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How prepared are the K-12 teachers of tomorrow to inspire the next generation of young mathematicians? In this webinar for the edWeb.net Adaptive Math Learning community, attendees learned how essential it is for pre-service teachers to learn, develop, and model the Standards for Mathematical Practice to improve learning for their future students. Ben Braun, Associate Professor of Mathematics at the University of Kentucky, and Tim Hudson, Senior Director of Curriculum Design at DreamBox Learning, discussed ways to ensure that pre-service teachers start their careers understanding how mathematical proficiency requires more than simply content knowledge. Tim and Ben shared ideas for K-12 school leaders and mentor teachers who are responsible for new teacher induction, as well as, implications for college and university faculty teaching both math methods and content courses. They also discussed potential disconnects between pre-service content and methods courses and also eventual in-service expectations, while providing examples of math problems to engage pre-service and new teachers. View the webinar to better understand how to use the Standards for Mathematical Practice.

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- 1. Empowering Pre-Service & New Math Teachers to Use the Common Core Practice Standards Tuesday, August 26, 2014 Dr. Tim Hudson, Senior Director of Curriculum Design, DreamBox Learning Benjamin Braun, Associate Professor of Mathematics, the University of Kentucky Join our Adaptive Math Learning community: www.edweb.net/adaptivelearning
- 2. Join our community Adaptive Math Learning www.edweb.net/adaptivelearning • Invitations to upcoming webinars • Webinar archives and resources • Online discussions • CE quizzes for archived webinars The recording, slides, and chat log will be posted in the Webinar Archives folder of the Community Toolbox.
- 3. Webinar Tips • For better audio/video, close other applications (like Skype) that use bandwidth. • Maximize your screen for a larger view by using the link in the upper right corner. • A CE certificate for today’s webinar will be emailed to you 24 hours after the live session. • If you are viewing this as a recording, you will need to take the CE quiz located in the Webinar Archives folder of the Community Toolbox. • Tweeting? Use #edwebchat
- 4. Empowering Pre-Service & New Math Teachers to Use the Common Core Practice Standards August 26, 2014
- 5. Benjamin Braun, PhD Associate Professor of Mathematics, U of Kentucky Editor-in-Chief, American Mathematical Society blog “On Teaching and Learning Mathematics” Twitter: @BraunMath Tim Hudson, PhD Senior Director of Curriculum Design, DreamBox Learning Former K-12 Mathematics Curriculum Coordinator, Parkway School District Twitter: @DocHudsonMath
- 6. 1970-1990 • “Back to basics” movements in 1970s led to influential reports arguing in favor of balance between conceptual and procedural understanding: o A Nation at Risk (1983, NCEE) o An Agenda for Action (1980, NCTM) • NCTM Standards released in 1989.
- 7. 2000-2001 • Role of standards-based assessment increased in early 2000’s with No Child Left Behind. • At the same time, updated NCTM Standards released in 2000, and National Research Council report Adding It Up released in 2001.
- 8. 2004 • Must read: “The Math Wars,” Alan H. Schoenfeld, Educational Policy, Vol. 18 No. 1, January and March 2004, pp. 253-286. (PDF versions available online.)
- 9. NCTM divided proficiency into two categories in Principles and Standards for School Mathematics (2000) Content • Numbers and operations • Algebra • Geometry • Measurement • Data analysis and probability Process • Problem solving • Reasoning and proof • Making connections • Oral and written communication • Uses of mathematical representation
- 10. NRC emphasized five “strands” of proficiency in Adding It Up (2001) • Conceptual understanding: comprehension of mathematical concepts, operations, and relations • Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately • Strategic competence: ability to formulate, represent, and solve mathematical problems • Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification • Productive disposition: habitual inclination to see mathematics as sensible, useful and worthwhile, coupled with a belief in diligence and one’s own efficacy
- 11. Common Core Mathematical Practice Standards 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.
- 12. Implications for Pre-service Teachers? Many (if not most) pre-service teachers at the elementary, middle, and secondary levels do not have a robust set of mathematical practices, as this has not been part of their own educational experience.
- 13. Implications for Pre-service Teachers? This creates a disconnect between content and methods courses, and also between pre-service coursework and in-service curriculum and assessment expectations.
- 14. Implications for Pre-service Teachers? Teacher educators, including faculty teaching both methods and content courses, need to ensure that pre-service teachers enter the beginning of their careers with an understanding that mathematical proficiency extends beyond content.
- 15. Implications for Pre-service Teachers? Challenges to incorporating practices in pre-service teacher courses include balancing practices and content, effectively training college faculty (including adjuncts and TAs), and building quality connections between content and methods instructors so these courses articulate well.
- 16. Implications for Pre-service Teachers? Even in non-CCSS states, the NRC and NCTM reports of the past 30+ years have had a major impact on curriculum and assessment, so this matters even in non-CCSS states.
- 17. Implications for Year 1-3 Teachers? In situations where there has been inadequate Pre-Service training for mathematics teachers, new teachers need even more content-specific support.
- 18. Implications for Year 1-3 Teachers? Many schools, districts, and states do not have adequate mathematics curriculum leadership to support new math teachers in content-specific ways.
- 19. Implications for Year 1-3 Teachers? New teacher induction and PD often emphasizes other aspects of teaching instead of curriculum (i.e., classroom management, parent communication, or building culture).
- 20. Implications for Year 1-3 Teachers? What are math teachers hired to accomplish? What is mathematics? How do people learn mathematics?
- 21. Grant Wiggins What’s the job of a teacher? The crying need for a genuine job description. grantwiggins.wordpress.com 7-25-14
- 22. Grant Wiggins “A real job description would be written around key learning goals and Mission-related outcomes. • What am I expected to cause in students? • What am I supposed to accomplish? Whatever the answer, that’s my job.” grantwiggins.wordpress.com 7-25-14
- 23. Must Cause 4 Things in Learners 1. greater interest in the subject and in learning than was there before, as determined by observations, surveys, and client feedback grantwiggins.wordpress.com 7-25-14
- 24. Must Cause 4 Things in Learners 2. successful learning related to key course goals, as reflected in mutually agreed-upon evidence grantwiggins.wordpress.com 7-25-14
- 25. Must Cause 4 Things in Learners 3. greater confidence and feelings of efficacy as revealed by student behavior and reports (and as eventually reflected in improved results) grantwiggins.wordpress.com 7-25-14
- 26. Must Cause 4 Things in Learners 4. a passion and intellectual direction in each learner as determined by student-initiated pursuits, observations, surveys, and behavior. grantwiggins.wordpress.com 7-25-14
- 27. Does Your Mission Obligate Teachers to Achieve these Goals? Do Teachers Hired at Your School Know these are their Goals? Do Teachers Receive Feedback about how well they’ve Achieved these Goals? 1. greater interest than was there before 2. successful learning related to key goals 3. greater confidence and feelings of efficacy 4. a passion and intellectual direction grantwiggins.wordpress.com 7-25-14
- 28. Two ways to incorporate mathematical practices in teacher training and professional development
- 29. Using Writing Assignments Pre-service teachers need to write and discuss mathematical practices explicitly in content courses. This can’t be “left to the methods courses” to handle, there must be a dialogue.
- 30. Using Writing Assignments One of the best tools we have for this task is writing assignments. • Personal writing, e.g reflective essays • Expository writing, e.g. report on good contact/practices contact points • Critical writing, e.g. analyzing practices of peers
- 31. Example of a Personal Essay Prompt There are eight Standards for Mathematical Practice in the Common Core State Standards for Mathematics. Select at least three of these standards to consider. For each of the standards that you select, discuss a situation where you have observed one of your classmates demonstrating that practice in their work. This situation might have arisen from in-class group work, from working with a study group on homework, from hallway discussions of a problem, etc, but you should discuss a moment when your classmate was using one of these practices when working on a mathematical problem. You should explicitly connect each situation with the written description of the related practice standard given in the Common Core.
- 32. Low-threshold high-ceiling problems • Nothing is better than doing mathematics while receiving quality feedback for developing good mathematical practices, which teachers must have if they are to help others develop them. • Students need to engage with low-threshold-high- ceiling (LTHC) problems. • Open (unsolved) problems in math are a great source of LTHC problems!
- 33. K-5 level open problem #1 Fibonacci Primes • The Fibonacci numbers are 1,1,2,3,5,8,13,21,34,55,... obtained by adding the two previous numbers to get the next in the sequence. • OPEN QUESTION: Are there infinitely many prime Fibonacci numbers?
- 34. K-5 level open problem #2 Fermat Primes • The Fermat numbers are 2^(2^n)+1 for all non-negative integers n, e.g. 3, 5, 17, 257, 65537,... • OPEN QUESTION: Are there infinitely many prime Fermat numbers?
- 35. K-5 level open problem #3 Collatz Conjecture • Given a positive integer n, if it is odd then calculate 3n+1. If it is even, calculate n/2. Repeat this process with your new number. • Example: 1,4,2,1,4,2,1,4,2,1,... • Example: 5,16,8,4,2,1,... • OPEN QUESTION: If you start with any positive integer, does this process always end by cycling through 1,4,2,1,4,2,1,...?
- 36. K-5 level open problem #4 Erdos-Strauss Conjecture • OPEN QUESTION: For every positive integer n larger than 1, does there exist a solution to 4/n = 1/x + 1/y + 1/z using positive integers x, y, and z? • Example: 4/5 = 1/2 + 1/5 + 1/10
- 37. Higher-level open problems • Parity of the partition function • Irrationality of Euler-Mascheroni constant • Lagarias’s reformulation of the Riemann Hypothesis Many more are available at: http://en.wikipedia.org/wiki/List_of_conjectures
- 38. K-5 Level Closed Problem On day three of the bicycle race, Donald’s time was: 3 hours, 4 minutes, and 11 seconds. Keina’s time was: 2 hours, 58 minutes, and 39 seconds. How long was Keina finished before Donald crossed the finish line?
- 39. Donald & Keina 61 Hours Minutes Seconds 71 6 3 3 X 3 4 11 2 58 39 3 2 5 1 0 5 2 304 – 298 = ?
- 40. Oxford University 1992 44 academic pure mathematicians were asked to estimate (make reasonable guesses) for 20 multiplication and division problems (Ex. 482 x 51.2 and 546 ÷ 33.5) Strategy Frequency Used Use of fractions 40% Using “nicer” numbers 17% Rounding two numbers 16% Rounding one number 8% Factorization 8% Standard algorithms 4% Distributive Property 3% Computational Estimation Strategies of Professional Mathematicians, Dowker, Journal for Research in Mathematics Education, Vol. 23 ©1992
- 41. Oxford University 1992 44 academic pure mathematicians were asked to estimate (make reasonable guesses) for 20 multiplication and division problems (Ex. 482 x 51.2 and 546 ÷ 33.5) Strategy Frequency Used Use of fractions 40% Using “nicer” numbers 17% Rounding two numbers 16% Rounding one number 8% Factorization 8% Standard algorithms 4% Distributive Property 3% Computational Estimation Strategies of Professional Mathematicians, Dowker, Journal for Research in Mathematics Education, Vol. 23 ©1992
- 42. Oxford University 1992 “To the person without number sense, arithmetic is a bewildering territory in which any deviation from the known path may rapidly lead to being totally lost. The person with number sense…has, metaphorically, an effective ‘cognitive map’ of that same territory.” Computational Estimation Strategies of Professional Mathematicians, Dowker, Journal for Research in Mathematics Education, Vol. 23 ©1992
- 43. Summary • It is well-established that mathematical proficiency involves both practices and content. • All teachers need support in developing skillful approaches to teaching both mathematical content and practices. • An excellent way for pre-service and new in-service teachers to develop their understanding of the practices is to work on LTHC problems themselves, then reflect on the mathematical practices they used in their own work. • Everyone - teachers and students - benefit the most from receiving quality feedback when developing their content knowledge and mathematical practices.
- 44. Q & A
- 45. Reinventing the Learning Experience Intelligent Adaptive Learning™ Engine • Millions of personalized learning paths • Tailored to a student’s unique needs Motivating Learning Environment • Student Directed, Empowering • Gaming Fundamentals, Rewards Rigorous Elementary Mathematics PreK-8 • Reporting Aligned to Common Core State Standards, Texas TEKS, Virginia SOL, Canada WNCP, & Canada Ontario Curriculum Reports • Standards for Mathematical Practice
- 46. DreamBox Lessons & Virtual Manipulatives Intelligently adapt & individualize to: • Students’ own intuitive strategies • Kinds of mistakes • Efficiency of strategy • Scaffolding needed • Response time © DreamBox Learning
- 47. Robust Reporting © DreamBox Learning
- 48. Strong Support for Differentiation © DreamBox Learning
- 49. DreamBox supports small group and whole class instructional resources Interactive white-board lessons www.dreambox.com/teachertools © DreamBox Learning
- 50. Free School-wide Trial! www.dreambox.com
- 51. Thank you!
- 52. Join our community Adaptive Math Learning www.edweb.net/adaptivelearning • Invitations to upcoming webinars • Webinar archives and resources • Online discussions • CE quizzes for archived webinars The recording, slides, and chat log will be posted in the Webinar Archives folder of the Community Toolbox.
- 53. Thank you to our sponsor: www.dreambox.com
- 54. Stay tuned for information on upcoming webinars! Join our Adaptive Math Learning community for an invitation: www.edweb.net/adaptivelearning

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