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Using Investigations to Teach Some Undergraduate level

Mathematics Courses

Mohammed A. QaziDepartment of Mathematics

Tuskegee University

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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

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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

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Pre-Surveys• Experiences with mathematics in school and at

college coming in to this course:

Questions Percentage of class

I like to learn mathematics 19%

I lack confidence in mathematics 86%

I am often bored in a mathematics class

80%

No matter how hard I try, I simply have not been able to understand most mathematical concepts

79%

5

Pre-Surveys

• Think about how you have studied mathematics coming in to this course

Questions Percentage of class

Learning math is mostly about memorizing facts

82%

If I am unable to solve a math problem the first time I try it, I don’t try it again

77%

I have used manipulatives and technology to learn mathematics

41%

I expect my teacher to review for me before each test

79%

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Pre-Surveys

• Looking ahead into the future, what importance will you be giving to learning mathematics?

Questions Percentage of class

I would like to improve my knowledge in mathematics

91%

I would like to overcome my weaknesses in mathematics

92%

Acquiring good studying habits in mathematics is one of my goals

94%

I feel that in a right learning environment in which I am fully engaged in the learning process, I can do much better in mathematics

92%

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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?

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Common Student Difficulties

• Solve (x - 3)(x - 4) < 0

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

number?

2

3 4

x x

1 5

x+++++ +++++

?

77

01272 xx

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Common Student Difficulties

• Give students a working understanding of the number line.– Describe situations where rationals,

irrationals, positives and negatives can occur

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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”

50050060050.2 RP

500)500600(50.2 RP

67)( 2 xxxf

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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

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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.

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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

)50()( wwwA

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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

)50()( wwwA

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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?”

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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.”

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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]:

2 0 1

2.5 1 2

( ) 3 2 3

3.5 3 4

4 4 5

if x

if x

f x if x

if x

if x

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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)

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College Algebra

Scatter Plot

0

1

2

3

4

5

6

0 1 2 3 4 5 6

Number of Weeks

Am

ou

nt

Acc

um

ula

ted

One Straight Line

0

1

2

3

4

5

6

0 1 2 3 4 5 6

Number of WeeksA

mo

un

t A

ccu

mu

late

d

• Student group attempts:

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College Algebra

0

1

2

3

4

5

0 1 2 3 4 5 6

Number of Weeks

Amou

nt A

ccum

ulat

ed• Student refinement after some in-class discussion:

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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

5

)5()(

t

hth

5

)5()(

5lim

t

hth

t

216400)( tth

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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”

216400)( tth 2)( tbath

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Calculus

• Derivative of the sine function– Traditional approach:

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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

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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”

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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

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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?

xxf 1sin)(

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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”

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Calculus

• What would be the sign of the derivative? – “Always positive since tangent lines slope

upwards”

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Observations

• Many questions get generated

• Answering questions / questioning answers

• Better understanding

• Some improvement in grades

• No loss of rigour

• Less absenteeism

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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

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Before & After

Questions Percentage of class

Before

Percentage of class

After

I like to learn mathematics 19% 71%

I lack confidence in mathematics 86% 41%

I am often bored in a mathematics class

80% 23%

No matter how hard I try, I simply have not been able to understand most mathematical concepts

79% 35%

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Before & After

Questions Percentage of class

Before

Percentage of class

After

Learning math is mostly about memorizing facts

82% 21%

If I am unable to solve a math problem the first time I try it, I don’t try it again

77% 30%

I have used manipulatives and technology to learn mathematics

41% 100%

I expect my teacher to review for me before each test

79% 55%