Research in Mathematics Education: New Directions for Two-Year Colleges

1. Research in Mathematics Education: New Directions for Two-Year Colleges April Strom, Scottsdale Community College (AZ) Ann Sitomer, Portland Community College (OR) Mark Yannotta, Clackamas Community College (OR) Amy Volpe, Glendale Community College (AZ)

2. RMETYC New AMATYC Committee esearch in Purpose: To encourage and support quality research in mathematics education in two-year colleges, conducted by two-year college faculty. athematics ducation in wo- ear olleges

3. Guiding Questions • Why conduct research in mathematics education at community colleges? • Theory without practice is empty; practice without theory is blind (Kwame Nkrumah, 1966). • Schoenfeld (2000) says “Research in mathematics education has two main purposes, one pure and one applied: • Pure (Basic Science): To understand the nature of mathematical thinking, teaching, and learning; • Applied (Engineering): To use such understandings to improve mathematics instruction. • What are some examples of research studies and findings conducted by two-year college faculty?

4. Current Trends in Math Education • Cognitive Research: Focus on student reasoning • Quantitative and Proportional Reasoning (Thompson, 1994; Smith III & Thompson, 2008) • Covariational Reasoning (Carlson et al., 2002) • Advanced Mathematical Thinking (Rasmussen & Zandieh, 2005); Realistic Mathematics Education (Freudenthal, 1991) • Research-based curricula: Rational Reasoning Group (Arizona State); Abstract Algebra Group (Portland State)

5. Presentations • Talk 1: Experience as a researcher (Ann Sitomer) • Talk 2: Experience as a subject (Mark Yannotta) • Talk 3: Experience with research design (April Strom) • Talk 4: Experience with data snooping (Amy Volpe)

6. Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences Ann Sitomer Portland State University Portland Community College

7. Adult returning students in developmental mathematics courses Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences Placement tests measure what adult returning students recall – or do not recall – about school mathematics. Students who have been away from school often place into the most elementary mathematics classes offered by mathematics departments at community colleges. But to what extent do adult students return to school with mathematical competencies not measured by placement tests?

8. An exploratory study Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • To what extent do adult students return to school with mathematical competencies not measured by placement tests? • Contexts for the question • Teaching • Research • The study • Setting • Proportional reasoning • Data collection • Sample of student work on The Wage Problem • Refining the research questions and the design of the study

9. Contexts for the question Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences Practice My work as a teacher To what extent do adult students return to school with mathematical competencies not measured by placement tests? Research Out-of-school mathematical practices Research Adults learning mathematics

10. Contexts for the question: Research Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • Out-of-school mathematical practices • Over the last 30 years, researchers have studied both adults’ and children’s mathematics in contexts outside of school: • Liberian tailors (Lave, 1977) • Dairy workers (Scribner, 1984) • Young street vendors in Brazil (Carraher, Carraher, & Schliemann, 1985) • Female shoppers in the US (Capon & Kuhn, 1979; Lave, 1988) • Odds makers at the horse track (Ceci & Liker, 1986) • Bookies for an unofficial lottery game (Schliemann & Acioly, 1989) • Carpet layers (Masingila, Davidenko, & Prus-Wisniowska, 1996) • Apprentice iron workers (Martin, LaCroix, & Fownes, 2006; Martin & Towers, 2007) • Structural engineers (Gainsburg, 2007)

11. Contexts for the question: Research Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • Out-of-school mathematical practices • Over the last 30 years, researchers have studied both adults’ and children’s mathematics in contexts outside of school: • Liberian tailors (Lave, 1977) • Dairy workers (Scribner, 1984) • Young street vendors in Brazil (Carraher, Carraher, & Schliemann, 1985) • Female shoppers in the US (Capon & Kuhn, 1979; Lave, 1988) • Odds makers at the horse track (Ceci & Liker, 1986) • Bookies for an unofficial lottery game (Schliemann & Acioly, 1989) • Carpet layers (Masingila, Davidenko, & Prus-Wisniowska, 1996) • Apprentice iron workers (Martin, LaCroix, & Fownes, 2006; Martin & Towers, 2007) • Structural engineers (Gainsburg, 2007) A partial catalog of mathematical competencies developed outside of school

12. Contexts for the question: Research Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • Adults learning mathematics • The role of affect (Evans, 2000; Wedege & Evans, 2006) • Translation between worlds (Benn, 1997; Martin et al., 2006)

13. An exploratory study Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • To what extent do adult students return to school with mathematical competencies not measured by placement tests? • Contexts for the question • Teaching • Research • The study • Setting • Proportional reasoning • Data collection • Sample of student work on The Wage Problem • Refining the research questions and the design of the study

14. The study: Setting Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • The students participating in the study are enrolled in Basic Mathematicsat a community college in an urban setting. • Participating students were enrolled in four of the 12 sections of this course offered on campus Fall 2009. The participation rate in each section varied from about 20% to 75% of the students enrolled in a section. Basic Math, along with a developmental reading and a developmental writing course, are standard prerequisites for most lower division collegiate courses offered at the college.

15. The study: Setting What are the core ideas of the course? Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • Basic Mathematics • “Use fractions, decimals, percents, integer arithmetic, measurements, and geometric properties to write, manipulate, interpret and solve application and formula problems. Introduce concepts of basic statistics, charts and graphs.” Problem solving Proportional reasoning

16. The Study: Proportional reasoning Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • What is meant by proportional reasoning? • “I propose that proportional reasoning means supplying reasonsin support of claims made about the structural relationships among four quantities (say, a, b, c, d) in a context simultaneously involving covariance of quantities and invariance of ratios or products; this would consist of the ability to discern a multiplicative relationship between two quantities as well as the ability to extend the same relationship to other pairs of quantities” (Lamon, 2007, p. 638, emphasis added).

17. The Study: Proportional reasoning Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • Why examine proportional reasoning for this study? • “If people with little or no schooling really understand proportional relations in these contexts, or if highly schooled individuals who have difficulty understanding proportionality in school-type settings fail to exhibit such difficulty in informal learning contexts, then there is something important to be understood” (Schliemann & Carraher, 1993, p. 49).

18. The study: Data collection Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • Three types of data are being collected: • Participating student’s responses to a brief biographical survey • Participating students’ work done on a problemsolved collaboratively during the second class meeting of the term, as well as field notes taken while students collaborated on a solution to this problem. • Selected participating students’ responses to interview questions about their work on The Wage Problem, about the mathematics they have used outside of school, and on a selected tasks from the research literature on proportional reasoning.

19. An exploratory study Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • To what extent do adult students return to school with mathematical competencies not measured by placement tests? • Contexts for the question • Teaching • Research • The study • Setting • Proportional reasoning • Data collection • Sample of student work on The Wage Problem • Refining the research questions and the design of the study

20. Students’ written work on The Wage Problem Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • Two things to consider • What mathematics do the students know before they return to school? • Are there strategies or concepts that students are bringing to bear on The Wage Problem that suggest that a student is building on her/his out-of-school experiences? • A guiding principle • “These researchers [ethnographers] have consistently tried to understand mathematical problem solving in the same way as their subjects” (Eisenhart, 1988, p. 110).

21. Students’ written work on The Wage Problem Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences Page 8 of the submitted work with computational support on pages 9 and 10.

22. Observations Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences

23. Refinements Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences • Goals of data analysis from pilot study • Is there evidence of mathematical competencies with which adult students are returning to school? • If so, how might these competencies be categorized and further studied? • If not, what another type of experimental design might uncover these competencies? • What themes emerge from the data? How might these themes refine future research questions aimed at uncovering the mathematical knowledge embedded in adult’s life experiences? Do the emerging themes suggest questions that might help us understand how adults students’ mathematical knowledge interacts with the mathematics they are learning in their mathematics classes?

24. Thank you Exploring the Mathematical Knowledge Embedded in Adults’ Life Experiences Questions?

25. Math 299: A Bridge to University Mathematics Mark Yannotta Portland State University Clackamas Community College

26. Overview • Background on bridge courses and Math 299 • The Math 299 Class of 2009 • Some preliminary results from the data • Participant activity

27. My dissertation area: • What are the challenges and opportunities associated with developing and sustaining a mathematics bridge course in a community college setting?

28. What do we know about mathematics bridge courses? • About 40% of colleges and universities in the US offer a dedicated course • No consensus on content (although many people argue that proof should be integral) • Research supports that “bridging” does not occur in a single course • Community colleges might provide some new direction for these courses

29. PORTLAND STATE UNIVERSITYDEPARTMENT OF MATHEMATICS AND STATISTICSBA/BS DEGREE REQUIREMENTS Mth 251, 252, 253, 254: Calculus I-IV (16) Mth 256 or Mth 421: Differential Equations (4/3) Mth 261: Introduction to Linear Algebra (4) A GAP IN THE CURRICULUM Mth 311, 312: Advanced Calculus (8) Mth 344: Group Theory (4) Additional 21 - 28 credits of elective courses

30. The Evolution of Math 299

31. How did I get involved with this abstract algebra curriculum? • 2006: I took a topics course with Sean Larsen. I started thinking about ways to incorporate more research-based ideas into Math 299 when I taught it again. • 2007: Sean contacted me about being Co-Pi on a collaborative NSF grant. • 2008: The grant was funded and I began incorporating some of the materials into the class. • 2009: In conjunction with the grant, we used a modified version of the group theory curriculum in Math 299 and collected data at CCC.

32. Benefits of guided reinvention • Freudenthal (1991) argues that, “knowledge and ability, when acquired by one’s own activity, stick better and are more readily available than when imposed by others” (p. 47). • The students actively participate in developing symbols, notation systems, definitions, and theorems. Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Norwell, MA: Kluwer Academic Publishers.

33. The students in my 2009 class: • Consisted of 9 community college students aged 17 - 35 years (6 in the 18 - 24 age category) • 4 males, 5 females* • 4 math majors, 2 engineering majors, 1 music major, 2 undecided • Various math backgrounds and experience • 4 completed differential equations (2 had taken a different version of the transition course in 2008) • 1 completed calculus III • 3 completed calculus II • 1 completed college algebra

34. The classroom environment: • Low risk • An elective course • Little homework was required • Inexpensive (<$200 to take the course) • Graduate student grading (A, B, or Pass) • High level of participation • Participation protocols were established early in the term • Students were expected to be active participants in the class every day

35. Some preliminary results from the data • My students were VERY active in class discussions • They used sophisticated geometric reasoning and were resistant to moving toward more axiomatic reasoning • The were several moments throughout the term when students seemed empowered and took ownership of the mathematics • The exit interview data suggests that most of the students thought they had changed the way they thought about mathematics

36. Activity 1: Defining the symmetries of the Equilateral Triangle • Let F stand for a flip across the vertical axis and R stand for a 120-degree clockwise rotation. • In the right-hand column of the table, express each of your symmetries in terms of combinations of F and R.

37. Activity 2: Completing the Group Table

38. A few ideas for a sustainable transition course model at CCC • Teach in way that combines the acquisition of knowledge with opportunities for students to participate in meaningful and contextualized mathematical activity • Run a transition (bridge) course once a year • Alternate curriculum every year: • Y1: Group Theory Curriculum • Y2: Advanced Calculus Curriculum • Widen the net to include high school and reverse transfer students

39. A Case Study of a Secondary Mathematics Teacher’s Understanding of Exponential Function: A Theoretical Framework April D. Strom

40. “The greatest shortcoming of the human race is our inability to understand the exponential function.” Albert Bartlett (1976)

41. Literature Review • Multiplicative Conceptual Field Theory • Multiplicative Reasoning • Exponential function (splitting, multiplicative unit) • Recursion • Rate of Change • Covariational Reasoning

42. Research Questions • What conceptions does a secondary mathematics teacher hold about exponential growth and decay? • How effective are the current attributes of the developed exponential function framework in explaining the ways of thinking about the concept of exponential function? • How does an instructional unit - focused on exponential function - facilitate the development of a secondary mathematics teacher’s understanding of exponential function?

43. Theoretical Perspective • Cognitive constructivism: • the philosophical belief that learning occurs through experiences and action, rather than through knowledge passed on by others (Steffe & Thompson, 2000). • … focus on describing themental images of an individual to learn one’s ways of thinking and approaches to problem situations (Piaget, 1970).

44. Exponential Function Framework • In groups, discuss the following: • What are the critical elements of knowledge that students should know/build for understanding exponential functions? • What are the critical reasoning abilities that students should use for understanding exponential functions?

45. All 16 teachers 2 teachers (report on 1) Functions course All 16 teachers 4 teachers (Pilot) + Ben Methodology: Data Collection • 1 secondary mathematics teacher: “Ben”

46. Teaching Experiment Methodology • Steffe & Thompson (2000) Middleton, Carlson, Flores, Baek, & Atkinson (2004)

47. Exponential Function Framework: Notation and Language

48. Exponential Function Framework: Reasoning Abilities

49. Results: Thinking about Exponents • When simplifying expressions with integer exponents, Ben said: “the exponent tells you the number of base numbers to be multiplied together” Ex: 32 = 3 · 3 • So why not use this thinking for fractional exponents?

50. Thoughts about Fractional Exponents • “I know I’ve explained this before, but I don’t even remember what I said.” • “I’d just tell students that they should go back to the rules for whole number powers because the rules are the same for any type of power.”