a u t u m n 2 0 0 8 • 33

M ATHEM ATIC A L TOOL S FOR THE R E A L WOR LD: a Fou ndat ions L e a r ning e x per ience in Qua nt itat i v e r e a soning

K. Shane Goodwin—Department of Mathematics

Midstream in the transition to the BYU–Idaho Foundations program, I find it helpful to reflect on our progress with FDMAT 108,

Mathematics for the Real World, and articulate some of the challenges and insights that have emerged so far in designing the curriculum. I will summarize the philosophy and curricular objectives of this course, which fulfills the quantitative reasoning requirement in Foundations. In the process, I hope to illuminate some of the work behind the scenes by many colleagues in the Mathematics Department. In addition, I hope my description of the options for different groups of students will help advisors, department chairs, and deans to understand the direction we are taking in FDMAT 108.

We hope that students will now have a richer Foundations experience even if they choose the test-out option, to be described below. The test-out option allows students who will take more advanced quantitative reasoning courses in Foundations (such as college algebra, calculus, statistics, etc.) to still have a significant learning experience with the course objectives covered in Math 108. We have a long way to go in the refinement process of both the course and its test-out option, so this article has a tentative quality not too different from the confidence and sense of progress we members of the FDMAT 108 team have felt.

Although I have served as the Math 108 Committee Chair for more than ten years, Kent Bessey was the original team leader for FDMAT 108 , and I am indebted to him for getting us onto solid footing. In addition to his personal leadership, he made significant contributions to our FDMAT 108 resource web page. (To see these pages, which remain under development, see http://www.byui.edu/math/Foundations/foundations.htm.) This site serves both students and faculty involved in FDMAT 108 classes, with links to electronic resources, study guides, and general information about the class. It also introduces the quantitative reasoning requirements of Foundations to students who will choose the test-out option. Our other committee members, Jennie Youngberg, Richard Pieper, Jackie Nygaard (Mathematics Department), and Chris Andrews (Business Department) have provided valuable assistance and suggestions along the way. Finally, Paul Cox, the Mathematics Department chair, continues to play a vital role as we review and refine this Foundations course.

We hope that

students will now

have a richer

Foundations

experience.

3 4 • p e r s p e c t i v e — a u t u m n 2 0 0 8

Histor ic a L O v erv iew oF M at hem at ics Gr a duat ion R eQu ir ements

In the Winter 2001 issue of Perspective, I wrote about our transition to the new mathematics graduation requirement that coincided with the conversion from Ricks College to BYU–Idaho.1 I briefly borrow key points of that historical overview to inform newer faculty of the evolution of our G.E. mathematics curriculum.

Not until the late 1970s did Ricks College propose a mathematics requirement for graduation. At the time, only a handful of two-year colleges around the nation had such a requirement for graduation. In the fall of 1980, Math 100A, Arithmetic, became the graduation minimum for Ricks students. By the end of the decade, however, an accrediting team of the Northwest Association of Schools and Colleges challenged Ricks College to upgrade to a course that would be considered collegiate level. In response to that challenge, Math 101, Intermediate Algebra, was introduced in the fall of 1994 as the new mathematics graduation requirement. An ACT math score of 22 or better alternately filled the requirement. Throughout the remainder of the decade, these two options were available to students.

Even as we tried to help students successfully make their way through a Math 101 experience, another kind of class emerged. Math 103, a course based on problem solving with only a little algebra emphasis, was modified in 1994 to meet the needs of students transferring to other Idaho institutions. It fulfilled the Idaho math core requirements and provided an alternative to Math 101. In the fall of 1998, however, our department voted to discontinue Math 103 and create a more rigorous quantitative reasoning course, to be numbered Math 108. We proposed that it be closer to a college algebra level of rigor but still approachable for general education students.

At the conclusion of their ten-year accreditation visit in 1999, representatives of the Northwest Association of Schools and Colleges recommended “a review of what constitutes General Education classes—especially in regards to Math and Human Relations.”2 In response, Academic Council approved Math 108 as the new minimum graduation requirement for specialized, associate, and baccalaureate degrees at Brigham Young University–Idaho. This change took effect in fall semester 2001 coinciding with our transition from a two-year junior college to a four-year university. No longer would a certain ACT math score serve as a waiver from the quantitative reasoning course.

Seven years later we have arrived at a new crossroads, where Math 108 needs to take a more robust role in the new Foundations program, not only by serving as the minimum requirement for graduation, but becoming the quantitative reasoning requirement for all students—whether they

Seven years later

we have arrived at

a new crossroads.

a u t u m n 2 0 0 8 • 35

choose the classroom experience or the test-out option. The mathematics department will offer more advanced mathematics courses (such as calculus and statistics) that will pre-empt FDMAT 108 for students in some majors. Nonetheless, in the course of conversations with President Clark and other academic administrators, we decided that every student would have some form of quantitative reasoning experience, either in the classroom or by the individual test-out requirement.

A Qua nt itat i v e R e a soning Test-Ou t Op t ion

The test-out option offers students who take a higher Foundations mathematics course an opportunity to prepare independently for two multiple-choice examinations centered on financial mathematics and statistical literacy. For students who pass a college statistics course or AP statistics, the statistical literacy requirement is waived. Two takes of the examinations are allowed with a minimum score of 60 percent for successful completion. If a student cannot meet these requirements, he or she is asked to take the Math 108 course instead of pursuing the test-out option.

Upon passing the financial mathematics test, students move on to complete a life-planning spreadsheet project oriented around provident-living skills and concepts such as personal budgeting, investments, home mortgages, and vehicle loans. There are podcasts, tutorials, templates, and student tutoring to help in this phase of the test-out option. Based on a pilot study of both the life-planning project and financial mathematics examination, we estimated that most students should be able to complete the test-out option in about 30-40 hours of independent work. Much of this work can be accomplished off-track. In the future, as we develop an online Math 108 course, perhaps this entire test-out option could be accomplished off campus.

As we implement the program over the next several semesters, we plan to interview students who pursue the test-out option and look for ways to improve the resources at their disposal. We may also streamline the requirements. To strike a balance that will provide a significant learning experience in quantitative reasoning while removing a bottleneck to graduation, is a major concern for students, faculty, and administration that merits ongoing attention.

Cour se O v erv iew

While we have not been obliged to create our course from scratch (as have others), we feel that the transition to Foundations has significantly influenced the direction we are taking in quantitative reasoning. In line with the idea that Foundations classes should develop skills for both

We decided that

every student

would have some

form of quantitative

reasoning

experience.

3 6 • p e r s p e c t i v e — a u t u m n 2 0 0 8

university and lifelong learning, Mathematical Tools for the Real World (FDMAT 108) provides students an opportunity to develop quantitative reasoning skills that benefit them in subsequent coursework, careers, and their daily lives.

As a committee we identified five course objectives, listed below. We then elaborated a series of observable outcomes related to each of the overall objectives. (These outcomes are listed in Appendix 1).

1. Develop the necessary arithmetic and basic algebraic skills to succeed in daily life.

2. Understand the dangers of debt, the power of compound interest, and the advantages and disadvantages of various financial choices.

3. Analyze and critique real-world issues and arguments involving probability and statistics.

4. Explore the use of mathematical models in describing and making predictions about real-world phenomena.

5. Learn to competently evaluate logical arguments and resolve real-life problems that require quantitative reasoning.

A Consistent A pproach

In keeping with the guidelines we received from the Foundations oversight committee, we considered the following issues as we examined and revised Math 108:

• A standardized approach to curricular objectives and topics core;

• Departmental examination(s) synchronized with the course objectives;

• Application of the BYU–Idaho learning model;

• Learning materials (textbook, web resources, PowerPoint slides, etc.);

• Course technology (see A Shift in Technology below).

FDMAT 108

provides students an opportunity to develop skills that

benefit them in subsequent

coursework, careers, and

their daily lives.

Career Daily Life

Quantitative Reasoning

Skills

Future Coursework

a u t u m n 2 0 0 8 • 37

In the course of our committee discussions, we soon determined that our current textbook, Using and Understanding Mathematics: A Quantitative Reasoning Approach (Bennett and Briggs, 4th Edition, Addison-Wesley, 2007), is well suited to the spirit and objectives of the course. While we feel that our course is not necessarily textbook dependent, the committee believes that of all textbooks in the quantitative reasoning market, this one allows ready integration of learning model principles and gets to the heart of what our BYU–Idaho students need to know and do to be successful in their collegiate coursework, career, and family lives. Springing from the work at the University of Colorado in the early 1990s, this text is rich in student activities that hold students accountable for their own reading and preparation, appropriate examples that can be discussed in small or large group settings, homework sets designed for skill practice, and enrichment activities that facilitate the ponder/prove phase of the learning model. In this case, we believe we have an excellent text upon which we can build as we work to improve the quality of the student experience.

In both committee and departmental settings, we tried to identify a curricular core of topics and measurable outcomes that we felt best suited the overall experience we wanted BYU–Idaho students to have with quantitative learning. This has not been easy. It is very hard to let

“pet” topics go when deciding what knowledge and skills are essential and which ones are elective. It was during these discussions that we came to agreement on retaining financial mathematics and statistical literacy as our curricular foci. Eventually we agreed that approximately 75-80 percent of the semester should cover the same core topics chosen by the department, with the remaining 20-25 percent of the class time left to individual faculty discretion to work through student-centered projects or optional sections from the text.

Perhaps the greatest challenge we faced in reaching a consensus was agreeing to implement two departmental exams tied to the financial mathematics and statistical literacy units of the course. Until now, we had not developed standardized departmental exams for any specific unit within Math 108. In multiple department meetings, our teachers debated the pros and the cons, the logistical, pedagogical, and philosophical issues pertaining to standardized objective tests. We wondered whether striving for uniformity is necessary or even realistic. We addressed the timing, the grading, the weighting of the test scores, the analysis and revision of test items, and finally the fine line between academic consistency and academic freedom. These have not been not easy issues to resolve. In the end, we decided we would allow teachers the option of developing their own forms of assessment of the financial mathematics and statistical literacy units, provided they demonstrate to the FDMAT

It is very hard to

let “pet” topics go

when deciding what

knowledge and

skills are essential.

3 8 • p e r s p e c t i v e — a u t u m n 2 0 0 8

108 committee that their own exams adequately assess mastery of the course objectives.

Although one of our goals is to assure that students taking Mathematical Tools for the Real World have similar learning experiences, we have come to the conclusion that this does not necessitate the need for identical, franchise-style classroom environments that supposedly guarantee educational uniformity. Each teacher and each class is unique, and we believe this diversity complements rather than detracts from the learning model and the Foundations experience.

A ShiF t in TechnoLogy

Over the last ten to fifteen years, our faculty have invested a lot of time designing Math 108 content and assessment around the graphing calculator technology. The relative affordability, portability, and ease of use of the calculator made it the tool of choice in the Math 108 course. These calculators are robust enough to contain a time-value-of-money package as well as descriptive statistical tools. For some time we had run a modest rental program in the department as a way of helping students offset the cost of a new graphing calculator, giving Math 108 students first priority in renting Texas Instruments calculators for only $5 per semester. Unfortunately, this program could not possibly expand to meet the needs of more than 20 to 25 sections of Math 108 each semester, and many students have had to purchase new or used graphing calculators as a result.

In the last year and a half, however, we have discussed the possibilities of of adopting a different technology: spreadsheet software. As we learned more about President Clark’s laptop initiative, it seemed to dovetail with our changing needs in the Mathematics Department. Dropping the requirement of expensive graphing calculators also responds to the president’s imperative of “reducing the relative cost of the educational experience.” Furthermore, we believe that spreadsheet skills will be much more transferable to future college courses, careers, and family settings than those based on graphing calculator technology. Spreadsheets lend themselves better to case studies, individual and group projects, simulations, web data mining, and so on. It could be argued that spreadsheets are to quantitative literacy as the word processor is to writing.

Adopting this spreadsheet approach posed some challenges of its own, however. Consider the following issues:

• training each other on spreadsheet features, roadblocks, potential, etc.;

• dealing with the difficult issues of assessment since spreadsheets most likely will not be a feasible tool of choice in traditional testing in our testing center;

• letting go of graphing calculator activities carefully crafted by faculty;

This does not

necessitate the

need for identical,

franchise-style

classroom

environments.

a u t u m n 2 0 0 8 • 39

• deciding how to design spreadsheet projects along with their grading rubrics;

• dealing with the disparity in student ability and mastery of spreadsheet skills;

• finding the time to provide adequate resources for students to acquire these spreadsheet skills through independent work with podcasts, tutorials, etc.;

• deciding what type of calculator technology to require in place of the graphing models; and

• managing the logistics of both student laptops and Ricks 245 computer lab for students.

We have encouraged our faculty to take a leap of faith this fall semester and start using spreadsheet projects and perhaps smaller-scale spreadsheet homework problems. Because Microsoft Excel is available across campus and OpenOffice is available in the public domain, we intend to use these software platforms. We decided to require only a scientific calculator (usually $15 or less), which can be used in both the classroom and the testing center. During a department workshop this summer, we trained each other on several financial spreadsheet projects (see examples in Appendix 2). We intend to share ideas, templates, handouts, successes and failures with one another throughout the semester. Traditionally, we try to do such things during our Wednesday “hot chocolates,” at lunch, and around the figurative water cooler. At the moment, many faculty feel both excitement and trepidation at the thought of implementing these ideas (curricular, pedagogical, and technological). We will doubtless learn more about what works well—or doesn’t—in coming semesters.

Su mm a ry

For those involved in FDMAT 108, like others involved in Foundations across campus, we sense there is much work still to come, and we will make mistakes along the way. Nonetheless we hope that whether our students enroll in the class or choose the test-out option, they will gain practical skills in finance and statistics. We invite questions, comments, and suggestions from faculty and students in helping us promote a higher degree of numeracy among our students. Our overarching goal is to provide them with a rich learning experience that blends nicely with the third part of the University’s mission statement, preparing for lifelong learning, employment, and roles as citizens and parents. While painful at times, our examination and refinement of Math 108 should strengthen the course and help provide students with quantitative learning experiences that will improve the quality of their lives. •

Many faculty feel

both excitement

and trepidation

at the thought of

implementing these

ideas.

4 0 • p e r s p e c t i v e — a u t u m n 2 0 0 8

Notes

1 Shane Goodwin, “Quantitative Reasoning: Our New Mathematics Requirement,”

Perspective,1.2 (Winter 2001), 55-64.

2 Ibid., 56.

3 Summary of objectives and outcomes for Math 108.

A ppendi x 1 : Object i v e s a nd Ou tcomes For FDMAT 108

1. Develop the necessary arithmetic and basic algebraic skills to succeed in daily life.

Students will be able to:• Properly apply unit conversions in their problem solving techniques.• Calculate appropriate absolute and relative changes for percentage work.• View percentages in terms of fractions, changes, and comparisons.• Apply the of versus more than rule with percentages.• Demonstrate the difference between percentage and percentage point.• Solve simple one and two-step equations.• Algebraically solve for the different variables structured within an applied formula.

2. Understand the dangers of debt, the power of compound interest, and the advantages and disadvantages of various financial choices.

Students will be able to:• Construct a reasonable budget based on income, savings, and expenses and demonstrate the importance of taking control of one’s personal finances.

• Demonstrate the astonishing power of compound interest and the role it plays in investments.

• Describe the difference between annual percentage rate and annual percentage yield.

• Calculate both the annual rate of return and total return percentage on an investment.

• Calculate the future value of both lump sum and annuity investments as well as for other variables such as present value and time.

• Demonstrate competence with the mathematics of installment loans by constructing amortization schedules using spreadsheet technology.

• Discuss the pros and cons of paying extra principal on a long- term loan.

• Gain facility with describing the relationship between principal

a u t u m n 2 0 0 8 • 41

and interest in the context of both debt and investment.• Discuss the key elements of liquidity, risk, and return and their relationship to investing.

• Carefully explain the mathematics and dangers of consumer cred it and how to plan properly for living within one’s means.

• Describe the marginal income tax brackets of our nation’s tax code and how calculations are made based upon deductions and exemptions.

• Calculate the FICA tax on wages, its history and impact on our retirement system.

• Describe the difference between a tax credit and a tax deduction using examples.

• Demonstrate an understanding of the difference between our national debt and deficit.

3. Analyze and critique real-world issues and arguments involving probability and statistics.

Students will be able to:• Discuss coherently the potential for abuse and misuse of statistics and be able to critique, at their level of ability, the validity of statistical studies found in professional journals.

• Describe the abuse of statistical graphs often found in the media.• Describe the role of randomness, placebos, single and double- blinded studies, control vs. experimental groups and other key concepts found in inferential statistics.

• Point out the difference between correlation and causation and how this impacts our statistical literacy.

• Calculate and interpret the mean, median, mode, range, and standard deviation of a data set.

• Calculate and interpret the five-number summary, z-scores, percentiles, and outliers of data.

• Construct histograms, box plots, pie charts, scatter plots and demonstrate pros and cons.

• Form a basic assessment of the role of hypotheses, margin of error, and statistical inference.

• Calculate the margin of error for proportional data and construct a 95% confidence interval.

• Show rudimentary understanding of statistical significance and the role it plays in a study.

• Identify normal, uniform, positively, and negatively-skewed distributions.

• Appropriately apply the 68-95-99.7 Rule to both data and population contexts.

4 2 • p e r s p e c t i v e — a u t u m n 2 0 0 8

• Gain facility with the Complement, And, Or, and At Least Once Rules from basic probability.

• Describe and interpret expected value, the Law of Large Numbers, and the gambler’s fallacy.

• Discuss, in a cogent manner, the mathematical, social, and moral issues that arise in lottery and other gambling activities.

• Discuss risk and the role it plays in our society and the decisions we make.

4. Explore the use of mathematical models in describing and making predictions about real-world phenomena.

Students will be able to:• Discuss the role of mathematical formulae, graphs, and numeric

tables in modeling real-world issues such as finance, populations, probabilities, etc.

• Describe the specific differences between linear and exponential growth.

• Construct exponential models to accommodate forecasting future values of a variety of phenomena.

• Apply the Rule of 70 to calculate doubling time and half-life periods for modeling.

5. Learn to competently evaluate logical arguments and resolve real-life problems that require quantitative reasoning.

Students will be able to:• Identify propositions, negations, conditionals, converses,

inverses, and contrapositives.• Distinguish between deductive and inductive arguments.• Assess the validity and soundness of arguments.• Apply George Polyá’s four-step problem-solving guidelines to quantitatively-based problems.

• Gain facility in the selection of appropriate tools such as the calculator and spreadsheet in the solving of quantitative real-

world problems.

a u t u m n 2 0 0 8 • 43

A p p e n d i x 2 : Two sa mpL es oF M at h 108 st udent spr e a dsheet projects (introductory a nd mor e a dva nced).

4 4 • p e r s p e c t i v e — a u t u m n 2 0 0 8