1. What is Scholarship of Teaching and Learning: Working examples Curtis D. Bennett Loyola Marymount University

2. What is the Scholarship of Teaching and Learning? Four Scholarships (Boyer, 1990) Discovery Integration Application Teaching (and Learning) Scholarship is: Public Subject to Critical Review Accessible for exchange and use. Thinking about teaching as a scholarly inquiry.

3. Situating the Scholarship of Teaching and Learning (SoTL) Teaching Tips Math ed research SoTL

4. Stages of Scholarship of Teaching and Learning A teaching problem. Frame/refine the problem into a researchable question. Investigate and gather evidence in a systematic way. Draw Conclusions. Go Public.

5. Teaching Problem – types of questions. What works? Does this method work in improving student learning. What is? What are the students’ doing, thinking, etc. when they approach the class, problem, etc. What could be? A vision of the possible: often an example of what can be achieved with one student or in one class.

6. A teaching problem. Driven by a question/concern/desire I wish my students would read their textbook. (Laura Graff, Dustin Culhan, and Felix Marhuenda-Donate, College of the Desert) Only 15% of our pre-algebra students successfully complete the basic skills math sequence. (Jay Cho – Pasadena City College) Students have difficulty moving beyond examples to argumentation/proof? (CB)

7. College of the Desert

8. College of Desert Teaching Problem When students enroll in liberal arts classes, they expect to attend lectures and take notes, read the textbook, and study questions the instructor provides to help them prepare for exams, but … these same attitudes and expectations do not seem to apply to math class.

9. Researchable questions Will outlining math textbooks get students in the habit of using their textbooks, i.e., read them effectively beyond getting problems from them? Will being in the habit of using their textbooks make them more likely to make their own connections and become more active learners?

10. College of the Desert The Classroom Change Students performing under 75% on first test are required to include “chapter outlines” of each section. (Variant of “think aloud” taught at Leadership Institute in Reading Apprenticeship). Intention: Teach students how to read book.

11. College of Desert The Evidence Copies of the student outlines Student Test Scores (i.e., grades) Student Retention Rates Student Interviews

12. College of the Desert Outcomes (expected) Test scores went up for some students. Higher retention rates. Students became more adept at identifying key concepts over the term. Students found outlines helped them complete their homework more quickly.

13. College of the Desert Unexpected Outcomes More personal relationship with students as students apparently had more trust in instructors. One student said she “wanted to finish the book over the summer” Greater retention because students were not as afraid to learn material on their own when they missed class - Less likely to give up.

14. Student interviews

15. College of the Desert Going Public KEEP Toolkit (www.cfkeep.org) - the KEEP Toolkit, developed by the Carnegie Foundation for the Advancement of Teaching provides a compact way to make results available to others.www.cfkeep.org http://www.cfkeep.org/html/stitch.php?s=14 832740290866&id=34947815104339 But easier if you go to KEEP toolkit and go to Gallery then Community Colleges then Mathhttp://www.cfkeep.org/html/stitch.php?s=14 832740290866&id=34947815104339

16. Why do the Scholarship of Teaching and Learning? The intentional gathering of evidence makes us much more aware of what is happening in the classroom. Avoid pedagogical amnesia. Creates richer knowledge for others to build on. It’s fun!

17. My teaching problem(s) In a first proof course for math majors, students accept multiple examples as sufficient proof that a claim is true. How do we help them move forward on this? Getting students to improve their mathematical skills is hard, particularly when it comes to problem solving/explaining. (They seem to “get it” or not.)

18. Researchable Question I needed to know what was going on in the brains of the students: a what is question. What do students at various levels consider acceptable evidence for a claim? What do students do as they problem solve?

19. Evidence (layered) Student Faculty Survey Later: –“Proof Aloud” with 12 students –Focus group with 5 of 12 students –Faculty “Proof Aloud.”

20. Result of Survey

21. Can there exist a counterexample to a proven statement

22. Faculty Explanation ‘Convinced’ does not mean ‘I am certain’… …whenever I am testing a conjecture, if it works for about 5 cases, then I try to prove that it’s true Sometimes we find holes in proofs.

23. We needed more and better evidence Proof aloud (Built off of a think aloud) –Investigate a statement (is it true or false?) –State how confident, what would increase it –Generate and write down a proof –Evaluate 4 sample proofs –Respond - will they apply the proven result? –Respond - is a counterexample possible?

24. Proof-aloud Task and Rubric Elementary number theory statement –Recio & Godino (2001): to prove –Dewar & Bennett (2004): to investigate, then prove Assessed with Recio & Godino’s 1 to 5 rubric –Relying on examples –Appealing to definitions and principles Produce a partially or substantially correct proof Rubric proved inadequate

25. Richer Representation/Rubric Needed Student progression toward proficiency –Using a K-12 classroom-based expertise theory (P. Alexander across all academic domains) Typology of mathematical knowledge –6 cognitive components (R. Shavelson in science) –2 affective components (C. Bennett & J. Dewar)

26. Proof generation/Problem Solving: complex tasks Requires several types of knowledge –Mathematical content of statement –Appropriate logical procedures –How to write/explain Can be approached in many ways –Involves strategic choice of method Requires persistence in face of uncertainty Writing explanations/proving requires additional knowledge and motivation to produce a polished result

27. Multi-faceted Student Work Insightful question about the statement Poor strategic choice of (advanced) proof method Exhibit advanced mathematical thinking, but had undeveloped proof writing skills Confidence & interest influence performance Even our expert was concerned before knowing the task!

28. Proof-aloud results Compelling illustrations –Types of knowledge –Strategic processing –Influence of motivation Greater knowledge = poorer performance Both expert & novice behavior on same task

29. How do we describe all of this? Expertise Theory (P. Alexander, 2003) Typology of Scientific Knowledge (R. Shavelson, 2003)

30. School-based Expertise Theory: Journey from Novice to Expert 3 Stages of expertise development Acclimation or Orienting stage Competence Proficiency/Expertise

31. Typology: Mathematical Knowledge Two Affective Dimensions (Alexander, Bennett and Dewar): –Interest: What motivates learning. –Confidence: Dealing with not knowing. Six Cognitive Dimensions (Shavelson, Bennett and Dewar): –Factual: Basic facts –Procedural: Methods –Schematic: Connecting facts, procedures, methods, reasons. –Strategic: Heuristics used to make choices. –Epistemic: How truth is known. –Social: How truth/knowledge is communicated.

32. Mathematical Knowledge Expertise Taxonomy

33. Confidence: Proficient/Expert Stage Students spend a great deal of time on a problem and try more than one approach before going to text or instructor. Students are accustomed to filling in the details of a problem. Can solve multi-step problems. Students will disbelieve answers in the back of the book if the answer disagrees with something they feel they have done correctly.

34. Using the Taxonomy Analyzing work of students from other classes and institutions. Helps guide my teaching of problem solving in all my classes.

35. Solving quadratics In solving problems on quadratic equations, what types of knowledge are necessary. Can you categorize. –Factual: Quadratic formula, what is a quadratic, what does a missing term mean, factoring. –Algorithmic: Completing the square, factoring methods, applying QF. –Schematic: How do all of these fit together. –Strategic: Which method should you choose when? –Epistemic: How do you know check to see that you have the right answer?

36. Scholarship of Teaching and Learning Idea is to investigate teaching issues in rich systematic ways to help with understanding. Teachers investigating their practice with the notion of communicating to other teachers. It is important that it happens at all levels with all students.

37. How do you get started Start with a teaching project or problem Frame a question that you would like answered. Think about what evidence would help you answer the question. Gather the evidence (if you’re like most of us, you will gather too much – that’s okay) Analyze Present