Quantitative Literacy Through Community-Based Group Projects
An Emerging Model
Thomas M. Zachariah, Department of Mathematics, Suzanne Larson, Department of Mathematics, Jacqueline M. Dewar, Professor of Mathematics, Loyola Marymount University, Los Angeles, California
This "emerging" model consists of the incorporation of SENCER strategies into an existing general education course (Math 102-Quantitative Skills for the Modern World) that is required for all students at Loyola Marymount University except those majoring in STEM disciplines, or other fields requiring further study in math and statistics (such as education, psychology, or business).
The new element in the course was the incorporation of a semester-long group project that explores the power of mathematics to address a specific problem or question that directly impacted members of the campus or local community. The ten topics developed in the 2005-6 academic year ranged widely, from an analysis of Social Security receipts and payments under the current system and under President Bush's proposed system, to a comparison of on-campus and off-campus living expenses. Three of the projects were student designed, including an analyses of the sufficiency and safety of campus parking, an investigation of college students work hours and their effect on academic work, and calculations of student loan debt and its effect on the financial future of graduates.
The specific mathematical operations used in the course vary depending on the nature of the project, but a checklist of mathematical content areas has been developed to guide the project design process and to help students track what mathematical knowledge will be needed to complete their investigation. Unit conversions, the use and abuse of percentages, absolute and relative error, statistical inference, margin of error, and the creation of computerized spreadsheets and graphs are some of the mathematical concepts presented in the course that were used by students in their independent projects.
What mathematics is covered and how is it linked to the civic content?
The following list, entitled "What Mathematics Should We Do For Our Project?" includes all the mathematical topics discussed in the course and describes the various ways in which these topics might be used in a typical investigation of a local civic issue. Of course, the exact topics any particular group uses will vary depending on the nature of their project. Students can use the list to track their progress through the course, and it should be especially useful to students who are designing their own project. When instructors introduce each stage of the model project in class, they can refer to the list to indicate which of the topics might be especially pertinent for that stage of any of the student projects. It could be turned into a checklist or form the basis for a final reflection by the students on what mathematical topics have underpinned their investigations.
What mathematics should we do for our project?
- Working with Units and Converting Units: Would the results found in sources or calculated by your group be put in better perspective in different units? If so, identify more appropriate units, and calculate the unit conversions. (For example, it would be better to report the average speed of a Greenland's Jakobshavn Isbrae glacier in miles per year than in miles per hour.)
Uses and Abuses of Percentages
- Counts vs. percentages: Will reporting percents, rather than just a count (number of things) be more enlightening? (For example, when comparing auto thefts that occurred on various sized college campuses, it would be more informative to report percentages of cars that were stolen from each of the campuses in a certain year than just the number of car thefts that occurred on each campus.)
- Percentages to describe change: Is there a quantity that has changed over time? Calculate the percent change of this quantity.
- Percentages for comparisons: Are there two quantities that you would like to compare? Calculate the percent that one quantity is more than the other quantity, or calculate the percent that one quantity is of the other.
- Misuses of percentages: Have you found any misuses of percentages in any of your sources? If so, show where/how the misuse occurs. Calculate the correct percentage.
Putting Numbers in Perspective, Significant Digits
- Perspective through comparisons: Are there any very large or very small numbers involved that could be better "made sense of" through comparisons? If so, calculate at least one meaningful comparison. (For example, if LMU's bookstore found next fall, there would be a $125,000 increase in the cost of ordering textbooks, it would help us to understand the impact of this increase by calculating that this represents an average increase of $235,000/8300 = about 28 dollars per LMU student.)
- Absolute, relative error: Can you calculate the absolute or relative error to any measured quantities involved?
- Combining and reporting measured numbers: Are you reporting results of measurements with an appropriate (and not misleading) number of significant digits? When adding, subtracting, multiplying or dividing measured numbers have you reported the results with the appropriate number of significant digits?
Power of Compounding
- Simple interest: Are there financial accounts involved that earn simple interest? Are there quantities (of any kind) involved that will grow over time at a rate proportional to the original quantity? If so, use the simple interest formula to predict account balances or quantities at various (later) times.
- Compound growth: Are there financial accounts involved that earn compound interest? Are there quantities (of any kind) involved that will grow over time at a rate proportional to the quantity in the previous time period? If so, use the compound interest formula to predict account balances or quantities at various (later) times.
- Annual percentage yield: Can you calculate the actual percentage by which a balance (quantity) will increase in one year?
Savings Plans and Investments
- Savings Plan Formula: Are there any savings plans (annuities, retirement savings plans) involved for which it would be helpful to predict account balances at (various) future times? Or do you want to know what regular savings will yield a desired amount at a future date? If so, use the savings plan formula. You may want to use the formula to predict account balances under various assumptions about interest rates, quantities deposited etc.
- Total and Annual Return: Is there an investment or quantity that has grown from some original quantity to a later quantity? It may be of interest to calculate the relative change in the investment and the annual percentage yield that would give the same overall change in the investment (quantity). That is, calculate the total return and annual return for the investment.
- Liquidity, Risk, and Return: Are there investments involved? If so, evaluate the investments for their liquidity, risk, and return.
Loans and Credit Cards
- Loan Payment Formula: Are there (installment) loans involved? If so, find the required monthly payment using the terms of the loan. Could the terms of the loan vary - such as the interest rate or the life (length) of the loan? If so, recalculate the required monthly payment using the various terms.
- Amortization Tables: Would it be helpful to be able to track the loan balance and interest paid over the life of a loan? If so, create an amortization table showing the payment date, payment amount, payment amount of interest, payment amount of principal, and account balance for each month of the loan.
- Predicting Credit Card Bills: Do credit cards play a role in your project? Can you calculate the amount due at various times for a certain credit card (assuming a certain previous balance, interest rate, and method of charging interest)? Given a certain credit card balance and desired timeframe to pay off the entire balance, use the loan payment formula to calculate the amount that must be paid each month in order to pay-off the credit card balance.
- Loan Payment, Prepayment Strategies: If there are (installment) loans involved, calculate the effect on the loan payment, total interest paid over the life of the loan and the length of the loan by (voluntarily) paying an extra amount each month.
- Fundamentals of Statistical Studies: Will you be carrying out a statistical study? If so, develop the basic steps for your statistical study: (i) develop the precise goal and population for your study; (ii) determine how you will choose a representative sample for your study; (iii) determine what raw data will you collect and what sample statistics you will calculate; (iv) determine what inferences you can make about the population parameters from the sample statistics; and (v) look back and evaluate your study and its results.
- Sampling Methods: Will you be choosing a sample as part of your statistical study? If so, carefully consider possible methods of choosing a (representative) sample and the feasibility of each method. Will you use simple random sampling, systematic sampling, convenience sampling, quota sampling, stratified random sampling etc?
- Should You Believe a Statistical Study? Have you evaluated the results of statistical studies you reference by considering the eight guidelines in our text for evaluating a statistical study? Have you evaluated the methods you use in your own survey project against the eight guidelines in our text for evaluating a statistical study?
Statistical Tables and Graphs: Which of the following
types of statistical tables and graphs will best
summarize the data you have collected - frequency tables, bar graphs, pie charts, histograms, line charts? After deciding on appropriate data categories (or bins), use a spreadsheet program to construct statistical tables and graphs from your results.
Characterizing a Data Distribution
- Measures of center: Can you summarize your data by calculating a mean, median, and/or mode value? Are there outliers in your data? If so, what effect do the outliers have on the mean, median, mode values?
- Shapes of distributions: What features does the distribution of your data have? Is there more than 1 peak to the data distribution? Is the distribution symmetric or skewed? Does it show large or small variation?
- Measures of Variation: Did you use a median value to describe the center of your data? If so, compute the quartiles and the five number summary as a measure of the variation in your data. Did you use a mean value to describe the center of your data? If so, compute the standard deviation as a measure of the variation in your data. Give interpretations of what your measures of variation tell about the data.
Normal Distribution: Are there any normal distributions
of data involved? If so, is it appropriate to
calculate standard scores and percentiles corresponding to certain data values?
Central Limit Theorem: What does the Central Limit
Theorem say about the distribution of sample
proportions or sample means taken from many different samples? What does this say about sample proportions or sample means you have calculated?
- Statistical significance: Does any of your background research indicate that certain measurements are "statistically significant?" If so, give some interpretation of what it means to say they are statistically significant.
- Margin of error, confidence intervals: Have you calculated a mean or percentage (proportion) using data values of a sample collected from a large population? If so, determine the appropriate confidence level(s), then calculate the associated margin of error and confidence interval, and finally give an interpretation of the meaning of each confidence interval.
- Income Taxes: Are there tax implications involved with the financial aspects of your project? If so, calculate the taxes an individual owes under the (various) financial actions projected.
Computer Spreadsheets: Are there forms, charts, graphs
or financial calculations you can create or
perform using a spreadsheet?