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Using Investigations to Teach Some Undergraduate level Mathematics Courses Mohammed A. Qazi Department of Mathematics Tuskegee University. Courses Targeted. Emphasis is on one section of College Algebra (MATH 107) Each section has a mix of majors, with an average of 10 -12 education majors

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Using Investigations to Teach Some Undergraduate level Mathematics Courses Mohammed A. QaziDepartment of MathematicsTuskegee University

courses targeted
Courses Targeted
  • Emphasis is on one section of
    • College Algebra (MATH 107)
      • Each section has a mix of majors, with an average of 10 -12 education majors
      • Generally 40 - 50 students
      • Mostly Freshman
    • Calculus and Analytic Geometry I (MATH 207)
      • Majority Engineering & Science majors, a few mathematics majors
      • Generally 40 - 50 students
why these courses
Why These Courses?
  • Unsatisfactory failing rate
    • College Algebra: 30%
    • Calculus and Analytic Geometry: 25%
  • Courses contain basic material that students must master
  • Resources are readily available to implement Investigations for many of the course concepts
pre surveys
Pre-Surveys
  • Experiences with mathematics in school and at college coming in to this course:
pre surveys5
Pre-Surveys
  • Think about how you have studied mathematics coming in to this course
pre surveys6
Pre-Surveys
  • Looking ahead into the future, what importance will you be giving to learning mathematics?
pre survey
Pre-Survey
  • Other questions:
    • What mathematics courses did you take grade 9 -12?
      • Pre-Algebra: 77%
      • Algebra I: 97%
      • Algebra II: 96%
      • Algebra III: 54%
      • Geometry: 95%
      • Pre-Calculus: 31%
      • Calculus: 9%
      • I don’t remember: 11%
    • Did you go to school in East-Alabama?
    • Do you know anything about cooperative learning?
common student difficulties
Common Student Difficulties
  • Solve

(x - 3)(x - 4) < 0

  • Where is on the number line?
  • Is a positive number or a negative number?

+++++

+++++

x

x

x

1

3

?

4

5

common student difficulties9
Common Student Difficulties
  • Give students a working understanding of the number line.
    • Describe situations where rationals, irrationals, positives and negatives can occur
common student difficulties10
Common Student Difficulties
  • Other Difficulties
    • Many students have a hard time making the difference between
      • Factoring
      • Expanding
      • Solving
      • Evaluating
        • Example: Evaluate at x = 8.
    • Use of parenthesis
    • Making the difference between “there is a point c” and “for every point c”
what are we doing different
What are we doing different?
  • Goal:
    • Engage students by turning them into active learners and help them make sense out of the concepts
      • Created modules for some of the concepts containing activities that are investigative in nature and help students discover and appreciate these concepts from the two courses through
        • Interactive Mathematics Program (IMP)
        • Connected Mathematics Program (CMP)
        • Algebra in Motion and Calculus in Motion
        • Cooperative Learning
college algebra
College Algebra
  • Traditional Approach to teaching the concept of Functions
    • A function f from a set X into a set Y is a rule that assigns each element in X to precisely one element in Y. f(x) is called the image of x under f. The image f(x) is the unique element in Y that corresponds to the element x in X. The set X is called the domain of f. The range of f is the set of elements in Y that are associated with some element in X.
college algebra13
College Algebra
  • A rectangular feeding pen for cattle is to be made with 100 meters of fencing.
    • If W represents the width of the pen, express its area A in terms of W.
    • From practical considerations, what restrictions must you put on W?

w

college algebra14
College Algebra
  • How would the domain of the function change if we were to remove the context from the problem?
  • Ask students to write the definition of domain in their own words
  • Study the formal definition of domain
college algebra15
College Algebra
  • Questions generated by students:
    • “Is there something like domain for the y-values?”
    • “Does the 625 mean that the largest pen you can construct with 100 m of fencing material is 625m^2?”
    • “So, does that mean to get the maximum possible area you construct a pen of width 25m… So you get a square pen?”
college algebra16
College Algebra
  • Student comments on this activity
    • “I enjoyed working on this activity. It was a lot of fun. What helped was that I could visualize what I was being asked to do.”
    • “My math teacher in school taught functions and I could never understand the concepts. With the help of this activity I can now visualize what domain and range are and know how to find them. The formal definitions don’t look that mysterious anymore either.”
college algebra17
College Algebra
  • Piece-Wise Functions
    • Graph

Students cannot get organized and give up very easily.

  • Alternatively, introduce the topic using an activity [IMP: Small World Isn’t It]:
college algebra18
College Algebra
  • Tyler is saving money so that he can purchase a basketball uniform
  • One Friday afternoon, right after getting his weekly allowance, Tyler puts some of the allowance into his piggy bank and counts the savings so far
  • Tyler has $2.00 that Friday (including the amount he just added) and he will add $0.50 to his savings every Friday afternoon from then on
  • Question: Draw a graph showing accumulated savings versus time elapsed for Tyler for a five week period (assuming that he adds his savings to his piggy bank each Friday and does not spend any of the saved money)
college algebra19
College Algebra

Student group attempts:

college algebra20
College Algebra

Student refinement after some in-class discussion:

calculus
Calculus
  • Differentiation
    • Activities from IMP Unit “Small World, Isn’t It”
      • A helicopter drops a bundle of supplies.
      • Distance fallen by the bundle:
      • The bundle can survive a fall of 165m/h
      • QUESTION: Does the bundle survive the fall?
        • Leads students to study expressions like
        • Making connections with the concept of limits:
        • Students learn analytical approach to computing derivatives and also see the geometric interpretation
calculus22
Calculus
  • A question generated by students:
    • What is the maximum height that a helicopter can fly at so that the bundle can be dropped without damage?
      • Now look for
        • What will “a” and “b” be?
          • “b” has to be -16! Only “a” will change
          • Some proceed by trial and error
          • Set-up appropriate equations to find “a”
calculus23
Calculus
  • Derivative of the sine function
    • Traditional approach:
calculus24
Calculus

Somewhat more investigative:

  • Use Geometer’s Sketchpad to animate the behaviour of the tangent line
  • Ask pertinent questions to get students engaged
  • Make an educated guess at what the derivative might be
calculus25
Calculus
  • Possible questions to ask:
    • What would be the domain of the derivative?
    • Do we need to find the derivative on the whole real line or can we restrict ourselves?
    • What would be the derivative at x = 0?
      • “It will be 1, I think, since the tangent line seems to be y = x”
    • What would be the derivative at the peaks and valleys bottoms?
    • What would be the value of the derivative at \pi?
      • “Looks like it’s going to be -1 since the tangent lines at x = 0 and at x = pi are orthogonal, and slopes of orthogonal lines have negative reciprocals of each other”
      • “Tangent at x = pi is parallel to line y = -x. Two parallel lines have same slope, So derivative at pi is -1”
calculus26
Calculus
  • Is the sign of the derivative the same throughout or does it change?
  • Is there a relationship between the sign of the derivative and the intervals of increase and decrease of the original function?
  • Estimate the derivative at a few values of x
calculus27
Calculus
  • Derivative of the inverse sine function
    • Analytical approach using implicit differentiation

Somewhat more investigative

  • Questions for discussion:
    • Examine the graph of
    • What properties do you think the derivative of will have?
calculus28
Calculus

Prompts:

  • What would be the domain of the derivative?
    • Responses:
      • “[-1,1] since there is no graph outside [-1,1]”
      • “But -1 and 1 cannot be included since graph of f(x) = sin x at -pi/2, pi/2 have horizontal tangents. By reflection, the tangent lines must be vertical at -1, 1. But that means that there should be no derivative at -1, 1. So the domain is (-1,1)”
  • What would be the value of the derivative at the origin?
    • Responses:
      • “0 because the graph is like that of f(x) = x^3”
      • “1 using reflexion and because derivative of f(x) = sin x is 1 at origin”
calculus29
Calculus
  • What would be the sign of the derivative?
    • “Always positive since tangent lines slope upwards”
observations
Observations
  • Many questions get generated
  • Answering questions / questioning answers
  • Better understanding
  • Some improvement in grades
  • No loss of rigour
  • Less absenteeism
observations31
Observations
  • Problems making the connections
  • Still some who are too passive and don’t want to work
  • Constantly reflect back on what you have learnt so that concepts settle in the mind
  • Tracking students