1. MATH 2303/2304/3305/3308 Workshop Aug 20, 2013 Bell Hall 130A, UTEP

2. Let’s Get to Know Each Other! Introduction UTEP Math Sciences Art Duval Art Duval Gregory Allison Gregory Allison Jeremy Rameriz Jeremy Rameriz Kien Lim Kien Lim Vodene Schultz Vodene Schultz UTEP Teacher Education Joyce Cashman Joyce Cashman Song An Song An EPCC Mathematics Eduardo Urquidi Eduardo Urquidi Fernando Falcon Fernando Falcon Ruben Carrizales Ruben Carrizales Ruth Ordaz Ruth Ordaz Pearson Publisher Diana Baniak Diana Baniak

3. Agenda (Proposed) 9:00 amBreakfast 9:00 amBreakfast 9:30 amIntroduction & Challenges 9:30 amIntroduction & Challenges 9:40 amEffective Use of Textbook 9:40 amEffective Use of Textbook 11:15 amMy Math Lab & Online Resources 12:30 pm Lunch 1:30 pmTimeline 1:30 pmTimeline 2:00 pmAssessment & Collaboration 2:00 pmAssessment & Collaboration 3:30 pmEnd 3:30 pmEnd

4. Agenda (Actual) 9:00 amBreakfast 9:00 amBreakfast 9:35 amIntroduction & Challenges 9:35 amIntroduction & Challenges 10:15 amCore Ideas in Section 6.4 11:15 amMy Math Lab & Online Resources 12:20 pm Lunch 1:00 pmQuestion 4 on Activity 6M 1:00 pmQuestion 4 on Activity 6M 2:55 pmObjectives of Activity 6N Resources for Section 6.4 2:55 pmObjectives of Activity 6N Resources for Section 6.4 3:35 pmEnd 3:35 pmEnd

5. Objectives Enhance our pre-service teachers’ conceptual understanding and mathematical thinking Get to know each other Get to know each other Know more about textbook and its resources Know more about textbook and its resources Share ideas (e.g., how to use textbook optimally; what and how to assess) Share ideas (e.g., how to use textbook optimally; what and how to assess) Collaborate and share resources Collaborate and share resourcesIntroduction

6. What Challenges Do We Face as Instructors of These Courses? Discuss!

7. What Challenges Do We Face as Instructors of These Courses? Math anxiety Lack of content mastery Procedure oriented Don’t see the point of learning the content Don’t want to know the conceptual underpinnings Pass without understanding How do they get so far? Last math course was taken many years ago (no math course in senior year) Too much emphasis on testing. No time for deep learning. Non-thinkers Irrelevant

8. One Big Challenge! “Doing mathematics means following rules laid down by the teacher, knowing mathematics means remembering and applying the correct rule when the teacher asks a question, and mathematical truth is determined when the answer is ratified by the teacher.” (Lampert, 1990, p. 31) Existing Beliefs

9. Pedagogical Content Knowledge Common Content Knowledge (CCK) Specialized Content Knowledge (SCK) Knowledge of Content and Students (KCS) Knowledge of Content and Teaching (KCT) Subject Matter Knowledge Knowledge at the mathematical horizon Knowledge of curriculum Mathematical Knowledge for Teaching

10. Notes about the Beckmann Text Explaining Why Core Concepts CCSSM Standards for Mathematical Practices Arithmetic Operations Visual Representations In-class Activities Practice Exercises vs. Problems Chapter Summaries IMAP Videos

11. Standards for Mathematical Practice in Common Core State Standards in Mathematics (CCSSI, 2010, p. 6-8) Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning

12. One More Thing about Teaching Teaching for Conceptual Understanding Teaching for Mathematical Thinking Teaching with Grace The Lesson of Grace in Teaching by Francis Su The Lesson of Grace in Teaching

13. Effective Use of Textbook Discuss one section of the text (Section 6.4 for M1351/2303/3305/3308) Core ideas In-class Activities Homework for students (problems to assigned, pages to read, etc.) Resources

14. Core Ideas for Section 6.4 Discussion 1. What important math ideas can students learn from Sec6.4? Two conceptualization of fraction division: repeated-subtraction model in 6.4 & sharing equally in 6.5 Different techniques for dividing fractions: o Common denominator (divide numerator) o Divide across (divide numerator, divide denominator) o Invert-multiply Division as inverse multiplication Common denominator strategy is meant for the “how many groups” perspective (it’s easier to see how many times the divisor fits into the dividend if we have common unit fractions; e.g., 2 one-sixths goes into 3 one-sixths 1½ times)

15. Core Ideas for Section 6.4 Discussion 2. What habits of mind can we foster? To solve the problem we first need to manipulate (e.g., make the denominators the same) See the structural similarity with division involving whole numbers (see Fig. 6.13 on page 250 in Beckmann) Have a sense of magnitude (on a number line), and not just numbers/symbols (i.e., foster number sense) Thinking reversibility (inverse operations) 3. What ideas can we learn from analyzing Section 6.4? Making students uncomfortable is necessary Experiencing the need for common denominator

16. MyMathLab & Online Resources We went over the following: Signing in at http://www.mymathlab.com/http://www.mymathlab.com/ Creating and downloading “Student Registration Handout” for students to enroll in your course (the pdf contains your specific Course ID) Creating a course, creating homework, managing gradebook Tools for Success (e.g., e-manipulatives, IMAP videos, downloads, handouts, internet resources for each chapter) Practice/Homework/Test items in mymathlab tend to be procedure-oriented whereas problems in the text tend to be conceptual Students can purchase book from www.myPearsonStore.com with a discount code (see next slide).

17. GET 20% OFF YOUR PURCHASE ISBN-13:9780321901200 (Loose Leaf, $94.00 before taxes) or ISBN-13:9780321901231 (Hardcover, $132.00 before taxes) Beckmann, Mathematics for Elementary School Teachers 4 th ed. with MyMathLab access code when you use the following coupon code: utepmath2013 To redeem this special offer, go to www.myPearsonStore.com and enter the coupon code during checkout to save. (Note: Code is case sensitive. Offer good through 12/15/13) *Member discount is limited to items purchased in a single transaction from www.myPearsonStore.com and excludes VangoNotes, CourseSmart, iPhone Apps, and access codes. Offer cannot be combined with other discounts. Please refer to our Terms of Use on our Customer Care page at www.myPearsonStore.com for more details on member discounts and free shipping terms.

18. LUNCHLUNCHLUNCHLUNCH

19. Work on Item 4 in Section 6.4 Individual Work & Group Work

20. Why is this question challenging? (we skipped the part on writing a word problem) Group Presentations The divisor is a fraction The divisor is larger than the dividend (“how many group” view doesn’t really work because there is not even 1 group)

21. What solutions were presented? Group Presentations Common denominator strategy: 1/3 ÷ 3/4 = 4/12 ÷ 9/12 = 4/9 It was conceptually difficult to explain why answer is 4/9, even with the support of a diagram. Invert-multiply strategy: 1/3 ÷ 3/4 = 1/3 × 4/3 = 4/9 This strategy relies on a standard procedure without really explaining the meaning of “1/3 ÷ 3/4” or how many ¾ are in 1/3.

22. What solutions were presented? (con’t) Group Presentations Make the divisor a whole number and use sharing-equally model: 1/3 ÷ 3/4 = 4/3 ÷ 3 = 4/9 This avoids the meaning of 1/3 ÷ 3/4. Proportional reasoning: 3/4 cups make 1 batch 1/3 cups make x Batch Scaling both quantities by a factor of 4/9. This strategy transforms a fraction division problem into a missing-value problem involving a proportion.

23. What solutions were presented? (con’t) Group Presentations Division as inverse-multiplication: 1/3 ÷ 3/4 = A A x 3/4 = 1/3 This view of division is most suitable for division fraction where the divisor is greater than the dividend because what the question is really asking is “what fraction of ¾ are in 1/3” instead of “how many times of ¾ fit into 1/3”.

24. How could we synthesize all these solutions? (i.e., how can a teacher provide closure) One strategy is to ask students to share what they have learned; the teacher records their ideas, elaborating or providing additional ideas as needed. Multiple ways to solve the problem o Using the invert-multiply procedure o Changing it to a missing-factor problem in a multiplication o Changing it to a proportion-type problem (may use double number line or a table) o Common denominator o Making the divisor a whole number o Making both divisor and dividend whole numbers: a/b ÷c/d = ad ÷ bc

25. How could we synthesize all these solutions? (i.e., how can a teacher provide closure) Continue to elicit from students/participants: “How many groups” view is hard when the divisor is greater than dividend. Extending basic concept (“how many groups” view like 8 ÷ 2) into a non-intuitive situation (like 1/3 ÷ 3/4).

26. Class Activities 6M & 6N (Skipped) 1.What do you think are the learning objectives for 6M? 2.What do you think are the learning objectives for 6N? 3.Which activities should we use? Why? a.6M b.6N c.Both 6M and 6N d.Neither 6M and 6N

27. What is the purpose of activities? Activity 6N Q4 is to show student the equivalence between the invert- multiply form, 5/7 × 3/4, and the division form 5/7 ÷ 3/4. Q2 is to explain why we can divide across by interpreting the division problem as a missing-factor problem. Activity 6M Q2 is to highlight the importance of attending to the referent. 1 ÷ 2/3 = 1 unit of 2/3 (divisor) with a remainder of 1/6 of the unit of 1 (dividend).

28. Class Activities 6M & 6N

29. Homework for Section 6.4 (Skipped) 1.Which pages should we assign students to read? 2.Which items should we assign? a.Practice Exercises for Section 6.4 b.Problems for Section 6.4 3.What else should/can we assign for homework to substantiate student conceptual understanding and/or mathematical thinking?

30. Resources for Section 6.4 What resources can we use to enhance student learning of the materials in Section 6.4? IMAP Videos – Students’ Reasoning with Fractions o Elliot: Explaining his reasoning for 1½ ÷ 1/3 http://media.pearsoncmg.com/cmg/pmmg_mml_shared/flash_video_player/player. html?video=aw/IMAP/video/0321 http://media.pearsoncmg.com/cmg/pmmg_mml_shared/flash_video_player/player. html?video=aw/IMAP/video/0321 o Trina: Learning the invert-multiply procedure http://media.pearsoncmg.com/cmg/pmmg_mml_shared/flash_video_player/player. html?video=aw/IMAP/video/0380 http://media.pearsoncmg.com/cmg/pmmg_mml_shared/flash_video_player/player. html?video=aw/IMAP/video/0380 TIMSS Videos – Japanese Lesson Study o After learning dividing across 9/20 ÷ 3/5 = (9÷3)/(20÷5), students struggled with 9/30 ÷3/7 http://hrd.apec.org/index.php/Lesson_Study_Video_for_Let%27s_Think_About_Ho w_To_Multiply_and_Divide_Fractions:_Students%27_Presentation_%282_of_5%29 http://hrd.apec.org/index.php/Lesson_Study_Video_for_Let%27s_Think_About_Ho w_To_Multiply_and_Divide_Fractions:_Students%27_Presentation_%282_of_5%29 Online Manipulatives

31. Alignment with TExES Competencies We can administer a pre-test with TExES items so that students get a sense of TExES and start preparing for it by taking math courses with the goal of doing well in the TExES exam. The pre-tests were given out. They can also be downloaded from our wiki at http://t-t-t.wikispaces.com/Resources-2303-1350. http://t-t-t.wikispaces.com/Resources-2303-1350 This PowerPoint presentation is posted at http://t-t-t.wikispaces.com/Workshop+Materials. http://t-t-t.wikispaces.com/Workshop+Materials

32. THE END