Designing and Implementing Conceptual Calculus

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Designing and Implementing Conceptual Calculus

• CMC - SS
• November 7, 2003
• Karen Payne
• Session 230

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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

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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.  

4
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.  

5
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.

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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

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Technology to consider including

• Motion Detectors
• Graphing Calculators
• Excel Spreadsheets
• Geometers Sketchpad (v. 4.0)

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Technology touched on today

• Motion Detectors
• Geometers Sketchpad (v. 4.0)

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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 dont 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.

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What story do graphs tell?
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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?

Distance from m.d.
time
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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.

   
 
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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?

Position
A
time
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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

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Why Motion Detectors?

• Kinesthetic experience reinforces the story
  behind the graph
• Combats the Graph as Picture misconception

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Relating Position and Velocity Graphs

• Act03RemoteControl CMC version.gsp

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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?

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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.

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A good conversation starter (see handout p. 15)

•  
•  
• velocity
•  
•   time
• What can you tell from this graph? What cant
  you tell from this graph? What does the point of
  intersection mean? Can you tell which car
  traveled the furthest distance?
•  
•  

A
B
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Area under the curve

• Time v. Velocity Graph (see handout p. 16)
• Velocity
• (ft/sec.) 1
• 1 4 10 Time (in sec.)

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

• Velocity
• (ft/sec.) 1
• 1 4 10 Time (in sec.)

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

• Velocity
• (ft/sec.) 1
• 1 4 10 Time (in sec.)

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

• Velocity
• (ft/sec.) 1
• 1 4 10 Time (in sec.)

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What about area below the axis? (see handout p.
17)

•  
•  
• Velocity
• ( ft/sec) 1
• 0
• 2 4 10
• Time (in sec.)

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Mathematical Big Ideas

• Meaning of area under the curve in context
• Ways of estimating Riemann sums, trapezoidal
  estimations
• Integral notation

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

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• Questions?

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Contact Information

• Karen Payne
• kpayne_at_sciences.sdsu.edu
• Office (619) 594 3970
• Fax [phone]
• Presentation can be found (sometime next week)
  at
• pdc.sdsu.edu