Description of a Foundations of Calculus Course for Teachers CMC-Asilomar December 2005 Karen Payne Aguilar

Outline for the talk Justification for and background of the course Share class activity examples, including connection to important calculus concepts Comments from participating teachers Unexpected outcomes Question and answer time

From the “Mathematical Education of Teachers,” by CBMS Additional coursework that allows prospective middle grades teachers to extend their own understanding of mathematics, particularly of the mathematics they are preparing their students to encounter, will also be required.We suggest that this second type of coursework contain at least one semester of calculus if a course exists that focuses on concepts and applications.

From the “Mathematical Education of Teachers,” by CBMS Additional coursework that allows prospective middle grades teachers to extend their own understanding of mathematics, particularly of the mathematics they are preparing their students to encounter, will also be required.We suggest that this second type of coursework contain at least one semester of calculus if a course exists that focuses on concepts and applications.

From the “Mathematical Education of Teachers,” by CBMS …carefully designed instruction that engages students in collaborative investigations rather than passive listening to their teachers, will produce deeper learning and better retention of mathematics as well as improved social and communication skills. Calculator and computer tools have suggested new ways of teaching school and collegiate mathematics, encouraging laboratory-style investigations of key concepts and principles.

Brief background of the course : Create a “Foundations of Calculus” course for teachers who may or may not have previously taken calculus Incorporate class activities to develop deep understanding of fundamental calculus concepts –instantaneous rate of change –accumulation of area under a curve

Technology to consider including… –Motion Detectors –Graphing Calculators –Excel Spreadsheets –Geometer’s Sketchpad (v. 4.0)

Technology touched on today… –Motion Detectors –Geometer’s Sketchpad (v. 4.0)

“Why did you take this class?” “I decided to take this class because even though I did well in my calculus class in H.S. I never (did) and still don’t understand what calculus is.” “Have been asked to teach calculus several times and have been hesitant so I want to brush up on my underlying understanding of calculus to eventually teach it.” “The application of (motion) detectors and geometer sketchpad appealed to me.” “I wanted to take this class because mathematically I feel a little like a fraud because I only know ‘kid’ math and not ‘real’ math.”

What story do graphs tell?

A Motion Detector Example What graph is created by this walk? –Start close to the motion detector. Walk away from it for 3 seconds then stop for 4 seconds. Then walk towards it again for 3 seconds. What walk would create this graph? time Distancefrom m.d.

Another Motion Detector Example How would you make the following Time vs. Distance from Motion Detector graphs? At your tables, discuss the walks needed to produce the graphs.

Use your results to predict… What walk would create the graph below? What is the significance of the point of inflection? time Position A

Mathematical Big Ideas from Motion Detector Activities… Total Distance v. Position graph Positive/negative velocity Significance of horizontal line in a distance graph, in a velocity graph Point of Inflection

Why Motion Detectors? Kinesthetic experience reinforces the “story” behind the graph Combats the “Graph as Picture” misconception

Relating position and velocity graphs Act03RemoteControl Asilomar.gsp

Area under the curve Time v. Velocity Graph Velocity 1 (ft/sec.) Time (in sec.)

Time v. Velocity Graph Velocity 1 (ft/sec.) Time (in sec.) How far does the walker travel between 4 and 10 seconds?

Time v. Velocity Graph Velocity 1 (ft/sec.) Time (in sec.) How far does the walker travel during the first four seconds?

Time v. Velocity Graph Velocity 1 (ft/sec.) Time (in sec.) How far does the walker travel between 10 and 15 seconds?

Time v. Velocity Graph Velocity 0 (ft/sec.) Time (in sec.) What happens now?

Time v. Velocity Graph Velocity 0 (ft/sec.) Time (in sec.) What happens now? Signed area.

Mathematical Big Ideas: Meaning of area under the curve in context Ways of estimating: Riemann sums, trapezoidal estimations Integral notation

Comments from teachers I am still struggling with the implications for my practice: when to find the time to do the work of exploring the software and wording the activity so the students will get the “aha” of the activity. Idea of the area under the curve equalling, say, the distance traveled, is rather amazing!

Comments from teachers The computer work was really hard to maneuver and I think I would have gotten more out of it by watching than doing. While working on the Geometer’s Sketchpad I found my focus going beyond the level of my early years of producing accurate, yet time consuming graphs…”

Unexpected Outcomes Expected Participants: -Teachers pursuing supplemental credentials in math -Middle and High school teachers seeking to revisit underlying calculus concepts of learn how technology can support this. Actual participants 7 elementary (some with degrees in math) 2 middle 3 high school 12 total

Unexpected outcomes Lots of cross grade and school level conversations about graphing: -High school and elementary in same room -Graphing happens in elementary! -How to deal with discrete graphs

Questions?

Valuable Resources Exploring calculus with GSP Sawyer, W.W. (1962). What is calculus about? Washington, DC: Mathematical Association of America. The CBMS “Mathematical Education of Teachers” document “Describing Change Module,” Reconceptualizing Mathematics: Courseware for Elementary and Middle Grade Teachers contact Judy Sowder for info.

Contact Information Karen Payne Aguilar Presentation can be found at: pdc.sdsu.edu