Vision - Potential
Vision Within Every Instructor - Potential Within Every Student
Newsletter of the HBCU College Algebra Reform Consortium*
Number 63, September 2005
www.ContemporaryCollegeAlgebra.org

Contents:
[1] School Bells are A-Ringing [2] Mathematics - Biology [3] "Renewing College Algebra," an MAA Program [4] CSM's Prediction for China's Automotive Production [5] Queries [6] Small Group Activity [7] Notices

[1] School Bells are A-Ringing

(Reprinted from September 2002 issue of the Vision - Potential Newsletter.)

Something is in the air, much more than a passage from one season to another - blood thickens (or thins), anticipation grows as does apprehension, confidence waxes and wanes like ocean waves, lists are made and then torn up, etc. It is the beginning of another school year and whether we are in our first, second, third, or fourth decade of teaching, it is an exciting time with rolling emotions. Our challenge is awesome! We are developing students to think quantitatively in all fields, rather than leading a narrow, skill oriented training program. College algebra provides the springboard to quantitative work in all fields as well as in the workplace. Our challenge, our mission, our goal is to arrange meaningful experiences resulting in personal growth for all students in terms of helping them become confident and competent problem solvers. Our objective must be to help prepare students to think clearly and deeply about quantitative issues in a world that is becoming more complex and more uncertain. Although developing skills in "problem solving in the modeling sense" is difficult to teach and even more difficult to assess, these are the essential life skills our students need.

Helping students learn how to learn and helping them develop habits of mind for learning, need to be primary concerns in the preparation of every lesson. Some examples of skills that address these concerns include:

a. Sketch a picture to illustrate the situation.

b. Iterate the paradigm "TRY something (e.g., guess), note the errors, modify the approach to reduce the errors, and try again" until an acceptable conclusion is obtained.

c. Ask: Does my answer make sense in terms of the original setting?

d. "What-if" an exercise to obtain a conceptual understanding of the problem underlying the exercise.

e. Look for examples of similar situations in other settings.

The Contemporary College Algebra program is very much in sync with the recommendations of The Mathematical Association of America (MAA). A recent Report on Introductory Courses to the MAA's Committee on the Undergraduate program in Mathematics states "A clear and dominant focus of all of these courses should include the following goals:"

a. Create confident and competent problem

solvers

b. Emphasize learning with understanding

c. Support the actual mathematical needs of

other disciplines

d. Provide life-time relevance to students

e. Employ student-centered pedagogy

[2] Mathematics - Biology

Technology has opened the door to a "New Biology," one that is highly integrative with mathematics, the physical sciences, information technologies. Several people have said that we have moved from the "Information Age" into the "Age of Biology." Biologist Eric Lander, a leader in the Human Genome Project, speaks of biology as information written in a code that will require sophisticated mathematical algorithms for deciphering. Leah Edelstein-Keshet in her article "Adapting Mathematics to the New Biology" in Math & Bio 2010, published by the MAA, wrote

The "new biology" is largely a biology based on data, models, and mathematics. Thus to understand modern biology, students need increasing breadth of knowledge in mathematics, and deepening sophistication in its use. All first-year biology students need to master certain essential concepts from mathematics, including how to model simple biological systems; how to read and understand graphical information (e.g., scatter plots, histograms, pie charts); how to use units, dimensions, and scaling; and how to sketch simple curves. In addition, familiarity with several more advanced topics is now also essential: unlimited growth ( exponential, log plots); periodic behavior; linear versus nonlinear behavior; dynamic behavior, steady states, and stability; discrete and continuous systems; rates of change.

Professor Keshet's remarks reinforce our belief that the Contemporary College Algebra program is on the right track by emphasizing graphical, numerical, and symbolic representations in data analysis, functions, and modeling. Her lists of topics is very similar to those we have heard from business schools and schools of nursing as well as from several departments such as economics, history, sociology, in addition to mathematics and the sciences.

[3] "Renewing College Algebra," an MAA Program

The National Science Foundation has funded the Mathematical Association of America's (MAA) program to "renew" college algebra. The two-year program under the leadership of Bill Haver (author of the proposal), Norma Agras, and Nancy Baxter Hastings will support eleven colleges and universities to pilot refocusing their college algebra programs. In addition, a cross-institutional research study will be conducted to gauge the success of the pilot programs. A three-day workshop was held last August prior to MathFest for teams from each of the eleven colleges. Four authors were invited to discuss their programs and texts: Bruce Crauder (Functions and Change: A Modeling Approach to College Algebra), Sol Garfunkel (College Algebra), Beverly Michael (Explorations in College Algebra), Don Small (Contemporary College Algebra: Data, Functions, Modeling) . Each institution will offer pilot and control (traditional) sections of college algebra second semester of this year and first semester of next year.

The institutions involved are:

University of Arizona

Essex Community College

Florida Southern College

Harrisburg Area Community College

Mesa State University

Southwest Missouri State University

North Carolina A&T State University

University of North Dakota

University of South Carolina

South Dakota State University

Southeastern Louisiana University

The MAA's endorsement and support for this program is a strong endorsement for the need to refocus college algebra courses. Bill Haver is Chair of the MAA's Committee on Curriculum Renewal Across the First Two Years (CRAFTY), a subcommittee of the MAA's Committee on the Undergraduate program in Mathematics. CRAFTY has established refocusing college algebra as its primary initiative.

[4] CSM's Prediction for China's Automotive Production

CSM, a worldwide automotive market forecaster, predicts that China's automobile and truck production will continue to grow linearly and that 2011 production will be roughly double its 2005 production of 4.2 million vehicles. In comparison, the prediction for Japan is that its production will remain approximately constant at 9.8 million vehicles per year and the U.S. production will grow from approximately 11.6 million to near 12.3 million vehicles over this time span. Assuming that the forecast for China's production is correct, predict how many vehicles China will produce in 2010. As a follow-on question, approximate the ratio of China's production of vehicles to the U.S. production of vehicles in 2010. (Assume a linear growth in the U.S. production.) For a follow-on, follow-on question, develop a linear model for the yearly ratio of China's production of vehicles to the U.S. production for the period 2005 - 2011.

[5] Queries

1. How Large a Hole Should I Dig?

   Recently I bought a pear tree from Hilt's Nursery. The container holding the ball (roots, peat moss, and dirt) of the tree was in the shape of a bucket. The diameter of the circular top of the bucket was 12 inches. The lady at the nursery told me to dig a hole that was twice the volume of the bucket and the same depth as the bucket. How large a hole should I dig? (What should be the diameter of the top of the hole?)
   
   

2. You are interested in designing the largest area that can be enclosed with 100 feet of fencing. For each of the following, be prepared to explain your reasoning and to justify that your answer is in fact a maximum.

   1. How large a rectangular area can be enclosed with a 100 foot fence?

   2. How large a circular area can be enclosed with a 100 foot fence?

   3. How large a triangular area can be enclosed with a 100 foot fence?
      

[6] Small Group Activity

For this activity, divide the class into pairs of students, have them work the following problem, then call on different pairs to describe what they did and to explain their reasoning. Ask how many pairs used a graphical approach? Call on different pairs to present their follow-on problem. This activity could be used to lead into a discussion of the reasonableness of a demand curve being decreasing and a supply curve being increasing (both curves representing functions of price). Another discussion topic emanating from this activity is the interpretation of the intersection point of the supply and demand curves as an equilibrium point. What does equilibrium mean? What happens when the price is changed from the equilibrium price? For example, is the demand higher or lower than the supply when the price is less than the equilibrium price?

Problem. One summer day, Brenda decides to sell apples from her Dad's apple tree to the people walking along her street. She decides that the price of an apple should depend on how many apples she plans to sell. (The more apples she sells, the higher she has to climb in the tree to pick them and thus the amount of her work per apple increases.) She also realizes that as the price per apple increases, fewer people will buy apples. Therefore her supply function is increasing and her demand function is decreasing. Suppose her supply function is and her demand function is where, in both functions, denotes the price of a apple. Thus if Brenda thought she could only charge 5 cents per apple, she would just pick 5 apples (the low hanging fruit); while at 15 cents per apple, she would be willing to pick 45 apples. On the demand side Brenda expects that at 5 cents apiece, there is a demand for 80 apples, but at 10 cents per apple the demand is only for 60 apples.

Your tasks are to:

a. Determine the price at which Brenda

would "sell out" while satisfying the

demand.

b. Determine how much Brenda would

make at the price in part a.

c. Develop a follow-on problem.

[7] Notice

1. Laurette Foster (Prairie View A&M), Dorothy Hunter (retired, Huston-Tillotson College), and Don Small (U.S. Military Academy) conducted a minicourse, Refocusing College Algebra, during 2005 MathFest in Albuquerque, NM. The participants (15) engaged in small group activities, Fun Projects, lots of discussion, and several Q and A sessions.
   

2. The next issue of the Vision-Potential Newsletter will appear in October 2005. Deadline for contributions to the October Newsletter is Monday, October 4, 2005. Opinion articles, suggestions for writing assignments, small group in-class activities, small group out-of-class projects, Queries, announcements, etc. are welcomed. Please send material to Don Small, don-small@usma.edu.
   

3. Subscribe to this Newsletter
   
   

* Supported by the National Science Foundation and the U.S. Military Academy.