Preparing Students for Elementary Statistics or Math for Liberal Arts

Mary ParkerAustin Community College

www.austincc.edu/mparker/jmm10/January 14, 2010

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Purpose of the Course

MATD 0385, Developing Mathematical Thinking, is designed to give students practice in the appropriate habits of mind to help them succeed in college-level mathematics courses.

It does not include enough algebra to be appropriate for students preparing for College Algebra.

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Purpose of this talk

To give you ideas about how to find / adapt materials to offer an alternative course to Intermediate Algebra, if you wish to do so.

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Outline of this talk

1. Characteristics of our students2. Choosing our topics and materials3. Discussion of our approach to “Exponential

Growth” in some detail4. Conclusions5. What I didn’t talk about. (Ask me.)

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Our students

Most of the students who enroll in MATD 0385, Developing Mathematical Thinking, at ACC are students who made C’s in Elementary Algebra and/or who have failed Intermediate Algebra (often multiple times.)

(Students can qualify to go on to Elementary Statistics, Math for Liberal Arts, or Math for Measurement with an Assessment Test score equivalent to approximately the end of Elementary Algebra. )

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Our students’ attitudes and characteristics

Most students in the class• Are grateful to have an alternative to

Intermediate Algebra.• Feel sure that math is supposed to be taught

in a “rote” fashion.• Are not skilled at organizing their homework

and study time.

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Preparing for later courses

None of the courses for which we are preparing these students are organized in a way that a “rote” learning method is going to be very successful.

We believe it is essential to get these students into better “habits of mind” before they go into college-level math.

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Choosing our topics

While we did make an initial topic list before we started looking at possible texts, we were most interested in finding materials which supported teaching in a less “rote” manner than typical algebra classes.

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Initial Topic List: Overarching Ideas

• Reading and Thinking - we want this throughout the course• Multi-step problems

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Initial Topic List: Specific topics• Linear models to a level similar to College Algebra• Introductory level for Exponential and Logarithmic

Models• Maybe some Quadratic Models• Maybe some Rational models - especially

concentrations and percentages• Percentages• Graphs, Tables, Data Interpretation• Logic - especially dealing with conditionals• Order of operations as needed to use a scientific

calculator or spreadsheet.

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How we looked for course materials

• Agreed at the beginning of our search that the point is to choose topics and materials that have a reasonable chance of changing the students’ habits of mind. The topics covered are not as important.

• We agreed to rely on custom publishing and put together chapters from various books.

• Looked in the Maricopa Modules and also at typical books for a Math for Liberal Arts course.

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Maricopa Modules

• Developed in the 1990s with some support from an NSF-ATE grant

• Focus on activities and real data. • Fifteen modules covering all levels of developmental

math.• Not sequential.• Generally, three modules is about right for one

course.• http://www.maricopa.edu/m2c/

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Our materials

• Problem Solving and Logic (from Pirnot book)• Exponential Growth and Decay (Maricopa

module)• Systems (Maricopa module)• Data and Graphs (Maricopa module)

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How we use the materials

• We do not follow the materials as closely as we would in a typical algebra class.

• We feel free to skip things that did not seem to be increasing understanding to spend more time on others.

• We do not complete the published material for any of the modules. We found that finishing about half or two-thirds of the lessons in a Maricopa module was adequate to give the students the experiences we wanted.

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Exponential Growth and Decay

• Distinguish between linear growth and geometric/exponential growth. – From tables, finding differences and ratios of

successive y-values.– Graph and see pattern.

• Write a formula to describe the growth using the common ratio.

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Exp. Growth (and Decay) continued 1

• Discuss doubling and halving. Write formulas.

• Go from doubling each year to doubling every fifteen years.

• Go from this to the common ratio. 115( ) 300 2 300 1.06667

nn

A n

15( ) 300 2n

A n

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Exp. Growth (and decay) continued 2

• Go from – table to formula– description in words to formula– formula to description in words– formula to graph– graph to formula– graph to description in words, using doubling

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Exp. Growth (and decay) continued 3

• Includes other bases, such as 3, 10, 1/3, etc.• Includes doubling over fractional lengths of

time.

• Compare exponential formulas with polynomial formulas, such as and

• Understand the need for and use of the geometric mean, .

3y x3xy

0.3( ) 300(2) nA n

pq18

Exp. Growth (and decay) continued 4

• Work with the world population dataset from year 0 to 1999, which is best modeled by several different exponential functions for different time periods.

• Here the x-values are not evenly spaced, thus we extend the geometric mean idea from the previous lesson to find the growth factor per unit time period.

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Finding the growth factor

Example: Suppose the bacteria population grew by 35% in two hours, growing from 100 to 135. Let f be the growth factor from one hour to the next.

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(0) 100 the initial population

(1) 100 the population after one hour

(2) 100 the population after two hours

p

p f

p f

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Finding the growth factor continued 2

Solve the equation for f

.

The growth factor is 1.162 so our formula for the bacteria population, p(n), after n hours is

2

2

100 135

1.35

1.162

f

f

f

( ) 100(1.162)np n 21

Why this topic (and why early in the course?)

• Important topic in understanding math in real-world applications

• New topic to the students• Good for multiple methods of understanding• Algebraic, but relatively easy• Starts with linear relationships (constant difference)

and shows the change to get exponential relationships (constant ratio.)

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Why this topic before Systems?

• Easy graphs. (Graphing in Systems module was much harder for the students because of looking at two formulas at once.)

• Multiple types of examples, such as population growth, radioactive decay, earning interest. (Staying on the one complex problem in the Systems module felt very restrictive and frustrating to the students.)

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Later in the Exponential Module

We did not have time to do these:• Logarithms- definition and developing a sense

for logarithms. • Semi-log graphs• Overpopulation• “The Land and Its Limits,” “Resources and

Limits”

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Fall 2009 Grades for my class

A 2

B 4

C 4

D 3

F 1

W 4

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Conclusions

Most of the students and I thought the class was a success.

(It remains to be seen how they will do in the later classes.)

They have met their requirement for developmental math and will not be “churning around” in that system any longer. That is a success in several ways.

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Some people ask

“Are students better-prepared for the college-level course by succeeding in Intermediate Algebra than by this course? “

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What should we ask?

That’s not the right question!

Here’s what we think:Is there a population of students whose success

will be improved by attempting this course more than by attempting Intermediate Algebra?

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Success in what?

All of these are interesting.• Success in persisting in college long enough to

learn something useful in their chosen area.• Success in finishing their required college-level

math course• Success in completing their intended degree• Success in feeling good enough about math to

support/encourage their children to learn math.

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Should my school offer such a course?

• Think about math skills needed in those college-level courses.

• Think about “habits of mind” needed in those.• Is using Intermediate Algebra as a

“gatekeeper” a good strategy to promote mathematical understanding in the US population?

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I didn’t talk about: (Ask me)

1. Why four modules, not three, or six?2. Why didn’t we write our own modules?3. Why didn’t we use all modules from same

publisher?4. What did we do about technology? (Only

scientific calculator) Why?5. What are some details about what are in our

other modules?31

I didn’t talk about: (Ask me) continued 1

5. What are some details about what are in our other modules?

6. How similar is the preparation needed for our three college-level courses – in topics and in level?

7. What percentage of our students do we expect to take this kind of course?

8. How do we prepare teachers to teach it?

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End of the talk

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