Page 1Developing an Intuitive Grasp of Exponential Functions from Real World ExamplesDavid M. RheamDepartment Of Physics, SUNY-Buffalo State College, 1300 Elmwood Avenue,Buffalo, NY 14222, [email protected] manuscript was completed as a requirement for PHY 690: Masters Project at SUNY-Buffalo State College Department of Physics, under the supervision of Dr. Dan MacIsaac. Dr. David Abbott also contributed comments and insights.Page 2AbstractAlbert A. Bartlett (1976) stated “The greatest shortcoming of the human race is our inability to understand the exponential function” (p.394). It is unsurprising that many of our high school students have difficulty grasping concepts involving exponentials (Weber, 2002). One main reason for this is that students enter our classrooms with their own ideas about the waythings grow and decay from what they have seen in their own life experience. Unfortunately, most of what they see is not exponential. It is extremely difficult to change deep-seated student beliefs. Students cannot simply be taught a formula or shown a graph dealing with exponential growth and decay and be expected to understand how it works. Rather, students need to build their own understanding of new concepts (Alagic and Palenz, 2006). Real life experiences in ourclassrooms in which students can explore exponential functions, select representations and make connections can make their learning more meaningful (Greeno & Hall, 1997). Not only are exponential functions essential to mathematics they are also embedded in the sciences and provide a model for representing growth and decay in real world phenomena (Strom, 2006). Here I describe several possible classroom experiences that can help students discover exponential functions and then connect them to some important topics in the realm of physics. I also take an in-depth look at compound interest as a physical example of e; which is the essentiallink to move from understanding discrete exponential functions in our classroom examples to grasping the continuous exponential functions that are explored in physics.Page 3Developing an Intuitive Grasp of Exponential Functions from Real World ExamplesIntroductionAlbert A. Bartlett (1976) stated “The greatest shortcoming of the human race is our inability to understand the exponential function” (p.394). It is unsurprising that many high school students have serious difficulty grasping concepts involving exponentials (Weber, 2002). One main reason for this is that students enter the classroom with their own extensive naïve observations and ideas about the way things grow and decay from their own life experiences. Students see the way they themselves develop, or plants and animals around them grow, or watchthe way a candle shrinks as it burns and they formulate ideas about how growth and decay happen. Unfortunately, most of what students see is not exponential but linear growth and decay instead. The concept of linear growth is then reinforced at school as students study linear relationships as a central theme in algebra. Students have many real life and classroom experiences with linear growth and, because of this, many students still revert back to linear representations when they first start to deal with exponential growth (Alagic & Palenz, 2006). It is extremely difficult to change deep-seated student beliefs. Students cannot simply be taught a formula or shown a graph dealing with exponential growth and decay and be expected tounderstand how it works. Rather, students need to build their own understanding of new concepts (Alagic and Palenz, 2006). Conventional lessons on functions use a correspondence approach that begins by establishing a rule that connects x-values and y-values, usually in the form an equation. However, research shows it is often more powerful to use a covariational approach, where students first work to fill in the table of x-values and y-values by an operation they create using the context of a real life problem (Confrey and Smith, 1994). These real life experiences in classrooms in which our students can explore exponential functions, selectPage 4representations and make connections, can make their learning more meaningful (Greeno & Hall,1997). The Nation Council of Teachers of Mathematics Principles and Standards has recognizedand emphasized the importance of function development using real world examples as well (NCTM, 2000). Not only are exponential functions essential to mathematics, they are also embedded in the sciences and provide a model for representing growth and decay in real world phenomena (Strom, 2006). Here I describe several possible classroom experiences that can help students discover exponential functions and then connect them to some important topics in the realm of physics. I also take an in-depth look at compound interest as a physical example of e; which is the essential link to move from understanding discrete exponential functions in our classroom examples to grasping the continuous exponential functions that are explored in physics.Classroom Examples of Exponential Growth: Chessboards and Rice, Paper FoldingWhen introducing the topic of exponential growth a simple approach is best, by restricting arithmetic to the familiar: multiplication, division, addition, and subtraction (Goldberg & Shuman, 1984, p.344). With this in mind, one of this simplest ways to think of exponential growth is something that has a constant doubling period. A good introduction to the idea of a doubling period and the power of exponentials is a lesson on the famous story of “The King’s Chessboard” (Birch, 1988). Here a man requests, as a reward, to have one grain of rice for the first square and then asks the king to double it for each of the squares on the chessboard. A short version of this story can be found online at http://www.cs.berkeley.edu/~vazirani/algorithms/chap8.pdf (paragraph 4).Have students think independently about the strange request and whiteboard predictions about how much rice they think it would be. Ask students whether they think the man would beRice on a Chessboard050001000015000200002500030000350000 5 10 15 20SquareGrains of RicePage 5better off taking 10,000 grains of rice per day, or some other linear relationship instead (Alagic &Palenz, 2006, p.644). Then have students start to act out the man’s request in groups of three or four having one student in each group record the
View Full Document
Unlocking...