cherry

Approaches to Quantitative Reasoning

Robert Cherry

 

      I would like to raise a particular issue: How does the objective of teaching quantitative reasoning make a difference in the choice of presentations topics?  In particular, I am interested in the difference between quantitative literacy as reflected in the CUNY rising junior exam – being able to interpret and utilize statistical presentations – and the development of problem solving skills.

 

I.                    Quantitative Literacy – The Construction and Use of Tables

 

TABLE 1

Employment Status of the Civilian Noninstitutionalized Population, 16 to 64 Years Old,

by Sex and Race, November 1999

 

 

Population

(000)

Labor Force

Participation Rate (%)

Employ-ment

Rate (%)

Unemploy-ment

Rate (%)

Full-Time

Rate

(%)

Involuntary

Part-Time

Rate (%)

All

   176,106

77.1

74.2

4.0

74.4

1.7

White

 

 

 

 

 

 

Men

71,914

85.2

82.4

3.1

80.5

1.2

Women

72,891

71.3

68.8

3.5

64.8

1.9

Black

 

 

 

 

 

 

Men

10,099

75.2

69.1

8.2

79.7

3.0

Women

12,126

71.4

66.2

7.3

72.7

2.9

Labor Force Participation Rate = share of population that is working or actively seeking work

Employment Rate = share of population working

Unemployment Rate = share of labor force that is unemployed

Fulltime Rate = share of those employed working full-time year round

Involuntary Part-time Rate = share of those working who are employed part-time but

                                                desire full-time employment 

 

     The objective for students is two-fold:

 

1. Understand definitions of terms and how analyses are influenced by them.

    In particular, the way of measuring “actively seeking work” is critical to the definition of who is in the labor force; if you are not in the labor force, you won’t be counted as unemployed.  There will also be some issue as to how we define who is working, reflected in the last category: involuntary part-time employed.

      How might you define “actively seeking employment” and count in the unemployment rate those who are working part-time but desire full-time employment? Think about other (non-economic) variables that are used in public policy debates that are sensitive to how we define them. 

 

2. The table groups individuals by race and gender.  Identify where group differences appear and hypothesize the reasons for them.

     Identify data in the table that reflects gender and/or racial differences.  Try to hypothesize why these differences exist.  Since some measures are interrelated with other(s), differences in one measure might explain differences in another measure.

 

II.  Problem Solving Skills

A. Weighted Averages

     Often, we are confronted with a small set of subgroups that sum to the total.  In economics, this phenomenon occurs with wage data.  For example, we have wage data for various demographic subgroups of black and white workers.  If we know what is happening with each of the subgroups – how they are changing over time, what are the black/white differences – what can we say is happening with the aggregate?

 

1) Basic Calculation

     At the most basic level, many people instinctively take simple averages.  For example, I state that black college graduates in the South earn $30,000 annually while those outside the South earn $50,000 annually.  Then I ask students, “With this information can you tell me the national average annual earnings of black college graduates?" Many will immediately say $40,000.  Some realize that they need more information, but their first response is invariably that they need to know the numbers in each region.  While this is correct, of course, they need to know the numbers so that they can identify the SHARE of black college graduates in each region. 

     Once I give the students one of the shares - say 30 percent of black college graduates live in the South - most can then proceed to find the national average.  (For some, it is not immediately clear that if it is 30 percent for the South, it must be 70% for the non-South.)

0.3(30,000) + 0.7(50,000) = $44,000

 

 

2) Problem Solving

The most basic problem solving for students is to see that they are solving a single equation with one unknown.  In this way, they can begin to see how if I change which of the variables is the unknown they can still solve it.  That is, the weighted average links a set of four variables: (p)(Ws) +  (1-p)( Wns) = A

where p= share of black college graduates living in the South

Ws and Wns = annual earnings of black college graduates in the South and non-South

A = national average annual earnings of black college graduates.

 

a) Simple Static Problem.

If p=0.3, Ws  = $30,000, how much must Wns be for A to equal $47,500?

 

    If students conceptually grasp what they are doing, they can apply this knowledge to a host of other situations.  Of course, the simplest would be if instead of wages, we had grades on the midterm and final, where p equals the weights for the two exams (that make up the entire grade).  Then after the midterm, students can calculate how much they need on the final to reach a target course grade.

    A second example might be drawn from baseball.  Instead of wages, let us look at batting average, where p equals the current share of total seasonal at bats.  Once we know the current batting average, we can compute what the individual’s batting average must be over the remaining share of seasonal at bats for him to attain a target season batting average.

b) Further Conceptualization.

Beyond the details of the calculations, we can better understand a number of dynamics that can occur: How is it possible over time for every region’s wages to increase but the national average to decrease?  How is it possible for the wages in all occupations to have increased between 1975 and 1993 but the average wage across all occupations to have decreased?

Understanding these examples can enable students to apply these concepts in other economic contexts.

      

c) More Complex Dynamics.

Above we looked at changes in one population.  Often, however, we are interested in relative changes.  In particular, let us look at relative earnings either by race or gender.

How might it be possible for black women to earn more than white women in every occupational category while the average wage of black women is less than that of white women?  Why might we expect that the very success of the welfare to work strategy of the last decade would necessarily lower the female/male earnings ratio?

How is it possible that between 1950 and 1970, the national black/white ratio among men rose but there was little change in any regions in the black/white ratio?

B. Linear Extrapolations

    Many relationships that exist in life are either linear or can be linearly approximated.  

 

1)      Budget Calculations

 

Suppose we have a fixed amount of income to spend on a variety of goods, and we know the unit prices of each.  If there are two goods, we have a linear budget line:

     (Pa)A + (Pb)B = I

where  Pa and Pb = the unit price of goods A and B, respectively

            A and B = the quantity of goods A and B, respectively

and       I = total income available to expend on these two goods.

 

Setting up the “budget” equation allows students to more systematically determine the choices they have available and the trade-offs that must be made when shifting purchases from one good to another.

2) Projections 

Businesses expect there to be an inverse relationship between the price they set and the demand that they can expect, holding all other influences constant: D = 100 -  2(P)

where D= the quantity demanded and P=  unit price.

 

Using this relationship, students can project by how much the firm must lower its price if it desires to sell an additional 10 units.

3) Finding intercepts

 

     Students can understand the financial examples above.  They are less comfortable with relationships that are based on statistical models.  For example, we can hypothesize that there is an inverse relationship between the percentage change in GDP – the economic growth rate – and the change in the unemployment rate.  That is, when economic growth is low (or negative), we can expect unemployment to rise, but when economic growth is high, we can expect unemployment to decline.

 

      DUN = 1.08 – 0.4(%DGDP)

 

where  DUN = change in the unemployment rate

        %DGDP = economic growth rate

 

a)  Here, the Y-intercept has an economic meaning.  If %DGDP=0, then DUN=1.08; that is, if there was no growth, we would predict that the unemployment rate would increase by 1.08 percentage points.

 

b) If we have some forecast for next year’s growth rate, we can now predict the change in the unemployment rate as well.  For example, most economic forecasts predict that the growth rate over the next year will be around 1.5 percent.  Plugging that in:

 

DUN = 1.08 – 0.4(1.5) = 1.08 – 0.60 = 0.48

 

This indicates that we can anticipate the unemployment rate to grow by 0.48 percentage points.  Can you think of any reasons why the unemployment rate is increasing even though the amount of production is increasing?

 

 c) While the Y-intercept is part of the equation of a line, we can also find the X-intercept.  In the current problem, this means finding out how fast the economy must grow in order to keep the unemployment rate constant.  In particular, this would mean setting DUN=0 and solving for (%DGDP).  In economics, this has a particular meaning – it is called the sustainable growth rate and is used for important projections.

 

      Another place in which finding the X-intercept is important is in any discussion of “means tested” programs where benefits decline as income rises.

 

d) The food stamp program provides monthly income to needy families.  It provides a maximum of $300 monthly, but when income rises above $8,000 these benefits are reduced with a 2 percent phase-out rate: i.e. for each additional $100 monthly benefits are reduced by $2.  This then is reflected in the linear relationship:

        F   =  300 – 0.02(I)     where I>$8,000  and F= monthly food stamp benefits

 

At what income level do families become ineligible for food stamps?

 

e) A similar situation develops with the earned income credit program: a program that provides income to low-income families depending upon their wage income.  In particular, a single head of household with two children and income of $13,000 currently receives $4000 from the government under this program.  At that income, however, the phasing out of benefits begins.  In particular, for each additional $100 of (wage) income, the families' EITC benefits are reduced by $21.

 

At what income level do single heads with two children lose eligibility? 

C.   Using Probabilities to Construct Expected Values

 

    Students have been exposed many times in their educational experience to probabilities, including independent and conditional probabilities.  However, they are uncomfortable using them to develop expected values and use them in decision making.  For example, I offer my students a series of bets.  To try and construct expected values, I state: “ I will flip this coin for any of you willing to pay me to participate in the following bet: If it comes up heads you receive nothing; however, if it comes up tails, you will receive $100.  How much would you be willing to pay me to participate in this bet?”

The focus is on students seeing that the expected value is $50 by looking at the outcome if the coin flip is repeated many times.  (We explore the issue of “risk aversion” – individuals would not necessarily be willing to pay up to the expected value of $50 because of uncertainty when the bet is only done once or a few times, combined with the fact that losing $50 is more costly than the benefits of an additional $50, since money has declining value).  This then enables me to develop a decision rule: Invest in an activity as long as its cost is less than or equal to its expect value.  The next offer I make to students is the following: “Let us suppose that I have the ability to be a despot and can make you the following offer: You can pick one card from a full deck of cards.  If it comes up the Ace of Spades, I can kill you; any other card, and I give you a million dollars.  Would you accept that bet?

 

   Of course, this raises some emotional issues about society's role in allowing individual choice.  For example, we don’t allow people to sell their body parts (or their children).  More broadly, ethical values limit individual ability to engage in “rational choice."  However, with some constraints, we do allow individuals such as firefighters to engage in risk taking behavior.

 

     We can then translate this benefit-cost analysis into using real world observations concerning choices made.  Here I use the example of the Triangle shirtwaist factory fire – something very dear to the secular Jewish tradition.  In my recent book, Who Gets the Good Jobs?  I state:

Ric Burn’s 1999 documentary on the history of New York . . . chose to use the 1911 Triangle shirtwaist factory fire, in which 146 workers lost their lives, as a context for his discussion of the economic consequences of immigration.  Michael Wallace and Edwin Burroughs, in the Pulitzer Prize winning book, Gotham: The History of New York City, articulated the program’s main thesis: immigrants were hapless employees unable to resist nightmarish exploitation in brutal, dehumanizing garment factories; the fire was laissez-faire competitive capitalism at its worst.  This provoked New York Times television critic, John Tierney to write a column (11/18/99) entitled “A 1911 Fire As Good TV, Bad History,” characterizing the program as “a documentary not burdened with the less vivid facts.”1

Tierney presented the views of pro-market economists who claim that New York City immigrant garment workers were generally treated fairly and did not undermine the wages or working conditions of native-born workers.  Moreover, whereas Burns’ documentary asserted that government regulations were necessary and effective instruments in improving the welfare of workers, the pro-market economists Tierney quoted reached the opposite conclusion: government legislation was unproductive and threatened the welfare of workers.”

     In particular, Tierney relies on the economic notion of compensating wage differentials by noting that unskilled garment workers – those killed in the fire – were paid 8 percent more than comparably skilled workers in less dangerous jobs.  (Wallace and Burroughs were wrong in suggesting that garment workers were super-exploited; the compensating wages equalized the level of exploitation experienced by all unskilled immigrant workers in 1911.)

 

      Getting back to the problem solving, by comparing wage differentials among comparably skilled workers and the differential death rates at these disparate jobs, we can impute the value workers place on their lives.  As long as workers are freely able to choice between the riskier and less risky jobs, compensating wage differentials will be created to attract workers to the riskier job.  Eventually, there will be a sufficient wage differential so that all firms will be able to compete effectively for employees.

 

     Suppose that those working in the dangerous garment firms were paid $5400 annually

and experienced a death rate of 0.03 percent.  In contrast, comparably skilled workers working at safer jobs were paid $5000 annually but experienced a death rate of only 0.02 percent.

 

       This problem is slightly different since the benefits from taking the better job are known -- $400 annually – but the risk of death is reflected in expected costs.  Expected cost equals the increased likelihood of death times the value of life.  Students can then calculate that as long as individuals value their lives at less than $40 million, the expected cost of the riskier job is lower than the financial benefits; and if they are not risk adverse, they will choose to work at the riskier job.

 

Summary Remarks

 

     This paper discussed two distinct aspects of quantitative reasons: understanding table and graphical data presentations and using basic high school math for problem solving.      This paper looked at tables where each cell represented the value of a particular economic variable for a particular demographic group.  This matrix presentation of data is typical.  Other examples are earnings for different demographic groups at different educational attainment levels; or test scores for students at different grade levels by school attended.

 

      Data can also be presented for discrete variables in the form of bar charts or for continuous variables in functional relationships.  When functional relationships are used, their slopes often have important meanings.

 

        Problem Solving Skills were also explored.  Weighted averages where shown to have value in making basic calculations and in understanding simple static and more complex problems where changes in weights crucially affect outcomes.  This was followed by examples of how linear extrapolations could be used in simple budget problems and judging when eligibility ends for means tested programs, like food stamps.  Finally, it was shown how expected probabilities can be used to infer the value individuals place on their lives.  In all these case, a working knowledge of basic mathematics  can enable students to solve basic problems and understand statistical outcomes related to their everyday lives and academic pursuits.

 

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[1] John Tierney.  1999. “A 1911 Fire as Good TV, Bad History.”  NY Times (Nov 18): B1.