Designing and Implementing Conceptual Calculus

CMC - SSNovember 7, 2003

Karen PayneSession # 230

Outline for the talk• Justification for such a course

• Brief background of the course

• Share class activity examples, including connection to important calculus concepts

• 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

Dista

nce

fro

m

m.d

.

Another Motion Detector Example…(see handout p. 11)• 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…(see handout p. 11)

•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 CMC version.gsp

Worthwhile mathematical investigations• What is happening to the position

graph when the velocity graph is…– Increasing? Decreasing? – Positive? Negative?

• How does the position graph look when the velocity is at a relative maximum? A relative minimum?

• When the position graph is horizontal, what is true of the velocity?

Comment related to the “Predict the Trace” Activity…

• “I guess I underestimated the importance of letting students really struggle with making sense of what is actually happening and how it corresponds to a graph.”

 

 

velocity 

  timeWhat can you tell from this graph? What can’t you tell from this graph? What does the point of intersection mean? Can you tell which car traveled the furthest distance?

 

 

A good conversation starter (see handout p. 15)

A B

Area under the curveTime v. Velocity Graph (see handout p. 16)

Velocity(ft/sec.) 1

1 4 10 Time (in sec.)

How far does the walker travel between 4 and 10 seconds?

Velocity(ft/sec.) 1

1 4 10 Time (in sec.)

How far does the walker travel during the first four seconds?

Velocity(ft/sec.) 1

1 4 10 Time (in sec.)

How far does the walker travel between 10 and 15 seconds?

Velocity(ft/sec.) 1

1 4 10 Time (in sec.)

What about area below the axis? (see handout p. 17)•  •  • Velocity• ( ft/sec) 1

•0

» 2 4 10

• Time (in sec.)

Mathematical Big Ideas:

• Meaning of area under the curve in context

• Ways of estimating: Riemann sums, trapezoidal estimations

• Integral notation

Valuable Resources (also listed p. 18)

• Exploring calculus with GSP• What is calculus about? by W.W. Sawyer,

MAA, 1962.• The CBMS “Mathematical Education of

Teachers” document  http://www.cbmsweb.org/MET_Document/index.htm 

• “Describing Change Module,” Reconceptualizing Mathematics: Courseware for Elementary and Middle Grade Teachers contact Judith Leggett for info. (619) 594 – 5090

Questions?

Contact Information

Karen Paynekpayne@sciences.sdsu.edu

Office: (619) 594 – 3970Fax: (619) 594 – 0725

Presentation can be found (sometime next week) at:

pdc.sdsu.edu