TEAC MATI MEA CHING HEMATICS NINGFULLY Solutions for Reaching Struggling Learners David H. Allsopp LouAnn H. Lovin Sarah van Ingen --- #### FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) # Teaching Mathematics Meaningfully ## Solutions for Reaching Struggling Learners ### Second Edition by **David H. Allsopp, Ph.D.** University of South Florida Tampa ##### LouAnn H. Lovin, Ph.D. James Madison University Harrisonburg, Virginia and ##### Sarah van Ingen, Ph.D. University of South Florida Tampa Baltimore • London • Sydney Excerpted from Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, Second Edition by David H. Allsopp, Ph.D.,LouAnn H. Lovin, Ph.D., & Sarah van Ingen, Ph.D. --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) **Paul H. Brookes Publishing Co.** Post Office Box 10624 Baltimore, Maryland 21285-0624 USA www.brookespublishing.com Copyright © 2018 by Paul H. Brookes Publishing Co., Inc. All rights reserved. Previous edition copyright © 2007. “Paul H. Brookes Publishing Co.” is a registered trademark of Paul H. Brookes Publishing Co., Inc. Typeset by Absolute Service, Inc., Towson, Maryland. Manufactured in the United States of America by Sheridan Books, Inc., Chelsea, Michigan. All examples in this book are composites. Any similarity to actual individuals or circumstances is coincidental, and no implications should be inferred. Purchasers of Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, Second *Edition, are granted permission to download, photocopy, and print the forms and activities found in* Appendices A–D for educational and professional purposes. This material may not be reproduced to generate revenue for any program or individual. Photocopies may only be made from an original book. Unauthorized use beyond this privilege may be prosecutable under federal law. You will see the copyright protection notice at the bottom of each photocopiable page. **Library of Congress Cataloging-in-Publication Data** Names: Allsopp, David H., author. | Lovin, LouAnn H., author. | van Ingen, Sarah, author. Title: Teaching mathematics meaningfully: solutions for reaching struggling learners / by David H. Allsopp, Ph.D., University of South Florida, Tampa, LouAnn H. Lovin, Ph.D., James Madison University, Harrisonburg, Virginia, and, Sarah van Ingen, Ph.D., University of South Florida, Tampa. Description: Second edition. | Baltimore: Paul H. Brookes Publishing Co., [2018] | Includes bibliographical references and index. Identifiers: LCCN 2017027635 (print) | LCCN 2017033234 (ebook) | ISBN 9781598575590 (epub) | ISBN 9781598575637 (pdf) | ISBN 9781598575583 (pbk.) Subjects: LCSH: Mathematics—Study and teaching (Elementary) | Mathematics—Study and teaching (Middle school) | Mathematics—Study and teaching (Secondary) | Attention-deficit- disordered youth—Education. | Learning disabled teenagers—Education. Classification: LCC QA13 (ebook) | LCC QA13 .A44 2018 (print) | DDC 371.9/0447—dc23 LC record available at [https://lccn.loc.gov/2017027635](https://lccn.loc.gov/2017027635) British Library Cataloguing in Publication data are available from the British Library. 2021 2020 2019 2018 2017 10 9 8 7 6 5 4 3 2 1 --- ## FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) # Contents ### About the Activities and Forms ... v ### About the Authors ... vii ### Preface ... ix ### Acknowledgments ... xv 1 Critical Components of Meaningful and Effective Mathematics Instruction for Students with Disabilities and Other Struggling Learners ... 1 ### I Identify and Understand the Mathematics 2 The Big Ideas in Mathematics and Why They Are Important ... 15 3 Children’s Mathematics: Learning Trajectories ... 41 ### II Learning the Needs of Your Students and ### the Importance of Continuous Assessment 4 Barriers to Mathematical Success for Students with Disabilities and Other Struggling Learners ... 69 5 Math Assessment and Struggling Learners ... 97 ### III Plan and Implement Responsive Instruction 6 Making Flexible Instructional Decisions: A Continuum of Instructional Choices for Struggling Learners ... 137 7 Essential Instructional Approaches for Struggling Learners in Mathematics ... 155 8 Changing Expectations for Struggling Learners: Integrating the Essential Instructional Approaches with the NCTM Mathematics Teaching Practices ... 217 9 Mathematics MTSS/RTI and Research on Mathematics Instruction for Struggling Learners ... 239 iii --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) iv Contents 10 How to Intensify Assessment and Essential Instructional Approaches within MTSS/RTI ... 247 11 Intensifying Math Instruction Across Tiers within MTSS: Evaluating System-Wide Use of MTSS ... 269 ## IV Bringing It All Together 12 The Teaching Mathematics Meaningfully Process ... 281 ## References ... 299 **Appendices** A Take Action Activities ... 313 B ARC Assessment Planning Form ... 337 C Peer-Tutoring Practice Activity ... 341 D Using a Think-Aloud ... 343 E Case Study ... 345 ## Index ... 375 --- ## FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) # About the Authors ### David H. Allsopp, Ph.D., Professor of Special Education, College of Educa- tion, University of South Florida, 4202 East Fowler Avenue, EDU 105, Tampa, Florida 32620 Dr. Allsopp is Assistant Dean for Education and Partnerships in addition to being the David C. Anchin Center Endowed Chair and Director of the David C. Anchin Center at the College of Education at the University of South Florida. He is also Professor in the Department of Teaching and Learning—Special Education Programs. Dr. Allsopp holds degrees from Furman University (B.A., Psychology) and the University of Florida (M.Ed., Learning Disabilities; Ph.D., Special Education). Dr. Allsopp teaches at both the undergraduate and doctoral levels, and his scholarship revolves around effective instructional practices, with an emphasis on mathematics, for students with high-incidence disabili- ties (e.g., specific learning disabilities, attention-deficit/hyperactivity disorder, social-emotional/behavior disorders) and other struggling learners who have not been identified with disabilities. Dr. Allsopp also engages in teacher edu- cation research related to how teacher educators can most effectively prepare teachers to address the needs of students with disabilities and other struggling learners. Dr. Allsopp began his career in education as a middle school teacher for students with learning disabilities and emotional/behavioral difficulties in Ocala, Florida. After completing his doctoral studies at the University of Florida, Dr. Allsopp served on the faculty at James Madison University for 6 years. He has been a member of the faculty at University of South Florida since 2001. ### LouAnn H. Lovin, Ph.D., Professor of Mathematics Education, Department of Mathematics and Statistics, James Madison University, 800 South Main Street, MSC 1911, Harrisonburg, Virginia 22807 Dr. Lovin began her career teaching mathematics to middle and high school stu- dents before making the transition to Pre-K through Grade 8. For over 20 years, she has worked in elementary and middle school classrooms. Then and now, Dr. Lovin engages with teachers in professional development as they implement a student-centered approach to teaching mathematics. At the time of this publica- tion, she focused her research concerning teachers’ mathematical knowledge for teaching on the developmental nature of prospective teachers’ fraction knowledge. She has published articles in Teaching Children Mathematics, Mathematics Teaching *in the Middle School, Teaching Exceptional Children, and the Journal of Mathematics* *Teacher Education. She coauthored the Teaching Student-Centered Mathematics* *Professional Development Series with John A. van de Walle, Karen Karp, and Jenny* vii --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) viii About the Authors Bay-Williams ( Pearson, 2013). Dr. Lovin is an active member of the National Coun- cil of Teachers of Mathematics, the Association of Mathematics Teacher Education, and the Virginia Council of Teachers of Mathematics. ## Sarah van Ingen, Ph.D., Assistant Professor of Mathematics Education, Depart- ment of Childhood Education and Literacy, College of Education, University of South Florida, 4202 East Fowler Avenue, EDU 202, Tampa, Florida 33620 Dr. van Ingen codirects the innovative and nationally recognized Urban Teacher Residency Partnership Program. In this role, she partners with Hillsborough County Public Schools’ teachers and administrators to improve the learning of both elementary students and prospective elementary teachers. She also teaches courses in mathematics education and teacher preparation at the undergraduate, masters, and doctoral levels. Dr. van Ingen holds a bachelor’s degree from St. Olaf College, a master of arts in teaching from the University of Tampa, and a doctoral degree from the University of South Florida. She was elected into membership in Phi Beta Kappa and was the recipient of the prestigious STaR fellowship in mathematics education. She taught mathematics for many years in urban, inclu- sive middle school classrooms before her work at the university level. Dr. van Ingen’s research agenda lies at the intersection of equitable math- ematics education and clinically rich teacher preparation. Her research interests include teachers’ use of research to inform practice, the use of mathematics con- sultations to meet the mathematics learning needs of students with exceptionali- ties, and the implementation of integrated STEM lessons in K–5 classrooms. She regularly publishes and presents her research to audiences who work in math- ematics education, special education, and teacher preparation. She is the principal investigator and coprincipal investigator for federally funded research and is active in leadership in her professional organizations. --- ## FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) # Case Study This appendix provides a case study in which two teachers, Ms. Thompson and Mr. Hart, apply the Teaching Mathematics Meaningfully Process with their strug- gling learners. The teachers implement each step of the process. The contents are organized as follows: #### • Purpose • Meet Ms. Thompson, Mr. Hart, and Their Students #### • Identify and Understand the Mathematics #### • Continuously Assess Students #### • Determine Students’ Math-Specific Learning Needs • Determine Struggling Learners’ Specific Learning Needs #### • Plan and Implement Responsive Instruction #### • Take Action ### PURPOSE The purpose of this case study is to provide you with a way to visualize how two teachers, an elementary general education math teacher and a special edu- cation teacher, might work collaboratively to utilize the Teaching Mathematics Meaningfully Process. We first introduce you to the teachers and their students. Then, we describe how the two teachers implement each of the five components of the process. Our goal is to provide you with an applied context for making sense of this process and to illustrate the types of decision making that will help you design instruction and interventions that are responsive to your students’ needs. Throughout the case study, marginal icons are included to indicate activities and decisions that illustrate specific Essential Instructional Approaches (EIAs), National Council of Teachers of Mathematics (NCTM) Effective Mathemat- ics Teaching Practices (MTPs), and anchors for intensifying instruction within multi-tiered systems of supports/response to intervention (MTSS/RTI). Each icon includes a number indicating which EIA, MTP, or anchor for intensifying instruction that it denotes. A key to these icons is provided next. Use the icons as you read to see how various elements of instruction discussed throughout the book are applied and integrated within the teachers’ classroom planning and instruction. --- FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully | Key | | | --- | --- | | Essential Instructional Approaches (EIAs) EIA | 1. Teach systematically. 2. Develop and explicitly share learning intentions. 3. Make instructional decisions that are student-centered and based on meaningful data. 4. Teach mathematical fluency. 5. Teach the language of mathematics through vocabulary development and discourse. 6. Provide many response opportunities with feedback. 7. Emphasize use of mathematical practices. 8. Utilize visuals. 9. Use different appropriate grouping structures. 10. Teach students to be strategic in their approach to mathematics. 11. Situate mathematics within meaningful contexts that help students to develop abstract reasoning. | | NCTM(2014b) Effective Mathematics Teaching Practices(MTPs) MTP | 1. Establish mathematics goals to focus learning 2. Implement tasks that promote reasoning and problem solving 3. Use and connect mathematical representations 4. Facilitate meaningful mathematical discourse 5. Pose purposeful questions 6. Build procedural fluency from conceptual understanding 7. Support productive struggle in learning mathematics 8. Elicit and use evidence of student thinking | | Anchors for Intensifying Instruction within MTSS/RTI(IAs) IA | 1. Purposeful Content Focus 2. Formative Assessment—Identifying What Students Know, Don’t Know,and Why 3. Explicitness and Teacher Direction 4. Teach Math Metacognition 5. Opportunities to Respond 6. Amount of Time 7. Teacher-Student Ratio | MEET MS. THOMPSON, MR. HART, AND THEIR STUDENTS Ms. Thompson is an elementary general education teacher who teaches fifth grade. She has been teaching elementary school students for 8 years; she spent 5 of those years teaching fourth and fifth grade. She has 22 students (10 boys and 12 girls) in her class. Fifteen are white/Caucasian, five are African American, one is Mexican American, and one is Korean American. Ms. Thompson’s class includes five students identified as having disabilities. Four are identified as having learning disabilities and receive special education services through the Individuals with Disabilities Education Improvement Act (IDEA) of 2004 (PL 108-446). One is identified as having attention-deficit/hyperactivity disorder (ADHD) and is supported through a Section 504 accommodation plan. In general, the students not identified with disabilities perform at grade level or above. Three additional students, MEET MS. THOMPSON, MR. HART, AND THEIR STUDENTS Excerpted from Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, Second Edition --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 347 who are not identified as having disabilities, sometimes struggle with math and reading. Two of these students are English language learners. Mr. Hart is a special education teacher who works with Ms. Thompson in a consultation and facilitation role, helping her support the needs of her students with disabilities, particularly in reading and mathematics. Mr. Hart co-teaches with Ms. Thompson during core instruction and is responsible for providing intensive instructional support for students who have the most difficulty in meet- ing core standards. Mr. Hart also provides input to the teams for Grades 3–5 regarding supplemental instructional support for students receiving exceptional student education (ESE) services. In Ms. Thompson’s class, the morning begins with a 60-minute block of core mathematics instruction, followed by a 120-minute reading or English language arts block. During an additional 50-minute block after lunch, all students engage in some type of supplemental or intensive instruction or enrichment for reading, mathematics, or both. Math standards in this state are closely aligned with the Common Core State Standards (CCSS). ## IDENTIFY AND UNDERSTAND THE MATHEMATICS For the Identify and Understand the Mathematics component, you will read how Ms. Thompson and Mr. Hart prepared themselves to teach the mathematical content. In essence, we pull back the curtain to show you the behind-the-scenes preparation that equips teachers with the mathematical understanding necessary for this process. Ms. Thompson and Mr. Hart go through three stages for this first component. First, they identify the relevant mathematics standards. Second, they look for and learn from an available trajectory that describes how students progress through various stages of learning related to the standards. Third, they consider the role mathematical practices have in the learning process with respect to the identified content. ## Math Standard Ms. Thompson and Mr. Hart’s first task is to identify the content that they will be teaching. Based on the curriculum map for fifth graders in their district, they are planning to teach a unit on multiplying multi-digit whole numbers using the stan- dard algorithm. The relevant CCSS standard for fifth grade is as follows (National Governors Association [NGA] Center for Best Practices & Council of Chief State School Officers [CCSSO], 2010). CCSS.MATH.CONTENT.5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm. In thinking about teaching this standard, Ms. Thompson knows that she and Mr. Hart need to unpack the included content. She also knows they need to consider how this content connects to what students have previously been exposed to in fifth grade as well as in earlier grades in order to ensure students have the prerequisite knowledge to engage successfully with this content. Keep- ing in mind where their students will be headed in future mathematics lessons, Ms. Thompson and Mr. Hart also look to related standards, both within the --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) 348 Appendix E grade level and beyond, to help them be purposeful in making decisions about instructional tasks and about how to leverage students’ current mathematical ideas. Ms. Thompson thinks about how these standards connect to other fifth-grade standards, including what students have already been exposed to this year and what they will be expected to learn later in the year, and she shares her thoughts with Mr. Hart. With respect to Number and Operations in Base Ten, their stu- dents have worked on the following standard to further their understanding of the place value system (NGA Center for Best Practices & CCSSO, 2010): CCSS.MATH.CONTENT.5.NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Ms. Thompson notes that the remaining fifth-grade standards for Number and Operations in Base Ten are related to two other mathematical ideas: 1) developing division strategies with whole numbers and 2) developing the four operations (addition, subtraction, multiplication, and division) with decimals to hundredths using place-value ideas, properties of operations, and relationships between the various operations. Ms. Thompson and Mr. Hart look at what their students were exposed to in fourth grade related to multiplication of multi-digit numbers. In particular, they consider the following fourth-grade CCSS standard (NGA Center for Best Practices & CCSSO, 2010): CCSS.MATH.CONTENT.4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equa- tions, rectangular arrays, and/or area models. Ms. Thompson knows the standard algorithm for multi-digit multiplication is one of the most difficult algorithms; students are very error prone when using this algorithm, especially when they have not had ample opportunities to work with multiplication strategies based on place-value concepts and representations such as area models. Both Ms. Thompson and Mr. Hart are aware of the importance of developing students’ procedural knowledge from conceptual understanding (NCTM, 2014b). This leads them to recognize that before they work to help stu- dents develop the standard multi-digit multiplication algorithm, they will need to revisit this related fourth-grade standard to ensure students have developed a rigorous conceptual understanding of multiplication. With the insights Ms. Thompson and Mr. Hart have developed by considering not only the target standard but also how it relates to other standards from the previous and current grade levels, they are establishing a better sense about how to build on students’ prior knowledge to understand and apply the new fifth- grade multiplication standard. Mr. Hart appreciates the way Ms. Thompson goes deeper in thinking about the content and related learning intentions they have for their students, including connecting the math their students will be learning to Excerpted from Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, Second Edition by David H. Allsopp, Ph.D.,LouAnn H. Lovin, Ph.D., & Sarah van Ingen, Ph.D. --- FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully that which they have already experienced. This helps Mr. Hart think about what prerequisite content, both at the grade level and below it, he will need to emphasize when he provides more intensive instruction to his students. He knows this content must connect to core math standards. Related Learning Trajectory Related Learning Trajectory Ms. Thompson knows the usefulness of learning trajectories in providing a road map for how student thinking progresses and how to sequence learning experiences to maximize learning of a mathematical concept or skill. She searches online for multiplication trajectories and finds one that describes a progression of students’ multiplicative reasoning and strategies. Table E.1 includes the more sophisticated ways of reasoning multiplicatively in this learning trajectory, more likely to be exhibited by older elementary students. (For the full trajectory, see the section on multiplicative reasoning in Chapter 3.) Ms. Thompson is also aware of several learning trajectories she can find online when she is working on other mathematical concepts, such as https://www.turnonccmath.net and http://www.numeracycontinuum.com/continuum-chart. Table E.1. The upper levels of multiplicative reasoning demonstrated by students | Level 3: Transitional multiplicative strategies(see Figure E.1a) | At this level, students demonstrate an increasingly robust capability of reasoning with multiples as their use of groups becomes more sophisticated. They no longer have to count each group by ones. Strategies such as using area models and open arrays are used to reason through multiplicative situations. | | --- | --- | | Level 3.3: Repeated abstract composite grouping | Students are aware that a number can be both composite and unitary at the same time, but at this level, students can only think of one of the numbers in a multiplication situation (one of the factors) in this way. For example, with $3\times4$, students are able to consider the 4 as both a composite unit and unitary at the same time, but they only think of the 3 as unitary—as a way to count the number of fours. They can see the 4 as consisting of 4 single units (unitary) but can also see (or make sense of) the 4 as one “thing” (a composite unit).For $3\times4$, they would reason $4+4+4$ (4 three times). | | Level 4: Multiplicative strategies(see Figure E.1b) | At this point, students can reason about multiplication and division using the more sophisticated strategies that rely primarily on numerical representations such as partial products,the distributive property,and doubling and halving of quantities. | | Level 4.1: Multiplication and division as operations | At this level, students can coordinate two composite units in the context of multiplication or division.For example,with a task such as six groups of four,the student is aware of both6and4as abstract composite units.The 6 can be used as a count of the groups of four but can also be considered its own composite unit.As a consequence,the commutative property of multiplication makes sense to the student.The student is able to immediately recall and quickly derive many of the basic facts for multiplication and division. | --- | Later transitional strategies | | | | | --- | --- | --- | --- | | Area model(less reliant on needing to see every square unit) | Open area model | | Open area model | | 4×6=244 | 7×12=70+14=847 | | 23×45=800+120+100+15=1035405203 | | | 70 | 14 | 800100 | | | 12015 | | | 7\times12=70+14=84 4\times6=24 120+100+15=1035 23\times45=800+ b | Multiplicative strategies | | | | --- | --- | --- | | Known or derived facts8×6=48because4×6=24and we need to double that | Commutative property6×8=8×6 | Powers of 103×500=3×50×10=3×5×10×10=15×100=1500 | | Associative property(4×6)×5=4×(6×5)=4×30=120 | Doubling and halving18×3=9×6=54halfof183doubled | Distributive property8×12=8(10+2)=8(10)+8(2)=80+16=96 | | Partial products23$\frac{x\cdot45}{15}$ (5×3)100 (5×20)120 (40×3)$\frac{+800}{1035}$ (40×20) | Standard algorithm1$\frac{1}{2}$23$\frac{x\cdot45}{115}$$\frac{+920}{1035}$ | | \cdot4\times6=24\cdot 6\times8=8\times6 1035 1035 Figure E.1. Examples of strategies students use to engage in multiplicative reasoning: later transitional strategies (a) and 3\times500=3\times50\times10 (4\times6)\times5 18\times3=9\times6=54 =120 \begin{array}{c}{\frac{\times\ 5}{1\ 1}}\\ {\frac{\ 1\ 020}{\ 10\ 335}}\end{array} --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 351 From studying this trajectory, Ms. Thompson and Mr. Hart come to under- stand that students who are ready to develop the standard algorithm for multi- digit multiplication should be using the later (more developed) multiplicative strategies seen in Level 4 of the learning trajectory, which are based on the various properties and on strategies such as doubling and halving (see Figure E.1B). They should also already be proficient in using the partial products algorithm. Students who are not there yet may be using the later transitional strategies that rely on an area model, or they may just be developing proficiency with the partial products algorithm. Some may also exhibit even less sophisticated reasoning that relies on skip counting or inefficient additive strategies. (See Chapter 3 for information per- taining to these lower levels of reasoning.) Ms. Thompson and Mr. Hart decide to use this learning trajectory to help them identify the level of sophistication of their students’ reasoning. For students whose assessment results indicate less sophisti- cated reasoning, interventions will need to be used. ## Math Practices Ms. Thompson and Mr. Hart remember that their students not only need to learn the math content within the target standards but also need to learn how to mean- ingfully engage with this content in different ways through the Common Core Eight Standards for Mathematical Practice (see Chapters 2 and 7). Textbox E.1 shows these practices as a reference. As Ms. Thompson thinks about each of the math practices, she realizes many could be appropriately utilized in conjunction with the target standard. To make things more manageable, she identifies two she wants to explicitly emphasize within her instruction. (These practices are bold in Textbox E.1.) Ms. Thompson determines that one practice, Look for and make use of structure (NGA Center for Best Practices & CCSSO, 2010), fits well because her students will be learning to mul- tiply multi-digit whole numbers by relating their understanding of area models, the distributive property, and place value to the standard algorithm. She chooses a second practice, Construct viable arguments and critique the reasoning of others (NGA Center for Best Practices & CCSSO, 2010), because she wants to help her students **Textbox E.1.** Common Core Eight Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. **Construct viable arguments and critique the reasoning of others.** 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. **Look for and make use of structure.** 8. Look for and express regularity in repeated reasoning. From Common Core State Standards Initiative. (2010). Common Core State Standards for mathematics. Retrieved from [http://www.corestandards](http://www.corestandards) .org/assets/CCSSI_Math%20Standards.pdf; reprinted by permission. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Excerpted from Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, Second Edition by David H. Allsopp, Ph.D.,LouAnn H. Lovin, Ph.D., & Sarah van Ingen, Ph.D. --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) 352 Appendix E become more comfortable and proficient with engaging in discourse, appropri- ately listening to and critiquing their peers’ arguments and reasoning. Because the standard multiplication algorithm requires a level of precision in order to ensure the proper digits are multiplied together and recorded appropri- ately, Mr. Hart suggests that they also emphasize with their students the practice of Attend to precision (NGA Center for Best Practices & CCSSO, 2010) when they write down the procedure. Mr. Hart knows this will be an important point of emphasis for several students who tend to be more impulsive and struggle with self-regulation strategies related to organization and time management during independent math work. ## CONTINUOUSLY ASSESS STUDENTS For the Continuously Assess Students component, Ms. Thompson and Mr. Hart work collaboratively to assess students’ specific learning needs related to the identified content standards and mathematical practices. First, Ms. Thompson reviews stu- dents’ available benchmark assessment data, using them to project which areas related to multiplication of whole numbers might require further formative infor- mal assessment. Then, Ms. Thompson and Mr. Hart create several assessment tasks to get at these areas. Last, they reflect on their students and possible ways to engage them in responding to the assessment tasks so that they can best deter- mine what their students know, don’t know, and why. ## Review Available Benchmark Assessment Data Ms. Thompson and Mr. Hart’s school district collects grade-level benchmark data three times during the school year. The multiple-choice benchmark assessments relate directly to the state standards and the end-of-year high-stakes test. Each math benchmark assessment is completed online and evaluates where students are in relation to the standards they have been exposed to when the benchmark assessment is administered (early September, early December, and mid-February). Each typically takes approximately 45–60 minutes to complete. School personnel use the first benchmark assessment in early September to make initial decisions about supplemental and intensive tiered instruction. The school also utilizes a commercial online curriculum-based measurement (CBM) progress monitoring tool that targets particular grade-level math concepts and skills. These measures are administered more often than the benchmark assessments—every 4 weeks to all students. They target a more specific subset of grade-level, CCSS domain–specific math concepts and skills (determined by grade-level teams). Each CBM assessment includes approximately 20–25 multiple- choice items and typically takes students 30 minutes to complete. Students receiv- ing supplemental or intensive math instruction in addition to core instruction are administered shorter, more focused, and more frequent CBM progress monitor- ing probes as needed during their supplemental and intensive instructional time. Given that it is late October, Ms. Thompson has data from the beginning year benchmark assessment and two CBM assessments for all students in her class. The school’s first benchmark assessment focused primarily on essential fourth-grade concepts and skills that are prerequisites for success in fifth grade. Ms. Thompson --- | | Most | Some | Few | | --- | --- | --- | --- | | Appears to know or understand | Has a conceptual understanding of multiplication as equal groupsCan use repeated addition to represent multiplicationUses known multiplication facts to derive unknown factsCan use an open area model to represent double-digit by double-digit multiplication and decomposes the tens and ones in the modelCan explain and appropriately utilize the commutative, associative,and distributive propertiesUses partial products(without a visual model)to solve double-digit multiplication problems | Uses the powers of 10 when appropriate to solve double-digit multiplication problems | Flexibly uses strategies such as doubling and halving to solve double-digit multiplication problemsUses partial products(with a visual model)to solve double-digit multiplication problemsUses an area model that shows all the square units to represent double-digit by double-digit multiplication and decomposes the tens and ones in the model | multiplication problems Figure E.2. Ms. Thompson’s summary of what her 22 students know, based on benchmark and curriculum-based measurement data through late October and organized by most students (80% +), some students (10%–15%), and few students reviews benchmark results related to prerequisites for the target math content. For her students currently receiving supplemental and intensive math support, she also has access to the progress monitoring data generated more frequently. her students currently receiving supplemental and intensive math support, she also has access to the progress monitoring data generated more frequently. Based on these data, Ms. Thompson creates a simple chart (see Figure E.2) to help her visualize what her students appear both to know and not know with respect to key prerequisite concepts and skills related to whole number multiplication. to key prerequisite concepts and skills related to whole number multiplication. Assessment Tasks Ms. Thompson and Mr. Hart utilize the information from Ms. Thompson’s benchmark and CBM summary table to think about which kinds of formative assessment to create. They aim to get a more in-depth understanding of their students’ thinking and level of understanding with respect to prerequisite concepts and skills that are foundational to multi-digit multiplication. Based on the benchmark assessment data, Ms. Thompson and Mr. Hart determine that the formative assessment should focus on three primary areas: flexible use of multiplicative strategies such as doubling, appropriate use of the properties of multiplication (commutative, associative, distributive), and accurate use of the partial products algorithm (numerical use of partial products). They agree it would be most efficient to develop a short assessment probe addressing these concepts and skills, to --- FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully The two teachers want to be sure that there are multiple tasks for each area of focus so that there are enough items to ensure they get an accurate appraisal of what students know and don’t know. (See Chapter 5 for more about developing informal formative assessments.) Furthermore, they want to be sure the tasks assess student engagement in their targeted mathematical practices: 1) Construct viable arguments and critique the reasoning of others, 2) Look for and make use of structure, and 3) Attend to precision (NGA Center for Best Practices & CCSSO, 2010). They decide to include some items that contain word problems (contextualized and applied problems) and some items that do not (see Textbox E.2). To this end, they create six noncontextualized tasks: two requiring knowledge of multiplication properties, two asking students to use flexible multiplication strategies, and Textbox E.2. Formative assessment created by Ms. Thompson and Mr. Hart to appraise what students know and don’t know about prerequisite multiplication concepts and skills about prerequisite multiplication concepts and skills 1. Use drawings or manipulatives to demonstrate why 3 × 4 = 4 × 3. 2. Darran thinks 5 × (2 × 6) is not the same as (5 × 2) × 6. Please explain why you agree or 2. Darran thinks 5 × (2 × 6) is not the same as (5 × 2) × 6. Please explain why you agree or disagree with Darran. 3. If you did not know how to multiply 5 × 14, which set of facts would help you find the 5\times1+5\times4 5\times10+5\times4 5\times1\times4 5\times10\times4 5 × 10 × 4 4. Xavier thinks that the product of 18 × 5 is the same as the product of 9 × 10. Do you agree or disagree with him? Explain why you agree or disagree. 5. Xiao and Maria both used partial products to solve 34 × 8. Look at their solutions. Explain why you think each solution is correct or incorrect: lem. (For students who finish early, ask them to solve using a different multiplication strategy.) 8. Niki collects stamps. She wants to buy an album to hold her stamps. One album holds 12 stamps on a page and contains 22 pages. Another album holds 15 stamps on a page and contains 18 pages. Niki wants to buy the album that holds more stamps. Which | Xiao: | Maria: | | --- | --- | | 34 | 34 | | $\frac{\times8}{32}$ | $\frac{\times8}{32}$ | | +240/272 | +24/56 | + 240 + 24 272 --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 355 two requiring use of the partial products algorithm. They also include two word problems that require students to apply relevant multiplication strategies. The two teachers estimate that it will take most students between 20 and 30 minutes to complete the assessment. ## Student Responses As students finish, Ms. Thompson and Mr. Hart review students’ responses to the informal assessment to get a sense of what they know and what they don’t know, and a sense of possible error patterns that may indicate students’ misconceptions about important underlying math concepts. Figure E.3 shows Ms. Thompson’s and Mr. Hart’s summary of their students’ responses, which they will use to determine their students’ specific mathematical learning needs and then to plan and implement responsive instruction. ## DETERMINE STUDENTS’ MATH-SPECIFIC LEARNING NEEDS For the Determine Students’ Math-Specific Learning Needs component, Ms. Thompson and Mr. Hart use student response data to determine their students’ math- specific learning needs. This involves three important activities, which all rely on the evalu- ation of student responses to the assessment tasks. First, based on the responses, Ms. Thompson identifies students’ positions on the related learning trajectory (from the Identify and Understand the Mathematics component). She then determines students’ misconceptions, as well as what students know (knowledge strengths) and don’t know (knowledge gaps) with respect to the identified mathematics. Finally, Ms. Thompson and Mr. Hart determine together which mathematical ideas they need to target specifically that will support students to further develop their math- ematical knowledge and skills along the identified learning trajectory and toward the identified standard(s). ## Identify Each Student’s Position on the Identified Learning Trajectory Based on students’ responses, the majority of students appear to be functioning at Level 4 of the learning trajectory, meaning they can demonstrate reasoning about multiplication using the more sophisticated multiplicative reasoning strategies, which rely primarily on numerical representations that incorporate properties, such as the associative and distributive properties. Most do not rely on the area model to use the partial products algorithm. Some students have demonstrated a limited understanding of key areas, such as the prompted use of multiplicative strategies such as doubling and halving and reliance on open area models to accurately complete the partial products algorithm. Based on this evidence gathered through assessment, Ms. Thompson judges these students to be functioning at Level 3 of the trajectory. Two students’ assessment performance indicates they are likely functioning at a lower level of the learning trajectory, possibly Level 2, because they need to rely on using an area model showing all the square units to make sense of the par- tial products algorithm. They demonstrated a solid understanding of the associa- tive and commutative properties but not the distributive property, so they could be making the transition from Level 2 to Level 3. --- FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully | Teacher:Ms.ThompsonClass/period:1st period | | | | | --- | --- | --- | --- | | Areas for the assessment | On target(Write names.) | Limited(Write names.) | Insufficient(Write names.) | | Flexible use of multiplicative strategies such as doubling(Level 4)Task 4 | Most students | Tommy S.Jerome B.Felisha T.NOTES:The students are able to use doubling or halving when prompted but at a very slow pace. | Steve A.Tamika W.NOTES:The students use calculation,not doubling or halving,to determine equivalence(e.g.,uses18×5and9×10,sees they are both90).The students are unable to use the strategy,evenwhen prompted. | | Appropriate use of the properties of multiplication(commutative,associative,distributive)(Level 4)Tasks1,2,and3 | Commutative property:All students | | | | Associative property:All students | | | | | Distributive property:Most students | | Distributive property:Steve A.Tamika W.NOTES:The students use multiplication instead of addition(e.g.,5×14=5×10×4). | | | Accurate use of the partial products algorithm(numerical use of partial products)(Level 4)Tasks5,6,7,and8 | Most students | Tommy S.Jerome B.Felisha T.NOTES:The students are able to recognize when partial products are used correctly(Task5)but not able to use the strategy accuratelywithout relying on an open area model(with and without contexts)(Tasks6,7,and8). | Steve A.Tamika W.NOTES:The students are unable to accurately identifyor use the partial products algorithm,evenwhen usingan open area model.WhenMr.Hart sketched an openarea model for Task5,these students seemed confused as to what the dimensions of thearea model represented.Theywanted to put those numbersinside the area model,notalong the edges.They mayneed to rely on an area model thatshows every square unit. | Excerpted from Teaching Mathematics Meaningfully: Solutions --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 357 ## Evaluate Prior Knowledge, Strengths and Gaps, and Misconceptions Most students indicated a readiness for learning the standard algorithm for multi-digit multiplication because they could reason using the partial products algorithm in a purely numerical representation and explain their reasoning. They could also demonstrate a flexible use of multiplicative strategies, including doubling and halving, as well as the associative, commutative, and distributive properties of multiplication. Three students could appropriately use the properties of multiplication but still relied on open area models to support their reasoning with the partial products algo- rithm. They also struggled with using and knowing when to use a doubling and halving strategy. It appears these students rely heavily on decomposing multi- digit numbers based on place value, so they are not looking for alternative number rela- tionships to exploit. Although this strategy will help enhance their number and com- putation sense, it should not hinder their progress toward becoming fluent with the partial products algorithm and the standard algorithm for multi-digit multiplication. Two students demonstrated appropriate use of the associative and commutative properties, but they demonstrated issues with the distributive property as well as the partial products algorithm, even when supported with open area models. Based on their confusion about what the numbers represented when Mr. Hart used an open area model to illustrate the partial products, it appears they may have some knowl- edge gaps in the concept of area. These two students also struggled with knowing when and how to use doubling and halving strategies. For all five of these students, Ms. Thompson and Mr. Hart noted that they want to think about how to address the different knowledge and skill gaps as they plan and implement instruction. ## Target Math Ideas for Instruction For most of the students, Ms. Thompson and Mr. Hart will target the mathemati- cal ideas closely associated with the identified math standard: Fluently multiply *multi-digit whole numbers using the standard algorithm (NGA Center for Best Practices* & CCSSO, 2010). In particular, they want to reinforce the relationship between the partial products algorithm, the area model, and the standard algorithm. Embedded in these three constructs is the important idea of the distributive prop- erty. Two students in particular, Steve A. and Tamika W., demonstrated insuf- ficient evidence of appropriate use of the distributive property, so Ms. Thompson and Mr. Hart make note that they need to include this property as a target math idea for these students. Given the significance of this property to multiplication, they decide to also include it as a target idea for whole-class instruction. In addi- tion, Ms. Thompson and Mr. Hart note that they will need to help Steve A. and Tamika W. enhance their understanding of area before they can be expected to work productively on multiplication in general. ## DETERMINE STRUGGLING LEARNERS’ SPECIFIC LEARNING NEEDS For the Determine Struggling Learners’ Specific Learning Needs component, Ms. Thompson and Mr. Hart work to determine the kinds of barriers that are likely affecting their struggling learners and making learning mathematics dif- ficult. They first identify the mathematics performance traits they have observed --- FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully with their struggling learners, then focus on determining the possible learning characteristics and curriculum factor barriers that may be contributing to these Identify Observed Performance Traits Identify Observed Performance Traits Using the form illustrated in Figure E.4, Ms. Thompson and Mr. Hart note the performance traits they have observed with most or some students in the first period class or with individual students. They note that most students demonstrate knowledge and skills for some math domains and not others or for certain standards within particular math domains and not others. Therefore, for that performance trait, they check off the box labeled Most. They also note that certain groups of students have consistently demonstrated two other performance traits, “The student is able to compute or engages in problem solving accurately but at a very slow pace” and “The student avoids engaging in certain mathematical tasks.” In the Some column, they write these students’ names in the boxes next to these two performance traits. Finally, Ms. Thompson and Mr. Hart also note individual students (two or fewer) who demonstrated still other performance traits (e.g., “The student demonstrates faulty mathematical thinking or ineffective strategies Tamika W. | Teacher:Ms.Thompson and Mr.Hart Class/period:Ist period | | | | | --- | --- | --- | --- | | Mathematics performance traits | Most(√) | Some(Write names.) | Individual(Write names.) | | The student demonstrates knowledge and skill for some mathematical domains and not others,or for certain standards within a domain and not others. | √ | | | | The student demonstrates faulty mathematical thinking or ineffective strategies when problem solving. | | | SteveA.TamikaW. | | The student is able to compute or engages in problem solving accurately but at a very slow pace. | | TommyS.JeromeB.FelishaT. | | | The student has difficulty with generalizing knowledge and skills to other mathematical concepts,skills,and contexts. | | | TommyS.JeromeB. | | The student demonstrates mathematical abilities at one point in time but then is unable to demonstrate the same abilities later. | | | SteveA.JeromeB. | | The student avoids engaging in certain mathematical tasks. | | TommyS.JeromeB.SteveA.TamikaW. | | --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 359 when problem solving,” “The student has difficulty with generalizing knowledge and skills to other mathematical concepts, skills, and contexts,” “The student demonstrates mathematical abilities at one point in time but then is unable to demonstrate the same abilities later”). In the Individual column, they write these students’ names in the boxes for these learning traits. In reviewing their notes, the two co-teachers develop a picture of the kinds of math performance traits demonstrated by most of their students, by some stu- dents, and by only one or two students. Ms. Thompson and Mr. Hart can also eas- ily see which students are demonstrating more math performance trait difficulties than others. This information now provides them with reference points to begin thinking about what potential learning characteristic and curriculum factor bar- riers they will want to consider when planning and implementing instruction. In considering potential learning characteristic barriers, Ms. Thompson and Mr. Hart think about their students as they review their notes on perfor- mance traits (see Figure E.4). Four of Ms. Thompson’s students have identified disabilities and therefore have individualized education plans (IEPs): Steve A., Tamika W., Tommy S., and Jerome B. In addition, one student identified with ADHD, Felisha T., has a Section 504 accommodation plan. Each of these students is listed in either the Some or Individual column (see Figure E.4) as demonstrat- ing more than one performance trait. Ms. Thompson and Mr. Hart know that, for these five students, they will need to consider all the typical learning character- istics of struggling learners when thinking about how these characteristics might be associated with their performance traits. Ms. Thompson finds it helpful to have Mr. Hart as a collaborator because he helps her to better understand the disability-related needs of her students with identified disabilities: Steve A., Tamika W., Tommy S., and Jerome B. Tamika W. also has an identified speech-language impairment, related to diffi- culties articulating particular sounds when she speaks. Tommy S. and Jerome B. also are identified as having ADHD—Tommy S. with the primarily inatten- tive type ( distractibility) and Jerome B. with the combined type (inattention and impulsivity/ hyperactivity). Felisha T., who does not have an IEP but has a 504 accommodation plan, is, like Tommy S., identified as having the primarily inattentive type of ADHD. Ms. Thompson and Mr. Hart decide to create a table that shows the students with identified disabilities, their particular disabili- ties, and specific information about particular cognitive, social-emotional, and behavioral issues— as documented in the students’ IEP and cumulative folders, and also based on the connections Mr. Hart made based on his knowledge of disability- related learning needs. Table E.2 shows the teachers’ notes. With this information at hand, Ms. Thompson and Mr. Hart can now start thinking about what learning characteristics could be contributing to their stu- dents’ math performance traits. They go back to their “most, some, individual” notes (see Figure E.4) and consider each student, the performance trait, and the information gathered for each student with an identified disability (see Table E.2). For example, they note that Steve A. demonstrates three performance traits (in addition to the one most of Ms. Thompson’s students demonstrate). For each of Steve A.’s performance traits, the two teachers consider his cognitive, social- emotional, and behavioral needs related to his disability and which learning char- acteristics are likely contributing to his performance trait difficulties. Table E.3 shows an example of their thinking. --- FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully Table E.2. Ms. Thompson's and Mr. Hart's notes about their students with identified disabilities Table E.3. Example of Ms. Thompson’s thinking about Steve A., his performance traits, and potential | Student | Performance trait | Potential learning characteristic barriers | Potential curriculum factor barriers | | --- | --- | --- | --- | | Steve A. | The student demonstrates faulty mathematical thinking or ineffective strategies when problem solving. | Metacognitive thinking disabilities Knowledge and skill gaps | | | The student demonstrates mathematical abilities at one point in time but then is unable to demonstrate the same abilities later. | Memory disabilities—memory retrieval (and working memory?) | | | | The student avoids engaging in certain mathematical tasks. | Math anxiety Learned helplessness | | | --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 361 ## Consider Possible Learning Characteristic Barriers For the first performance trait, Ms. Thompson and Mr. Hart quickly realize that Steve A.’s difficulties in making connections between certain math ideas and applying effective strategies align closely with one of the learning characteristic barriers, metacognitive thinking disabilities. They also suspect that Steve A. has some gaps in his mathematical knowledge base, confirmed by his low scores on math benchmark testing. Ms. Thompson thinks this could also be a contributing factor. With the second performance trait, Mr. Hart focuses on Steve A.’s memory retrieval difficulties, so he suspects that memory disabilities likely contribute to his pattern of being able to demonstrate knowledge of a concept or skill at one point in time but not at another point. Mr. Hart also wonders whether working memory could be a factor in this; during instruction, Steve A. appears to under- stand parts of the concept being taught but not others. The teachers note this with a question mark. For the third performance trait, both teachers realize that, given the difficulties Steve A. has, he probably shuts down when confronted with math- ematics tasks he does not believe he can complete successfully. Steve A. also often raises his hand for help with math, even when he has the knowledge and skill to complete the task, so they think learned helplessness is potentially playing a role as well. Ms. Thompson and Mr. Hart use the same process to identify the learning characteristics that are most likely contributing to the difficulties of Tamika W., Tommy S., Jerome B., and Felisha T. ## Consider Possible Curriculum Factor Barriers As Ms. Thompson and Mr. Hart continue to think about each of their struggling learners, they consider the five curriculum factors and how any of these might be barriers to their students’ math success and might contribute to the math per- formance traits they demonstrate. Table E.4 shows their thinking for Steve A. in connection to their notes about potential curriculum barriers. As Ms. Thompson thinks about the math curriculum she uses, she thinks about how certain characteristics thereof might contribute to her students’ difficul- ties. For the first performance trait, Ms. Thompson reviews a few lessons from the teacher’s edition of the math textbook. She notices that although each lesson has a segment related to conceptual understanding, little direct connection is made between the concept and the procedures emphasized during the rest of the lesson. In some ways, these two aspects of the lesson—conceptual understanding and reasoning, and procedural understanding and proficiency—seem to be treated as separate sections. It makes sense to her that this might contribute to Steve A.’s tendency to use inefficient strategies when solving problems; it may also explain why he is sometimes off base in connecting what he is doing to why he is doing it. Therefore, Ms. Thompson suspects that the textbook’s limited emphasis on inte- grating conceptual understanding with procedural proficiency is a curriculum factor barrier for Steve A. In thinking about the second performance trait, Ms. Thompson considers the memory difficulties Steve A. can experience. This makes her wonder whether the curriculum allows Steve A. to fully store what he learns about a new math concept or skill and have enough opportunities to apply or practice it. This in turn makes --- FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully Table E.4. Ms. Thompson’s and Mr. Hart’s thinking about Steve A., his performance traits, potential learning | Student | Performance trait | Potential learning characteristic barrier | Potential curriculum factor barrier | | --- | --- | --- | --- | | Steve A. | The student demonstrates faulty mathematical thinking or ineffective strategies when problem solving. | Metacognitive thinking disabilities Knowledge and skill gaps | Level of emphasis placed on the integration of conceptual understanding with procedural proficiency | | The student demonstrates mathematical abilities at one point in time but then is unable to demonstrate the same abilities later. | Memory disabilities—memory retrieval (and working memory?) | Instructional pacing | | | The student avoids engaging in certain mathematical tasks. | Math anxiety Learned helplessness | Lack of utilizing effective mathematics practices for struggling learners across instructional tiers in multi-tiered systems of supports(MTSS) | | her consider whether the instructional pacing is appropriate for Steve A. Given his memory disabilities, the pacing might be too rapid for Steve A. to fully learn and become proficient with newly introduced concepts. He may be able to demonstrate what he understands in the moment, but when he is asked to apply it later in the lesson or on another day, he cannot effectively retrieve this learning from memory because he did not have enough opportunities to apply his learning in order to make retrieval automatic. It is also possible that the lesson’s instructional pace was faster than Steve A.’s ability to process the information efficiently, affect- PLAN AND IMPLEMENT RESPONSIVE INSTRUCTION PLAN AND IMPLEMENT RESPONSIVE INSTRUCTION At this point, Ms. Thompson and Mr. Hart have worked fully through the first four components of the Teaching Mathematics Meaningfully Process. They have pace was faster than Steve A.’s ability to process the information efficiently, affecting his working memory. As Ms. Thompson thinks more deeply about Steve A., his performance traits, learning characteristic barriers, and potential curriculum factor barriers, she begins to realize there are some disconnects between the instruction emphasized in the math textbook and his learning needs. Ms. Thompson hypothesizes that this could be a reason for Steve A.’s hesitation to engage in certain mathematics activities: He has not adequately learned them. So, Ms. Thompson notes utilization of effective instructional practices for struggling learners as another important potential curriculum factor barrier for Steve A. Excerpted from Teaching Mathematics Meaningfully: Solutions --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 363 teacher and that of a special education teacher—as they worked through each aspect of the two related components, Determine Students’ Math-Specific Learning *Needs and Determine Struggling Learners’ Specific Learning Needs.* For the Plan and Implement Responsive Instruction component, you will read how Ms. Thompson and Mr. Hart take what they learned so far and use it to plan and implement their instruction in ways that respond to their struggling learners’ needs. This final component of the Teaching Mathematics Meaningfully Process includes the following steps: • Developing a math instructional hypothesis (see Chapter 5) to guide planning • Planning for and implementing effective instructional practices (see Chapters 7–8) • Reflecting and revising instruction based on student performance data (see Chapter 5) As you read about how Ms. Thompson and Mr. Hart engage in this process, you will learn how they identify instructional hypotheses to address the needs of the students whose formative assessment results indicated knowledge and skill gaps. Also, you will learn how the two teachers plan and implement instruction that is organized at three levels based on how MTSS/RTI is implemented in their school: • Less intensive (whole-class, differentiated core instruction at Tier 1) • More intensive (small-group, supplementary instruction at Tier 2) • Even more intensive (intensive instruction at Tier 3 for a few students) As noted previously, the school has a daily 50-minute period devoted to pro- viding students with additional instructional time in reading and mathematics, used either for more intensive instruction or for extension and enrichment. Dur- ing this time, general education teachers, such as Ms. Thompson, provide supple- mentary instruction (Tier 2) for students who need it; special education teachers, such as Mr. Hart, and a math coach provide even more intensive instruction (Tier 3). Teachers either alternate days for reading and mathematics instruction or split the 50-minute period in half to address each content area. With this information in mind, we focus on how Ms. Thompson and Mr. Hart plan and implement instruction for struggling learners at each level or tier. Please note the icons in the right margin that highlight how these teachers’ instruction incorporates certain practices discussed throughout the book: EIAs (Chapter 7), anchors of instruction that can be intensified within MTSS/RTI (Chapter 10), and the MTPs (Chapter 8). ## Instructional Hypothesis Reflecting on students’ responses from the formative assessment tasks, Ms. Thompson and Mr. Hart determine that most have the prerequisite knowl- edge and skills to achieve the overall learning intention, multiplication of multi- **3** digit numbers using the standard algorithm. However, five students struggle with different prerequisites. Although Ms. Thompson and Mr. Hart do not believe an instructional hypothesis is needed to guide instruction for most of their students, --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) 364 Appendix E they decide instructional hypotheses would be helpful for these five. For the three students (Tommy S., Jerome B., and Felisha T.) who cannot accurately use the partial products algorithm without relying on an area model, Ms. Thompson and Mr. Hart develop the following instructional hypothesis to support planning and instruction: ### Given two multi-digit numbers to multiply: *Students are able to recognize situations that involve multiplication and accurately* use the partial products algorithm when using an open area model. *Students are unable to use the partial products algorithm without relying on an area* model . . . because they have difficulty keeping track of the numerical partial products without the visual cues provided with the area model. For the two students (Steve A. and Tamika W.) who are struggling with the distributive property and partial products algorithm, Ms. Thompson and Mr. Hart develop the following instructional hypothesis: ### Given two multi-digit numbers to multiply: *Students are able to use the associative and commutative properties for multiplication.* *Students are unable to use the distributive property or partial products algorithm* even with an area model . . . because they do not understand the relationship between the numbers in the partial product and their distribution within an area model. Ms. Thompson and Mr. Hart use these two instructional hypotheses to guide their instructional planning and teaching as they differentiate and intensify their instruction across tiers for these five students. ## Planning and Implementation As Ms. Thompson and Mr. Hart start to plan their instruction based on the two instructional hypotheses, they consider how the intensification of instruction for those who need it aligns with how their school employs MTSS/RTI (see the intro- ductory paragraph for more about this component of the Teaching Mathematics Meaningfully Process). Ms. Thompson and Mr. Hart begin to plan a sequence of teaching and learning activities that will be used several days. They keep in mind that, because the students have already had lots of experiences using base-ten materials and area models to think about multiplication and have connected these ideas to partial products, the primary goal is for students to develop the written **6**record for the standard algorithm for multi-digit multiplication. Because most of these students are proficient with the partial products algorithm, Ms. Thompson and Mr. Hart agree that instruction related to the target standard should begin by connecting the two algorithms, partial products and standard, using con- crete or representational (i.e., semi-concrete, such as an area model) models first. **8** They decide to focus the first few lessons on using area models in conjunction **1**with the written record. Once students are able to demonstrate and articulate --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 365 **4**an understanding of what is happening with the models, they will move to the written algorithm without relying on the area model. They decide to start with a word problem to reinforce how multiplying multi-digit numbers applies to the real world. After working with both the standards and the learning trajectory, Ms. Thompson and Mr. Hart articulate the following overall and long-term learn- **2** ing intentions for their students as follows: students will be able to (1) conceptu- ally understand the mathematical ideas related to multiplying multi-digit numbers **1**using the standard algorithm, (2) multiply multi-digit whole numbers with pro- ficiency, and (3) make sense of and solve word problems that involve multi-digit multiplication of whole numbers. ## Differentiated Whole-Class Core Instruction (Less Intensive, Tier 1) Both Ms. Thompson and Mr. Hart agree that using a systematic instruction **1** approach is important for all their students. In particular, it will allow them to evaluate student learning after each lesson and determine what to emphasize in the next. Based on the assessment data gathered, Ms. Thompson and Mr. Hart decide to use parallel tasks, one with smaller numbers and one with larger num- **1**bers, during the initial whole-class lesson. Parallel teaching, in which two teachers teach two different groups the same content, at the same time, in differentiated ways, is a co-teaching model that supports differentiating instruction within whole-class or large-group contexts (Friend & Cook, 1996). Doing this will lower **7**the teacher–student ratio for those students who are struggling with the math- ematics content. For those struggling with the distributive property and the area model, the task with a single-digit number multiplied by a double-digit number will reduce the number of partial products on which they must focus. For students with working memory difficulties, such as Steve A., this strategy will lessen the cognitive load necessary to complete the task—making it more likely that they will be able to process the numbers accurately and complete the necessary cogni- tive actions. This strategy could also help to alleviate students’ potential anxiety about attempting something new. The teachers decide on these word problems as **2**their parallel tasks: • In the front section of the school auditorium are 6 rows where 45 students can sit in each row. How many students can sit in the front of the auditorium? • In the front section of the school auditorium are 23 rows where 45 students can sit in each row. How many students can sit in the front of the auditorium? Students are asked to draw the corresponding rectangular area, mark off the area that aligns with each of the partial products, and then complete a written record of the multiplication, using the partial products on a recording sheet that **3**has base-ten columns identified (see Figure E.5). Ms. Thompson and Mr. Hart pro- vide base-ten grid paper (see Figure E.6) to Steve A. and Tamika W. This paper not **8** only explicitly shows the individual squares (as opposed to an open-area model where these squares are implied), but also organizes the squares into groups of 10 rows and 10 columns. This structure is intended to eliminate the need to count all the individual squares and also supports students’ partitioning the numbers being multiplied into their respective place values. --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) 366 Appendix E Th H T O 2 7 40 5 × 4 5 1 3 5 20 800 100 1 0 0 2 8 0 7 280 35 + 8 0 0 1 2 1 5 **Figure E.5.** Place value recording sheet and an open area model showing the partial products for 27 × 45. At this point in the lesson, it is time to introduce the written record for the standard algorithm. Ms. Thompson and Mr. Hart recognize that based on stu- dents’ learning needs, some students will need less guidance and others will need more. For those students needing less guidance, their task is to consider the written record they have created using partial products and another one given to them that uses the standard algorithm along with the corresponding area model. **2**Together, they will work to figure out the written record of the standard algorithm (i.e., what the numbers mean and where they came from, and why numbers are **6**recording in particular places). Ms. Thompson and Mr. Hart provide these stu- dents with a sequence of written records, as seen in Figure E.7, so the students can **8** see how and when numbers are introduced into the written record. Ms. Thompson and Mr. Hart know they need to offer more support to some **1 3**students by starting with a simpler problem and being more explicit. For these students, they use the first word problem that results in 6 × 45. After students have completed the partial products algorithm and corresponding area model for the computation (see Figure E.8), Mr. Hart works with the small group, asking a **5 5**series of focused questions to help them relate the two written records, such as the following: **3** *Where is the 30 in the partial products and in the area model? What numbers were used* *to get 30? Where and how is the 30 recorded in the standard algorithm? Why is the* *3 recorded above the 40 in the 45?* He knows he needs to use strategies such as visual cuing to help students make these connections (see Figure E.9). Mr. Hart highlights the three tens in the partial products algorithm and uses an arrow to show the connection to the **8** regrouped three tens in the standard algorithm. The use of visuals to help students make connections between mathematical ideas is an effective instructional practice for struggling learners because it helps students focus on important features of mathematical ideas and tasks despite attention difficulties or cognitive processing impairments. Mr. Hart works with this small group of students on additional mul- tiplication of single-digit numbers by double-digit numbers, eventually leading to multiplication problems involving two double-digit numbers. **7 1**Gradually scaffolding content in order to help students succeed, with less diffi- cult content expectations initially and then with more difficult expectations later on, supports struggling learners to be willing to take risks. This reduces the likelihood --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 367 **Figure E.6.** Base-ten grid paper. (Various open source sheets can be found via an Internet browser search.) that they will engage in learned helplessness. This practice also supports the needs of students who process information more slowly so they can become proficient with less demanding mathematics tasks before moving to more demanding tasks. Ms. Thompson and Mr. Hart are also sensitive to the idea that with both the partial products and the standard algorithm, it is imperative that students still --- 1 2 1 5 23\times45. 5 be precise in the language used. For example, when students who are finding the product of 23 × 45 multiply the 2 and 5, they should say “20 times 5” so that it is more apparent to the student why he or she writes the 1 in the hundreds place. Supplemental Small-Group Instruction (More Intensive, Tier 2) The three students who currently receive supplemental mathematics instruction— Tommy S., Jerome B, and Felisha T.—demonstrated they were ready to move toward understanding and becoming proficient with the standard algorithm for multiplication. For this reason, Ms. Thompson and Mr. Hart agree that, although they will likely need some additional instruction on concepts and skills covered during whole-class core instruction, they mostly will benefit from additional response and practice opportunities in order to become proficient with these concepts and skills. Ms. Thompson decides to begin each supplemental instruction session at a smallgroup table by engaging students in a pre-instructional “check” activity, in which she presents several prompts related to these core concepts and skills and asks 2students to quickly respond on individual dry-erase boards. As students respond, Ms. Thompson notes any error patterns or apparent misconceptions and writes them in a small journal she uses to informally track students’ progress during supplemental instruction. (She finds using a supplemental instruction journal this way helps her to efficiently plan from day to day and pinpoint where to target instruction.) Based on her observations during this pre-instructional check, Ms. Thompson then determines which content or mathematical practices her students need more support in understanding. She communicates to students her learning intentions for the session and how these relate to what she observed during the pre- instructional 2 check. 2 7 0 Excerpted from Teaching Mathematics Meaningfully: Solutions --- | Partial Products | | | | Standard | | | | --- | --- | --- | --- | --- | --- | --- | | H | T | O | | H | T | O | | × | 4 | 5 | | × | 34 | 5 | | | 6 | | | | | | | +2 | (3) | 0 | | 2 | 7 | 0 | | 4 | 0 | | | | | | | 2 | 7 | 0 | | | | | Figure E.9. Place value recording sheet comparing the algorithms for partial products and the standard algorithm for $ 6 \times45. $ Knowing her students and considering the nature of the mathematics content standard, Ms. Thompson decides to emphasize the three additional instructional intensification anchors, Explicitness and Teacher Direction, Teach Math Metacognition, and Opportunities to Respond. She believes that her students will benefit from moderate levels of intensification for the first two of these anchors and a high level of intensification for Opportunities to Respond. (This is an example of differentiating intensity among the seven instructional anchors; see Chapter 10 for discussion about differentiating the intensity levels of these within MTSS/RTI.) She decides this because she knows that to build appropriate levels of proficiency, her students will likely need some modeling or reteaching and greater opportunities to respond and practice. She understands that Tommy S., Jerome B., and Felisha T. need more instructional time devoted to responding and practice than possible during whole-class core instruction. these materials to help students visualize its important features, repeating the visual cuing practices she and Mr. Hart used during whole-group instruction. When students need support in recalling steps for completing the standard multiplication algorithm or solving multiplication word problems, Ms. Thompson 10 teaches the use of explicit learning strategies (see Chapter 8 for examples) that 10 teaches the use of explicit learning strategies (see Chapter 8 for examples) that support students’ memory recall and help them build metacognitive awareness. For example, she noted that when students were not provided the place value recording sheet (see Figures E.5–E.9), some forgot or did not connect the place value of digits they regrouped using the standard algorithm. So, she taught these students the FIND strategy, which helps them identify and remember the place value of digits in multi-digit numbers (see Figure E.10). The FIND strategy helps students independently recreate the place value template that was used in differentiated whole-class core instruction. This in turn allows them to respond independently while reinforcing thinking about the place value of digits when using the standard algorithm. Over time, Ms. Thompson will fade students’ use of the teaches the use of explicit learning strategies (see Chapter 8 for examples) that support students’ memory recall and help them build metacognitive awareness. For example, she noted that when students were not provided the place value recording sheet (see Figures E.5–E.9), some forgot or did not connect the place value of digits they regrouped using the standard algorithm. So, she taught these students the FIND strategy, which helps them identify and remember the place value of digits in multi-digit numbers (see Figure E.10). The FIND strategy helps students independently recreate the place value template that was used in differentiated whole-class core instruction. This in turn allows them to respond independently while reinforcing thinking about the place value of digits when using the standard algorithm. Over time, Ms. Thompson will fade students’ use of the --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) 370 Appendix E Th H T O 1 **Find the columns (space between digits)** 4 2 **Insert the ts.** × 2 9 **Name the place values.**1 3 7 8 **Determine the value of each digit.** 8 4 0 1 2 1 8 Students use the FIND Strategy to monitor the value of digits as they complete the standard multiplication algorithm. **Figure E.10.** Example of the FIND Strategy (Mercer & Mercer, 2005) being utilized to monitor the place value of digits when completing the standard multiplication algorithm. To help her students build their levels of proficiency, Ms. Thompson incorpo- rates multiple response opportunities during both instruction and practice. For **6** example, she utilizes the individual white boards to ensure all students in the group respond to her questions and prompts. She incorporates the use of instruc- **5**tional board games (see Chapter 7) for practice. ## Individualized Instruction (Even More Intensive, Tier 3) Two students, Steve A. and Tamika W., exhibited an underdeveloped understand- ing of the concept of area and its relationship to multiplication. Both have been identified as students who need more intensive math instruction in addition to core instruction. Although Ms. Thompson and Mr. Hart have attempted to differ- entiate their practice within whole-class planning and instruction to better sup- port struggling learners’ needs, they know Steve A. and Tamika W. need even more intensive support compared to the students receiving Tier 1 instruction or supplemental Tier 2 support. Based on the instructional hypothesis the two teachers developed from their informal assessment, Mr. Hart plans to organize his instruction according to three goals, which he will apply in each daily 50-minute intensive session with Steve A. **1 2**and Tamika W. First, he knows they have gaps in their knowledge base related to the standard targeted in whole-class core instruction. They will need explicit systematic instruc- tion in related foundational concepts and skills: the area model, the distributive prop- erty, and the partial products algorithm. Second, he knows his students will need multiple response opportunities and practice with these concepts and skills to build their proficiency and fluency. Third, Mr. Hart knows that Steve A. and Tamika W. will need pre-instructional support in order to benefit from whole-class core instruction. Therefore, Mr. Hart organizes each intensive instructional session according to these three areas of focus as follows: 1. During the first 20–25 minutes, he focuses on the foundational ideas: area, distributive property, and the partial products algorithm. He decides to begin with the foundational concept of area and how it relates to making sense of multiplication, ideas with which these students are still struggling. Excerpted from Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, Second Edition by David H. Allsopp, Ph.D.,LouAnn H. Lovin, Ph.D., & Sarah van Ingen, Ph.D. --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 371 2. During the next 15–20 minutes, he engages his students in practice opportu- nities related to one of these foundational concepts. 3. During the last 10 minutes or so, he preteaches content he and Ms. Thompson will cover during the next whole-class core instruction class period or peri- ods. He does this so Steve A. and Tamika W. have a preview of what they will be learning the next day and how it relates to what they are learning during Tier 3 instruction, and they can begin learning the content. Mr. Hart believes doing this will better prepare Steve A. and Tamika W. for the next day’s core instruction so that they can engage in the whole-class lesson more successfully. *Area Model Instruction* Mr. Hart decides to emphasize the same three instruc- tional intensification anchors used by Ms. Thompson with her supplemental Tier 2 group: Explicitness and Teacher Direction, Teach Math Metacognition, and Opportuni- *ties to Respond. However, in contrast to Ms. Thompson, Mr. Hart greatly intensi-* fies the anchors Explicitness and Teacher Direction and Teach Math Metacognition in addition to Opportunities to Respond (another example of differentiating intensity among the anchors). He begins with single-digit multiplication scenarios, such as the following, and has the students create corresponding rectangular areas on **1**grid paper to represent the scenarios. *Ms. Thompson wants to buy a rug for the classroom that is 5 feet by 6 feet. How much floor* *space (area) will the rug cover?* Mr. Hart begins with scenarios that are easy to model concretely in the class- room. For this scenario, students can utilize a tape measure, the classroom floor **8 3** tiles that each measure 1 square foot, and 1 foot by 1 foot paper squares to repre- sent, think about, and solve area problems. Mr. Hart first models this using think- **10 4** alouds that show his thinking about how much floor space the rug in the scenario will cover. Then, he invites students to do the same and to justify why the area model they created is appropriate and how it accurately represents how much space the rug will cover. Next, he poses different area scenarios using feet and inches and challenges Steve A. and Tamika W. to determine the different areas using the tape mea- sure and square paper “tiles.” He also asks them to mark the individual feet or inches on the paper squares and to justify their response. After each response, the students then represent the same area using their grid paper, labeling the **6** feet or inches and identifying the total area represented on the grid paper inside. Mr. Hart checks both students’ responses on the grid paper and provides spe- **5 8**cific positive reinforcement and corrective feedback. He asks each student to iden- tify the relationships between the area on the grid and the scenario. Once they are able to identify these relationships, he asks them to write the corresponding multiplication equation. As the students demonstrate proficiency with scenarios that involve continu- ous quantities, Mr. Hart begins to use scenarios that involve discrete quantities (i.e., ones students count). This is done to help his students generalize multiplica- tion to discrete quantities using an array structure. For example: *In a classroom, there are 4 rows of 5 desks. How many desks are in the classroom?* --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) 372 Appendix E Again, Mr. Hart models an example first, using think-alouds, with index cards representing the desks. As he did with continuous quantities, he begins with discrete-quantity scenarios that are easy to model concretely using an array. For example, Steve A. and Tamika W. can use index cards to create the 4 × 5 array. **3**Mr. Hart first models this using think-alouds about how to align the index cards in rows and columns. To help Steve A. and Tamika W. track the number of each, he marks cards with red and blue highlighters to identify the four rows and five columns. Then, he asks them to do the same and to justify why their array model is appropriate and how it accurately represents the total number of desks in the scenario. Next, he poses different combinations of rows and columns, challenges Steve A. and Tamika W. to work in the same way to create the appropriate array for each, and prompts them **2**to justify their response. After each response, the students represent the same area, using their grid paper in much the same way they did earlier when working with continuous quantities, including writing the multiplication equation underneath. **3 6**Mr. Hart checks both students’ responses on the grid paper and provides spe- **5**cific positive reinforcement and corrective feedback. He asks each student to iden- tify the relationships between the array model with the index cards, the area on the grid, and the equation (e.g., the 4 in the equation is represented by the four rows, the 5 in the equation represents the five columns, the 20 in the equation represents the total number of squares in the area). He does this to make sure that Steve A. and Tamika W. are making the connection between area and multiplication. Next, Mr. Hart moves to problems that involve the multiplication of a single- digit number by a double-digit number. He addresses these students’ misunder- standings about the distributive property by helping them partition into tens and ones the part of the area model corresponding to the double-digit number. He fol- lows a process similar to that used for problems involving single-digit numbers only. As Mr. Hart works with Steve A. and Tamika W. over time, he takes opportu- nities to explicitly connect what they are doing to the core content already covered during whole-class instruction because he wants to continuously help his students make these connections. For example, he shows a problem from a prior whole- class session as an example (see Figure E.8) and demonstrates how the single-digit by single-digit area models they have been working with (e.g., 4 × 5) relate to the partial products area model (e.g., 6 × 45) by pointing out how both have rows and columns. In other words, he relates 4 × 5 (four rows of five) to 6 × 45 (six rows of 45). *Practice* Because he has only two students and he wants to keep them moti- vated as they practice, Mr. Hart decides to utilize both instructional games and self-correcting materials (see Chapter 7). For example, he thinks the instructional game Pig would be good for practicing single-digit multiplication of whole num- bers (see Figure E.11). For the Pig game, Mr. Hart decides to have Steve A. and Tamika W. roll two dice to generate two single-digit numbers, represent the area of the resulting rect- angle using grid paper, label its rows and columns, identify the area (and product) by writing the number inside the rectangle, and write the corresponding mul- tiplication equation underneath. Mr. Hart monitors and provides feedback and coaching as needed. After the session, Mr. Hart reviews Steve A.’s and Tamika W.’s **6** responses to evaluate their progress and inform planning for the next session. --- # FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) Case Study 373 *Purpose: Provides students with computation practice* *Materials: Grid paper and pencil; two dice, with a pig sticker on one face of one of the die. (If you do not have* a pig sticker, purchase a pink dot sticker from an office supply store and draw a pig face on it.) *Directions: Students work in pairs and take turns. On each turn, a student rolls two dice at the same time. He or* she uses the numbers rolled as dimensions of a rectangle. The student draws the rectangle on his or her grid paper. The student multiplies the two numbers and writes the product inside the area of the rectangle. The student keeps a running total of the areas he or she finds. When the pig is rolled, the student has to deduct 20 from his or her total area. Play continues until one student reaches a designated total area (e.g., 100). *Extension 1: Each student rolls three dice at a time to create a one-digit and a two-digit number. Use different* colored dice to designate the one-digit number versus the two-digit number. For example, use one white die to generate the one-digit number and use two green dice for the two-digit number. Let the student determine how to use the digits from the dice to create a two-digit number (e.g., 32 or 23). The choice made will provide some insight into the student’s number sense and strategy awareness. *Extension 2: Each student rolls four dice at a time to create 2 two-digit numbers. Use two white dice for one* two-digit number and two green dice for the other two-digit number. Again, let the student determine how to use the digits from the dice to create a two-digit number. **Figure E.11.** Pig Math instructional game. (Source: Mercer & Mercer, 2005.) **2** *Pre-instruction for the Next Whole-Class Instruction Session* During this por- **3 3**tion of more intensive instructional time, Mr. Hart emphasizes the instructional intensification anchor Explicitness and Teacher Direction, using an explicit instruc- tion process, LIPP, for connecting what students will learn: • **Linking the whole-class learning intention to what Steve A. and Tamika W.** are learning about an area model in their even more intensive (Tier 3) session • **Identifying the learning intention they will focus on during subsequent** whole-class instruction • **Providing a rationale for why the upcoming learning intention is important** and relevant to their lives • **Previewing one or more foundational ideas related to it** Mr. Hart likes the mnemonic LIPP because it helps him remember the four areas to focus on for each pre-instruction session. ## Reflect/Revise Ms. Thompson and Mr. Hart meet regularly to both reflect on and revise their instruction. They do this by evaluating student performance data and through their ongoing observations. This includes students’ levels of engagement during instruction; at-the-moment diagnostic interviews and error pattern analyses the teachers complete with students based on their responses during instruction; and other informal formative assessments, such as weekly class mini-quizzes and --- ## FOR MORE, go to [http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully](/content/Teaching-Mathematics-Meaningfully/index.html) 374 Appendix E school-wide continuous progress monitoring data from benchmark and CBM assessments as appropriate. (This is all part of the Continually Assess Students com- ponent of the Teaching Mathematics Meaningfully Process.) Ms. Thompson and Mr. Hart concentrate on three questions as they reflect on their instruction across tiers: What is working? What is not working and why? How can we improve what we are doing? Because they continually keep these three questions in mind as they teach, they find that they do not need lots of time when they meet to reflect and revise; reflection becomes a habit of mind, so they already have concrete ideas before they meet. Ms. Thompson and Mr. Hart agree it is wonderful to be able to collaborate for several reasons: They share common per- spectives, but each teacher also has another perspective to pull from. Because nei- ther can be present for instruction across all three instructional tiers, collectively they feel that collaboration helps them have a much better handle on all students’ performance and related mathematics learning needs, particularly those who are struggling with mathematics. They also find that they are much better prepared when their grade-level team meets to review student performance data to make instructional decisions related to MTSS/RTI at the grade level and for individual students. # TAKE ACTION We designed this case study to illustrate how teachers can integrate the compo- nents of the Teaching Mathematics Meaningfully Process. Our illustration is simply one way the process can be carried out; the potential variations and adaptations are limitless and depend on the needs and characteristics of specific students and teachers. Throughout this book, we challenged you to put each component of the decision-making process into action. Our final challenge for you is to integrate the components by putting the entire process into action in your own classroom. Because of the complexity of this task, you may find it useful to work with a partner and talk through, or write down, how you envision each step playing out in your classroom. Then, go for it! It may feel time consuming and labor inten- sive, but we encourage you to see the entire process through from beginning to end. Pay close attention to how you and your students respond. Were the results useful? What were the effects on student learning and engagement? Which parts of the process worked well and which need to be tweaked? If you were fortu- nate enough to have another teacher collaborate with you throughout the process, what were your interactions like? How could they be strengthened in the future? Were there any points of tension or misunderstandings that could be addressed? Finally, we want to conclude this book by honoring you for the time you have taken to improve your mathematics instruction. Struggling students and students with special education needs historically have not received equitable mathematics instruction. The efforts you have made to read and apply the research-supported strategies in this book are clear indicators of your commitment to meeting the mathematics learning needs of all students. Thank you for this commitment. The outcome—improvements in mathematics teaching and learning for struggling students—is surely worth the effort! --- Abstract-level understanding, assessment centers and, 125 Abstract-Representational-Concrete (ARC) assessment assessment and, 103–104 examples of, 105f, 106f MDA and, 103, 127 Struggling learners and, 104–108 Academic self-concept, 105 Access assessment and, 122–124 core instructions as, 275 equity and, 10 providing for, 162 UDL and, 249 Accommodations IDEA 2004 and, 251 learning barriers and, 95 for testing, 289 Accuracy in understanding advanced acquisition stage of learning and, 117 initial acquisition stage of learning and, 117 scaffolding instruction for, 201 Acquisition stages, see Advanced acquisition stage of learning; Initial acquisition stage of learning Acronyms, see Mnemonics Actions That Will Help You Implement Each Step of MDA activity, 133 Activities Bifocal Vision for Math Teaching as, 38–39 explicitness instruction and, 145 finding the area as, 119f Incorporating Effective Teaching Practice into a Lesson Plan as, 237 learning trajectory as, 64–65 MTSS/RTI Instructional Tiers as, 246 as peer-mediated, 143t, 206, 341–342 Take Action Activities as, 96, 133, 153, 216, 245, 266–267, 278, 374 see also Instructional strategies and practices Adaptations illustration of, 103t for nonresponders, 276–277 Adaption stage of learning overview of, 116 teaching strategies for, 120 understanding and, 118–119 Adaptive reasoning, 31 procedural fluency and, 163 see also Reasoning and proof skills Addends, 230f Adding It Up, National Research Council (2001) report, 31, 162 Abstract-level understanding, assessment centers and, Advanced acquisition stage of learning, 60, 63t accuracy in understanding in, 117 Abstract-Representational-Concrete (ARC) assessment overview of, 116 Affective learning network, 250 Algebra building foundation for, 24–26, 25t instructional games for, 183f operations and, 17–19 solution drawings of, 198f struggling learners and, 26, 60–61 Algebraic thinking, 78 development of, 184 instructional games for, 183f operations and, 17–19 Algorithmic fluency examples of, 165–171, 167f, 169f procedural fluency and, 228, 229–230 Alternative procedures fractions and, 21–22 for multiplication, 20–21, 112 Amount of Time anchor, 263–264 Answers-only versus reasoning and answers, 29f Application fluency, procedural fluency and, 171–172, 228, 230 ARC, see Abstract-Representational-Concrete assessment ARC Assessment Planning Form, 107 ARC assessment response sheet, 126f, 127f Area model instruction, 371 Area problems, 119f Arithmetic properties, 18 Assessment, student, 2f access and, 122–124 CRA instruction and, 125 definition of, 97 diagnostic interview and, 112–113, 128 error pattern analysis as, 108, 255 fractions and, 60, 62t information from, 107 instructional accommodations and, 251 instructional decisions and, 288 literature support and, 6 methods for, 276 MTSS/RTI and intensifying of, 247–267 of prior knowledge, 227 purpose and process of, 107 the SOLO taxonomy and, 173 struggling learners and, 97–133 students response to, 356f summary of, 290 Adding It Up, National Research Council (2001) report, types of, 98t word problems for, 125 see also Mathematics Dynamic Assessment (MDA); Monitoring and charting performance Assessment-related constructs, struggling learners and, 115–124 Associative property, multiplicative strategy, 55f, 350f --- Authentic contexts CRA instruction, 195 explicit instruction and, 215 identifying, 125 instructional practices and, 78, 161, 214 students interests and, 213f, 215 Barriers curriculum factors and, 9, 71f, 95, 296–298 information on, 292 learning characteristics as, 293, 361 special education and, 8 struggling learners and, 36, 69–96 understanding and, 60–61, 77–78 Base-10 system, 17 CCSS domain, 253–254 conceptual understanding with, 227f concrete materials for, 197f grid paper for, 367f operations and, 19–21 regrouping with, 228f Behaviors of struggling learners, 70t, 72t, 74, 76–77, 289, 359 Benchmark Assessments, 101 Bifocal Vision for Math Teaching activity, 38–39 Big ideas CCSS and, 17 content strands and, 17, 38–39 importance of, 15–39 teacher self-examination and, 38–39 Case study, 345–374 CBM, see Curriculum-based measurement CCSS, see Common Core State Standards CGI, see Cognitively Guided Instruction Charting performance, see Monitoring and charting performance Child versus adult views, 7, 41 Children’s mathematics, learning trajectories for, 41–65 Choices, 121–122 Class Mathematics Student Interest Inventory Form, 213f Classroom instruction, see Whole-class instruction Classroom-Based Formative Assessments, 102 CLD students, see Culturally and linguistically diverse students Cognition, see Metacognition Cognitive interview, 226t Cognitively Guided Instruction (CGI), 46–49 Collaborative approach, 271 color-coding, 222, 256 Common Core State Standards (CCSS), 4, 5–6 adaption illustration of, 103t associated skills cluster as, 284t big ideas and, 17, 24 Eight Standards for Mathematical Practice in, 102, 159, 171, 191t NCTM process standards and, 31–32, 32t, 37t Number Operations in Base Ten as, 253–254 Common error patterns, 110–111, 356f Communication in classroom, 1 disconnect in, 89 impact of learning characteristics on, 84 process standards and, 29–30, 115 Commutative property, as multiplicative strategy, 55f, 350f Compensation strategy, 229t Computational fluency error pattern analysis, 109 as fundamental skill, 18–19, 165 Index Concepts, 16, 42 fractions and, 58–61, 63 learning objectives and, 212 skills and, 143t, 207, 287 Conceptual knowledge, 76 lack of, 105 symbols and, 198 Conceptual understanding, 31 instructional choices and, 226 instructional decisions and, 226 procedural fluency and, 93, 163, 172–174, 218t, 223–230 vocabulary knowledge and, 176 Concrete-level understanding assessment and, 105 modeling and, 195–196 Concrete-Representational-Abstract (CRA) instruction assessment and, 104 Explicitness, instructional levels of and, 195, 202, 203–204t instructional programs, 128 Scaffolding and, 243t sequence of, 192–195, 201f Behaviors of struggling learners, 70t, 72t, 74, 76–77, 289, studies on, 241 understanding and, 199 visual cues and, 194f Concrete-semiconcrete-abstract (CSA), 104 Connections between ideas graphic organizers and, 211f impact of learning characteristics on, 231 process standard as, 30–31, 285 representations and, 218–223, 236 scaffolding and, 210 Construct viable arguments and critique the reasoning of others, 286, 354 Content big ideas and, 17, 38–39 complexity levels, 144t expectations for, 159 geometry as, 24 learning intentions and, 191 Children’s mathematics, learning trajectories for, 41–65 learning needs and, 239 learning trajectory and, 65 Class Mathematics Student Interest Inventory Form, 213f measurement and data as, 22–23 overview of, 5 proficiency stage of learning and, 31, 94 CLD students, see Culturally and linguistically diverse Continuous assessment, 2–3, 6–7 case study and, 352–355 instructional decisions and, 287 Continuum of instructional choices, 138–142 application of, 145–153 making choices across, 142–145 scaffolding across, 149–150 Continuum of learning, 61, 121, 117f see also Learning, stages of Cooperative learning groups CLD students in, 91 as group instruction, 83 teacher directed instruction and, 152, 205 use of, 140f Core beliefs, 76 Core instruction, 240f, 249f for all students, 272–273, 275 differentiated instruction and, 273–274 flexible grouping with, 204–208 Counting, 44–45 Commutative property, as multiplicative strategy, 55f, manipulatives and, 45, 51 opportunities for, 233 strategies for, 48–50 types of, 47t CRA, see Concrete-Representational-Abstract instruction --- Index Cuing attention and memory disabilities and, 82 choices as, 121–122 explicitness and, 152 multisensory, processing disabilities and, 87–88 tools for, 207 see also Mnemonics Culturally and linguistically diverse (CLD) students, 89–91 cultural differences among, 90–91 funds of knowledge and experiences from, 250–251 Culturally responsive materials, 214 Curriculum considerations and instructional reforms, 92 Curriculum factors as barriers, 9, 71f, 95 297–298 success and, 91–93 Curriculum-based measurement (CBM), 6, 101 Data from benchmark assessments, 287, 352–353 methods for use of, 276 from summative assessments, 271 Data analysis content strands as, 22–23 database from, 212 decision making and, 239, 253 Decision making, 34 data for, 239, 253 information and, 97 instructional process of, 281 Describing wheel, graphic organizer, 177f Diagnostic Assessments of Achievement, 101 Diagnostic interview, 112–113 Diagnostic interviews, information from, 128–129 Diagrams for problem solving, 219 Differentiated instruction core instruction and, 273–274 determining need for, 263 instructional intensity and, 265–266 planning and, 247–251 whole class instruction as, 365–368 Disability-related characteristics, 80–88 Disconnections, communication, 89 Discrete independent variable, 241 Discrete learning, see Concrete-level understanding Diverse learners, see Struggling learners Division explicit trade algorithm for, 170f as operation, 54 for struggling learners, 58 Documents, on instructional strategies and practices, 37t Doing mathematics, see Process standards Domains, as standards, 5, 285 see also Content Double-digit addition, strategies for, 225f, 227 229f Drawings Algebra solutions in, 198f as assessment tool, 30f concrete materials and, 201 kinesthetic cues and, 197 representational-level learning with, 129, 219 strategies for, 199f, 200f Dynamic assessment, see Mathematics Dynamic Assessment (MDA) Educational contexts, 241 Educational materials, see Materials Effective Teaching Practices for General and Struggling Index 377 Efficiency, developing of, 118 EIAs, see Essential instructional approaches Emergent stage of learning, 59, 62t Engaged dialogue assessment strategy, 113 Engagement, 70t application and, 171 expectations for, 89 in mathematical discourse, 179–180 response opportunities and, 182–185, 259 struggling learners and, 151–153 funds of knowledge and experiences from, 250–251 student practice and, 148, 285–286 word walls for, 176 Curriculum considerations and instructional reforms, English language learners, 1 linguistic differences for, 90 mathematical discourse, 179–180, 224f native language use for, 176 RD/MD, as related for, 88 Equity, access and, 10 Error pattern analysis assessment and, 108, 255 computational fluency, 109–111 mistakes for, 233f observations and, 129, 164 Essential instructional approaches (EIAs) case study and, 345–346 Language and, 174–180 MTPs integration with, 217–237, 220f, 224f, 232f research support for, 241–245 Essential Instructional Approaches (EIAs) struggling learners and, 155–216, 156t, 270t tiered instruction and, 248 Evaluating an Assessment Against NCTM Standards activity, 133 Evaluation activity for, 278 model for, 277 MTSS and, 269–278 of performance, 157 of prior knowledge, 291 see also Monitoring and charting performance Expectations of content, 159 for engagement, 89 for productive struggle, 231 for representations work, 219 for struggling learners, 217–237 Experiences of students CLD students and, 90 generalization of, 18–19 see also Prior knowledge Explicitness, instructional levels of activities and, 145 Documents, on instructional strategies and practices, authentic context and, 215 characteristics of, 140f CRA instruction and, 195, 202, 203–204t cuing and, 152 examples of, 139f to implicitness as continuum, 141–142 instructional practices and, 143t, 151, 152f teacher direction and, 255–257 in think-aloud strategies, 221 vocabulary practices and, 175–178 Expressive response assessment and, 120–122 examples of, 123f, 124f in practice activities, 190f Excerpted from Teaching Mathematics Meaningfully: Solutions Extension, see Adaption; Generalization Facts fluency and, 43, 50t, 63 Effective Teaching Practices for General and Struggling as known or derived, 55f --- 378 Index FASTDRAW strategy peer-tutoring activity with, 342 problem solving and, 208 word problems and, 260t, 262 Feedback as corrective, 202 as effective, 235 instruction and, 158 during modeling, 35 response opportunities with, 180–189, 232f, 259 during scaffolding, 152 see also Monitoring and charting performance Figurative composite units, 54, 56t, 59f FIND strategy, 258, 370f Fluency with algorithms, 165–171, 167f, 169f in application, 171–172 of computation, 165, 166f EIAs and, 162–164 facts and, 43, 50t, 63 in proficiency and maintenance stages of learning, 118 with whole numbers, 25t Formative assessments in case study, 354t decision making and, 253 hypothesis and, 131–132 identifying for, 254–255 rubrics as, 114 Summative versus, 101–102 use of, 226 Foundational knowledge cognitive interview and, 226t identifying and understanding as, 1–2, 3, 4, 64, 282 standards relation to, 253 Fractions alternative procedures and, 21–22 assessment and, 60, 62t concepts and, 58–61, 63 fluency with, 25t instructional program recommendations and, 61 representation of, 59–60 SOLO taxonomy, 173 Frayer Model graphic organizer, 177, 178 Generalization, experience of, 18–19 Generalization stage of learning overview of, 116 teaching strategies and, 120 understanding and, 118–119 Geometry content strands and, 24 measurement and, 25t Goals of instruction, 179 instructional decisions and, 288 learning objectives and, 218t of modeling, 156 see also Learning, stages of Graphic organizer visuals, 193f Graphic organizers, 176–178 connections between ideas and, 211f Group instruction anchors for, 266 cooperative learning as, 83 differentiated instruction and, 204 instructional decisions for, 148–149 prior knowledge and, 150 scaffolding and, 148f as supplemental, 368 whole class instruction as, 127 Index High-Stakes Tests, standardized and, 100 Hypotheses, instructional, 67f formative assessment information for, 131–132 information for, 283 reasoning and, 42–43 IDEA, see Individuals with Disabilities Education Improvement Act of 2004 (PL 108-446) Identifying and understanding appropriate procedures, 224 authentic contexts, 125 concepts or skills for practice, 207 content preparation for, 347 with disabilities, 234, 360t formative assessments and, 254–255 as foundational, 1–2, 3, 4, 64, 282 information structure, 85 instructional decisions and, 283 learning intentions, 156 mathematical areas for, 284 performance traits, 292–293t, 358–359 place on learning trajectory, 291, 355–356 tasks, 233 IEP, see Individualized education program IES, see Institute of Education Sciences Illustration adaptations in, 103t FIND strategy in, 258f, 370f instruction with visuals, 256 procedural fluency in, 164f proficiency in, 163f tiered instruction in, 240f, 249f, 272f Implementing the 11 Essential Instructional approaches activity, 216 Implementing the Essential Instructional Approaches activities, 216 Implicitness, levels of instruction characteristics of, 140f, 142f continuum as explicitness to, 141–142 examples of, 139f instructional practices and, 143t, 151, 152f student directed instruction and, 151–153 Incorporating Effective Teaching Practice into a Lesson Plan activity, 237 Independent practice, 157, 182 see also Practice opportunities Individual Mathematics Student Interest Inventory Form, 211–213 examples of, 212f Individualized education program (IEP), 123 Individualized instruction, 370–371 Individuals with Disabilities Education Improvement Act (IDEA) of 2004 (PL 108-446), 242 accommodations and, 251 Informal data collection form, 186f Information, 3 from assessment, 107 about barriers, 292 decision-making and, 97 diagnostic interviews as, 128–129 for hypothesis, 283 identifying structure of, 85 for instruction, 108, 112, 132 passive learning and, 77 about student’s stages, 46 about student’s understanding, 122 Initial acquisition stage of learning accuracy in understanding in, 117 authentic contexts and, 214 overview of, 116 Excerpted from Teaching Mathematics Meaningfully: Solutions 77 --- Institute of Education Sciences (IES), 61 Instruction feedback loop for, 158 focus for, 274–275 goals of, 179 ideas for, 244t, 292, 357 information for, 108, 112, 132 intervention and, 276 language and, 250–251 scaffolding and, 146f, 148f, 150f seven anchors for, 252f, 265f special education and, 92 student’s ideas inform, 64 with visuals illustration as, 256 see also Instructional strategies and practices Instructional approach to mathematics, 208–211 positive outcomes with, 243t see also Essential instructional approaches (EIAs) Instructional Approaches activity, 153–154 Instructional decisions conceptual understanding and, 226 along continuum, 153–154 flexibility in, 137–154 group instruction and, 148–149 MDA and, 130–131, 291 performance data and, 271, 287 process for, 281 real time making of, 182 as student-centered, 162 Instructional games practice opportunities with, 183f tips for making, 184f Instructional hypotheses, see Hypotheses, instructional Instructional intensity, 240f, 249f differentiating, 265–266 increasing levels of, 252f, 272f MTSS and, 269–278, 275t Instructional materials, see Materials Instructional Pacing, 92–93 Instructional program recommendations CRA instruction and, 128 fractions and, 61 number and operation sense, 159 practices for, 182 problem solving and, 28 research and, 242 SOLO taxonomy and, 173 substandards and, 158 Instructional reforms and curriculum considerations and, 92 Instructional strategies and practices authentic contexts and, 78, 161, 214 cooperative learning groups/peer tutoring and, 205 documents on, 37t EIAs and, 155 explicitness or implicitness and, 143t, 151, 152f games/self-correcting materials and, 183f, 184f, 185f general strategies as, 42 learning stages and, 44–45 literature support for, 9 meaningful contexts for, 212 modeling and, 34, 48, 217 as multidimensional, 241 National Mathematics Advisory Panel on, 137 problem-solving, 27–28, 32–33 receptive and expressive response formats, 131 research base to improve, 244–245 scaffolding and, 145–147, 201 school-wide, 269–270 for struggling learners, 1–11, 32–33, 91–94, 239–246 see also Assessment; Monitoring and charting Index Instructional time, 243t, 244t Instrumental understanding, 26 Interactive learning, 202 Interest inventories, 216 Intermediate stage of learning, 60, 62t Intervention instruction and, 276 research on, 264 Inverse operations, 168f Kinesthetic cues drawings and, 197 for processing disabilities, 87 Knowledge and skill gaps instructional intensity and, 254 for struggling learners, 78–80 Language EIAs and, 174–180, 224f instruction and, 250–251 symbols and, 86, 89–90 understanding and, 199 Language development, mathematics achievement and, 79, 81 Language-based processing difficulties, 87 Learned helplessness, learning characteristic as, 8, 74–77, 231 Learning deep, teaching for 4 determing of, 158–159 memory and, 81 Instructional hypotheses, see Hypotheses, instructional mistakes for, 233 through practices, 27 stages of, 44–46, 59–60, 115–119, 116t Learning characteristics as barriers, 293, 361 performance traits and, 95f, 297f struggling learners, 8–9, 71–88, 145 Learning disabilities, 214 see also specific disabilities Learning intentions, 156 content and, 191 EIAs and, 158–161 examples of, 160f sharing, 156, 160–161 Learning needs content and, 239 Instructional reforms and curriculum considerations EIAs and, 155 grouping based on, 205 of students’ with disabilities, 92, 251 Learning objectives, concepts and, 212 cooperative learning groups/peer tutoring and, 205 Learning standards, see Instructional program recommendations Learning trajectory, 7–8, 281 capabilities during, 50, 233 games/self-correcting materials and, 183f, 184f, 185f identifying place on, 291, 355–356 mathematics and, 41–65 relation of, 34 standards and, 286 Lesson plans, see Planning Line segments, 178f Line symmetry, 24 Linear equations, algorithm for, 168f Linguistic differences, 89–90 Literature support for assessment, 6 for differentiated instruction, 204 for instructional practices, 9–10 for peer-mediated learning, 206 for representations, 220 --- 380 Index Maintenance stage of learning fluency in, 118 overview of, 116 Manipulatives concrete-level understanding, 195–196 counting and, 45, 51 materials as, 23 Mastery of basic facts, 49–51, 50t, 165 demonstration of, 108 Materials as concrete, 196f, 197f instructional games/self-correcting materials, 183f, 184f, 185f manipulatives as, 23 ten frames as, 20f Math anxiety, 80, 289 Math practices, 285–286, 372–373 Math standards, 284–285 see also Common Core State Standards (CCSS) Mathematical discourse, 89 engagement in, 179–180, 218t, 243t for English language learners, 179–180, 224f Mathematical practices development of, 27 emphasis on use of, 189–192 Mathematics achievement Diagnostic Assessments of Achievement and, 101 language development and, 79, 81 Mathematics difficulties, see Reading difficulties and Mathematics difficulties (RD/MD), as related Mathematics Dynamic Assessment (MDA) ARC assessment and, 103 assessment through, 125–128 conducting, 128–130 instructional decisions, 130–131, 291 overview of, 125 results of, 130f Mathematics learning productive struggle in, 231–235, 236 writing and, 180 Mathematics processes adult versus child views in, 7, 41 see also Process standards Mathematics vocabulary categories of, 175 EIAs and, 174–180 visuals cues and, 194f Math-specific learning needs, determining of, 3, 7 assessment tasks and, 288, 290–291 instructional decisions and, 283 MDA, see Mathematics Dynamic Assessment Meaningful connections, metacognitive disabilities and, 84–85 Meaningful contexts, abstract reasoning development in, 211–215 Measurement geometry and, 25t units of, 58–59 Measurement and Data, content as, 22–23 Memory, working and learning with, 81–82 Memory disabilities retrieval and, 81 summary of, 72t, 80 Metacognition strategy examples and, 28–29, 35–36, 85 Index Mistakes, see Error pattern analysis Mnemonics strategy instruction and, 82–83 struggling learners use of, 262 visual strategies for, 209f Model with mathematics (NGA Center for Best Practices & CCSSO, 2010), 171 Modeling at abstract level, 198–204 CCSS and, 191t concrete-level understanding and, 195–196 feedback during, 35 instructional games/self-correcting materials, 183f, goal of, 156 instructional strategies and practices, 34, 48, 217 overview of, 34 process standards and, 34 at representational level, 196–198 of thinking, 208 Models for evaluation, 277 problem solving with, 222 self-monitoring and, 206 Modes of input, 85–86 Modifications, see Adaptations Monitoring and charting performance case study example of, 358t strategies for, 128 systemic teaching and, 157 techniques for, 188f, 189f Mathematics difficulties, see Reading difficulties and Motor integration disabilities, 87, 110 MTSS, see Multi-tiered systems of supports Multiplication add-on strategy in, 77 alternative procedures for, 20–21, 112 as operation, 54 repeated addition process for, 256f for struggling learners, 58 Multiplicative reasoning, 43, 51–58, 63 strategy examples of, 52f, 53f, 54f, 56t, 350f see also Reasoning and proof skills Multi-tiered systems of supports (MTSS), 9 assessment and, 98t, 100–101 characteristics of, 269–274, 270t flexible grouping and, 205 instructional intensity and, 269–278, 275t number sense and, 93–94 RTI and, 239–246 Multi-tiered systems of supports (MTSS)/Response to intervention (RTI) case study and, 345–346 intensifying assessment and, 247–267 Multi-tiered systems of supports (MTSS)/Response to Meaningful connections, metacognitive disabilities and, intervention (RTI) instructional Tiers activity, 246 Meaningful contexts, abstract reasoning development National Council of Teachers of Mathematics (NCTM) on access and equity, 10–11 on assessment, 97 curriculum content strands and, 37t definition of standards and prompts by, 99t Effective Mathematics Teaching Practices by, 218t process standards and, 5–6, 31–32, 114 tiered instruction and, 248 see also Instructional program recommendations National Mathematics Advisory Panel final report (2008) by, 24–26, 242 on instructional practices, 137 National Research Council (2001), 242 NCTM, see National Council of Teachers of Mathematics NCTM Mathematics Teaching Practice (MTPs), 345–346 EIAs integration with, 217–237, 220f, 224f, 232f --- FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully Nonresponders, adaptations for, 276–277 Number sense common errors and, 111 definition of, 15 development of, 184, 225 MTSS and, 93–94 Number sequences, 45–46 student capabilities and, 47–48t Numbers and Operations algebraic thinking and, 17–19 base-10 system and, 19–21 connections between, 18 as content strand, 15 division as, 54 error patterns and, 110 fractions and, 21–22 instructional program recommendations for, 159 multiplication as, 54 MTSS and, 93–94 order of, 159 place value and, 20 see also Number sense Numeracy, 63 research on, 184 Objects, see Concrete-level understanding; Manipulatives Observations from diagnostic interviews, 130 error pattern analysis and, 129, 164 OGAP, see Ongoing Assessment Project Multiplicative Framework Ongoing Assessment Project (OGAP) Multiplicative Framework, 51–55 Operations Algebra and, 17–19 in Base Ten as CCSS, 253–254 Base-10 system and, 19–21 division as, 54 instructional program recommendations for, 159 multiplication as, 54 place values and, 20 see also Numbers and Operations Opportunities for improvement, 11 Index Parallel, 178 Part-part-whole strategies, 50 Passive learning productive struggle and, 231 struggling learners and, 77–78 Patterns fluency and, 164 of performance, 107 problem solving and, 109 search for and use of, 18–19, 36 see also Common error patterns Peer-mediated learning activities for, 143t, 341–342 grouping structures for, 206–208 Peer-tutoring, 206, 341–342 Perceptual multiples, 52, 56t Performance evaluation of, 157 level of understanding and, 108 pattern of, 107 Performance charting, see Monitoring and charting performance Performance data instructional decisions and, 271, 287 struggling learners and, 244t Performance Rubric, 186f Performance traits, 8 identifying observance of, 292–293t, 358–359 impact of learning characteristics and barriers, 95f, 297f record of, 294f struggling students and, 69, 70t Perseverance, 235 Phonological processing, effect on, mathematics learning, 79 Pictorial representations, 220, 221f Place values algorithms and, 167, 169 multi-digit numbers, 258f operations and, 20 problem solving and, 110–111 recording sheet for, 368f, 369f Planning differentiated instruction and, 247–251 instructional intensity, 266 for success, 248 whole-class instruction and, 128 Planning Intensive Instruction Using Instructional Anchors activity, 267 Polygon, 178f Positive reinforcement, 157, 290 Practice opportunities Objects, see Concrete-level understanding; Manipulatives for counting, 233 engagement and, 148 instructional games for, 183f providing for, 35, 120, 175 OGAP, see Ongoing Assessment Project Multiplicative structured language experiences and, 1 teacher support levels and, 120, 229 visual diagrams for, 222 Pre instruction/anticipatory set, 158 Precision, 35, 222, 354 Preservice teachers, see Teachers Principles and Standards for School Mathematics (NCTM), 4 problem solving and, 27–28 process standards and, 5–6, 27–28 representation and, 220f Principles to Actions (NCTM) effective formative assessment definition by, 102 MTPs, 217, 270 Prior knowledge activation of, 33, 89, 121, 201, 214 assessment of, 227 gaps and strengths in, 288, 291, 357 group instruction and, 150 lack of, 274 response cards and, 182f, 183f Problem solving, 5 drawings and, 197 FASTDRAW strategy and, 208 impact of learning characteristics on, 84, 121 instructional practices and, 28, 32–33 mixing problem types and, 235 models of, 222 NCTM and, 27–28 patterns and, 109 place values and, 110–111 reasoning for, 234 strategy examples of, 32, 70t, 209f, 210t structured dialogue sheet for, 181f Procedural fluency, 31 accuracy requirements and error patterns for, 110 building of, 228–230 conceptual understanding and, 93, 163, 172–174, 218t, 223–230 for understanding, 225 Procedure-first instruction, 225 Process standards communication as, 29–30, 115 --- Process standards—continued modeling and, 34 NCTM and, 5–6, 31–32, 114 proof skills as, 28–29f, 36 representation and, 30, 114 Processing disabilities summary of, 73t, 296t types of, 85–88 Productive disposition, 31, 32 procedural fluency and, 163 Productive struggle in learning mathematics, 231–235, 236 Proficiency stage of learning content strands and, 31, 94 fluency in, 118 illustration of, 163f overview of, 116 teaching strategies and, 15–16 understanding and, 147–149, 281 see also Monitoring and charting performance Progress Monitoring Assessments, 100–101 Prompts for math writing, 180f purpose of, 226t questions or, 187 recognition, 190f for students, 114, 115, 172 Index RD/MD, see Reading difficulties and mathematics difficulties, as related Reading, research on, 264 Reading difficulties and mathematics difficulties (RD/ MD), as related, 88 Reading disabilities, 88 summary of, 73t, 296t Reason abstractly and quantitatively, 33 practice of, 286 Reasoning and proof skills development of, 49 hypotheses and, 42–43 for problem solving, 234 process standards as, 28–29f, 36 strategy examples for, 50t, 161 Receptive response assessment and, 120–122 examples of, 124f practice activities and, 190f Recognition recognition prompt, 190f response opportunities versus, 187–189 Record, 294f for place values, 368f, 369f teachers’ notes as, 294f, 360t Reflections: How to Support the NCTM Teaching Practices with EIAs activity, 237 Reforms, see Instructional reforms Regrouping errors, 110–111 Reinforcement, see Feedback Relational understanding, 19, 26 Repeated abstract composite grouping, 54, 56t Repeated addition process, 256f Representation of fractions, 59–60 learning characteristics and, 74–75f, 86 of mathematics, 172 multiplicative reasoning and, 55f process standards and, 30, 114 of solutions, 86f types of, 220, 221f Representational-level understanding assessment centers and, 125 drawings and pictures for, 129 modeling for, 196–198 Representations, categories for, 219 Research, 10 on basic fact automaticity, 165 on error patterns, 109 to improve instruction practices, 244–245 on math intervention, 264 Productive struggle in learning mathematics, 231–235, on mathematics education, 93 on numeracy development, 184 about special needs, 242 on struggling learners, 30 support for EIAs and, 241–245 on working memory, 81–82 Response cards, 182, 183f Response formats assessment from, 288–290 receptive and expressive as, 131 recognition-type of, 187 Response opportunities anchor, 252f, 259–263 feedback and, 180–189, 232f, 235, 259 providing for, 181, 182–185, 227 see also Practice opportunities Response to intervention (RTI) assessment and evaluation with, 100–103 MTSS and, 239–246 see also Multi-tiered systems of supports (MTSS)/ Response to intervention (RTI) Responsive instruction, planning and implementing of, 3–4, 9–10 Reading difficulties and mathematics difficulties (RD/ case study and, 362–374 guidance for, 298 hypotheses as guide for, 283 Retrieval skills, 81 Role playing demonstration, 113 Rounding, 167 RTI, see Response to intervention Rubric examples of, 114f, 115f for fluency development, 174t as formative assessments, 114 Scaffolding connections and, 210 across continuum of instructional choices, 149–150 with emphasis, 146f, 147–149 examples of, 146f, 148f, 150f feedback during, 152 group instruction and, 148f instructional strategies and practices, 145–147, 201 tiers as, 271–272 visual cues as, 262t Schema-based instruction, 192 Schematic representations, 220, 221f School performance, see Mathematics achievement School-wide practices, 269–270 SEAL, see Stages of Early Arithmetic Learning instruction Selective attention, see Attention disabilities Self-correcting materials, practice opportunities with, 185f Self-evaluation/monitoring, metacognition and, 85 Self-monitoring, 206 example of strategy for, 209f Self-observation, 11 Self-reflection inventory, 153–154, 216, 237, 246, 278 Self-regulation, 243t --- FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully Seven anchors model, 252f Sharing, learning intentions, 156, 160–161 Skills, 16 application of, 171 assessment of, 106 cluster of, 284t concepts and, 143t, 207, 287 discrimination as, 83 SOLO, see the Structure of the Observed Learning Outcome taxonomy Solving linear equations, 168f Special education instruction and, 92 learning barrier accommodation in, 8 MTSS and, 273 practices in, 244 Special education teachers, see Teachers Stages of Early Arithmetic Learning (SEAL) instruction, 43–48 Standard algorithm double digit addition with, 225f, 227, 229f as multiplicative strategy, 55f, 350f Standard procedures, 21, 28 Statistics, processes and, 23 Story problems, see Problem solving; Word problems Strategic competence, 31, 163 Strategy instruction addition strategies and, 48, 50 mnemonics and, 82–83 problem-solving and, 32 see also Cuing Strengths, 11 Structure of assessment, 289 of evaluations, 277 grouping and, 204–208, 265 of information, 85 intensive instructional sessions, 259 language experiences and practice opportunities with, 1 for recording, 294f SOLO taxonomy as, 173t standards, 284 use of, 35, 140f Structure of the Observed Learning Outcome (SOLO) taxonomy, 173t Structured dialogue cue sheet, 181f Struggling learners algebra and, 26, 60–61 ARC assessment, 104–108 assessment for, 97–133 assessment-related constructs for, 115–124 attention disabilities of, 33, 83–84 barriers for, 36, 69–96 changing expectations for, 217–237 choices continuum for, 137–154 curriculum considerations for, 202t diagnostic interviews for, 112–113 EIAs and, 155–216, 156t, 270t engagement and, 151–153 error pattern analysis, 109–111 fractions and, 22 graphic organizers for, 177 instruction for, 1–11, 32–33, 91–94, 239–246 intention importance for, 160 learning characteristics of, 8–9, 71–88, 145 metacognitive disabilities and, 33, 69 mnemonics use by, 262 multiplication and division for, 58 research on, 30 response opportunities for, 184, 187 Index Index 383 time for, 263 visuals use for, 192, 219, 221 word problems for, 260 Struggling learners, specific learning needs of, 2–3, 7–9 assessment tasks and, 288 case study and, 357–362 instructional decisions and, 283 performance traits and, 292 Student directed instruction characteristics of, 140f continuum as teacher directed to, 139–141 examples of, 139f, 146f, 147 implicitness and, 151–153 Student responses, performance data from, 185–187 Student-centered instruction instructional decisions and, 162 Stages of Early Arithmetic Learning (SEAL) instruction, student-directed and, 137–138 student-directed versus, 137–138 Students interests, authentic contexts of, 213f, 215 Students with disabilities barriers to success for, 69–96 identified as, 293, 360t learning needs of, 92, 251 testing accommodations for, 289 Substandards program recommendations and, 158 skills cluster and, 284t Subtechnical words, 175 Subtraction algorithms and, 167, 168 with understanding, 173 Success, 1 barriers to, 69–96 with core instruction, 274 curriculum factors and, 91–93 determining criteria for, 159–160 growth mindset for, 234 math anxiety and, 80 MTSS and, 277 planning for, 248 teaching systemically for, 155 Summative assessments, data from, 271 Formative versus, 101–102 Structure of the Observed Learning Outcome (SOLO) Supplementary instruction at elementary level, 252 MTSS and, 205, 240f, 249f, 272f, 275 response opportunities and, 259 time for, 264 Support cuing as, 121 determining appropriate levels of, 156 teacher features of, 91, 179, 263, 290 Symbolic words, 175–176 Symbols conceptual knowledge and, 198 language and, 86, 89–90 representations as, 18–19, 81, 250 Systemic instruction framework phases of, 156, 157f see also Teaching systemically Teach Math Metacognition anchor, 257–258 Teacher directed instruction characteristics of, 140f cooperative learning groups and, 152, 205 examples of, 139f, 146f, 147 explicitness, instructional levels of and, 255–257 to student directed as continuum, 139–141 --- FOR MORE, go to http://www.brookespublishing.com/Teaching-Mathematics-Meaningfully Teachers knowledge for, 240 notes record by, 297f, 360t in special education, 294 to student ratio, 252f, 264–265 Teaching for deep learning, 4 forest and trees analogy and, 15, 36–38 incrementally, 77 math metacognition, 257–258 measurement, 23 vocabulary, 174–176 Teaching Mathematics Meaningfully Process as case study, 345–374 examples of, 2f, 10, 67f, 248f, 279f overview of, 281–298 Teaching strategies generalization stage of learning and, 120 proficiency stage of learning and, 15–16 see also Instructional strategies and practices Teaching systemically, 155–158 Teaching to mastery, see Mastery Technical words, 175 Ten Frame, 20f Testing, 289 Think-aloud strategies, 82, 113, 208 explicitness in, 221 use for story problems, 343–344 Thinking strategies, see Metacognition; Strategy instruction Tools appropriate use of, 35–36, 235 for cuing, 207 drawings as, 30f representations as, 219 Traditional regrouping algorithm, 258 Transitional multiplicative strategies, 52–55, 56t, 350f Triangle as term, 177, 178f Index Triangle as term, 177, 178f UDL, see Universal Design for Learning Understanding accuracy in, 117–119 algorithms and, 169 barriers and, 60–61, 77–78 CRA instruction and, 199 demonstration of, 105 early numeracy and, 63 information about student’s, 122 learning intentions, 161 MDA and levels of, 130f performance and, 108 procedural fluency for, 225 subtraction with, 173 see also Assessment; Learning Units of measure, see Measurement Universal Design for Learning (UDL), 204 core instruction with, 273 planning and instructional framework of, 249–250 Universal Screeners, 100 Value in learning, 212 Variables, 168f, 241 structured dialogue cue sheet for, 181f Visual cues CRA instruction and, 194f for mnemonics, 209f as scaffold, 262t in strategy instruction, 219 Visual diagrams as explicit, 223f as nonexplicit, 222f Visual models, 22 Visual processing disabilities, 86–87 see also Processing disabilities Visual representation, 220 Visual spatial processing difficulties, 87, 296t Visual vocabulary word strategy, 178f Visuals cognitive framework as, 210 utilization of, 192–204, 256f, 257f Vocabular y EIAs and, 174–180 word problems and, 78–79 Wait time, providing, 82, 88 What Works Clearinghouse (WWC) practice guide, 61 Whole-class instruction data collection for, 186f differentiated instruction for, 365–368 groups for, 127 planning for, 128 Word problems assessment with, 125 CGI and, 48–49 FAST DRAW strategy and, 260t one variable equations, 260t, 261t, 262t reasoning for, 234 story problems as, 172t, 221f, 222f think-aloud strategies, 343–344 vocabulary and, 78–79 Word walls, 176–178 Writings, mathematics learning with, 180