A Radical Approach To Calculus David Bressoud Electronic Math - PowerPoint PPT Presentation

A Radical Approach To Calculus David Bressoud Electronic Math Education St. Paul, MN Seminar MIT April 17, 2018 A pdf file of this PowerPoint is available at Conference Board of the www.macalester.edu/~bressoud/talks Mathematical Sciences

Understanding Calculus Through Its History A Guide for Teachers and Students David M. Bressoud Princeton University Press 2018

NCTM Research Compendium , to appear this year 1 19 Understanding the Concepts 2 of Calculus: Frameworks 3 4 and Roadmaps Emerging 5 6 From Educational Research 7 8 9 10 sean larsen 11 Portland State University, Portland, Oregon 12 karen marrongelle 13 14 Portland State University, Portland, Oregon 15 david bressoud 16 Macalester College, Saint Paul, Minnesota 17 karen graham 18 19 University of New Hampshire, Durham 20 21 C alculus is a foundational course for most dis- related to calculus. For example, the recent national study 22 ciplines in science and engineering around by the Mathematical Association of America (Bressoud, 23 the world. It lies at the heart of any modeling Mesa, & Rasmussen, 2015 ) focused on identifying char- 24 of dynamical systems and often is used to sig- acteristics of college calculus programs that contribute 25 nal whether a student is prepared for advanced to student success as measured by retention and changes 26 mathematics, science, and engineering, even in attitudes. Other work has explored issues related 27 when such courses do not explicitly build on calculus to the rapid growth of the Advanced Placement Calcu- 28 (Bressoud, 1992 ). At the same time, calculus is a barrier lus program in the United States (Keng & Dodd, 2008 ; 29 to the academic progress of many students. Across the Morgan & Klaric, 2007 ). Törner, Potari, and Zachariades 30 United States, 28 % of those enrolled in postsecondary ( 2014 ) provide an overview of curricular evolution in 31 calculus 1 (typically consisting of di fg erential calculus) calculus in Europe at the secondary level. There has 32 receive a D or F or withdraw from the course (Bressoud, also been research on students’ readiness to learn calcu- 33 Carlson, Mesa, & Rasmussen, 2013 ). Only half earn the B lus (Carlson, Madison, & West, 2015 ). Finally, there has 34 or higher that is taken as a signal that one is prepared for been research focused on calculus instructors. This work 35 the next course, and many of these, despite their grade, includes investigations focused on instructors’ percep- 36 are discouraged from continuing (Bressoud et al., 2013 ). tions of instructional approaches (Sofronas et al., 2015 ), 37 New challenges have arisen, from the movement of cal- relationships between teaching practices and content 38 culus ever earlier into the secondary curriculum in the coverage concerns (Johnson, Ellis, & Rasmussen, 2015 ), 39 United States to the pressure to drastically reduce fail- and the professional development of graduate students 40 ure rates (Bressoud, 2015 ). Meeting these challenges will (Deshler, Hauk, & Speer, 2015 ). 41 require the research community to develop better under- Schoenfeld ( 2000 ) noted that research in math- 42 standings of how students negotiate this subject, where ematics education has two purposes. The fj rst is a pure 43 the pedagogical obstacles lie, and what can be done to research purpose, “To understand the nature of math- 44 improve student success. ematical thinking, teaching, and learning,” and the sec- 45 In the interest of assuring the coherence of this ond is an applied purpose, “To use such understandings 46 chapter, and to provide an appropriate level of detailed to improve mathematics instruction” (p. 641 ). It makes 47 attention to the work we discuss, we concentrate our sense to organize this chapter around these two pur- 48 attention on the research focused on students’ under- poses for two reasons. First, such an organization will 49 standing of calculus content. However, we are compelled allow us to explicitly shine a light on applied research. 50 to fj rst acknowledge the wide variety of important edu- It is critical that we do so because calculus is a key part 51 cational research that has been done on other issues of science, technology, engineering, and mathematics 52 526 1ST PAGES 14398-19_Ch19.indd 526 11/14/16 5:33 PM

Traditional order of four big ideas: 1. Limits: as x approaches c , f ( x ) approaches L 2. Derivatives: slope of tangent 3. Integrals: area under curve 4. Series: infinite summations

Problems Traditional order of four big ideas: 1. Limits: as x approaches c , f ( x ) approaches L 2. Derivatives: slope of tangent • Leads to assumption that f cannot oscillate around or equal L when 𝑦 ≠ 𝑑 3. Integrals: area under curve • x -first emphasis makes transition to 4. Series: infinite summations rigorous definition difficult • Difficult to prove theorems that rely on definition of limit • Belief that if lim '→) 𝑔 𝑦 = 𝑐 and -→. 𝑕 𝑧 = 𝑑 , then lim lim '→) 𝑕 𝑔 𝑦 = 𝑑

Solution Traditional order of four big ideas: 1. Limits: Algebra of Inequalities 2. Derivatives: slope of tangent Build from bounds on approximations 3. Integrals: area under curve 3 3 3 8 Leibniz series 1 − 4 + 6 − 7 + … = 4. Series: infinite summations 9 Justified because each partial sum differs 8 from 9 by less than absolute value of next term.

Mike Oehrtman Clearcalculus.okstate.edu

( ) = sin 9 − t 2 v t

Problems Traditional order of four big ideas: 2. Derivatives: slope of tangent 3. Derivatives: slope of tangent • Derivative becomes a static number • Students have difficulty making the 4. Integrals: area under curve connection to average rate of change 5. Series: infinite summations • Makes it difficult to understand derivative as relating rates of change of two connected variables

Solution Traditional order of four big ideas: 2. Derivatives: Ratios of Change 3. Derivatives: slope of tangent Focus on function as a relationship between two linked variables 4. Integrals: area under curve 5. Series: infinite summations Derivative connects small changes in one to small changes in the other

Sketch the graph of volume as a function of height.

Indian astronomy: 𝜄 Arclength measured in minutes Circumference = 60 < 360 = 21,600 Radius = 3438 ~ AD 500, Aryabhatta showed that for small increments ∆ sine ∆ arclength ~ cos 𝜄

Problems Traditional order of four big ideas: 3. Integrals: area under curve 2. Derivatives: slope of tangent • Students don’t see integral as accumulator “I don’t understand how a distance can be 3. Integrals: area under curve an area.” 4. Series: infinite summations • Leads to difficulties interpreting definite integral with variable upper limit, critical to understanding the Fundamental Theorem of Integral Calculus • Don’t retain definition of definite integral as limit of Riemann sums

Wagner, J.F. (2017). Students’ obstacles to using Riemann sum interpretations of the definite integral 1 st -year physics students see Riemann sums as either irrelevant or simply a tool for approximating definite integrals. 3 rd -year physics majors cannot justify why the following produces the area under y = x 3 from 0 to 2. P P = 1 = 16 N 𝑦 4 𝑒𝑦 4 𝑦 9 S 4 − 0 = 4. Q Q See launchings.blogspot.com April, 2018

Solution Traditional order of four big ideas: 3. Integrals: Accumulation 2. Derivatives: slope of tangent START with accumulator functions, i.e. Riemann sums with variable upper limit, 3. Integrals: area under curve ' leading to ∫ 𝑢 4 dt . This accumulates up to x 4. Series: infinite summations Q the quantity whose rate of change is t 3 . Students are easily led to discover that rate of change of this function is x 3 . Leads to FTIC.

http://patthompson.net/ThompsonCalc/

Problems Traditional order of four big ideas: 4. Series: Infinite Summations 2. Derivatives: slope of tangent • Students view series as sums with a LOT of terms 3. Integrals: area under curve • Convergence tests become arcane rules with 4. Series: infinite summations little or no meaning

Solution Traditional order of four big ideas: 4. Series: Sequences of Partial Sums 2. Derivatives: slope of tangent Taylor polynomials rather than Taylor series 3. Integrals: area under curve Prefer emphasis on Lagrange error bound (as extension of Mean Value theorem) rather 4. Series: infinite summations than convergence tests. 𝑔 𝑦 = 𝑔 𝑏 + 𝑔 X 𝑏 𝑦 − 𝑏 + 𝐹 𝑦, 𝑏 𝐹 𝑦, 𝑏 = 𝑔 𝑦 − 𝑔(𝑏) = 𝑔′(𝑑) 𝑦 − 𝑏

Traditional order of four big ideas with right emphasis: 1. Limits: Algebra of Inequalities 2. Derivatives: Ratios of Change 3. Integrals: Accumulation 4. Series: Sequences of Partial Sums

Preferred order of four big ideas with right emphasis: 1. Integrals: Accumulation 2. Derivatives: Ratios of Change 3. Series: Sequences of Partial Sums 4. Limits: Algebra of Inequalities A pdf file of this PowerPoint is available at www.macalester.edu/~bressoud/talks