Improving Mathematical Problem Solving in Grades 4 through

Improving Mathematical Problem Solving in Grades 4 through 8 REL Appalachia Bridge Event November 6, 2013 Louisville, KY

Welcome & Overview Dr. Justin Baer Director, REL Appalachia CNA

What is a REL? • A REL is a Regional Educational Laboratory. • There are 10 RELs across the country. • They are administered by the U. S. Department of Education, Institute of Education Sciences (IES). • A REL serves the education needs of a designated region. • It works in partnership with the region’s school districts, state departments of education, and others to use data and research to improve academic outcomes for students. 3

What is a REL? 4

REL Appalachia’s Mission • Meet the applied research and technical assistance needs of Kentucky, Tennessee, Virginia, and West Virginia. • Conduct empirical research and analysis. • Bring evidence-based information to policy makers and practitioners: – Inform policy and practice – for states, districts, schools, and other stakeholders. – Focus on high-priority, discrete issues and build a body of knowledge over time. www. relappalachia. org 5

Welcome • Thanks to our partners: – Kentucky Association of School Administrators – Kentucky Center for Mathematics – Kentucky Department of Education • Please follow @REL_Appalachia on Twitter, and use the hashtag #RELAP when tweeting about this event! • Make sure to pick up your certificate of attendance before you leave at the end of the day! 6

Agenda 9: 40– 10: 30 Introduction to the Practice Guide 10: 30– 11: 00 Teaching Mathematical Problem Solving 11: 00– 11: 10 BREAK 11: 10– 12: 00 Identifying Challenges: Table Discussion and Group Work 12: 00– 12: 20 Lunch & Keynote Address 12: 20– 12: 50 Lunch & Networking 7

Agenda 12: 50– 1: 40 Applying the Practice Guide’s Five Key Recommendations: Breakout Sessions (Block 1) 1: 40– 1: 50 BREAK 1: 50– 2: 40 Applying the Practice Guide’s Five Key Recommendations: Breakout Sessions (Block 2) 2: 40– 2: 45 BREAK 2: 45– 3: 15 Next Steps in Improving Students’ Problem Solving / Taking It Back to the Classroom 3: 15– 3: 30 Closing Remarks 3: 30– 3: 45 Stakeholder Feedback Survey 8

Goals • Build participants’ knowledge of the importance of math problem solving in grades 4 through 8. (Why does it matter so much? ) • Engage participants in a meaningful and practical discussion of the evidence-based research in the Practice Guide. (How do we know what works? ) • Provide participants with evidence-based strategies that can be implemented in their schools. (How can we use this material in our classrooms to help students? ) 9

Goals • Plan for making use of the research-based practices to improve instruction, student achievement, and students’ preparation for more rigorous mathematics courses in middle and high school. • Facilitate networking/idea exchanges among participants, especially around improving mathematics instruction. • Provide participants with materials that can inform their efforts in improving students’ mathematical problemsolving ability. 10

Introduction to the Practice Guide “Improving Mathematical Problem Solving in Grades 4 through 8” Dr. John Woodward Dean, School of Education University of Puget Sound

Where Can I Find This Guide? • http: //ies. ed. gov/ncee/wwc/Practice. Guide. aspx? sid=16 • Doing What Works website: http: //dww. ed. gov/Math-Problem. Solving/topic/index. cfm? T_ID=41 12

What Are Practice Guides? Practice guides provide practical research-based recommendations for educators to help them address the everyday challenges they face in their classrooms and schools. • Practice guides include: § § § • Concrete how-to steps Rating of strength of evidence Solutions for common roadblocks The What Works Clearinghouse currently has 14 Practice Guides 13

Evidence Rating • Each recommendation receives a rating based on the strength of the research evidence. § Strong: high internal and external validity § Moderate: high on internal or external validity (but not necessarily both) or research is in some way out of scope § Minimal: lack of moderate or strong evidence, may be weak or contradictory evidence of effects, panel/expert opinion leads to the inclusion in the guide 14

Evidence Rating Recommendation Level of Evidence 1. Prepare problems and use them in whole-class instruction. Minimal 2. Assist students in monitoring and reflecting on the problem-solving process. Strong 3. Teach students how to use visual representations. Strong 4. Expose students to multiple problem-solving strategies. Moderate 5. Help students recognize and articulate mathematical concepts and notation. Moderate 15

Challenging Issues for the Panel • One definition of problem solving – Common agreement: § Relative to the individual § No clear solution immediately (it’s not routine) § It’s strategic – Varied frameworks § Cognitive: emphasizing self-monitoring § Social Constructivism: emphasizing community and discussions 16

Challenging Issues for the Panel • How much time should be devoted to problem solving (per day/week/month) – It’s not a “once in a while” activity – Curriculum does matter – Sometimes it’s a simple change § 4 + 6 + 1 + 2 + 9 + 8 averages to 5. What are 6 other numbers that average to 5? 17

Challenging Issues for the Panel • A script or set of steps describing the problem-solving process. – What we want to avoid: § Read the problem. § Select a strategy (e. g. , draw a picture). § Execute the strategy. § Evaluate your answer. § Go to the next problem. 18

Challenging Issues for the Panel • The balance between teacherguided/modeled problem solving and student-generated methods for problem solving. – Teachers can think out loud, model, and prompt. – Teachers can also mediate discussions, select and revoice student strategies/solutions. 19

Recommendation 1 • Prepare problems and use them in whole -class instruction. § Include both routine and non-routine problems in problem-solving activities. § What are your goals? § Greater competence on word problems with operations? § Developing strategic skills? § Persistence? 20

Recommendation 1 • There are many kinds of problems. – Word problems related to operations or topics: § I have 45 cubes. I have 15 more cubes than Darren. How many cubes does Darren have? – Geometry/measurement problems. – Logic problems, puzzles, visual problems. How many squares on a checkerboard? 21

Recommendation 1 • Prepare problems and use them in wholeclass instruction. § Ensure that students will understand the problem by addressing issues students might encounter with the problem’s context or language. § Linguistic issues are a barrier. § Cultural background is a big factor. 22

Ensure That Students Will Understand the Problem A yacht sails at 5 miles per hour with no current. It sails at 8 miles per hour with the current. The yacht sailed for 2 hours without the current and 3 hours with the current and then it pulled into its slip in the harbor. How far did it sail? yacht? slip? harbor? 23

Recommendation 1 • Prepare problems and use them in wholeclass instruction. § Consider students’ knowledge of mathematical content when planning lessons. § Sometimes it’s appropriate to have students practice multiple problems in the initial phase of learning. § Concept of division, unit rate proportion problems. § Sometimes it is appropriate to have a more inquiryoriented lesson with only one or two problems 24

Recommendation 2 § Assist students in monitoring and reflecting on the problem-solving process. § Provide students with a list of prompts to help them monitor and reflect during the problem-solving process. § Model how to monitor and reflect on the problemsolving process. § Use student thinking about a problem to develop students’ ability to monitor and reflect. 25

Recommendation 2 § This is what we want to AVOID. § Read the problem (and read it again). § Find a strategy (usually, “make a drawing”). § Solve the problem. § Evaluate the problem. 26

Provide Prompts or Model Questions § What is the story in this problem about? § What is the problem asking? § What do I know about the problem so far? § What information is given to me? How can this help me? § Which information in the problem is relevant? § Is this problem similar to problems I have previously solved? 27

Provide Prompts or Model Questions (continued) § What are the various ways I might approach the problem? § Is my approach working? If I am stuck, is there another way I can think about solving this problem? § Does the solution make sense? How can I check the solution? § Why did these steps work or not work? § What would I do differently next time? 28

Recommendation 3 § Teach students how to use visual representations. § Select visual representations that are appropriate for students and the problems they are solving. § Use think-alouds and discussions to teach students how to represent problems visually. § Show students how to convert the visually represented information into mathematical notation. 29

Cognitive Load: Problem Solving Through Words Alone Eva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with? 30

Draw a Picture? Eva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with? 31

Problem Representation Tape Diagram vs. Pictures Eva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with? 32

Tape Diagrams as a Tool Eva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with? 33

Tape Diagrams as a Tool Eva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with? 34

Tape Diagrams as a Tool Eva spent 2/5 of the money she had on a coat, then spent 1/3 of what was left on a sweater. She had $150 remaining. How much did she start with? $150 = 2/5 of the money. That means 1/5 = $75 5 x 1/5 = 5/5, or the whole amount, so 5 x $75 = $375 Eva started with $375. 35

Recommendation 4 § Expose students to multiple problem-solving strategies. § Provide instruction in multiple strategies. § Provide opportunities for students to compare multiple strategies in worked examples. § Ask students to generate and share multiple strategies for solving a problem. 36

Recommendation 5 § Help students recognize and articulate mathematical concepts and notation. § Describe relevant mathematical concepts and notation, and relate them to the problem-solving activity. § Ask students to explain each step used to solve a problem in a worked example. § Help students make sense of algebraic notation. 37

Practice Guide— Classroom Application Ms. Karen Campbell Dr. Tim Gott Ms. Kelly Stidham

Practice Guide—Classroom Application Karen Campbell Math Consultant Kentucky Center for Mathematics 39

Practice Guide—Classroom Application Kentucky Core Academic Standards • Where do we stand in implementing the new standards? • Is everyone on the same page? 40

Practice Guide—Classroom Application • Teachers know that it’s good practice to teach mathematical problem solving, but actually being able to implement this concept is another story. • Teachers need to learn how to combine the topic and skills into deeper problem-solving units. But how do you get from point A to point B? 41

Practice Guide—Classroom Application Teaching a problem versus completing a task: • Problem solving is less about the problem and more about the solving. You have to teach the solving part. • Problem solving teaches us to have multiple options for finding our way to a solution. 42

Practice Guide—Classroom Application • Teachers need to also teach analysis! • Teachers need to use a variety of visual representations! • Teachers know what they need to do (for the most part), but reinforcing these ideas is critical. • Connecting both sides of the brain… 43

Practice Guide—Classroom Application Math is a foreign language. 44

Practice Guide—Classroom Application Tim Gott Director, The Carol Martin Gatton Academy of Mathematics and Science in Kentucky Bowling Green, Kentucky 45

Practice Guide—Classroom Application • Mathematics is more than multiplication tables and percentages. • We know what problem solving looks like. It’s exploration, it’s using tools, it’s inquiry based. • We must integrate problem solving into the mathematics curriculum. 46

Practice Guide—Classroom Application Problem solving is the conduit for making things relevant. 47

Flow of Mathematical Concept Development* Experiential Kinesthetic, Tactile, Hands-On Manipulatives, Experiments, Physical tasks, Field trips, Measurements, Demonstrations Pictorial Visual, Spatial Pictures, Drawings, Graphs, Diagrams Concrete representations, Sketches Verbal Auditory, Linguistic Words, Critical vocabulary, Sentences, Phrases Both written and spoken Symbolic Abstract, Generalized Formulas, Equations, Expressions Variables, Symbols, Structures * Courtesy of Dr. Tim Gott 48

Practice Guide—Classroom Application The more strategies a student has for finding the answer, the more that student can choose the right strategy. 49

Practice Guide—Classroom Application Math doesn’t have to be entirely symbolic. Problem solving conceptualizes these equations into real-world, tangible ideas. 50

Practice Guide—Classroom Application Kelly Stidham Math Strategist Northern Kentucky Cooperative for Education @kastidham #RELalgebra kelly. stidham@nkces. org 51

Practice Guide—Classroom Application Statewide shift to Kentucky Core Academic Standards • It’s exciting, but difficult. • Strategies are aligned with standards of effective teaching, but the challenge comes in implementation 52

Practice Guide—Classroom Application Changing Perspective: Memorization Strategy-based fluency 53

Practice Guide—Classroom Application How can teachers translate the standards from an outcome to the vehicle through which they are solved? 54

Practice Guide—Classroom Application Problems create critical thinking, but where can teachers find quality content? • Working together. • Sharing resources. 55

Practice Guide—Classroom Application What does it mean to be a “Problem Solver”? • Students need time to get deeper into the content, strategies. • A number of students believe they “aren’t good at math” – but problem solving can change that. • Redefine the role of the student, focus on personal growth through the mathematical process. 56

Practice Guide—Classroom Application • Visual representations are an incredibly important part of learning how to do math. • Being able to see mathematical concepts in different ways helps break down the procedures. 57

Practice Guide—Classroom Application The solution isn’t about just addressing these modalities (area and length x width visualizations pictured above), but much more about giving students the space to find meaning and understand how things have the same relationship. 58

Identifying Challenges Table Discussion & Group Work Lydotta Taylor REL Appalachia

Identifying Challenges Please assign a table leader who will report out to the larger group at the end of this session. 60

Discussion Question • What role do you play in your school? – Teacher. – Administrator (Principal, etc. ). – Curriculum Coach. 61

Discussion Question • How do you use problem solving outside the classroom? 62

Identifying Challenges Work with your tablemates through each question. At the end of this session, you will be asked to identify the top three challenges your table faces when teaching problem solving. 63

Identifying Challenges • How does teaching problem solving fit into your mathematics curriculum? • What obstacles do you face in bringing the recommendations in the Practice Guide to your classroom to help students? • Do you feel you have time to dedicate to whole classroom instruction of problem solving? How can you spend more time on these areas? 64

Identifying Challenges • What challenges do your students face when they work through the context of specific problem-solving exercises? • What barriers do you face in teaching your students how to represent problems visually? • How can you work with your colleagues – other teachers, administrators, curriculum coaches – to strengthen your own teaching of problem solving in mathematics? 65

Identifying Challenges • Where do you find the best mathematics resources? Are they proven, or aligned with the research in the Practice Guide? • Teachers: Does your administration support the professional development you need? • Administrators: Do you support the professional development your teachers need? • How can you use the recommendations from the Practice Guide to improve students’ problem solving? 66

Discussion Question • What are the top three challenges you’ve faced in teaching problem solving? 67

Identifying Challenges Connecting the Dots: Responses from Content Experts 68

LUNCH Please grab a boxed lunch and return to your tables! • Keynote Address • Networking 69

Keynote Address Ms. Felicia Cumings-Smith Kentucky Department of Education

A Statewide Perspective Felicia-Cumings Smith Associate Commissioner, Office of Next Generation Learners, Kentucky Department of Education @Ky. Deptof. Ed 71

Networking Get to know your tablemates! • Where do you work? What’s your role and your specific challenges in teaching, school leadership, or other role in a school or district? • Are you on Twitter? Follow @REL_Appalachia and share your own Twitter handle with your tablemates! • Were you familiar with the Practice Guide before today? • What resources do you use to support mathematics instruction? Can you share resources that align with the evidence-based recommendations in the Practice Guide? • How do you plan to use the Practice Guide moving forward in your classroom or school? • What is your favorite success story from teaching math? 72

Applying the Practice Guide’s Five Key Recommendations Breakout Sessions

Applying the Recommendations Please report to your breakout session (chosen during registration). • Algebra, Kelly Stidham – Wilson Room. • Fractions, Karen Campbell – Mc. Creary Room. • Statistics, Sue Peters – Taylor Room. • Students Struggling with Mathematics, John Woodward – Jones Room. 74

Next Steps in Improving Student’s Problem Solving Taking It Back to the Classroom

Where Do We Go from Here? 76

Discussion Question • Who can be your ally to support your implementation of the recommendations from the Practice Guide? 77

Discussion Question • How can you involve parents to continue students’ learning at home? 78

Discussion Question • How can you use examples of problem solving in the classroom? 79

Where Do We Go From Here? We encourage you to set specific goals about how you will use the recommendations from the Practice Guide to improve problem solving in your classrooms and schools! 80

Closing Remarks & Stakeholder Feedback Survey Dr. Justin Baer Director, REL Appalachia CNA

Closing Remarks • Thank you for spending your day with REL Appalachia! • Join us on Wednesday, December 4, at 4 p. m. EST for a follow-up webinar, Identifying and Assisting Students Struggling with Mathematics. For more, please visit http: //www. relappalachia. org/newsevents/events. • Before you leave, please complete the Stakeholder Feedback Survey. This allows us to produce better future events, catering to your specific needs! 82

Stakeholder Feedback Survey In return for your completed Stakeholder Feedback Survey, we will give you a certificate of attendance from the Kentucky Department of Education, demonstrating that you have earned professional development credit. Thank you! 83