Moving Toward a More Effective Learning Environment in First Year Calculus

Javier Garza, Tarleton State University

March 30, 2012

At the end of the semester I would be happy if

Deeper understanding/Learn 21 More confident 2 Attend class 1

Make a good grade 19 Richer passion for math 2"Earn a grade that I'm not

ashamed of" 1Prepared for Calculus

II/Physics 10 Independence 2"I have not created a hatred

for math" 1Real world problem solving/practical use 9 Improve study habits 2 Graduation 1

Do better than last time I took it 5

Meet other students interested in math 2 Learn to Learn 1

Meet the challenge/tried my best 4 Refresher 1

"Calculus isn't scary anymore" 1

Do well on tests/final 4 Fun 1"If I don't fall asleep during

class" 1

Specific topics 3 Help other students 1"I hope to be noticed on how

I feel about math" 1

N=32It helps me to be a better

teacher 1

Beginning of Semester Survey•17 questions•Responses: CRD Group (N=65)* and Calculus I Group (N=15)* 78 in CRD logged in to survey though only 65 provided consistent responses.•1 = Strongly Agree 2 = Agree 3 = Neutral 4 = Disagree 5 = Strongly Disagree•Of 78 in CRD group, 63 (84%) were enrolled in RDG 301 (Intro. to Children’s Lit)•A/D Gap is the difference between % reporting 1 or 2 and % reporting 4 or 5

CRD Cohort (F11)

Cal I (F11)

Cal I (Sp 12)

I think this course will be intellectually stimulating.

Mean 1.88* 1.53Agree/Disagree

Gap+82 +93

This is my first time to take this course. Mean 1.68***

3.2 1.75**

Agree/Disagree Gap

+82 -7 +75

Only a course grade of “A” in this course is acceptable to me.

Mean 2 1.24**Agree/Disagree

Gap+76 +60 +100

CRD Cohor

t (F11)

Cal I (F11)

Cal I (Sp 12)

I can contribute valuable thoughts and ideas to a discussion in this area of study.

Mean 1.92* 2.53Agree/Disagree

Gap+82 +46

The requirements of this course scare me.

Mean 2.94* 3.53Agree/Disagree

Gap+1 -40

I do not need this course content in everyday life.

Mean 4.09* 3.73Agree/Disagree

Gap-81 -60

I find this subject area to be interesting. Mean 1.75**

2.2

Agree/Disagree Gap

+86 +73

I think this course will be fun. Mean 2.02* 2.47Agree/Disagree

Gap+74 +53

0 20 40 60 80 100 1200

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f(x) = 0.5800368301934 x + 15.1531862801788R² = 0.731655665411245

Fall 11Test 1

Fall 11Test 2

Fall 11 Test 3

Sp 12Test 1

Sp 12Test 2

Sp 12Test 3

N 33 29 28 25 25 23

Avg 73.4 65 61.6 75.3 70.2 77.0

StDev 19.1 21 26.4 18.2 13.8 14.3

WHAT’S EATING YOU?Arum, R., & Roksa, J. (2011). Academically adrift: Limited learning on college campuses. Chicago, IL: The University of Chicago Press.

• 45 percent of students "did not demonstrate any significant improvement in learning" during the first two years of college. • 36 percent of students "did not demonstrate any significant improvement in learning" over four years of college.

Deslauriers, L., Schelew, E., & Wieman, C. (2011). Improved learning in a large-enrollment physics class. Science 332(6031), 862-864. Abbr. Abstract: We measured the learning of a specific set of topics and objectives when taught by 3 hours of traditional lecture given by an experienced highly rated instructor and 3 hours of instruction given by a trained but inexperienced instructor using instruction based on research in cognitive psychology and physics education. The comparison was made between two large sections (N = 267 and N = 271) of an introductory undergraduate physics course. We found increased student attendance, higher engagement, and more than twice the learning in the section taught using research-based instruction.

Merrow, J. (2005). Declining by degrees: Higher education at risk [Motion picture]. United States: Learning Matters, Inc.

“The lecture method is a process whereby the lecture notes of the instructor get transferred to the notes of the students without passing through the brains of either”.

Eric Mazur: Confessions of a Converted Lecturer

Some References

Schneps, M. H., & Sadler, P. M. (1988). A private universe: misconceptions that block learning [ Motion picture]. United States: Corporation for Public Broadcasting. Available at http://www.learner.org/vod/vod_window.html?pid=9. 

Carlson, M. P., & Rasmussen, C. (Eds.). (2008). Making the connection: research and teaching in undergraduate mathematics education (MAA Notes #73). Washington, DC: Mathematical Association of America.

 Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs,

values and goals. Annual Review Psychology, 53, 109-132.

 

Some More References

Finkel, D. L. (2000). Teaching with your mouth shut. Portsmouth, NH: Boynton/Cook.

 Newcombe, N. S., Ambady, N., Eccles, J., Gomez, L., Klahr, D.,

Linn, M., Miller, K., & Mix, K. (2009). Psychology’s role in mathematics and science education. American Psychologist, 64(6), 538-550.

Nilson, L. B. (2010). Teaching at its best (3rd ed.). San Francisco, CA: Jossey-Bass.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan.

This is not about “change” for the sake of change

My goal is to provide students a transformational experience subsequent to which they believe themselves to be independently equipped and empowered to “do” mathematics and to engage in meaningful mathematical dialogue with others, and are intrinsically inclined to engage in the struggles of problem solving.

Indicators of success: • Students attend class (based upon intrinsic interest); • Students complete the homework with a high level of mastery• Students constructively engage in cooperative/collaborative

learning activities • Students perform well on exams• Students take more mathematics coursework• Students are successful in “subsequent coursework”• Students are interested in independent study

Classroom – Justify their attendance; encourage, empower, equipOffice – InvitingHome – Learning and evolution toward independence

Some strategies implemented

Utilized Enhanced WebAssign for online homework (10% of grade; objective questions), integrating dynamic content and support features

Required written homework assignments (10% of grade; problems requiring justification)

Provided daily course calendar (comfort through explicit demonstration of course design)

Provided grader with rubric for grading written homework (intentional, constructive feedback)

Tailored solutions (Stewart) provided for exam practice problems

Replaced curves with extra credit serving as final exam review

More strategies

Administered beginning of semester and end-course (after grades assigned) survey online; compared results to those of other classes

Developed and administered Calculus Concept Inventory (mention Goldstein) at beginning and end of course

Reviewed math education research (Carlson) to bone up on common misconceptions among calculus students and develop activities (example, bottle problem)

Spent over a week to shore up prerequisite knowledge (in particular, notion of function)

Rigidly enforced no-late homework policy

More strategies Recorded item-by-item results for each

student on each test to inform opportunities for emphasis next semester.

Sent email to every individual student several times a semester to notify of progress and express support/concerns as needed

Scheduled office hours based upon student availability rather than my convenience.

Recommitted to use of CAS (Mathematica) to facilitate dialogue and enhancement of conceptual understanding

More strategiesProvide more in-class opportunities for students to make

connections between multiple representations of calculus concepts and discuss (Teach with my mouth shut) Example: Provide function graphs on grids and ask them to

sketch derivatives by estimating derivative values Provide simple (constant, linear) function graphs on grids and

ask them to complete a table providing area under the graph on interval [0,t]. Make connection between definite integral and antiderivative.

Directed learning activity connecting prior knowledge of relationship between graphs of f(x) and f(x)+k, and the family of antiderivatives of a function

Provide direction fields for y’=f(x) and ask them to sketch y; make connection to antiderivatives.

Provide table that facilitates their understanding of inherent structure of chain rule application

28. The course improved my understanding of the subject area. 1.18

36. It was necessary for me to effectively utilize my graphing calculator to successfully solve problems in the course. 1.3

5. Having a solid background in this area will be useful for me in the future. 1.36

8. I think this course was intellectually stimulating. 1.36

41. The instructor was sufficiently accessible via email. 1.36

45. The tests served as good assessments of my mastery of Calculus concepts. 1.36

22. I demonstrated an assertiveness in exchanging ideas that is different from my previous behavior. 2.27

43. The Supplemental Instruction leader served as an effective source of academic support for me this semester. 2.27

37. It was necessary for me to effectively utilize Mathematica to successfully solve problems in the course. 2.36

4. I will need this course content in everyday life. 2.45

17. Uses the same sort of activities that any other teacher would use. 2.55

24. My experiences resulted in differences in how I analyzed, discussed, and shared ideas with others. 2.55

23. I had an experience in which dialogue with my teacher or peers resulted in a change in my ideas or beliefs. 2.73

42. The mathematics clinic served as an effective source of academic support for me this semester. 2.82

Next steps

Assist Supplemental Instruction leader in developing activities

Change daily MO so that students are expected to complete problems requiring lower level thinking skills independently, outside of class, without benefit of prior coverage (evolution)

“Uncover material”: More effectively utilize class time to enrich conceptual understanding, make connections, facilitate dialogue

Learn while listening more and “telling” less Integrate in-class presentations at board

Thank you!!

garza@tarleton.edu