBER SENSE-BASED STRATEGIES 2 ACKNOWLEDGEMENTS I would like to thank and acknowledge all those who have supported me throughout this endeavor. I thank my committee for their expertise and their careful advice. I also thank the professors and staff at Weber State University for a quality education that supported the development of this project and that will support me throughout my career. And, of course, I thank my parents for their constant love and support. NUMBER SENSE-BASED STRATEGIES 3 TABLE OF CONTENTS LIST OF FIGURES ............................................................................................................ 5 ABSTRACT ........................................................................................................................ 6 NATURE OF THE PROBLEM.......................................................................................... 7 Literature Review............................................................................................................ 8 The Importance of Number Sense in Learning Mathematics ..................................... 8 A New Emphasis on Number Sense ......................................................................... 11 The Ineffective Incorporation of Multiple Mathematical Strategies ........................ 18 Summary ....................................................................................................................... 20 PURPOSE ......................................................................................................................... 22 METHOD ......................................................................................................................... 23 Procedure ...................................................................................................................... 23 PROGRAM CREATION.................................................................................................. 25 Platform........................................................................................................................ 25 Course Materials ........................................................................................................... 25 Articles ...................................................................................................................... 25 Web Pages ................................................................................................................. 26 Videos ....................................................................................................................... 26 Assignment ............................................................................................................... 27 Quizzes ...................................................................................................................... 28 NUMBER SENSE-BASED STRATEGIES 4 Surveys ...................................................................................................................... 28 Modules........................................................................................................................ 29 Module 1 ................................................................................................................... 29 Module 2 ................................................................................................................... 31 Module 3 ................................................................................................................... 32 Module 4 ................................................................................................................... 34 Module 5 ................................................................................................................... 35 Module 6 ................................................................................................................... 36 FEEDBACK AND ADJUSTMENTS .............................................................................. 38 Feedback ....................................................................................................................... 38 Adjustments .................................................................................................................. 38 NEXT STEPS ................................................................................................................... 40 SUMMARY ...................................................................................................................... 41 REFERENCES ................................................................................................................. 42 NUMBER SENSE-BASED STRATEGIES 5 LIST OF FIGURES Figure 1. The CMI Framework Learning Cycle ............................................................ 16 Figure 2. Screenshot of pre- and post-course survey ..................................................... 29 Figure 3. Screenshot from Module 1 with attachment of the article “Strategies Are Not Algorithms” by Ian Whitacre and Donna Wessenberg.............................. ......... 30 Figure 4. Screenshot of free online resources from Module 2 ...................................... 31 Figure 5. Sample questions from the 5-question assessment quiz in Module 3 ............ 33 Figure 6. Screenshot of a video from Module 4 explaining the importance of using number sense-based language when modeling standard algorithms ......................... 34 Figure 7. Screenshot of a video from Module 5 explaining the conceptual principles found in the lattice method ........................................................................................ 35 Figure 8. Pair of questions from the error analysis assignment in Module 6 ................ 37 NUMBER SENSE-BASED STRATEGIES 6 ABSTRACT Number sense is an essential component of quality mathematics education, but many teachers do not themselves possess sufficient number sense to effectively facilitate the growth of number sense in their students. This project is a professional development course available on Canvas that aims to enhance the number sense of upper elementary teachers in Weber School District. The course contains six modules that include articles, web pages, videos, quizzes, surveys, and other resources. All materials focus on developing the number sense of upper-elementary teachers in an effort to enhance the number sense instruction available to students in the district. NUMBER SENSE-BASED STRATEGIES 7 NATURE OF THE PROBLEM Number sense is an imperative prerequisite to the fluent use of standard algorithms (Fuson & Beckmann, 2012). Even in the earliest years of schooling, research shows that the presence or lack of number sense significantly influences a student’s future mathematics success (Locuniak & Jordan, 2008). Students who have number sense are more confident when working with numbers, are flexible in their thinking, and often invent their own computational procedures and strategies (Martinie & Coates, 2007). To address the importance of number sense, many current curricular programs place an emphasis on the development of number sense in elementary-age learners. The Common Core State Standards for Mathematics (CCSS-M) also focus on this key aspect of mathematics learning, recognizing that true standard algorithm fluency comes in later years after the establishment of number sense (Fuson & Beckmann, 2012). There are many concrete strategies used to teach mathematical operations (Fuson & Beckmann, 2012). Many of these strategies are more time-consuming than a standard algorithm but can be utilized to develop number sense as students progress toward standard algorithm fluency. While standard algorithms are efficient ways to arrive at correct answers, alternative strategies - including strategies invented by students themselves - can help build number sense in elementary students (Carpenter, Franke, Jacobs, Fennema, & Empson, 1998). Invented strategies lead to better number sense, mathematical understanding, and the ability to extend knowledge (Carpenter et al., 1998). Students show a better understanding of place value and mathematical operations when allowed to use invented strategies before being introduced to algorithms (Whitacre & Wessenberg, 2016). NUMBER SENSE-BASED STRATEGIES 8 However, while alternative methods to standard algorithms are growing in popularity, it seems that many teachers do not know how to effectively implement these strategies in their instruction. Alternative methods are often taught as if they were algorithms: encouraging students to memorize many methods to solve the same problem and pick the one they like best, rather than reason through problems and invent their own strategies (Whitacre & Wessenberg, 2016). Rather than building number sense that allows students to manipulate numbers in creative and strategic ways, allowing them to perform standard algorithms fluently while understanding why they work, these methods become algorithms in and of themselves (Fuson & Beckmann, 2012), just more steps for students to memorize with no real connection to the grounded foundation of mathematics. Often, these additional methods are not tied to the standard algorithms at all. Instead, the methods detract from and even contradict each other and the standard algorithms. While students can benefit from comparing solution methods for the same mathematics problem side by side (Rittle-Johnson & Star, 2009), instruction of this type is not widely utilized. Literature Review The Importance of Number Sense in Learning Mathematics Procedural and conceptual teaching approaches in the United States. At the turn of the twentieth century, the increases in both the number of U.S. schools and in the state requirements for children to attend schools led to a marked increase in the number of students in U.S. classrooms (Library of Congress, 2015). This increase in the number of students caused educators to contemplate the most efficient ways to prepare these children for life and the workforce. Educational leaders such as William Heard Kilpatrick and Edward L. Thorndike advocated for practical forms education (Klein, 2003). These NUMBER SENSE-BASED STRATEGIES 9 educational leaders and their followers believed that mathematics was not necessary, or even possible, for all to learn (Klein, 2003), and that mathematics instruction to children should be systematic and purely procedural (Berry III & Ellis, 2005). Thus emerged an educational culture of teacher modeling, drill and practice, and standard algorithms. Many educational reforms left their mark on the twentieth century. Reforms that were widely accepted and widely perpetuated were constructed within what Berry III and Ellis (2005) described as the procedural-formalist paradigm (PFP). Within this paradigm, instruction focuses on teacher demonstration, student imitation, memorization, and practical application (Berry III & Ellis, 2005). But, according to Ma (2010), focusing on these methods was outside children’s cognitive development. Ma suggested that the removal of the “reasoning system” (p. xi) from mathematics teaching stemmed from a discrepancy between the students’ cognitive abilities and the way in which this system was presented (2010). Recently, however, a more concept-based paradigm has emerged among mathematics educators, known as the cognitive-cultural paradigm (CCP; Berry III & Ellis, 2005). Within this paradigm, students work within “logically organized and interconnected concepts that come out of human experience, thought, and interaction” (Berry III & Ellis, 2005, p. 12). While, like with the PFP, instruction within the CCP emphasizes procedural knowledge, the CCP focuses more on critical thinking in connection to procedural knowledge while the PFP focuses more on the rote memorization of skills to be performed (2005). While efficiently and proficiently performing practical mathematical skills is important in school and the workforce, concept-based instruction that focuses on building NUMBER SENSE-BASED STRATEGIES 10 number sense is invaluable (Fuson & Beckmann, 2012). Instead of standing alone, procedure-based instruction and concept-based instruction should exist together. As Klein (2003) states, “More fundamentally, the separation of conceptual understanding from basic skills in mathematics is misguided. It is not possible to teach conceptual understanding in mathematics without the supporting basic skills, and basic skills are weakened by a lack of understanding” (p. 202). For students to learn within the CCP, teachers must teach within it (Berry III & Ellis, 2005). Number sense and success in mathematics. The presence or lack of number sense significantly influences a student’s future mathematics success (Jordan, Glutting, & Ramineni, 2010; Locuniak & Jordan, 2008). In a 2008 longitudinal study, Locuniak and Jordan found that 84% of students who were indicated as possessing number sense in kindergarten did not show calculation fluency difficulties in second grade. The study also showed that 52% of students who did show number sense difficulties in kindergarten showed calculation fluency difficulties in second grade. In a 2010 longitudinal study, first grade students with strong number sense were significantly more successful in mathematics as third grade students (Jordan et al., 2010). While educators want their mathematics instruction to prepare their students for real world experiences, many tend to instill a focus on answers more than processes, even though real life problem-solving requires creativity and flexibility of thinking (Bayazit, 2013; Martinie & Coates, 2007). Number sense allows elementary students to be more flexible in their thinking and manipulate numbers sensibly (Trafton & Thiessen, 2004). Carpenter et al. (1998) found that children who used their own invented strategies to solve mathematics problems NUMBER SENSE-BASED STRATEGIES 11 before using standard algorithms were more successful and made fewer systematic errors than those who used only standard algorithms. A New Emphasis on Number Sense Number sense and the Common Core. The Common Core State Standards for Mathematics (CCSS-M) emphasizes number sense in a way that highlights the importance of number sense in mathematics education (Myths vs. Facts, 2018). The Common Core State Standards (CCSS) official website states (Myths vs. Facts, 2018) The standards set a rigorous definition of college and career readiness not by piling topic upon topic, but by demanding that students develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly do. (para. 9) Number sense allows students to think more flexibly about their problem solving and to determine the reasonableness of their answers (Martinie & Coates, 2007). But while the CCSS-M are designed to increase problem solving and number sense in students, they do not dictate how these standards are taught (Myths vs. Facts, 2018). Bayazit (2013) found that, when solving textbook mathematics problems, students disregarded the real-life context to which the problems were related. For example, when asked how long it would take a woman to dry 9 kg of laundry if it took 25 minutes to dry 3 kg, the majority of students did not use problem solving skills, which would have led them to realize that it would not take any more time to dry more laundry. Instead, they assumed they needed to multiply 25 by 3, giving a solution of 75 minutes. While looking at this problem as a textbook mathematics problem, students overlooked the common NUMBER SENSE-BASED STRATEGIES 12 sense to be found within a real-life context. Bayazit suggests that this lack of problem-solving skills is likely due to limitations in textbooks and classroom instruction. Building number sense through alternative strategies. To combat a lack of number sense and reasoning skills, number sense should be built before students are exposed to standard algorithms (Fuson & Beckmann, 2012). Fuson and Beckmann also state that “standard algorithms are to be understood and explained and related to visual models before there is any focus on fluency.…Full fluency is not achieved until subsequent years” (2012, p. 28). These alternate strategies are shown to increase number sense in elementary students (Carpenter et al., 1998). Examples of alternative methods are numerous and adaptable. Fuson and Beckmann (2012) mention number decomposition, adding on, and place value drawing as strategies for adding multi-digit numbers; ungrouping with drawings or manipulatives as strategies for subtracting multi-digit numbers; area drawings, area models, partial products, and the lattice method as strategies for multiplying multi-digit numbers; and area drawings, area models, and rounding as strategies for dividing multi-digit numbers. These methods build number sense and an understanding of why numbers behave the way they do. Additionally, alternative methods can include invented strategies or invented strategies that have evolved into more general procedures. Invented strategies are problem-solving methods derived by students without explicit instruction (Carpenter et al., 1998). While standard algorithms, although efficient, have evolved over time and become removed from conceptual understanding, invented strategies are rooted in underlying mathematical concepts (Carpenter et al., 1998). A three-year longitudinal NUMBER SENSE-BASED STRATEGIES 13 study found that students who used invented strategies showed better base ten understanding, were more flexible problem solvers, and made fewer systematic errors than those who relied on algorithms (Carpenter et al., 1998). Teaching alternative strategies. Concrete strategies are beneficial to all students, not only those who struggle (Lynch & Star, 2013). In response to the increased focus on number sense-based learning, many curricula have emerged to support the CCSS-M within the context of conceptual reasoning. Engage New York (EngageNY). Engage New York (EngageNY)—which Great Minds has extended upon to create the program Eureka Math (“EngageNY Is Eureka Math,” 2017)—is a free P-12 mathematics curriculum created and maintained by the New York State Education Department (“Frequently Asked Questions,” 2013). The EngageNY mathematics curriculum focuses largely on base ten number sense as lessons and strategies build from concrete to abstract (“Grade 5 Mathematics Module 2,” 2013). When teaching how to multiply multi-digit whole numbers, students begin the unit by creating diagrams that depict the repeated addition interpretation of multiplication (“Grade 5 Mathematics Module 2,” 2013). For example, when solving the problem 21 x 5, students create a diagram illustrating 21 rows of 5 blocks. They use this image to separate the rows into more workable segments: one row of 5 and 20 rows of 5. Knowing that one row of 5 contains five blocks and that 20 rows of 5 contains 100 blocks, students determine the answer to 21 x 5 to be 105. This concept becomes slightly more advanced in the same lesson when students create area models that do not contain individual boxes but an empty, segmented box with the sides labeled to denote how many boxes would be present if drawn. (“Grade 5 NUMBER SENSE-BASED STRATEGIES 14 Mathematics Module 2,” 2013). For example, when solving the problem 21 x 5, students draw a rectangle labeled as measuring 5 units across the top. Down the side, the rectangle is segmented into two pieces. One piece represents one row of 5 units and the other much longer piece represents 20 rows of 5 units. Students use the skills they learned in the previous lesson to determine the area of, or number of units found in, each segment. Once again, students multiply 1 x 5 and 20 x 5 to find a total answer of 105. This general concept is used to connect the area model to the partial products algorithm. When solving the problem 31 x 23, students create an area model with 23 written as the measurement of the top and 31 (split into segments of 1 unit and 30 units) written down the side (“Grade 5 Mathematics Module 2,” 2013). Students determine the top and bottom segments to have an area of 23 and 690 respectively, and conclude that the total answer is 713. This strategy is connected to the partial products algorithm where the multiplicand and multiplier are aligned vertically. The value of the number in the ones place value of the multiplier, in this case 1, is multiplied by the entire multiplicand, producing a partial product of 23. Next, the product of the value of the number in the tens place value of the multiplier, in this case 30, is also multiplied by the entire multiplicand, producing a partial product of 690. The two partial products, aligned vertically, are added together to create a total product of 713. The model is further broken down when the multiplicand is decomposed entirely in addition to the multiplier (“Grade 5 Mathematics Module 2,” 2013). For example, in the problem 39 x 814, the multiplicand is decomposed into three segments of 800 units, 10 units, and 4 units. The multiplier is also decomposed into two segments of 30 units and 9 units. As students notice that the rows of segments correlating with the segment of NUMBER SENSE-BASED STRATEGIES 15 30 and the segment of 9 can be combined separately rather than adding up all six segments together, they make connections to the same process that takes place in the standard algorithm. They see that the standard algorithm is based on mathematical concepts just like the other strategies available to them. Comprehensive Mathematics Instruction (CMI). The Comprehensive Mathematics Instruction (CMI) Framework was developed through the collaborative efforts of professors from Brigham Young University’s Educational Leadership, Mathematics, Mathematics Education, and Teacher Education departments as well as representatives from five surrounding school districts (Tour WCSD, n.d.). In 2013, the CMI Framework was shown to have increased standardized-testing scores by as much as 16.5% (McKay School of Education, 2013). The CMI Framework is a task-based structure rather than a program or a prescription (Womack, 2011) based on the learning cycle shown in Figure 1. Recognizing that traditional teaching as well as inquiry- and discovery-based learning have their limitations, the CMI Framework provides opportunities for students to engage in, explore, and discuss mathematical concepts in ways that develop number sense and solidify understanding (Womack, 2011). The CMI Framework is diagrammed as a Learning Cycle with three phases: Develop Understanding, Solidify Understanding, and Practice Understanding (see Figure 1). Each phase of the Learning Cycle consists of a teaching cycle. Each teaching cycle is likewise divided into three stages: Launch, Explore, and Discuss. Womack (2011) explains NUMBER SENSE-BASED STRATEGIES 16 The Launch stage creates a context for the mathematics that will follow. The Explore stage provides a task or series of tasks within which students explore the mathematics individually or in groups. And the Discuss stage allows the entire class of students, collectively or individually, under the guidance of the teacher, to clarify, explain, justify, prove, and connect the mathematics just explored. (pp. 7- 8) While each teaching cycle is developed as an individual lesson, the Launch, Explore, and Discuss stages take place within each lesson (Womack, 2011). The teaching Figure 1. The CMI Framework Learning Cycle (Bahr, Hendrickson, & Hilton, 2009). NUMBER SENSE-BASED STRATEGIES 17 cycles build on each other to help students develop understanding, extend their ideas, and develop fluency. “The more fragile ideas become concepts, the strategies solidify into algorithms, and the representations become more thoroughly understood, making them useful tools” (Womack, 2011, p. 9). Dreambox Learning. Dreambox Learning is an online math program that provides a wide variety of concept-based activities that are strategically personalized to the abilities of each student (“Why Dreambox?,” 2012). Dreambox Learning aims to close gaps in “conceptual understanding, fluency, reasoning, and problem-solving skills” (2012, para. 4) while providing teachers with actionable data. While the program focuses on conceptual skills, the activities align with the Common Core, the Texas Essential Knowledge and Skills, and other standards (2012). Didax. Didax provides material and virtual resources applicable to a variety of mathematics topics (Didax, 2018). The virtual resources can be used by students and teachers for free. Virtual manipulatives include number lines, place value discs, unifix cubes, algebra tiles, and number lines, among others. The Math Learning Center. The Math Learning Center provides virtual resources that can be accessed through a web app, the Apple App Store, or the Chrome Store (Math Learning Center, 2019). Available apps include fraction bars, math vocabulary cards, number pieces, and a partial products finder. Illustrative Mathematics. Illustrative Mathematics is an online resource that allows teachers to find and download conceptual math tasks specific to grade level and core standards (“Content Standards,” 2015). The reasoning-based tasks include commentaries that outline common mistakes and ways to combat them, and a detailed NUMBER SENSE-BASED STRATEGIES 18 solution that walks students through the thinking process after they have had a chance to solve it on their own (“Elmer’s Multiplication Error,” 2014). Inside Mathematics. Developed by the Silicon Valley Mathematics Initiative, Inside Mathematics is an online resource that provides “non-routine” (“Problems of the Month,” 2014) problems of the month that work to promote a problem-solving culture in the classroom (2014). Each problem is divided into five levels of difficulty to allow all students an opportunity to increase their ability to work with complex mathematical problems. The problems are aligned with the Common Core standards. 5280 Math. Educator Jerry Burkhard developed 5280 Math as a platform to share his challenging mathematics activities with elementary and middle school teachers (Burkhart, 2018a). The activities, divided into early grades, middle grades, and later grades, provide some kind of visual or model to be analyzed (Burkhart, 2018b). Students are encouraged to ask themselves what they notice, what they wonder, and what they can create based on what they see (Burkhart, 2018b). The Ineffective Incorporation of Multiple Mathematical Strategies Misunderstanding the role of number sense in the curriculum. While many educators believe that number sense is an important part of learning mathematics, others still argue that students learn best through demonstration and practice without much focus on understanding (Trafton & Thiessen, 2004). Although there is evidence that conceptual and procedural knowledge support each other, teachers are often unclear about what their students should be able to do and how to balance skills and understanding (2004). Without clear guidelines, many teachers offer a “dualistic program” (p. 114) with one side of their curriculum teaching number sense and understanding and the other side NUMBER SENSE-BASED STRATEGIES 19 teaching procedural skills. In reality, conceptual and procedural learning should complement each other (2004). The lack of concept-based reasoning among U.S. teachers. Teaching mathematics requires knowledge and reasoning beyond what is required to model standard algorithms (Ball, Hill, & Bass, 2005; Ma, 2010). For example, teachers must be able to analyze errors and teach clearly so that students can understand the fundamental principles behind the operations (Ball, Hill, & Bass, 2005). In order to understand student responses and determine lesson objectives, teachers must draw on sound subject matter knowledge, not just pedagogical knowledge (Ma, 2010). However, teachers in the U.S. often lack concept-based reasoning ability in mathematics, making it more difficult to teach children number sense-based strategies (Ma, 2010). In a 1999 qualitative study, Ma interviewed 23 U.S. teachers and 72 Chinese teachers, finding that 83% of the U.S. teachers compared to only 14% of the Chinese teachers displayed only procedural knowledge of subtracting with regrouping (Ma, 2010). While manipulatives are often utilized in U.S. classrooms, the ways in which they are utilized are not always aligned with the intended objectives (Ma, 2010). For instance, Ma (2010) gives the example of teaching problems like 52 minus 25 or 23 minus 17 using manipulatives. These problems require regrouping to solve with the standard algorithm, but using single-unit manipulatives eliminates the need to regroup. Small and seemingly insignificant oversights like these remove the important connection between procedure-and concept-based instruction. There are many written or drawn teaching strategies used to build number sense before teaching the standard algorithm. These strategies help form connections between NUMBER SENSE-BASED STRATEGIES 20 the procedure-based and concept-based strategies. However, simply making students aware of multiple strategies is not enough, and some utilizations of written and drawn representations are more effective than others. For example, comparing strategies with different solution methods can be beneficial to learning, but U.S. teachers who use comparison in their lessons often have their students compare trivial differences between solution methods rather than have their students compare solution methods that differ in meaningful ways (Rittle-Johnson & Star, 2009). Even more, these strategies, which are meant to build number sense and give meaning to standard algorithms later on, are frequently taught as algorithms themselves (Whitacre & Wessenberg, 2016). Procedural knowledge is an important part of mathematics proficiency; however, procedures must be backed up by conceptual principles, which in turn should not be presented as more algorithms to memorize (2016). This inadequate mathematics instruction is not only a result of educators misunderstanding the purpose of multiple strategies; in many cases, textbooks are also to blame. Many textbooks follow a “strategy-of-the-day” approach, requiring students to follow the step-by-step procedures for strategies that should be building intuitive number sense (2016). Carpenter and colleagues (1998), speaking of the dangers of teaching invented strategies like steps to be memorized, stated, “If these strategies were the object of direct instruction, there would be a danger that children would learn them as rote procedures in much the way that they learn standard algorithms today” (p. 19). Summary Number sense is essential in building mathematical fluency and proficiency. When students possess sound number sense, they are more flexible and confident in NUMBER SENSE-BASED STRATEGIES 21 their mathematical reasoning, and their future success in mathematics is increased. Recently, mathematics education in the U.S. has increased focus on building number sense in students, and conceptual strategies have become more prevalent in U.S. classrooms. However, many educators do not possess the number sense themselves to teach these strategies effectively. As a result, conceptual strategies are often taught in ways that do not build number sense or connect procedural algorithms to concrete principles. NUMBER SENSE-BASED STRATEGIES 22 PURPOSE Number sense is an underlying prerequisite to mathematical computational fluency. Students with number sense are more confident with numbers and more flexible in their problem solving, leading to stronger computational skills. In an effort to increase number sense in elementary students, a wide range of conceptual strategies have become popular in U.S. classrooms. However, many teachers implement these strategies ineffectively. Often, teachers teach number sense-based strategies as algorithms with steps and rules to be followed rather than a cohesive set of options that follow the same rules and patterns. Additionally, teachers often fail to connect conceptual strategies to procedural algorithms. The purpose of this project was to provide accessible professional development to fifth grade teachers in Weber School District that clarifies how to effectively teach concept-based strategies used to multiply multi-digit whole numbers. Through a series of short videos, articles, quizzes, and other resources, this project highlighted the number sense-based connections between strategies and the importance of both conceptual and procedural knowledge. The project implemented strategies found in the EngageNY math program, which is the approved elementary mathematics program in the district. Aspects of the CMI Framework were implemented to provide examples of how the comparison between strategies and the combination of procedural and conceptual knowledge can be implemented. This project was built on Canvas, a platform that makes the project most useful to Weber School District. A focus group of upper-elementary math fellows, or full-time elementary educators who serve as mathematics leaders and coaches for Weber School District, reviewed and evaluated the materials for revision. NUMBER SENSE-BASED STRATEGIES 23 METHOD Number sense is essential in a student’s growth towards mathematical computational fluency. Number sense gives students more confidence and flexibility in their problem solving. In an effort to provide students with opportunities to improve their number sense, mathematics instruction has shifted towards more conceptual strategies. However, many educators do not know how to implement these computational strategies effectively. They teach number sense-based strategies as steps to follow rather than principles to manipulate, and they fail to show students the connections between conceptual strategies and procedural algorithms. This project was designed to provide professional development to fifth grade teachers in Weber School District by clarifying how to effectively teach strategies used to multiply multi-digit whole numbers. Through short videos, articles, and other resources, the professional development demonstrated how to connect conceptual and procedural strategies and algorithms and the need for both conceptual and procedural knowledge. The professional development was made available through Canvas, the district’s preferred platform for this resource. Strategies found in the district’s math program, EngageNY/Eureka Math, were implemented, as well as aspects of the CMI Framework. Procedure A Canvas course set up by Weber School District served as the platform for this project. I created and posted a series of short videos, articles, quizzes, surveys, and other resources discussing effective ways to incorporate conceptual and procedural strategies into whole number multi-digit multiplication. The professional development course included six modules focused on the following topics:  The importance of number sense in learning mathematics NUMBER SENSE-BASED STRATEGIES 24  The new emphasis on number sense in mathematics curricula  Teacher language supportive of conceptual understanding  Connections between the following conceptual and procedural strategies used to multiply multi-digit whole numbers: o Repeated addition o Drawings o Base ten blocks o Area models o Partial products o Standard algorithm o Lattice method  The analysis of student errors stemming from conceptual misunderstanding and the determination of appropriate interventions Upon completion, I made the professional development available to upper-elementary math fellows, or full-time elementary educators who serve as mathematics leaders and coaches for Weber School District. These fellows provided feedback on the course. I added resources and made changes based on the feedback received. NUMBER SENSE-BASED STRATEGIES 25 PROGRAM CREATION Platform I used a Canvas site set up by Weber School District as the platform for my professional development course. There were multiple reasons why Canvas was the chosen platform. In talking to Weber School District Curriculum Specialist Jennifer Boyer-Thurgood, we determined that Canvas would be the most useful platform for the district now and in the future. Additionally, teachers completing the course are able to monitor their progress and take end-of-module assessments on a Canvas course, while course administrators can monitor participants’ progress and the course’s overall participation without needing to be directly involved in the course instruction. Course Materials To meet the course objectives, I gathered articles and webpages and created videos, short article summaries, quizzes, and surveys. Articles I selected three articles for teachers to study as part of the course: “The Paradigm Shift in Mathematics Education: Explanations and Implications of Reforming Conceptions of Teaching and Learning” (Ellis & Berry, 2005), “Standard Algorithms in the Common Core State Standards” (Fuson & Beckmann, 2012), and “Knowing Mathematics for Teaching” (Ball, Hill, & Bass, 2005). Summaries. I included short self-written summaries of “Linking Mental and Written Computation via Extended Work with Invented Strategies” (Trafton & Thiessen, 2004), and Knowing and Teaching Elementary Mathematics (Ma, 2010). I also included a short self-written introduction to the importance of number sense-based instructional language. NUMBER SENSE-BASED STRATEGIES 26 Web Pages As part of the curriculum, I included several online articles, posts, and other resources. These include a website article entitled “Strategies Are Not Algorithms” (Whitacre & Wessenberg, 2016), a Washington Post article entitled “Today’s Math Vocabulary Exposes Generational Divide” (Chandler, 2012), a blog post entitled “The Myth of the Careless Error” (Applerouth, 2011), a link to an online error analysis activity (Stacey, 2003), a video entitled “Mathematical Knowledge of Teaching with Dr. Deborah Loewenberg Ball” (Mathematical Knowledge, 2018), links to two Common Core webpages: “Standards for Mathematical Practice” (2019) and “Mathematics Standards” (2019), and three online teaching resources: Illustrative Mathematics (Content Standards, 2015), Inside Mathematics: Problem of the Month (Problem of the Month, 2014), and 5280 Math (Burkhart, 2018a; 2018b). Videos I created a total of eleven short videos for this course. Each video was created with an iPad Air screen recorder. I used Notes Plus (Viet Tran, 2018), Base Ten Blocks Manipulative (Brainingcamp, 2018), and Google Sites to display the recorded information. Introductory video. I used an iPad Air to record myself introducing the objectives of and reasons for the course. This introductory video is found on the course’s home page. Supplementary videos. I made two videos meant to supplement other module materials. The first supplementary video, entitled Comparing Strategies Effectively, is found in Module 3 and is based on ideas from Rittle-Johnson and Star (2009). The second NUMBER SENSE-BASED STRATEGIES 27 supplementary video, entitled Procedural vs. Conceptual Language When Modeling Standard Algorithms, is found in Module 4. Module 5 modeling videos. The instructional materials found in Module 5 are comprised solely of short videos modeling multiple strategies and algorithms used to multiply multi-digit whole numbers. The first video, entitled Using Multiple Strategies Effectively, outlines the importance of teaching multiple strategies to enhance student number sense and the conceptual connections between all valid strategies and algorithms. The next seven videos focus on drawings and repeated addition, the use of base ten blocks, area models, partial products, the standard algorithm (part 1 and part 2), and the lattice method. Assignment The one assignment of the course is found in Module 6. The purpose of the assignment is to provide teachers an assessable form of error analysis practice. There are eight pairs of problems in the assignment. Each pair of questions focuses on one example of a common error made while multiplying multi-digit whole numbers. In the first question, teachers determine whether or not the error is likely the result of a conceptual misunderstanding. In the second question, teachers provide an example of an intervention that could be used to address the misunderstanding, if applicable. If an intervention is not applicable, the teacher explains why not. Only the first question of each pair is a multiple-choice question, meaning that it is the only one graded. However, an example answer for the second question of each pair is given after submission so teachers can compare their answers to a correct one. Students must score 8 out of 16 questions—or all of the multiple choice questions—correctly in NUMBER SENSE-BASED STRATEGIES 28 order to move on in the course. Students can retake the assignment as many times as necessary. Quizzes Because assessment quizzes need to be self-grading, all quizzes contain five multiple-choice questions. The quizzes focus on pertinent information found in the module resources relating to the module’s objective(s). Students must answer four questions correctly in order to move on to the next module. While students cannot view the correct answers after submitting, they can see which questions they missed and retake the quiz as many times as needed. Surveys The course begins and ends with a ten-question survey (see Figure 2). The questions focus on the teachers’ capabilities and understanding required to effectively facilitate the growth of number-sense in their students. The content of the questions directly aligns to the module objectives. NUMBER SENSE-BASED STRATEGIES 29 Modules I organized the instruction into six modules, each designed to take approximately one hour to complete. Each module includes resources designed to help teachers meet certain objectives. Module 1 The objective of the first module is for teachers to “understand the importance of number sense in elementary students and their teachers.” Resources include a website article (see Figure 3) entitled “Strategies Are Not Algorithms” (Whitacre & Wessenberg, Figure 2. Pre- and post-course survey. NUMBER SENSE-BASED STRATEGIES 30 2016), an article entitled “The Paradigm Shift in Mathematics Education: Explanations and Implications of Reforming Conceptions of Teaching and Learning” (Ellis & Berry, 2005), and a video entitled “Mathematical Knowledge of Teaching with Dr. Deborah Loewenberg Ball” (Mathematical Knowledge, 2018). A five-question quiz was at the end of the module assesses whether or not the teacher reaches the objective. Figure 3. Screenshot from Module 1 with attachment of the article "Strategies Are Not Algorithms" by Ian Whitacre and Donna Wessenberg (Whitacre & Wessenberg, 2016). NUMBER SENSE-BASED STRATEGIES 31 Module 2 The objective of Module 2 is for teachers to “recognize the emphasis on number sense found in the Common Core State Standards and mathematical curricula.” The module includes an article entitled “Standard Algorithms in the Common Core State Standards” (Fuson & Beckmann, 2012) and links to two Common Core webpages: “Standards for Mathematical Practice” (Standards, 2019) and “Mathematics Standards” (Mathematics Standards, 2019). A five-question quiz assesses to what degree the objective has been met. In addition, I included five online resources (see Figure 4) to enhance number sense-based instruction: Illustrative Mathematics (Content Standards, 2015), Inside Mathematics: Problem of the Month (Problem of the Month, 2014), and 5280 Math (Burkhart, 2018a; 2018b). Figure 4. Screenshot of free online resources from Module 2. NUMBER SENSE-BASED STRATEGIES 32 Module 3 The objective of Module 3 is for teachers to “recognize common gaps in the number sense of American teachers.” I included an article entitled “Knowing Mathematics for Teaching” (Ball, Hill, & Bass, 2005), a short summary from “Linking Mental and Written Computation via Extended Work with Invented Strategies” (Trafton & Thiessen, 2004), a short summary of “Knowing and Teaching Elementary Mathematics” (Ma, 2010), and a self-made video entitled Comparing Strategies Effectively and based on ideas from Rittle-Johnson and Star (2009). A culminating 5- question quiz ensures the teacher has achieved the objective (see Figure 5). NUMBER SENSE-BASED STRATEGIES 33 Figure 5. Sample questions from the 5-question assessment quiz in Module 3. NUMBER SENSE-BASED STRATEGIES 34 Module 4 The objective of Module 4 is for teachers to “recognize the difference between procedural and number sense-based language in teaching and modeling.” I included a short self-written introduction to the importance of number sense-based instructional language, a Washington Post article entitled “Today’s Math Vocabulary Exposes Generational Divide” (Chandler, 2012), and excerpt from “Knowing Mathematics for Teaching” (Ball, Hill, & Bass, 2005) first referenced in Module 3, and a self-made video (see Figure 6) entitled Procedural vs. Conceptual Language When Modeling Traditional Algorithms. I include a five-question quiz to assess teachers’ level of mastery of the objective. Figure 6. Screenshot of a video from Module 4 explaining the importance of using number sense-based language when modeling standard algorithms. NUMBER SENSE-BASED STRATEGIES 35 Module 5 Module 5 contains three objectives: 1) Teachers will be aware of key strategies and algorithms used to multiply multi-digit whole numbers. 2) Teachers will possess a deep understanding of the connections between these key strategies and algorithms. 3) Teachers will know how to implement these key strategies and algorithms in ways that build number sense in their students. The module contains eight short videos detailing how to effectively teach number sense through seven strategies included in EngageNY/Eureka Math for multiplying multi-digit whole numbers: drawings, repeated addition, base ten blocks, area models, partial products, the standard algorithm, and the lattice method (see Figure 7). These videos focus on the concepts behind each strategy and the underlying conceptual connections the strategies share. A five-question quiz assesses the teachers’ level of understanding. Figure 7. Screenshot of a video from Module 5 explaining the conceptual principles found in the lattice method. NUMBER SENSE-BASED STRATEGIES 36 Module 6 Module 6 contains two objectives: 1) Teachers will accurately analyze the thinking behind common procedural errors. 2) Teachers will know how to address common procedural errors through number sense-based instruction. I included another excerpt from “Knowing Mathematics for Teaching” (Ball, Hill, & Bass, 2005), which is also referenced in Modules 3 and 4; an excerpt from “Standard Algorithms in the Common Core State Standards” (Fuson & Beckmann, 2012), first referenced in Module 2; a blog post entitled “The Myth of the Careless Error” (Applerouth, 2011); and a link to an error analysis activity (Stacey, 2003). As a way for teachers to practice analyzing errors, I created a 16-question assignment (see Figure 8) requiring teachers to find the error in a multi-digit whole number multiplication problem and describe possible interventions that could be given to rectify the conceptual misunderstanding(s). A five-question quiz assesses the teachers’ level of mastery of the module’s content. NUMBER SENSE-BASED STRATEGIES 37 Figure 8. Pair of questions from the error analysis assignment in Module 6. NUMBER SENSE-BASED STRATEGIES 38 FEEDBACK AND ADJUSTMENTS Feedback Five upper-elementary math fellows (full-time elementary educators who serve as mathematics leaders and coaches for Weber School District) volunteered to review the course and provide feedback. Of the five math fellows, two provided feedback. I also received feedback from the members of my project committee. Reviewers found the course to be accurate and highly relevant, although one reviewer expressed that the course was more relevant for fourth and fifth grade teachers than for sixth grade teachers. Content-related feedback included the use of the term “traditional algorithm” rather than “standard algorithm” because the algorithm, though standard in the U.S., is not standard everywhere in the world. In addition, I received a suggestion to add Didax (2018) and The Math Learning Center (2019) to the free online resources included in Module 2. I also received suggestions relating to the layout of the course, such as hiding unnecessary tabs, giving more explicit instructions on the home page about how to use the course, and instructing teachers to mark each page as “done.” Adjustments I did not find it necessary to edit or re-record videos that included the phrase “standard algorithm” since this term is familiar among teachers who will access this course. I chose not to change the titles of the two videos focused on the algorithm in Module 5 because many resources in previous modules including articles and web pages used the term “standard algorithm.” Instead, I included the following disclaimer to the Module 5 Overview, Standard Algorithm | Part 1, and Standard Algorithm | Part 2. NUMBER SENSE-BASED STRATEGIES 39 The term "standard algorithm" is used [here] because of the frequency with which this term is used to label the algorithm in the United States. However, although this algorithm is standard in the U.S., it is not standard in many other countries around the globe. For this reason, the term "traditional algorithm" may be more appropriate. I added Didax (2018) and The Math Learning Center (2019) to the list of free online resources found in Module 2. I also made the suggested changes to the course layout. NUMBER SENSE-BASED STRATEGIES 40 NEXT STEPS Within the next eighteen months, Weber School District will begin a digital badging system to support teachers in more online professional learning opportunities. Teachers will have access to a library of both self-paced and facilitated courses available for relicensure or other certifications. When this system launches, this project will be included in the library of courses available to teachers. NUMBER SENSE-BASED STRATEGIES 41 SUMMARY The idea for this project stemmed from the lack of conceptual understanding in my own fifth-grade students. Several weeks after my students learned how to multiply multi-digit whole numbers, I realized that many of them possessed little or no conceptual understanding connected to the topic. Research on the topic of number sense-based instruction showed that a lack of number sense is a large issue in classrooms across the nation. In addition, it showed that a primary factor leading to students’ lack of conceptual understanding is a lack of conceptual understanding in their teachers. This revelation prompted the creation of a professional development course that could help teachers increase their own number sense in order to more effectively facilitate the growth of number sense in their students. The course includes scholarly articles, web pages, videos, practice activities, quizzes, and surveys. The course will be made available to teachers in Weber School District within the next 18 months when the district launches a new digital badging system for professional development. NUMBER SENSE-BASED STRATEGIES 42 REFERENCES Applerouth, J. (2011, March 23). The myth of the careless error. Available from https://www.applerouth.com/blog/2011/03/23/the-myth-of-the-careless-error/ Bahr, D., Hendrickson, S., & Hilton, S. C. (2009). 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