The Role of Calculus in the Transition from

High School to College Mathematics

T3 Chicago, IL March 4, 2017

ApdffileofthisPowerPointisavailableatwww.macalester.edu/~bressoud/talks

Mathematical Association of

America #15504984

David Bressoud St. Paul, MN

Conference Board of the Mathematical Sciences

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APCalculusExams

allexams bystudentsbeforegrade12 bystudentsbeforegrade11

Approximately 800,000 high school students enroll in calculus each year, more than half the 1.5 million who matriculate as full-time students in four-year undergraduate programs.

Source: The College Board

The Peacock-Tail Theory of AP® Calculus Bob Orlin

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FirstCollegeMathClassforthosewhotookCalculusinHighSchool(thousands)

CalcIIorhigher

CalcI,earnAorB

CalcI,earnCorlower

BusinessCalc,Stat,ornomath

Precalc,CollegeAlgebra,orlower

~150,000

~150,000

~100,000

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~250,000

These are rough estimates based on a variety of data sources.

K-12 Math

CollegeMath

Calculus

The traditional curriculum

Although calculus can play an important role in secondary school, the ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college.

MAA/NCTM Joint Statement, March 2012

1.  Students who enroll in a calculus course in secondary school should have demonstrated mastery of algebra, geometry, trigonometry, and coordinate geometry;

2.  The calculus course offered in secondary school should have the substance of a mainstream college-level course

MAA/NCTM Joint Statement, March 2012

3. The college curriculum should acknowledge the ubiquity of calculus in secondary school, shape the college calculus curriculum so that it is appropriate for those who have experienced introductory calculus in high school, and offer alternatives to calculus.

MAA/NCTM Joint Statement, March 2012

Course taking in High School By students in Calculus I at PhD Universities

percentage Mean grade (SD) Algebra II* ≤ 10th grade 77% 3.8 (0.5)

Precalculus* ≤ 11th grade 67% 3.7 (0.6)

Calculus ≤ 12th grade 67% 3.8 (0.5)

Statistics ≤ 12th grade 9% 3.6 (0.6)

* Does not count students who took an integrated curriculum

Characteristics of Calculus I Students at PhD Universities

Average high school math GPA 3.77

SAT scores: 3rd Quartile Median 1st Quartile

710 670 620

Agree that to succeed in Calculus I, must have taken it before. 49%

22.7%

24.4%

21.1%

21.1%

29.7%

29.6%

21.8%

22.0%

22.9%

23.7%

20.1%

19.5%

24.7%

22.3%

37.1%

37.5%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

researchuniversity

undergradcollege

mastersuniversity

twoyearcollege

A B C DFW

Final Grades as Reported by Instructors

14%

13%

25%

34%

26%

26%

34%

30%

26%

30%

18%

18%

34%

31%

23%

18%

0% 20% 40% 60% 80% 100%

noHScalc

APexam<3

HScalc/noAPexam

APexam≥3

A B C DFW

Confidence in mathematical abilities

Enjoyment of doing mathematics

Desire to continue the study of mathematics

Effect of Calculus I at PhD-granting universities on

Down 1/6 of a standard

deviation

50th percentile down to 44th

Down 1/3 of a standard

deviation

50th percentile down to 37th

Down half a standard deviation

50th percentile down to 31st

Switchers: students who started certain that they would go on to take Calculus II, but by the end of the term were no longer sure or had definitely decided not to continue.

Of those who started with intention to take Calculus II and who answered end-of-term survey (predominantly A and B students):

11% of men were switchers 20% of women

Switchers by grade in Calculus I. Women:

A: 10% B: 13% C: 24%

Men A: 6% B: 6% C: 12%

Switchers by intended major Women in Bio Science:

A or B: 19% C: 29% Men in Bio Science

A or B: 13% C: 26%

Women in Engineering: A or B: 4% C: 19%

Men in Engineering A or B: 2% C: 7%

Reasonforswitching Gender StudentsearningAorB

StudentsearningC

ToomanyothercoursesIneedtotake

Women 43% 33%

Men 42% 16%

Havechangedmajor Women 40% 43%

Men 33% 39%

TakestoomuchXmeandeffort

Women 33% 25%

Men 29% 26%

BadexperienceinCalculusI

Women 18% 53%

Men 19% 35%

Don’tunderstandcalculuswellenough

Women 18% 38%

Men 4% 26%

Gradewasnotgoodenough

Women 7% 15%

Men 0% 13%

Students could select more than one response.

Reasonforswitching Gender StudentsearningAorB

StudentsearningC

ToomanyothercoursesIneedtotake

Women 43% 33%

Men 42% 16%

Havechangedmajor Women 40% 43%

Men 33% 39%

TakestoomuchXmeandeffort

Women 33% 25%

Men 29% 26%

BadexperienceinCalculusI

Women 18% 53%

Men 19% 35%

Don’tunderstandcalculuswellenough

Women 18% 38%

Men 4% 26%

Gradewasnotgoodenough

Women 7% 15%

Men 0% 13%

Students could select more than one response.

University Professors: More emphasis on algebra and precalculus topics, less teaching of calculus, downplay role of pedagogy and applications. Stress “deep understanding.”

High School Teachers: See calculus as a way of strengthening algebra and precalculus skills, emphasize importance of group work and applications.

Wade, C. H., Sonnert, G., Sadler, P., Hazari, Z., & Watson, C. (2016). A Comparison of Secondary Mathematics Teachers’ and Mathematics Professors’ Views on Secondary Preparation for Tertiary Calculus. Journal of Mathematics Education at Teachers College, 7(1), 7-16.

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jects

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inCollegeCalculus

PreparaFonforCalculusComposite

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HSCalc

noHSCalc,CollegeGrade

HSCalc,CollegeGrade

Calculus in high school does improve grade in post-secondary Calculus I (by about half a grade).

Sadler, P. & Sonnert, G. (2017). The Path to College Calculus: The Impact of High School Coursework. Journal for Research in Mathematics Education.

Anything less than an A in Algebra I, Geometry, Algebra II, or Precalculus makes it unlikely that a student will earn B– or better in Calculus I at a college or university.

Sadler, P. & Sonnert, G. (2017). The Path to College Calculus: The Impact of High School Coursework. Journal for Research in Mathematics Education.

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ictedGrade

inCollegeCalculus

CutoffPreparaFonScoreforAssignmenttoPrecalculus

PrecalcboundaryNoPrecalcboundary

Source: Sonnert, G., & Sadler, P. M. (2014). The impact of taking a college precalculus course on students’ college calculus performance. International Journal of Mathematical Education in Science and Technology, 45(8), 1188-1207

VariaFon PhD(133)

MA(89)

Stretched out Calculus I 13 7 Calculus infused with precalculus 7 4 Stretched out Calculus I & II 6 1 Precalc contemporaneous with calculus

2 1

More time spent reading text è Lower grade

Barnett, M., Sonnert, G., & Sadler, P. (2014). Productive and Ineffective Efforts: How Student Effort in High School Mathematics Relates to College Calculus Success. International Journal of Mathematical Education in Science and Technology, 45(7), 996-1020.

“Developing strong study habits prior to attending college was found to reap benefits in college calculus … We suggest instructing all mathematics students in how to effectively study… Providing instruction on ‘best study practices’ may lead to a stronger mathematical foundation, more positive attitudes towards mathematics, and higher mathematics achievement.”

Barnett, M., Sonnert, G., & Sadler, P. (2014). Productive and Ineffective Efforts: How Student Effort in High School Mathematics Relates to College Calculus Success. International Journal of Mathematical Education in Science and Technology, 45(7), 996-1020.

In university classes, “students are expected, on their own incentive, to know about and seek out opportunities to clarify or address a learning gap or to engage in collaborative learning with peers … In high school, structured support was available primarily for struggling students; only those with serious learning problems were involved in the support systems.”

Burrill, G. (2017). Challenges in the transition from high school to post secondary mathematics. Chapter 8 in The Role of Calculus in the Transition from High School to Post Secondary Mathematics. Joint publication of MAA and NCTM.

“The cultures for teaching and learning mathematics in high school and in college have significant differences… Dialogue between instructors in the two communities can help high school teachers recognize the constraints and affordances of the university/college system and can help post secondary instructors become aware of and understand their students’ expectations for learning in a mathematics class.”

Burrill, G. (2017). Challenges in the transition from high school to post secondary mathematics. Chapter 8 in The Role of Calculus in the Transition from High School to Post Secondary Mathematics. Joint publication of MAA and NCTM.

ApdffileofthisPowerPointisavailableatwww.macalester.edu/~bressoud/talks