symphony math teacher guide

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1 symphony math teacher guide

2 Copyright and Trademark Notice 2012 Symphony Learning, LLC. All rights reserved. Symphony Math is a trademark of Symphony Learning, LLC. Director is a trademark of Adobe. Macintosh is a trademark of Apple Computer, Inc. MS Windows is a trademark of Microsoft Corporation. Symphony Math has been created with Adobe Director software. Published by: Symphony Learning, LLC. PO Box 5491 Hanover, NH Phone: support@symphonylearning.com Information in this document is subject to change without notice and does not represent a commitment on the part of Symphony Learning, LLC. The software described in this document is furnished under a license agreement or non-disclosure agreement. The software may be used only in accordance with the terms of the agreement. This document and the software described within it may not, in whole or in part, be copied, photocopied, reproduced, translated, or reduced to any electronic medium or machine-readable form other than that which has been specified herein without prior written consent from Symphony Learning, LLC. Acknowledgments: We thank the many math researchers, teachers and cognitive scientists for their research and practice in the field of mathematics education upon which this program is based. Version 6.01

3 contents symphony math 01 overview of the symphony math program...1 introduction... 1 Screener... 1 Benchmarker... 1 Intervention... 2 The Symphony Approach Efficient Implementation Using the Symphony Math Program... 6 Alternative Uses of Symphony Math symphony math screener....8 Introduction... 8 Defining Risk Screener Reports Three Testing Windows Each School Year symphony math benchmarker...13 Introduction Benchmarker Methodology Benchmarker Norm-Referenced Scores symphony math intervention...18 Introduction The Big Ideas of Mathematics Multiple Ways of Knowing Learning Through Thinking and Making Connections Dynamic Branching Allows Students to Learn at Their Own Level Comprehensive Reports Who Can Use Symphony Math? intervention scope and sequence...29 Introduction The Seventeen Stages of Symphony Math Big Idea #1: Quantity Big Idea #2: Parts-To-Whole Big Idea #3: Hierarchical Groupings Table of Content Overview of Symphony Math iii

4 Big Idea #4: Hierarchical Groupings with Parts-to-Whole Big Idea #5: Repeated Equal Groupings Big Idea #6: Repeated Equal Groupings with Parts-to-Whole setting up accounts and classes...44 Introduction...44 System Requirements for the Symphony Administration Panel...44 Logging in to the Symphony Math Administration Panel Symphony Dashboard Creating Student Accounts and Your Class Using the Import Tab to Create Student Accounts and Classes Introduction System Requirements installing symphony math intervention Introduction System Requirements Download and Install the Symphony Math Application Student Sign In Network Installation Instructions Set Up the School-To-Home Option district set up...58 Introduction Compatible Web Browsers for Administration Panel Configuring the Administration Panel Logging in to the Symphony Math Administration Panel Changing District Settings Creating School Accounts Creating Student Accounts Using the Import Tab to Create Student Accounts and Classes Installing the Symphony Math Intervention Application System Requirements Ensuring Student Access to the Symphony Web Servers Deciding Between Local and Network Installation installing intervention at home Introduction System Requirements Download and Install the Symphony Math Intervention Application Student Signin using symphony math intervention...73 Introduction The Student Signin Screen The Student Scoreboard Screen The Thinking Round The Help Button The Mastery Round Table of Content Overview of Symphony Math iv

5 11 the intervention dashboard...79 Overview of the Dashboard Status Over Past Four Weeks The Dashboard Categories Understanding the Gauges How to Remedy Common Implementation Issues The Overall Status Statistics Understanding the Progress Over Time Graphs Other Important Dashboard Features reports Overview of Reports Symphony Math Reports Selecting Reports Screener Reports Compare Groups: Screener Status Benchmarker Reports Intervention Reports managing user accounts Staff Access Managing School Accounts Within a District Creating Staff Accounts Managing Student Accounts Managing Classes symphony math intervention advanced settings Introduction Settings Panel Account Information Student Settings Network Settings Graphics and Sound Settings Administration technical troubleshooting Ensuring Student Access to the Symphony Web Servers Deciding Between Local and Network Installation Importing Students, Teachers, and Classes Fixing Program Slowness, Crashes, and Error Messages Symphony Math Intervervention Automatic Updates Symphony Math Intervention School-to-Home Use preparing for a new school year Option 1: Delete All Students and Re-Import School Roster Option 2: Update the Existing School Roster Table of Content Overview of Symphony Math v

6 17 frequently asked questions Assessment Using Symphony Math Intervention Symphony Math Intervention Installation & Setup Symphony Math Administration Panel Contacting Technical Support Appendices Appendix A: Symphony Math Intervention Scope & Sequence Appendix B: Symphony Math Intervention Word List Appendix C: Assessment Criteria and Specifications Appendix D: Research Base Table of Content Overview of Symphony Math vi

7 01 overview of the symphony math program introduction Symphony Math is a three-step program that identifies and supports students who need help with math. Each step in the Symphony Math process is supported by one of the program tools: Screener, Benchmarker, and Intervention. Together the three tools offer a framework for identifying students at risk for math failure, quantifying their rate of progress, and filling in gaps in their mathematical development. Symphony Math efficiently identifies students in need of additional support in mathematics, then provides the tools to track and support the effort to close the gap with their peer group. The program is suitable for students in kindergarten through grade eight. The assessments are designed to measure learning against the Common Core State Standards for Mathematics. The three tools can be used individually or together to support Response to Intervention. Screener Symphony Math Screener is a quick and accurate tool for identifying which students in a school (or district) are at-risk or borderline for math failure. With an average test time of five minutes, Screener quickly determines a student s at-risk status. Screener is given three times a year (Fall/Winter/Spring). Benchmarker Symphony Math Benchmarker is a computer adaptive test (CAT) that determines the instructional level of each student and quantifies their progress over the course of the school year. The average test time for Benchmarker is twenty minutes. The assessment provides a standard score, grade-level Chapter 01 Overview of the Symphony Math Program 1

8 equivalent, and percentile rank for each student. As students repeat the assessments a second and third time, Benchmarker provides scores that show the rate of student learning from one testing period to the next. Intervention Symphony Math Intervention helps address one of the primary causes of poor math performance: Weak foundation skills. The Intervention tool provides students with the experience of learning and thinking about the most important mathematical concepts through a systematic progression. The Symphony Approach Many students have not become proficient in math because they have not mastered the foundational concepts with sufficient depth. Symphony Math philosophy emphasizes the development of a deeper understanding of critical mathematical concepts. There are several pedagogic techniques the program employs to support the development of a profound understanding of mathematics: Symphony Math Screener and Benchmarker probe for in-depth understanding by using technology-enhanced test items. Symphony assessment problems challenge students to produce multiple correct solutions, demonstrate fluency, graph lines, use measurement tools, and solve for the exact answer, instead of providing only a simple, multiple choice. These technology-enhanced problems require students to demonstrate a deeper level of understanding to produce correct responses. Symphony Math Screener & Benchmarker are multi-skill assessments. They do not focus on one or two specific skills, but have a databank that contains several assessment problems for each Common Core State Standard from kindergarten through grade eight. Chapter 01 Overview of the Symphony Math Program 2

9 Symphony Math assessments are suitable for younger students. This allows educators to identify students at risk for math failure in the first month of kindergarten, instead of waiting until grade 4 when end-of-year grade 3 results become available. Instructional supports can be implemented early in the student s learning career, to close the gap with peers before it becomes unmanageably large. Symphony Math conforms to the Common Core State Standards (CCSS), and adopts the same commitment to in-depth learning and understanding. By aligning its assessments to the CCSS, Symphony Math enables schools to focus on assessing a narrow range of target concepts, at the in-depth and rigorous level specified by the CCSS. Symphony Math supports Response to Intervention, helping assess and develop indepth understanding of mathematics. Some screeners used for Response to Intervention only measure fluency, or focus on a single skill, and are not aligned to the depth and rigor of the Common Core State Standards. However, using Symphony Math minimizes the chance that an at-risk student might quickly learn a skill needed to pass the fluency screener (e.g. single digit addition) without acquiring the in-depth and rigorous understanding necessary to succeed in a CCSS curriculum. The Symphony Math Benchmarker provides growth scores for each student, class, grade, and school. Growth scores may be used to ensure that ALL students are making expected progress: With Benchmarker growth data at hand, schools can readily identify students who -- while not currently on the edge of the pass-fail bubble -- still are not showing appropriate advancement. Resources may therefore be allocated to help those students just beginning to fall behind, Benchmarker supports students who may not be making appropriate growth, as well as assisting those well below the pass-fail bubble who will need two to three years to achieve proficiency. Chapter 01 Overview of the Symphony Math Program 3

10 Efficient Implementation Having accurate, effective assessment and intervention tools is often not enough to successfully manage a large-scale, data-driven educational program. The tools must also be easy to use, and consume as little instructional time as possible. That s why efficiency of implementation forms another key component of Symphony Math philosophy. If the screening and benchmarking process takes too long, educators do not have sufficient time to teach their students during the intervention phase. Symphony Math employs the following features to offer an instructional support system that is easy to use and consumes as little instructional time as possible: Symphony Screener and Benchmarker run in a web browser. The student sits at the computer, enters the school account number, types his or her username and password, then begins the assessment immediately. There is no software to install. Student names, passwords and demographic data can either be entered manually via a web browser (for a small class), or the entire roster of student names and associated information can be imported for a school or district. The assessments are quick 5 minutes for Screener and 20 minutes on average for Benchmarker. The assessments are online, which means no printing or grading of tests. The assessments can be group administered, since most students can complete them independently. Audio narration supports non-fluent readers, and Spanish audio is also available. Results are immediately available to authorized personnel, from the classroom teacher all the way up to the district office. Chapter 01 Overview of the Symphony Math Program 4

11 Using the Symphony Math Program The three Symphony Math tools can be used individually or together as an integrated system. Individual use might consist of only using Screener to quickly identify at-risk students across a district. Benchmarker can be used as a stand-alone tool to measure student growth over the course of the school year. Intervention could be used as a school-wide math support program, perhaps where previous data has already identified that the majority of students are at risk for math failure, and Intervention should part of the core curriculum. Below is an outline of how Symphony Math can be implemented as an integrated system to identify and support students at-risk for math failure. Step 1: Screen All Students At the beginning of the school year, screen all students in kindergarten through grade eight with Symphony Math Screener. Students can login from desktop computers using a web browser. Screener takes about five minutes, and returns one of three outcomes for each student: Not at-risk the student is above the at-risk threshold. Borderline the student is very close to the at-risk threshold and should be given the longer Benchmarker assessment. At-risk the student is below the at-risk threshold and should be given the longer. Benchmarker assessment to confirm this result. Chapter 01 Overview of the Symphony Math Program 5

12 Students take the screener in the fall, winter and spring. Screener consists of eight test items that are at or near the at-risk threshold for the student s grade level. For example, if the risk threshold is set to the 20th percentile, a grade two student will see eight test items during their screening that are at or near the 20th percentile for second grade students. Immediately after the screening, teachers can access online reports that document which students are not at-risk, borderline, or at-risk. Step 2: Benchmark At-Risk and Borderline Students The second step in the Symphony Math process is benchmarking students with an at-risk or borderline status. Benchmarking each student identified as at-risk or borderline gives the student an opportunity to reject their status and demonstrate more of their mathematical knowledge. The benchmarking process also provides a baseline diagnostic score, which helps determine a student s instructional level, and charts the student s growth over the course of the school year. Step 3: Intervene with At-Risk and Borderline Students The third step in the Symphony Math process is implementing Intervention with students identified as borderline or at-risk. Intervention requires a minimum of 45 minutes of use each week, and is designed to help students fill in gaps in their foundational understanding of mathematics. This could be a part of a student s tier II or tier III intervention program in the RtI framework. Fidelity of implementation is vital, as students at-risk for math failure often need intensive and consistent practice with foundational concepts in order to develop mastery. Chapter 01 Overview of the Symphony Math Program 6

13 Alternative Uses of Symphony Math Some schools only use Symphony Math Intervention, forgoing screening and benchmarking. Their data-driven decision making has lead them to conclude that more than twenty percent of their entire school population is at-risk for math failure, and therefore needs intensive support at the tier I level. Many schools find it more efficient to administer Screener and Benchmarker together in their integrated form to all students: The 20-minute average test time for Benchmarker is not prohibitively long, and they want to track the growth of all students. This is the default setting for Symphony Math, where Screener and Benchmarker are given together in one 20-minute session. Ideally, each school or district will use the Symphony Math tools in the way that makes the most sense for their students, and which produces the most optimal outcomes. Chapter 01 Overview of the Symphony Math Program 7

14 02 symphony math screener Introduction Symphony Math Screener quickly and accurately identifies students at risk for math failure. Screener is especially useful for large schools or districts that want to screen all students, but do not have the time, or computer availability, for longer tests. The assessment time for Screener averages less than 5 minutes per student. Defining Risk The first step in implementing Screener is setting the cut score definition of at-risk. When a student is said to be at-risk for mathematics failure, there must be a definition of what at-risk means. In the context of Response to Intervention, at-risk is typically defined as a cut score based on the percentile rank of a student. The 25th percentile is a common cut score for the definition of at-risk: All students who are at or below the 25th percentile are deemed at-risk, and should be monitored more closely. In a school or district with many students performing below grade level, using the 25th percentile may be inappropriate. If a school screens all of its students using a cut score at the 25th percentile, then finds half of the students at-risk, the school could not easily provide progress monitoring and interventions for all of those students. In such cases, a lower cut score should be considered, one that is more likely to identify 15 to 20 percent of the students as being at-risk. On the other hand, for a school or district that has many students performing at or above grade level, a higher cut score may be appropriate. Some schools might use the 40th percentile cut score, because it effectively identifies the struggling learners of that particular school system. Chapter 02 Screener 8

15 Administrators can define the At-Risk threshold from the online Administration Panel. Schools that are not using a multi-tiered system may be more comfortable defining risk in terms of months below grade level. For example: At the beginning of the school year, a grade three student testing at the 2.1 grade level is 12 months below grade level. If risk is defined as 12 months below grade level, this student would be positively identified. The 12-Months-Below grade level setting can make more intuitive sense than the more abstract percentile setting. If a teacher knows that a student is one year (or 18 months, or 24 months) below grade level, it makes it very clear that a different instructional approach will be necessary. If a school or district wants to experiment with different risk definitions, we recommend assessing students with the longer Benchmarker assessment tool. After students have been benchmarked, the risk definition can be modified in the Administration Panel, and reports reviewed to determine the appropriateness of the risk definition. The reports will update dynamically and display the distribution of at-risk students each time Benchmarker receives new data. If only Screener is administered, changes to the risk definition will not be reflected until the next testing window. Chapter 02 Screener 9

16 Screener Reports Screener produces one of three outcomes for each student tested: Not at-risk: The student is above the at-risk threshold. If at-risk is defined as the 25th percentile, a student identified as not at-risk is at least five percentile points above the 25th percentile. Borderline: the student is very close to the at-risk threshold, and should be given the longer Benchmarker assessment to better determine his or her status. If at-risk is defined as the 25th percentile, a student identified as borderline is between the 20th and 30th percentiles. Borderline status results from a score that is plus or minus five percentile points from the at-risk cut score. At-risk: The student is below the at-risk threshold, and should be given the longer Benchmarker assessment to confirm the positive result. If the at-risk definition is set to the 25th percentile, a student identified is at-risk is at least below the 20th percentile, since the borderline status is used for plus or minus five percentile points. Chapter 02 Screener 10

17 Three Testing Windows Each School Year Symphony Math Screener may be administered to a student three times each school year. It is possible that a student not identified as at-risk after the fall screening will be identified as at-risk at the winter or spring screening, if the student did not make sufficient progress to keep up with the peer group. The exact date ranges of the testing windows may be defined in the online Administration Panel. Defining the date range ensures that students are not accidently tested before or after the desired test period. Administrators can define the starting date for each testing window from the on-line Administration Panel Screener Methodology: An Emphasis on Saving Time The design of Screener emphasizes quick and efficient assessments on a large scale. Screener provides very specific information (a student s at-risk status) resulting from a very short test (3-5 minutes). Screener also allows for custom cut scores to be designated in advance of the assessment. Screener achieves this efficiency by only administering test items that have a difficulty rating that is approximate to the cut score. If the cut score (definition of at-risk status) is set to 12-months below grade level, then a student will only see test items that are 12-months below his or her current grade level. If the student does well on these items, the test determines that she is above the cut score, and not at risk. If the student does poorly on these test items, she will be identified as at-risk. A mixed performance would result in a status of borderline. Chapter 02 Screener 11

18 At-Risk Borderline Students Students who are identified as at-risk or borderline by Screener should be given the longer Benchmarker assessment to confirm the at-risk status, and give the student more of an opportunity to demonstrate her math knowledge. It is possible that the short Screener assessment did not provide enough time for that student to become comfortable with the online testing interface. Or maybe the student was momentarily distracted during the screening. Benchmarker, with an average test time closer to 20 minutes, is more in-depth. After completing Benchmarker, the student s status in the screener reports will be updated if there are any changes to her status. Screener or Benchmarker Schools and districts have the option of configuring Symphony Math to provide an integrated screening and benchmarking assessment. This assessment session will output all of the data in Screener reports, as well as a standard score, grade level equivalent, percentile rank, and growth score. If a school or district has the time and computer capacity for the longer Benchmarker it may prefer this configuration in order to obtain more comprehensive data. The administration of Screener by itself is only for those interested in the shortest, most efficient mathematics screening process available. Chapter 02 Screener 12

19 03 symphony math benchmarker Introduction When a teacher begins the school year with a new class of students, most curricula are designed with the assumption that all of the students in the class are at the same general level of academic ability. In reality, the typical elementary classroom will reveal a wide range of student ability. There may be some students that are one or two years below grade level, in terms of academic knowledge and skills; other students may be above grade level. And of course there will be a middle group with skills and knowledge within the grade-level range. The challenge for the teacher is to engage and teach the specified curriculum to all of these students of varying levels of ability and learning readiness. How can the teacher quickly determine where each student is in relation to each other? How can the teacher ensure that each student is improving and benefiting from instruction over the course of the year? Symphony Math Benchmarker is designed to help answer these questions. Benchmarker is an online assessment that provides a standard score, grade-level equivalent, and percentile rank for each student three times a year. Twice a year Benchmarker generates a growth score. These scores supply the teacher with the means to better understand the needs and capabilities of each student. Benchmarker Methodology Benchmarker is a Computer Adaptive Test (CAT) that dynamically locates each student on a standardized scale of mathematical ability. The test is based upon an item bank of 900 test items that were specifically designed to measure progress against the Common Core State Standards (CCSS) for Mathematics. Benchmarker test items have been calibrated and organized on a scale of low difficulty to high difficulty; the program applies Item Response Theory algorithms which move the student to an easier item after an incorrect response, and to a harder item after a correct response. Chapter 03 Benchmarker 13

20 Each student begins Benchmarker with an assessment item that is two years below the student s enrolled grade level. Depending on the student s pattern of correct and incorrect responses, the test items will become more or less difficult. A student may see between 18 and 24 assessment items during a Benchmarker session. Benchmarker ends the assessment after it has seen enough responses to determine the student s location on the standardized scale of CCSS mathematics ability. Benchmarker Norm-Referenced Scores After students complete the test, Benchmarker outputs four performance scores: Scale Score, Percentile, Grade-Level Equivalency, and (after initial testing) Growth. Scale Scores All students receive a standardized score on a common vertical scale which reveals how the tested students performed in comparison to grade-specific national norms. Such standard scores are known for all grades, and the score averages for each grade are listed in the following table. Grade Mean Scale Score* Kindergarten 485 Grade Grade Grade Grade Grade Grade Grade Grade *Scale scores are for end of school year for each grade level Because students performance increases over the school year the table only shows data for the last month of instruction, typically the month of May. For instance, a fifth-grader who in May received a scale score of 865 performed above average on the test, while a sixth-grader with the same score is below average. And, because Symphony norms are based on nationwide data, this also means that this particular fifth-grader performed better than most other fifth-graders nationwide who were tested in May. The other grades are normed in the same fashion. While the table only shows the average scores for the end of the school year, the dynamic reports of Symphony Math use month by month tables to automatically provide the percentile rank for the month Benchmarker was administered. Because Symphony Benchmarker scores are expressed on a vertical scale, students scores can be readily compared across grades. We all expect that most first-graders have learned more than most kindergartners, most second-graders have learned more than most first-graders, and so on. With Symphony s scale scores, teachers can track performance regardless of assigned grade, allowing tracking of students across grades, thus supplying a detailed picture of their progress over time. Chapter 03 Benchmarker 14

21 Percentile A percentile rank tells what percentage of students in a specific group received lower scores than the student in question. The range is from Each student s score corresponds to a percentile relative to the distribution of other students scores in the same grade, for a particular month. The percentage of scores at or below a particular value defines scores percentiles. Students with score percentiles near 50 are at the average for the month in which the assessment was taken. Lower percentiles indicate slower overall growth, while higher values indicate accelerated overall growth relative to the nation at large. For instance, assume that a third grader obtains a score of 800 when tested in May. From our data we estimate that this particular score equals or exceeds that of 70% of all other third graders tested nationwide in May. Therefore, this third grader s percentile score is 70, or the 70th percentile. It is to be expected that students score percentiles will vary somewhat across the school year. Yet, while most students scale scores naturally rise over time, their positions within the national distribution and therefore their percentile score are often quite stable. Chapter 03 Benchmarker 15

22 Grade Level Equivalent Grade-level equivalents express students performance as being typical of some particular grade and month of instruction. The word typical in this context refers to the median or middle score (or 50th percentile).if a student receives a grade equivalent of 7.2, then this student s score corresponds to that of an average (median) seventh-grader after two months of instruction. Grade-Level Equivalents make it immediately clear whether a student is ahead, behind, or average with respect to his or her test performance. Growth Score Benchmarker generates a growth score after the second and third test administrations. The growth score is simply the recent scale score subtracted by an earlier scale score. For example, the winter growth score is determined by subtracting the fall scale score from the winter scale score. A student s growth for the entire school year is determined by subtracting the fall scale score from the spring scale score. Because these scores are based on an equal interval, vertical scale, Benchmarker growth scores can be used to compare growth among students, or among groups of students, such as classes and grades. Chapter 03 Benchmarker 16

23 Configuring Assessment Options Benchmarker can be administered in one of two ways. The online Administration Panel can be configured to give Screener and Benchmarker in one integrated assessment session, or as two separate assessments. In the latter case, only the students identified as at-risk or borderline by Screener take Benchmarker afterwards. For those electing to administer Benchmarker as a separate assessment from Screener, there is an additional option to have students take Benchmarker immediately after their screening (if they test positively for risk), or on a subsequent day. As with the Symphony Screener, there are three testing windows each year. The dates of these testing windows are the same for Benchmarker as for Screener, and can be configured in the online Administration Panel. Chapter 03 Benchmarker 17

24 04 symphony math intervention Introduction Symphony Math is an intervention program designed to help students develop a profound understanding of the most important mathematical concepts. Many students struggle to become proficient in math because they do not have the opportunity to master foundational concepts with sufficient depth. In an age when most curricula value covering a large number of topics, some students are falling through the cracks. They need more time and more practice working with the big ideas of mathematics in order to develop the proper foundation. Symphony Math is an educational software program that provides students with the experience of learning and thinking about the most important mathematical concepts. This experience provides the necessary foundation for a successful future of math learning. Symphony Math helps students achieve this solid mathematical foundation by implementing several key research-based pedagogic strategies. These strategies are recommended by leading experts in the field of mathematics education (see Research Base in the Appendix), and have improved student learning and understanding. The Big Ideas of Mathematics The conceptual sequence of Symphony Math consists of a tightly connected progression of the most important mathematical ideas. These underlying big ideas are important because they provide the foundation for later mathematical learning. The program helps students develop a profound understanding of the big ideas listed in this table. Chapter 04 Intervention 18

25 Mathematical Topic Underlying Big Idea Stages Number Quantity 1, 16 Addition and Subtraction Part-to-whole relations 2, 4, 6 Place Value Hierarchical groupings 3, 5 Multiplication and Division Repeated equal groupings 8, 13 Multi-digit Addition and Subtraction Fractions Hierarchical groupings coordinated with part-to-whole relations Repeated equal groupings coordinated with part-to-whole relations 7, 9, 11 12, 14, 15, 17 Symphony Math guides students through a carefully constructed sequence of these big ideas which are broken down into smaller concepts and presented in a developmental sequence. It s easier for students to work through the big ideas in smaller parts, since it can be difficult to internalize them all at once. Each concept provides the foundation for the subsequent concept, and later concepts are built upon and coordinated with earlier concepts. Symphony Math s conceptual sequence provides an in-depth learning experience because it gradually builds the complexity of problems to a more advanced level than students typically encounter. For example, to help students develop a deep understanding of Quantity (or number sense), the program guides students through the following sequence of concepts from Stage 1: Concept Example Solution Equal 3 =? 3 = 3 Greater 8 >? 8 > 2 Less 5 <? 5 < 7 Between 4 <? < 9 4 < 6 < 9 Missing comparison 2? 5? 7 2 < 5 < 7 Not equal 7? 7 10 A student does not move on to the next concept in the sequence until she has mastered the current concept. One concept follows logically from the previous concept. While a student is working on a new concept she sees review concepts that help support her learning of the new concept. This process helps the student connect new knowledge to previous knowledge. Numbers represent relative quantities, or amounts. This is the fundamental idea that students work on in Stage 1. Symphony Math challenges students to develop an internal number line so that they can quickly and easily see numbers as a network of related quantities. Solving more complex problems (e.g., 1 <? < 6, solved three different ways) leads to a more robust understanding for the student. Compare this to the more superficial understanding some students develop during elementary school, where numbers are just a sequence of memorized sounds and symbols. Counting is an important skill. However, it is only the beginning of understanding the number system. Chapter 04 Intervention 19

26 Multiple Ways of Knowing Integrated with the Symphony Math conceptual sequence are six distinct activity environments. These activities provide multiple representations of each concept in the program. Activity Manipulatives Manipulatives & Symbols Symbols Auditory Sentences Story Problems Mastery Round Purpose Conceptually understand what the concept looks like Explicitly connect symbols to their visual representations Understand the concept at the abstract level Learn the formal language of mathematics Extend understanding to real life problem solving Develop immediate recall of number relationships The use of different activities accomplishes several goals. First, it introduces concepts at the concrete (or visual) level. Students can see what the concept looks like, and this helps them develop a mental model of its meaning. In stage two, Intro to Addition & Subtraction, students work with manipulatives to internalize the Parts-to-Whole big idea that underpins addition and subtraction. In this stage, students encounter problems like the one below: Five bar must be inserted in this area. A three bar and a two bar are combined. Students must find a number bar that is the same length as the 3 bar and the 2 bar combined. Later, they will see a problem like the one below. The whole is on top and is represented by the yellow 5 bar. One of the parts is below and is represented by the blue 4 bar. The student needs to find the length of the missing part. In symbols, this problem would be represented as? + 4 = 5 Chapter 04 Intervention 20

27 The whole is five units long. A part is missing. (One unit long.) In this way students come to understand how to solve problems like =? and 4 +? = 9 using manipulatives. The process helps students understand the Parts-to-Whole big idea and its different variations. The use of a variety of activities also helps students make the connection between symbols and the concepts that the symbols represent. At a superficial level some students can memorize the counting sequence and basic number facts without appreciating their meaning. A student may understand that = 7 because 7 comes 2 numbers after 5. However, the student may not know what = 7 looks like concretely, or may not know why it is also true that = 7. The Manipulatives & Symbols activity helps with this issue by explicitly challenging students to connect symbols with concrete representations. A student will be given a number sentence such as =? and must construct the corresponding visual representation. Or, they will be given the problem with number bars and they must provide the corresponding symbols = 9 is given with number bars. Student must construct the corresponding problem with symbols. In this way, students explicitly connect symbols with visual representations and manipulatives with symbolic representations. This is the bridge from the concrete Chapter 04 Intervention 21

28 (manipulatives) to the abstract (symbols) and helps students connect their intuitive understanding of number relationships to the formal number system we use to represent them. The third activity uses only symbols. Now that the student has demonstrated proficiency with the manipulatives in the first activity and the meaning of symbols in the second activity, she gains proficiency with procedures in the the third activity. The Symbols activity presents problems with symbols, but the integer bars will appear automatically if a student makes a mistake or asks for help. The fourth activity emphasizes spoken language. Students hear the problem, then must construct it with symbols and solve it. This activity helps students learn the formal language of mathematics and connect it to the symbols and manipulatives from earlier activities. The fifth activity challenges students with story problems. Students are presented with a written word problem as shown below. Students must construct the corresponding number sentence and solve it. If they need to, they can press a button and have the story problem read out loud, with each word highlighted as it is read. This activity challenges students to extend their knowledge to real life problems. Story problems are traditionally quite hard for struggling math students. A student s previous in-depth work with manipulatives, symbols, and language helps provide the conceptual foundation and experience they need to succeed. Solution is constructed here. Students can press this button to hear the story problem. The sixth activity environment is the Mastery Round, shown on the next page. This is a fluency activity that helps students develop immediate recall of number relationships. Students are presented with number relationships problems with symbols or spoken language. They need to answer correctly before the problem disappears to demonstrate mastery. Only concepts and number relationships with which students have demonstrated proficiency in an untimed setting are presented during the mastery round. This avoids the too-common problem of encouraging students to memorize what they do not understand. Chapter 04 Intervention 22

29 Students need to solve the problem before the cube falls away from view. Chapter 04 Intervention 23

30 Learning Through Thinking and Making Connections The pedagogic style of Symphony Math emphasizes thinking, figuring out, and making connections. The program is designed to be used as a complement to the classroom learning experience. Students receive direct instruction and group learning in a classroom setting. The program provides the opportunity for individual practice at the developmental level of each student. The style of this practice encourages independent thinking and problem solving, and this is accomplished through the use of three important pedagogic strategies. Instructive Feedback Symphony Math encourages independent thinking by providing instructive feedback that reveals the nature of each incorrect response. For example, if a student answers 3 +? = 6 with a 2, the program immediately shows that a 3 bar combined with a 2 bar is not the same length as a 6 bar. The student solves this number sentence incorrectly. The number bars automatically appear to illustrate that a 3 bar and a 2 bar are not the same length as a 6 bar. Providing instructive feedback encourages students to deduce for themselves why an incorrect answer is not accepted. This is preferred to saying something like that s not right, try again because that approach often leads to guessing and not understanding why the answer is not correct. Using the Help Button Another way that Symphony Math encourages learning through thinking is the use of the Help button. The Help button provides scaffolding, which leads the student closer to the solution but does not give the answer immediately. Additionally, the scaffolding provided by the Help button does not directly explain the procedures for achieving the solution. Immediately providing the correct procedures or solution undermines the thinking and understanding of the student and encourages her to depend on the program for solutions instead of developing her own problem solving skills. If a student is confronted with the problem =?, she can press the Help button to activate scaffolding that will help her connect with her existing knowledge of the concept and number relationships. This maintains the emphasis on thinking and making connections. If she needs more help, she can continue to press the Help button. The sequence of the help scaffolding for the problem =? is shown in the table below. Chapter 04 Intervention 24

31 Help Button Activation Help Provided for the Problem =? 1st Show a near neighbor : = 8 2nd Show a second near neighbor : = 10 3rd 4th Show using number bars Show that the 9 bar is equal in length to the 8 and 1 bar The Help button provides hints and scaffolding that challenge students to make inferences and connect what they are learning to concepts they have already mastered. This helps guide the disposition of the student toward problem solving and independent thinking, not copying or guessing. In-Depth Problem Solving A third strategy that Symphony Math uses to encourage independent thinking is the use of specially designed problems that provoke thinking. For example, to master place value concepts students will be asked to solve a series of problems designed to help understand the base ten system. Students will combine numbers of different place values, such as =?. They will also be asked to create number sentences where the sum is provided but the addends are missing, such as? +? +? = 286. Each addend must correspond to the ones, tens and hundreds place value (e.g = 286). At the most difficult level, students need to provide three different solutions to this type of problem. Each stage in the program has specially designed problems that keep the focus on thinking and understanding. Students are not specifically told how to solve problems such as? +? +? = 431 or? -? = 3, but they logically deduce the solutions by connecting these problems to similar problems that they have previously mastered. Chapter 04 Intervention 25

32 Dynamic Branching Allows Students to Learn at Their Own Level Symphony Math works with each student at his or her developmental level. The dynamic branching of the program and detailed progression of the scope and sequence allows students to work within their developmental zones. An advanced student who has complete mastery of all of the program s concepts can complete the program in less than two hours. An intermediate student might move through the program to stage eight in less than an hour, but then spend several weeks working on stage eight because that is her specific area of need. A younger or struggling math student may only progress to stage two and then spend the rest of the school year working on Stage 2 concepts. The amount of time and practice that students need to understand mathematical concepts is not uniform. Symphony Math allows students to spend the time they need mastering foundational concepts. In addition, the program quickly moves students through the conceptual progression of the program to identify their area of need. Once the area of need has been identified, the program slows the progress until the necessary understanding has been achieved. Dynamic branching not only assesses students understanding vertically through the hierarchy of stages, but it also tracks students proficiency horizontally across the six different activity environments. In addition, Symphony Math tracks and branches students according to their mastery of specific number relationships. For example, the program tracks whether a student understands what 1/6 of 12 is, or 2/3 of 9. The program systemically records which number relationships have been mastered and which have not, and branches students accordingly. Another example is the number relationships for multiplication and division. Not only does the program track basic multiplication and division number relationships such as 2 x 3 and 8 4, but it also records progress with number relationships such as missing dividend (? 4 = 2), missing multiplicand (3 x? =12), and missing multiplier (? x 2 = 8). This detailed level of data tracking allows Symphony Math to find a student s area of need down to the specific concept, representation, and number relationship. The branching responds in a way that helps the student fill in these gaps in her learning while quickly moving through those areas that have been mastered. Chapter 04 Intervention 26

33 Comprehensive Reports With each day of student use, Symphony Math provides teachers with detailed reports that inform their classroom instruction. The program reports provide a big-picture look at where students are in their overall mathematical development. Detailed reports inform teachers of students progress with specific concepts and number relationships. The Symphony Dashboard alerts teachers to specific issues that need to be addressed. For example, the Dashboard may alert a teacher about a student who needs help with missing addends using symbols. Digging deeper into the reports might reveal that the student is specifically having difficulty with 9 +? = 16 and 5 +? =13. Chapter 04 Intervention 27

34 Who Can Use Symphony Math? The rigorous application of cognitive development principles in Symphony Math make it a suitable intervention program for a wide range of students. The program seeks to identify gaps in each student s mathematical understanding, and then present a series of problems designed to fill in those gaps. Grades Category Objective Recommended Use K - 2 Gifted Enhancement 15 minutes 2 times per week 1-4 On Grade Level 2-6+ Remediation Deepen understanding and support instruction Identify and fill in gaps in mathematical foundation minutes 3 times per week 20 minutes 5 times per week Gifted students in kindergarten through second grade may use Symphony Math to enhance their knowledge and move ahead to where their ability and motivation takes them. Students performing on grade level in grades one through four can use the program to deepen their understanding and to support teachers classroom instruction. Struggling students in grades two through six (and beyond) may use the program to identify gaps in their mathematical foundation and begin to fill them in. The age neutral interface of Symphony Math makes it comfortable for a wide age range of students to enjoy and benefit from the program. Chapter 04 Intervention 28

35 05 intervention scope and sequence Introduction Symphony Math is an intervention program designed to help students develop a profound understanding of the most important mathematical concepts. Symphony Math focuses on the big ideas of mathematics the foundational knowledge upon which all later math learning is based. The six big ideas developed in Symphony Math include: Quantity Parts-to-Whole Hierarchical Groupings Repeated Equal Groupings Hierarchical Groupings with Parts-to-Whole Repeated Equal Groupings with Parts-to-Whole. Each big idea consists of a network of related concepts which are sequenced throughout the seventeen stages of Symphony Math. Another important component of the Symphony Math Scope & Sequence is the multiple representations of each concept. The program uses a mixture of manipulatives, symbols, language, story problems, and auditory and symbolic fluency to help students make connections between representations and more fully understand each big idea. While this chapter lays out the sequence of stages and concepts, it is important to remember that each concept utilizes multiple representations throughout the program. See Chapter 1 for an overview of the different representations used in Symphony Math. Chapter 05 Scope and Sequence 29