TEACHING THIS OBJECTIVE
As you are teaching this objective, it is important to appreciate the students’ need for understanding why this is valuable. The experience that many students have in mathematics is one in which they memorized what to them were meaningless, computational algorithms and tried to match the memorized algorithm with each problem they confronted. Don’t treat quantitative literacy as something to be addressed at the end of a unit “if time permits.” Instead, start with the numerical questions inherent in the content and use them as motivation to use quantitative skills to find answers.
In addition, it is helpful for students if you are sensitive to the prevalence of math anxiety. There are exercises that can be used for students overwhelmed by a phobia of mathematics. See (17) and (18) for these resources.
To begin preparing a lesson addressing quantitative literacy, identify realistic scenarios requiring numerical information. Look at your own life and your profession. Many of us pay taxes – how do we fill out a tax form? Once it is completed, how can we analyze the data on the form to make decisions for next year? Professionally, when do you use data? Here are some examples in different disciplines:
Political Science/Law: What does it mean for a law to have a “disparate impact” on different populations (such as African Americans) and what does evidence of a disparate impact look like?
Education: How does one analyze standardized test results? What is the difference between normative and positive data? How do you make decisions based on this information?
History: What does the Russian census of 1897 tell you about Russian society on the eve of the revolution? How does that play into the way the Bolshevik regime evolved according to different scholars?
Music: What are the ratios among the frequencies of various pitches in the 12-note scale?
Mathematics: Why does our progressive tax code set marginal rates instead of average rates? What does this have to do with continuity?
Health Professions: What are the advantages and disadvantages of asking a patient to rate their pain on a scale of 1 to 10?
Business: How did Enron misuse its data and why did that lead to its collapse? What role did Arthur Andersen play in the misuse of data?
Consider addressing the following:
Quantifying situations often requires making assumptions. For example, maybe we assume that a particular type of model (exponential or linear) fits a situation. Or maybe we have to assume that the sampling distribution for our methodology is normal in order to draw conclusions from our data.
To illustrate, suppose we would like to estimate the world population in 100 years as part of an environmental study. If the world population grew by 1.2 billion people in the last hundred years, we might assume that it will grow by 1.2 billion in the next hundred. This would be an incorrect conclusion, since it assumes population growth is linear, when it is in fact exponential. However, even using an exponential growth assumption, we should be careful as the parameters that we use (such as the growth rate) are only useful approximations for short-term analysis.
Make these assumptions explicit, and debate their merits. The strongest quantitative conclusions will hold if the assumptions are mildly not true, but assumptions should also be used to limit our conclusions.
How we share our conclusions is as important as how we draw them. Students should be encouraged to think about their audience when sharing their quantitative work and their conclusions, and consider how to ethically make their point without overstating it. Provide students with examples from your profession where a case was made too strongly for a conclusion and discuss how it could have been better presented. There are lots of examples in the media where conclusions are incorrectly stated or exaggerated (see this amusing list of spurious correlations).
There are many interesting ethical issues that arise with data collection, analysis, and reporting. Many of them concern how to report your data to support your conclusions. Here are some questions you can ask:
- Is it appropriate to cherry-pick evidence to support a predetermined conclusion
- What kinds of inappropriate ways can charts be used?
Care needs to be taken when collecting and analyzing data. Everything from sample sizes to sampling methods should be considered and scrutinized. Case studies of poor methodology and overstated conclusions could be addressed and critiqued.
Data vs Modeling: Not a Zero-Sum Game
Mathematical modeling involves using a mathematical representation (typically an algebraic equation together with a graph) to examine quantitative relationships.
For example, we can use the equation:
FV = PV (1 + r)t
to model how the value of a fixed income investment (like a savings account) grows over time. In this example, FV is the future value of the account at time t while PV(present value) is the initial deposit and r is the interest rate. We could also represent this model with a graph where FV is on the vertical axis and t is on the horizontal axis.
Data and modeling go hand-in-hand. Traditionally, mathematical modeling is part of a course such as College Algebra while data analysis is part of Statistics. For quantitatively literate citizens, however, data is used to estimate a model. Students should learn how to start with data, find a model that fits well, and then use that model for prediction while avoiding inappropriate extrapolation.
If it fits within your instructional context, include numerical and quantitative matters in active learning activities. We want our students to appreciate and develop a disposition for using quantitative information, so embedding numerical work in our active learning activities and case studies has the potential make progress toward this goal. Within the mathematical community, Yoshinobu and Jones (19) argue that the benefits of active learning outweigh the costs of content coverage, while Kohen and Laursen (12) document the benefits of active learning in a systematic study.
Finally, consider the appropriate type of technology for your learning goals and your students’ abilities. Graphing calculators are powerful but may not be used by those in a particular profession. Alternatively, spreadsheets are commonplace and useful both professionally and personally. See instructional resource 6 below for best practices in using spreadsheets.