# Principles for Teaching QR

*"Quantitative literacy involves sophisticated reasoning with elementary mathematics more than elementary reasoning with sophisticated mathematics*." Lynn Arthur Steen in

*Achieving Quantitative Literacy*

**Jump down to**: Consider the social construction of numbers | Ask students to communicate | Use active learning | Don't forget that small can be beautiful

There are several principles for teaching QR that will help you and students remain on track to learn the kinds of skills you are interested in teaching them without getting buried in complicated calculations or endlessly long tasks.

- Emphasize accuracy relative to precision
- Consider the social construction of numbers--and then help students get past it
- Ask students to communicate about their quantitative work
- Use active learning to make students "get their hands dirty"
- Don't forget that small changes can have large impacts

## 1. Emphasize accuracy relative to precision

The real-world data which quantitative reasoning (QR) routinely takes as a starting point for analysis often contain significant measurement error either because what is measured isn't quite "right" or because the methodology inherently includes measurement error. And sometimes QR asks us to form back-of-the-envelope estimates without access to any specific data whatsoever! One important aspect of QR is working through this lack of precision to arrive at accurate solutions nonetheless. For example, in a discussion of all-day kindergarten initiatives you may ask students to estimate the cost of implementation in your own state. Students would need to combine a guesstimate of the number of 5-year-olds with a guesstimate of the cost of the additional half day of schooling to arrive at a reasonable figure without any precise inputs.

Encourage your students to embrace this reality and be prepared for some resistance. Many students immediately associate quantitative work with their experience in traditional math courses in which problems often have only two types of answers: exactly right and wrong. (Of course, in advanced mathematics the reality is far more complex, but that is not the typical student's experience.) As a result, they may initially stumble when asked to make rough estimates or to analyze the quality of someone else's back-of-the-envelope calculation.

a) there was no way their estimate was that accurate

b) determining the number of balls (to whatever decimal point) did not address the central question of the cost of the promotion.

Of course, data can be so imprecise that it effectively fails to answer our questions. For example, if the data only shows if a family's income is above or below $75,000 then a student will not be able to answer many important questions about class. Recognizing these limitations is certainly part of the QR discipline. But the point here is that there are many times when being "approximately accurate" is more valuable than being precisely wrong.

Require students to make rough estimates. Give them problems for which you intentionally withhold some information which they have to "guesstimate." Give assignments in which students collect samples of data, and then show them how to present their findings with appropriate recognition of the limitations implied by their data gathering.

## 2. Consider the social construction of numbers--and then help students get past it

In his book *More Damned Lies and Statistics* Joel Best explores the social process that determines what we count (or don't) and how. While Best does acknowledge that some numbers are intentionally slanted or cherry-picked, his focus is on the more mundane realities--that all variables have to be defined by someone, that our minds are pre-disposed to look for causal connections even though often a correlation is not causal, that numbers carry with them an authority that may be greater than the strength of the data.

When first encountering these facts students can easily slip into Twain's unconstructive cynicism referred to in Best's book title, concluding that numbers are always the worst of lies. If we leave them in this half-baked QR state of mind we have done them a great disservice. By engaging them in active learning, we can move through this early stage of learning to a better and more balanced perspective in which they take a critical perspective that questions numerical evidence while remaining open to its power. Force students to make data-based arguments of their own (whether in essays, spreadsheets or charts & graphs). In the process they will come to learn that social construction is inevitable; it is literally impossible to say anything quantitative without making many important choices. But they will also learn that there are better and less sound ways of making those choices.

## 3. Ask students to communicate about their quantitative work

To become fully quantitatively literate, students must be able to communicate their quantitative arguments. A first step in this process is being able to analyze the quantitative analysis presented by others...and newspapers provide great opportunities for that every day. But even after learning to see shortcomings in others' work, students can struggle to make good communication choices themselves. Because quantitative communication takes many forms, you can work on this in multiple contexts including:

- Quantitative writing assignments
- Activities based in spreadsheets (And here is a second collection of spreadsheet activities.)
- Asking students to create visual representations of data
- Oral presentations or posters

## 4. Use active learning to make students "get their hands dirty"

Studies show that most students learn more when engaged in active learning. Don't simply show them how to work with numbers; let them get involved. In disciplines in which it is appropriate, consider involving students in the data collection process. Here are a few ideas on how you might do that (with links to more detailed information on each pedagogical method):

- Teach with data or data simulations
- Role play in a case study
- Give students a data "jigsaw"
- Have students work through a context rich problem
- Work with data visualizations
- Give quantitative writing assignments

## 5. Don't forget that small changes can have large impacts!

Quantitative reasoning is a habit of mind--an instinct to consider how quantitative analysis might shed light on a problem. That habit can be exemplified and nurtured in small "moves" in the classroom.

QR does not have to be the focus of your entire course to contribute to the cause. One assignment, one course module, or a steady trickle of QR-exemplified work will help move us toward the goal. "Becoming a writer" is not a single event; it is a life-long process that will be supported across many courses. The same is true of the QR habit of mind. No one of us needs to try to accomplish everything--we need only do our part, the part that fits naturally within the contexts of our discipline and our course.

**An example by economics professor at Carleton College, Nathan Grawe**

"When I was teaching the classic

**"diamond-water paradox" in economics,**The diamond-water paradox was explained by Adam Smith. He noted that diamonds had a very high price and water was free even though the former had very little practical use while the latter was essential for life. The explanation for this is that the abundant supply of water drives down its price even though the total value of consumption is very high. For those familiar with demand curves, Smith pointed out that the area beneath the curve for all consumed units could be very large even though the height of the curve at the last unit consumed was low. I noted there were "many" K-12 teachers in America. And then I caught myself. I should demonstrate the care for precise quantitative evidence that I advocate. So, I asked my students to estimate the number of K-12 teachers while I looked the number up in the

*US Statistical Abstract.*After a minute or two, we compared answers. I explained a little about

**how to search the**The

*Abstract,**Abstract*is a compendium of data sources for the United States. It contains a wealth of data drawn from other sources that are cited and often linked through notes at the bottom of each table. This makes it a great gateway into data sources on many US topics. The

*Abstract*website provides links to spreadsheets as well as pdfs of each table. Don't overlook the link at the top of the page to the "PDF Version" of the

*Abstract*. This version includes introductions to each section which share details on the data sources. The pdf version also includes an indispensable index to aid data searches. they practiced estimation, and the whole exercise took less than 5 minutes."