Course Redesign to Deepen Early Mathematical Experiences of Pre-Service K-8 Teachers

1. Course Redesign to Deepen Early Mathematical Experiences of Pre-Service K-8 Teachers Presented by Ann Assad Jackie Vogel Jennifer Fillingim Audrey Bullock Austin Peay State University

2. The Problem – MATH 1410: Structure of Mathematical Systems I Low Student Success: 33.9 % DFW rate for Fall 2010 Lack of Carryover of Content to Methods Courses: Although the students express appreciation for methods that encourage thinking in MATH 1410, when they get into the methods courses, they seem to immediately revert to being very mechanical in their approach to teaching mathematics.

3. The Opportunity APSU Internal Grant: Revitalization for Student Success Initiative (RASI) Stipend for planning and implementation of course redesign Training and support for course development Travel for faculty development

4. The MATH 1410 Project Team Mary Lou Witherspoon Jackie Vogel Andy Wilson Ann Assad Jennifer Fillingim Ashley Whitehead

5. Goals of the Project Increase the student success rate in MATH 1410 in order to maintain a strong cohort of preservice elementary and middle school teachers. Improve the students' perception of the usefulness of the content in MATH 1410 and its implementation in the school classroom. Deepen students' understanding of the role of problem solving in the teaching and learning of mathematics.

6. Strategy Readings. Provide a selection of readings based on current research on best practice in mathematics education. Problem Solving Journal. Establish a set of problems that require higher-order thinking which students will solve and reflect upon in their journals. Assessment of Student Beliefs. Provide pre- and post-assessment of student beliefs about teaching.

7. Preparing for Change Select a beliefs instrument. Create a formative and a summative rubric for evaluating the journal. Identify the problems to be used in the journal. Identify the related best practice research.

8. Beliefs Instrument Sample Questions: Drill and practice exercises help students understand mathematics. SD D U A SA Calculator use should be integrated throughout the elementary school mathematics curriculum. SD D U A SA At the beginning of class, teachers should work every homework problem that a student asks about. SD D U A SA

9. Elementary school mathematics should include open-ended questions (questions with many reasonable solutions).SD D U A SA The role of the teacher in elementary school mathematics is to explain concepts. SD D U A SA In elementary school mathematics, teachers should not ask questions that have more than one correct response. SD D U A SA Hands-on materials are appropriate for all mathematics students in grades 4-6. SD D U A SA

10. Development of Rubric

14. Selection of Problems

15. Theater Problem RASI 2011-2012

18. Correct picture:

19. Incorrect Answer:

20. Incorrect Answer 2:

21. Continued:

22. Measurement Problem – 1420RASI 2011-2012 Determine the volume of a doughnut. Include the following: A picture of your doughnut sitting next to a metric ruler. Zoom in enough to be able to see centimeter marks so that someone looking at your picture could get a good idea of the diameter of the doughnut. Describe any assumptions you are making about the shape of the doughnut. Describe how you determined the volume of the doughnut. Clearly state why your approximation of the volume of the doughnut is reasonable by comparing it to objects of known volume.

23. I assumed that the doughnut and the hole through the center were cylinders. A way to find the volume of a cylinder is to calculate π x r2 x h. I did this for both the radius of the entire doughnut and for the radius of the doughnut hole. My calculations are as follows: • π x (4.5cm)2 x 3cm - π x (1cm)2 x 3cm = about 181.3cm3 for the volume of the doughnut. It is important that we say this is just an estimate though, because we measured and rounded numbers, and our doughnut isn’t really perfect. I think it is reasonable because I can put 181 mL of water into a bowl and see that it takes up about the same amount of space as my doughnut. It is also comparable to a little less than 2 flats from our base ten blocks. Correct Responses

24. Incorrect Responses • The length and width should be the exact same measurement, and those, multiplied by the height, should equal the volume, which should be around 100 or so, as the doughnut is approximately the size of a flat base 10 block manipulative used in class. • I wrapped my doughnut in plastic wrap and submerged it under water like displacement in biology. I took the water level before and after the submersion and determined that the volume of my doughnut is about 4 ml. This must be reasonable because I subtracted correctly.

25. Results: DFW RatesMath 1410

26. Results: Beliefs Instrument Each survey was scored by awarding a score of 1 – 5 for each response. Strongly Disagree ⟹ 1 Disagree ⟹2 Undecided ⟹3 Agree ⟹4 Strongly Agree ⟹5 Questions for which a positive response is undesirable were adjusted so that a positive response was changed to a negative response. For example, Strongly Disagree became Strongly Agree.

27. Results: Individual Beliefs For each respondent, a pre- and post- assessment score was calculated based on the average of the scores for each question. The average pre-test score was 3.293. The average post-test score was 3.407. A matched-pair t-test indicates that the average increase in scores of 0.114 is significant (n=65, α=0.005).

28. Results: Individual Questions An examination of average change in undesirable results ranged from +1.18 to +2.63. The changes for the other questions ranged from 0.11 to 0.267. This seems to indicate success in changing students’ undesirable responses, while desirable responses remained fairly stable.

29. Example Question 10 states, “Elementary school mathematics is a collection of computational procedures necessary for problem solving.” The average response to this question increased 2.53 points from 1.70 (disagree) to 4.23 (agree)

30. The MATH 1420 Project Team Mary Lou Witherspoon Jackie Vogel Andy Wilson Ann Assad Audrey Bullock

31. Results: DFW RatesMATH 1420

32. Results: Individual BeliefsMATH 1420 For each respondent, a pre- and post- assessment score was calculated based on the average of the scores for each question. The average pre-test score was 3.536. The average post-test score was 3.464, indicating an average decrease in scores of 0.072.

33. Challenges Persistent undesirable beliefs. Drill and practice exercises help students understand mathematics. At the beginning of class, teachers should work every homework problem that student asks about. Routine calculator use in the mathematics classroom causes students to forget how to compute.

34. Time. Instructors spent a great deal of time developing, implementing, and assessing the problem solving. In a large class, this could be a barrier to successful implementation of the journals and could threaten the sustainability of change to the course. Instructors need to find ways to integrate the journals more smoothly into the curriculum so that the benefits will be maintained without added burden to the instructor.

35. Extension to Subsequent Courses. A RASI grant has been obtained to continue the journal through MATH 1420. If MATH 1410 -1420 journals are successful, the researchers hope to extend them to other mathematics content and pedagogy courses. MATH 4100 – Teaching Mathematics: Grades K-3 MATH 4150 – Teaching Mathematics: Grades 4-6

36. A Community of Learners The ultimate goal is to create a masters level course for first year teachers that would continue the reflective journal into their teaching. This would provide a vehicle for the continued support of teachers as they become professional educators.

37. Fern’s Flower Shoppe In order to fulfill their orders, Fern’s Flower Shoppe tries to keep certain proportions of flowers in the inventory. Fern gives the following guidelines to her assistant Francine, and asks her to check to see if the conditions are met so that they’ll know what they might need to order:

38. Roses should make up 20% of all of the flowers. For every two roses, there should be three carnations. There should be as many lilies as there are carnations. There should be 5 more daisies than there are lilies. We keep 50 tulips in the inventory at all times. The rest of the flowers change; we just call these “seasonal.”

39. Francine does a quick check of all of the flowers and thinks that they should meet what Fern wants, but she’d like you to help her check. What amounts of flowers do you think should be in the inventory if Francine can tell you that there are 250 flowers total? Use pictures and mathematical reasoning to model and solve the problem. Solutions that rely on traditional calculations instead of pictures/reasoning will not receive credit. Explain how you got your answer in a short paragraph. Make sure you reference your picture/reasoning and how you used them to solve the problem in your explanation. Note: You will turn in your picture as part of your grade. The scoring rubric will be the same as for The Theatre Problem.