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NebraskaMATH A partnership to improve mathematics education in Nebraska Jim Lewis University of Nebraska-Lincoln Aaron Douglas Professor of Mathematics

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Why is this work important? U.S. competitiveness depends on dramatic improvements in the math and science education of its K-12 students. Rising Above the Gathering Storm, (National Research Council report 2007), identified four strategies for strengthening U.S. competitiveness. The first is: – Increase Americas talent pool by vastly improving K-12 science and mathematics education. The National Math Panel report asserts: – To compete in the 21 st century global economy, knowledge of and proficiency in mathematics is critical.

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How Are We Doing? The National Assessment of Educational Progress reports these data: 4 th Grade 8 th Grade Top (Mass)252299 Indiana243287 Nebraska239284 Nation239282 Bottom (Miss)227265

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Lets compare Nebraska with Indiana using NAEP data* 4 th Grade 8 th GradeIndiana Neb All students18 th 33 rd 18 th 27 th White 28 th 38 th 28 th 27 th Black 24 th 44 th 11 th 38 th Hispanic 24 th 40 th 13 th 41 st * NAEP data has 52 states including DC and DoD schools. Data for Black students are reported only for 46 states and only for 47 states for Hispanic students.

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Are our students proficient? In 2009, Nebraska reported that 92% of 8 th grade students met or exceeded Nebraskas proficiency standards.

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The realities of scale

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Our Belief: Good Teachers Matter There is much evidence that teachers are the most important variable in student learning – Studies have found that the effect size of good teaching is greater than any other variable, including students socioeconomic status. But Mathematics teaching is an extraordinarily complex activity involving interactions among teachers, students, and the mathematics to be learned in real classrooms. (National Math Panel, 2008)

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The Challenge We Face as We Teach Teachers What Mathematics do Teachers Need to Know and How Should They Come to Know Mathematics? – What does it mean to offer challenging courses and curricula for math teachers? – How do we help teachers translate the mathematics they come to know into classroom practice that leads to improved student learning?

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Can you compute 49 times 25? Of course: 49 x 25 245 + 980 1,225 Why might a fourth grader think the answer is 1,485?* 49 x 25 405 + 1080 1,485 * On May 4, 2010, Deborah Ball, Dean of the University of Michigan School of Education used this question as part of her testimony before the U.S. House of Representatives Education and Labor Committee. 5*(4+4)=40 or 5*4 + 4

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What is so difficult about the preparation of mathematics teachers? Our universities do not adequately prepare mathematics teachers for their mathematical needs in the school classroom. Most teachers cannot bridge the gap between what we teach them in the undergraduate curriculum and what they teach in schools. We have not done nearly enough to help teachers understand the essential characteristics of mathematics: its precision, the ubiquity of logical reasoning, and its coherence as a discipline. The goal is not to help future teachers learn mathematics but to make them better teachers. H. Wu, Professor, University of California, Berkeley

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What is so difficult ….? The mathematics taught should be connected as directly as possible to the classroom. This is more important, the more abstract and powerful the principles are. Teachers cannot be expected to make the links on their own. Get teachers to believe, that mathematics is something you think about - that validity comes from inner conviction that things make sense, that mathematical situations can be reasoned about on the basis of a few basic principles. The goal is to have teachers develop flexibility in their thinking, to be able to reason about elementary mathematics. Roger Howe, Professor, Yale University

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Educating Teachers of Science, Mathematics, and Technology New Practices for the New Millennium This National Research Council report recommends: a new partnership between K-12 schools and the higher education community designed to ensure high-quality teacher education and professional development for teachers.

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The Mathematical Education of Teachers Recommendations Teachers need mathematics courses that develop a deep understanding of the math they teach. Mathematics courses should – focus on a thorough development of basic mathematical ideas. – develop careful reasoning and mathematical common sense. – develop the habits of mind of a mathematical thinker and demonstrate flexible, interactive styles of teaching. The mathematics education of teachers should be based on – partnerships between mathematicians, mathematics education faculty and school mathematics teachers.

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At the University of Nebraska-Lincoln, our work is supported by: The Center for Science, Mathematics and Computer Education (permanent infrastructure) Math & Science Teachers for the 21 st Century Program of Excellence ($350,000/yr 2002-2015) $18,100,000 in NSF support – Math in the Middle Institute Partnership (2004-2011) – NebraskaMATH Partnership (2009 – 2013) – NebraskaNOYCE (2010 – 2016)

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The NebraskaMATH Partnership Principal Investigators – Jim Lewis, Mathematics – Ruth Heaton, Teaching, Learning and Teacher Education (TLTE) – Carolyn Edwards, Psychology and Child, Youth and Family Studies – Walt Stroup, Statistics – Ira Papick, Mathematics – Tom McGowan, TLTE – Barb Jacobson, Lincoln Public Schools

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NebraskaMATH K-12 Partners Grand Island Public Schools Lincoln Public Schools Omaha Public Schools Papillion-La Vista Public Schools Nebraskas Educational Service Units – Our work involves teachers from 101 school districts and 216 schools.

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NebraskaMATH Professional Development Math in the Middle Institute – A masters program for middle level (5-8) teachers Primarily Math – An 18-hour certificate program for K-3 teachers Nebraska Algebra – A 9-hour program for Algebra 1 teachers New Teacher Network – A 24-hour PD and mentoring program for new teachers Robert Noyce NSF Master Teaching Fellowships – A program for extraordinary master teachers Robert Noyce NSF Teaching Fellowships – A postbac masters and certification program

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Math in the Middle Institute Partnership A 25-month, masters program that educates and supports outstanding middle level teachers who will become intellectual leaders in their schools, districts, and ESUs. A major research initiative to provide evidence-based contributions to research on learning, teaching, and professional development. A special focus on supporting rural teachers, schools and districts

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Math in the Middle M 2 courses focus on these objectives: enhancing mathematical knowledge enabling teachers to transfer mathematics they have learned into their classrooms leadership development and action research

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Math in the Middle Courses Eight new mathematics and statistics courses designed for middle level teachers (Grades 5 – 8) including: – Mathematics as a Second Language – Functions, Algebra and Geometry for Middle Level Teachers – Experimentation, Conjecture and Reasoning – Number Theory and Cryptology for Middle Level Teachers – Using Mathematics to Understand our World Special math focused sections of three pedagogical courses: – Inquiry into Teaching and Learning – Curriculum Inquiry – Teacher as Scholarly Practitioner An integrated capstone course: – Integrating the Learning and Teaching of Mathematics

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Math in the Middle Instructional Model SUMMER Offer 1 and 2 week classes. Class meets from 8:00 a.m. - 5:00 p.m. 35 teachers – 5 instructors in class at one time. Substantial homework each night. End-of-Course problem set – Purpose – long term retention of knowledge gained. ACADEMIC YEAR Two-day (8:00 – 5:00) on- campus class session. Course completed as an on- line, distance education course using Blackboard and Adobe Connect. – Major problem sets – End-of-Course problem set – Substantial support available for teachers

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157 Math in the Middle Teachers

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Primarily Math Focuses on strengthening the teaching & learning of mathematics in grades K-3 Six course, 18-credit hour program leading to a K-3 Mathematics Specialist certificate – 3 mathematics courses – 3 pedagogy courses Optional 7 th course focusing on leadership On-going support in the form of study groups lasting 2 years after coursework

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Primarily Math Teachers

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Primarily Math Research What happens to student achievement as elementary buildings employ math specialists as coaches, to departmentalize math instruction, or to continue as general classroom teachers? What happens to teachers classroom practices during and after their participation in Primarily Math?

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Primarily Math Research Research design led us to recruit three cohorts of participants during 2008-2009 – First cohort began summer 2009 (classes met in Lincoln) – Second cohort began summer 2010 (classes in Omaha) – Third cohort begins summer 2011 (two institutes, one in Lincoln and one in Grand Island) Collect data from all three cohorts beginning summer 2009, continuing through summer 2013

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Primarily Math Research School building climate and degree of teacher networking Teacher Network Survey administered to elementary buildings in four core partner districts 2010, 2011, 2013 Survey allows principals to see where teacher collaboration & isolation are occurring Partnership with colleagues at Northwestern University

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Lewis Middle School

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Nebraska Algebra 9 hours of graduate coursework – Math 810T: Algebra for Algebra Teachers – EDPS 991: Cognition and Instruction for High School Algebra Teachers – TEAC 991: Field Studies in Mathematics Some districts are able to provide participants with an algebra coach – If a coach is not available, we provide a teacher mentor All teachers have a university mentor

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Nebraska Algebra Teachers

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Algebra for Algebra Teachers Objectives – To help teachers better understand conceptual underpinnings of school algebra – To leverage new understanding into improved classroom practice Pedagogy – Combines collaborative learning with direct instruction – Provide teachers with dynamic learning & teaching models Assessment – Individual & group presentations, written & historical assignments, mathematical analyses of curricula, extended projects, and a final course assessment

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Mathematical Knowledge for Teaching Teachers need specialized content knowledge: – Deep understanding of content – Representations and connections – Understand student thinking – Assess student learning – Make curricular decisions This type of knowledge is not typically gained through most pre-service mathematics programs (i.e., Ball, Thames & Phelps, 2008; NCTM, 2000)

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Mathematical Knowledge for Teaching Algebra Teachers need knowledge of mathematics that enables them to address a wide range of mathematical ideas and questions. For example, here are some questions that school algebra teachers might be asked. 1) My teacher from last year told me that I whatever I do to one side of an equation, I must do the same thing to the other side to keep the equality true. What am I doing wrong when I add 1 to the numerator of both fractions in the equality 1/2 =2/4 and get 2/2 = 3/4? 2) My father (who is very smart) was helping me with my homework last night and he said the book is wrong. He said that 4=2 and 4=–2, because 2 2 =4 and (–2) 2 =4, but the book says that 4–2. He wants to know why we are using a book that has mistakes.

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Nebraska Math & Science Summer Institutes

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In 2010 we offered 13 courses for math teachers. Examples include: – Algebraic Thinking in the K-4 classroom – Functions for Precalculus Teachers – Concepts of Calculus for Middle Level Teachers

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The habits of mind of a mathematical thinker Have you observed two people who appear to know the same facts but for whom there is a marked difference in their ability to use that information to answer questions or solve problems? Why? Do mathematical thinkers approach problems differently? And, if so, how do we develop the habits of mind of a mathematical thinker in teachers and assist them in cultivating this knowledge among their students? To study this question, we developed a working definition based on experience and the work of other mathematics educators (e.g., Cuoco, et al., Driscoll)

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Our Courses Emphasize Problems that Develop Teachers Mathematical Habits of Mind Goals: Give teachers experiences to develop their: Strategies for solving problems Flexibility in thinking An appreciation for the importance of precise mathematical definitions and careful reasoning Ability to explain solutions to others Persistence and self-efficacy

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The Chicken Nugget Conundrum Theres a famous fast-food restaurant where you can order chicken nuggets. They come in boxes of various sizes. You can only buy them in a box of 6, a box of 9, or a box of 20. Using these order sizes, you can order, for example, 32 pieces of chicken if you wanted. Youd order a box of 20 and two boxes of 6. Heres the question: What is the largest number of chicken pieces that you cannot order? For example, if you wanted, say 31 of them, could you get 31? No. Is there a larger number of chicken nuggets that you cannot get? And if there is, what number is it? How do you know your answer is correct? A complete answer will: i) Choose a whole number N that is your answer to the question. ii) Explain why it is not possible to have a combination of boxes of 6 and boxes of 9 and boxes of 20 chicken nuggets that add to exactly N pieces of chicken. iii) Explain why it is possible to have a combination that equals any number larger than N.

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Problematic Answers Explain why it is not possible to order exactly 43 pieces. Argument #1: You can not have any combination that adds to 43 because it cant evenly divide by 6, 9, or 20. It is not a multiple of 15 and it cant be evenly divided in half. Argument #2: You are not able to get the number 43 because none of the numbers add equally into that number. Explain why it is possible to have a combination that equals any number larger than N. Argument: Its possible to have a combination greater then 43. This is because you can buy all the multiples of the numbers. For example, if you buy 18, you can buy 36 and 70. Or if you by 20 you can buy 40, 60, 80, 100, etc.

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The Triangle Game (Paul Sally, U. Chicago) Consider an equilateral triangle with points located at each vertex and at each midpoint of a side. The problem uses the set of numbers {1, 2, 3, 4, 5, 6}. Find a way to put one of the numbers on each point so that the sum of the numbers along any side is equal to the sum of the numbers along each of the two other sides. (Call this an Equal Side Sum Solution.) – Is it possible to have two different Equal Side Sum Solutions? – Which Equal Side Sum Solutions are possible? – How can you generalize this game?

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Side Sum Solutions for Hexagons Side Sum 17: 3, 8, 6, 4, 7, 9, 1, 11, 5, 10, 2, 12 Side Sum 18: None Side Sum 19: 6, 2, 11, 5, 3, 9, 7, 4, 8, 10, 1, 12 And 4, 10, 5, 8, 6, 2, 11, 1, 7, 9, 3, 12 And 5, 11, 3, 9, 7, 4, 8, 10, 1, 6, 12, 2 And 3, 9, 7, 11, 1, 10, 8, 6, 5, 2, 12, 4 Side Sum 20: 7, 11, 2, 8, 10, 4, 6, 9, 5, 3, 12, 1 And 9, 3, 8, 5, 7, 11, 2, 12, 6, 4, 10, 1 And 8, 2, 10, 4, 6, 9, 5, 3, 12, 7, 1, 11 And 10, 4, 6, 2, 12, 3, 5, 7, 8, 11, 1, 9 Side Sum 21: None Side Sum 22: 10, 5, 7, 9, 6, 4, 12, 2, 8, 3, 11, 1

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Patterns with Minimums & Maximums Polygon Minimum Side Sum To find the next Minimum Maximum Side Sum To find the next Maximum Triangle9+312+3 Square12+215+4 Pentagon14+319+3 Hexagon17+222+4 Heptagon19+326+3 Octagon2229

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A Solution for an n-sided polygon, n odd General solution for an n-gon where n = 2k + 1, n odd For a Heptagon Solution, n = 7; k = 3 To find the vertices begin with 1, move clockwise by k each time, and reduce mod n. The midpoints begin with 2n between 1 and 1+k and move counterclockwise, subtracting 1 each time. For a heptagon, the Side Sum = 5k + 4. 1 14 4 8 7 9 3 106 11 2 12 5 13

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Impact on UNL and our partners Many people are involved in NebraskaMATH teaching and research UNL faculty23 Other collegiate faculty 7 UNL graduate students45 UNL undergraduates18 Master teachers38

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Making Our Work Public Please visit the Products section of our web site: http://scimath.unl.edu/MIM/http://scimath.unl.edu/MIM/ for information on Math in the Middle: Course Materials Teachers Expository Math Papers Teachers Action Research Papers

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