Developing a critical mathematical numeracy through real real-life word problems

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MARILYN FRANKENSTEIN
DEVELOPING A CRITICALMATHEMATICAL1
NUMERACY THROUGH REAL REAL-LIFE WORD
PROBLEMS
INTRODUCTION: ALL MATHEMATICAL WORD PROBLEMS ARE NON-NEUTRAL
A great honour was conferred on me a few years ago when right-wing conservative
Lynne V. Cheney (1998), former USA Vice-President Dick Cheney’s wife, trashed
my work because I stated that in mathematical texts, “A trivial application like
totaling a grocery bill carries the non-neutral message that paying for food is
natural.” (Frankenstein, 1983, p. 328). Contrary to Cheney’s claim that I would not
want students to solve problems totalling grocery bills, I certainly would want
students to solve such problems – comparing grocery bills in poor neighbourhoods
with those in rich neighbourhoods, for example, or problems comparing the costs
of the packaging with the costs of the food, for another example, or countless
mathematical investigations that could relate to issues of hunger and capitalism
where tens of millions of tons of surplus food rot for the profit of a few (Mittal,
2002) while approximately 40 million people die from hunger and hunger-related
illness every year and “available evidence indicates that up to 20,000,000 citizens
[living in the USA] may be hungry at least some period of time each month.”2
(National Council of Churches, 2007).
I argue that all real-life mathematical word problems contain non-numerical
“hidden” messages, and that, if those problems are presented as neutral, they can
stifle creative thought and questioning, by increasing the aspects of our society that
people take for granted.
Gill (1988) gives another example: mathematical texts that “neutrally” define
“profit” as the differen ce between the selling price and the cost price. In contrast,
she argues that a Marxist definition of profit as ultimately unpaid labour “suggests
that if the total of goods or capital in a social system is un equally divided between
people at different levels in the social-industrial hierarchy, exploitation is
necessarily taking place” (p. 122). In an analysis of real-life mathematical word
problems in Greek elementary school texts, Chassapis (1997) found that more than
70% of the examples, applications, and problems for the fifth and sixth grades were
“financial, and especially commercial, situations devoid of any pertinent social
relationships….to all appearances existing on their own beyond any human agency
and away from any space, time, or social structure….” (p. 26).
Moreover, the “hidden” messages do not just come from the subtexts of the
problem statements. MacKernan (2000, p. 45) contends that in England,
increasingly, teachers need “to start discussing what we are not allowed to teach.”
He presents many government statistics and suggests mathematical investigations
that could not take place in the classroom—questions, for example, about why the
UK is responsible for almost a fifth of the world’s arms exports.
There has, in recent years, been a push to include real-life mathematical word
problems in the curriculum. Verschaffel, Greer, and De Corte (2000) presented a

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comprehensive analysis of pedagogical concerns about practices involving word
problems that lead students to suspend their sense-making critical capabilities.
Contributions to this present volume extend that analysis in various directions. In
this chapter, I discuss how teachers can try to develop students’ critical political
capabilities through using mathematics word problems that challenge the
mainstream narrative of how our society works.
First, I’ll discuss some political concerns about other aspects of the non-neutral
“hidden” curriculum that results from particular selections of real-life data used to
create contrived and/or context-narrow word problems. In the body of the chapter,
I’ll suggest various categories of real real-life mathematical word problems,
illustrating how to go about creating such problems so they are presented in a
broad enough context for students to appreciate how understanding numbers and
doing calculations can illuminate meaning in real life. In conclusion, I will discuss
some pedagogical and political questions about the real-life use of real real-life
mathematical applications, returning to the issue of the non-neutrality of
knowledge, and addressing the question of teaching difficult, pessimistic
perspectives.
The main goal of a criticalmathematical literacy is not to understand
mathematical concepts better, although that is needed to achieve the goal. Rather it
is to understand how to use mathematical ideas in struggles to make the world
better. In other words, the question to be investigated about my
criticalmathematical literacy curriculum is not “Do the real real-life mathematical
word problems make the mathematics more clear?” The key research questions are
“Do the real real-life mathematical word problems make the social justice issues
more clear?” and, “Does that clarity lead to actions for social justice?”
PROBLEMS WITH REAL-LIFE MATHEMATICAL WORD PROBLEMS
Real-Life Mathematical Word Problems Without Real Meaning
Mathematical word problems are all too easy to satirize (Fig. 1).

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3
Figure 1: A satirical view of word problems
In a French study (IREM de Grenoble, 1980), a 7-year-old was asked the following
question: “You have 10 red pencils in your left pocket and 10 blue pencils in your
right pocket. How old are you?” When he answered: “20 years old,” it was not
because he didn’t know that he was 7 in real life, or because he did not understand
the relevant mathematical concepts. Rather it was, as Pulchalska and Semadeni
(1987, p. 15) conclude, because the unwritten social contract between mathematics
students and teachers stipulates that “when you solve a mathematical problem...
you use the numbers given in the story... Perhaps the most important single reason
why students give illogical answers to problems with irrelevant questions or
irrelevant data is that those students believe mathematics does not make any
sense”.
Pedagogically, that kind of “social contract” is a political problem—the political
implications of “educating” people to accept nonsensical statements uncritically in
order to “fit in” hardly need to be stated. Moreover, it is also politically
problematic even when mathematical word problems do not ask nonsensical
questions, but use real-life numerical data without real meaning, but only as
“window dressing” to practice a particular mathematical skill. First, when
assumptions about what are the “natural” conditions of real life (e.g., heterosexual
families) are used as the “window dressing” context for mathematical problems,
students who do not fit those “natural” categories are disrespected and/or made
invisible. Further, the “hidden curriculum” about what is “natural” gets reinforced,
making it less likely that students will question these taken-for-granted
assumptions. Second, the real significance of the “window-dressed” real-life data
is also hidden. When no better understanding of the data is gleaned through solving
the mathematics problem created from the data, using real-life data masks how
other mathematical operations, as well as other non-mathematical investigations,
could be performed that would illuminate those same data. It gives a “hidden
curriculum” message that using mathematics is not useful in understanding the
world—mathematics is just pushing around numbers, writing them in different
ways depending on what the teacher wants.

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Real-Life Mathematical Word Problems Without Real Real Context
There are, of course, curricula that contain real-life mathematical word problems
that involve using numbers to gain more information to help make real-life
decisions. However, often these problems assume everyone’s real-life context is
the same. By contrast, Greer, Verschaffel, and Mukhopadhyay (2007, p. 96) stated
that “if a decision is made to mathematize situations and issues that connect with
students' lived experience, then it brings a further commitment to respect the
diversity of that experience across genders, classes, and ethnicity”. Take the
following example:
It costs $1.50 each way to ride the bus between home and work. A weekly
pass is $16.00. Which is the better deal, paying the daily fare or buying the
weekly pass?
It was found that inner-city African-American students “transformed the ‘neutral’
assumptions of the problem – all people work 5 days a week and have one job –
into their own realities and perspectives” (Tate, 1995, p. 440). In their experience,
a job (such as cleaning) might mean making several bus trips every day and
working more than 5 days a week. If items of this type are used for assessment,
with assumptions about the "right" answer, the implications are clear, given that, as
Tate (1995, p. 440) puts it: “the underpinnings of school mathematics curriculum,
assessment, and pedagogy are often more closely aligned with the idealized
experience of the White middle class”.
Apple (1992, pp. 424-425) concludes that the NCTM Standards (1989) do not
address “the question of whose problem ... by focusing on the reform of
mathematics education for ‘everyone’, the specific problems and situations of
students from groups who are in the most oppressed conditions can tend to be
marginalized or largely ignored (see Secada, 1989, p.25).” The Standards do not
contain, for example, suggestions for mathematical investigations that would
illustrate how the current US government’s real-life de-funding of public
education, through funding formulas based on property taxes, creates conditions in
which the real-life implementation of the NCTM student-centered pedagogy is
virtually impossible except in wealthy communities (Kozol, 1991).
Real-Life Mathematical Problems Without Enough Real Context
Those “neutral” real-life mathematical word problems that do include a real-life
context like totalling grocery bills still omit the larger contexts of individual
economic differences within a system where a 1997 report from the US
Department of Agriculture declared that 11 million citizens, including 4 million
children, “live in households categorized as moderately or severely hungry.”
(Sarasohn, 1997, p. 14).
Other “neutral” real-life mathematical word problems involve numerical
descriptions that omit the larger contexts that created the reality of those
descriptions. For example, Multiplying People, Dividing Resources (Zero
Population Growth, 1994) contains a worksheet of real-life mathematical word

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5
problems designed to help students conceptualize large numbers. In the section on
“Explanations/Applications,” there is their “neutral” comment that:
When Columbus arrived in the Americas in 1492, there were probably 5
million Native Americans living in the area of the United States, and 57
million in the two American continents. World population at that time was
about 425 million, and did not reach one billion until approximately 1810. . . .
In 1994, the United States has approximately 260 million people within its
borders . . .
Hidden in this real-life context is the larger context of what happened to those
Native Americans. Although there is some academic debate about the number of
people living North of Mexico in 1492 (ranging from about 7 million to 18
million),
There is no doubt, however, that by the close of the nineteenth century the
indigenous population of the United States and Canada totaled around
250,000. In sum, during the years separating the first arrival of Europeans in
the sixteenth century and the infamous massacre at Wounded Knee in the
winter of 1890, between 97 and 99 percent of North America’s native people
were killed (Stannard, 1992, p. 432) .
For a real-life project-based example, Brown and Dowling (no date) comment
on a mathematics study card in England that asks students to do various
measurements in order to “study the problems of the disabled” (p. 23). They note
that if the activity were really about real-life study, students would be involved in
asking disabled people about their problems, instead of using their physical
situation as “window dressing” for measurement exercises. Further, they remark
that the larger institutional context is hidden: “The card is not about the ways in
which able-bodied society handicaps disabled people…No mention is made of
financial difficulties or difficulties relating to state ‘aid,’ ‘compensation,’
insurance, … nor does the card make any reference to physical or emotional pain,
social isolation…” (pp. 22-23).
REAL REAL-LIFE MATHEMATICAL WORD PROBLEMS3
Real real-life mathematical problems occur in broad contexts, integrated with other
knowledge of the world. I (Frankenstein, 1983) contend, along with Freire (1970;
Freire & Macedo, 1987) that the underlying context for critical adult education, in
this case criticalmathematical literacy, is “to read the world.” In that case,
mathematical skills and concepts are learned in order to understand the institutional
structures of our society.
Below are various categories of problems that, of course, overlap in different
ways. The overarching activity is gaining a better analysis of the issue through
understanding the meaning of the numbers, and gaining more knowledge about the

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issues through performing relevant calculations. The purpose of discussing the
examples in this manner is to show many types of situations in which numbers can
be used to make sense of the world, and then to make justice in the world.
Understanding the Meaning of Numbers
The real real-life mathematical word problems whose solutions involve
understanding the meaning of numbers focus on using different kinds and
arrangements of numbers (e.g., fractions, percents, graphs) to:
• describe the world
• reveal more accurate descriptions of the world
• understand the meaning of the sizes of numbers that describe the world
• understand the meanings that numbers can hide in descriptions of the world
• understand the meanings that numbers cannot convey in descriptions of the world
Understanding the meaning of the numbers is needed to understand the meaning of
these situations, situations that illu minate the way our world is structured.
Using Numbers to Describe the World
Example:
Although Helen Keller was blind and deaf, she fought with her spirit and her
pen. When she became an active socialist, a newspaper wrote that "her
mistakes spring out of the… limits of her development.” This newspaper had
treated her as a hero before she was openly socialist.
In 1911, Helen Keller wrote to a suffragist in England: “You ask for votes
for women. What good can votes do when ten-elevenths of the land of Great
Britain belongs to 200,000 people and only one-eleventh of the land belongs
to the o ther 40,000,000 people? Have your men with there millions of votes
freed themselves from this injustice?” (Zinn, 1980, p. 337).
Students are asked to discuss how numbers support Helen Keller’s main point
and to reflect on why she sometimes uses fractions and other times uses whole
numbers. Information about the politics of knowledge is presented as a context in
which to set her views, including class discussions about Keller’s militant answer
to the ed itor of the Brooklyn Eagle (Zinn, 1980, p. 338) and about why so many
children’s books ignore her socialist activism (Hubbard, 2002).
Using Numbers to Reveal More Accurate Descriptions of the World
Example: Students are asked to read articles that present numbers that counter
taken-for-granted assumptions that many view as “natural” facts about the world. I
use an editorial on “The wrong face on crime,” (Jackson, 1994) which gives
myriad “counter-intuitive” data such as “white Americans are ... three times more

REAL REAL-LIFE WORD PROBLEMS
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likely to be violently assaulted by another white person than by an African-
American;” and a newspaper article which shows that in spite of widespread belief
that “illegal”4 immigrants are robbing tax payers through their use of hospital
emergency rooms and public education, not only do “illegal” immigrants pay sales
and other such taxes, but they also pay over $6 billion in Social Security and about
$1.5 billion in Medicare taxes, without collecting any of the benefits from those
taxes (Porter, 2005).
Understanding the Meaning of the Sizes of Numbers that Describe the World
Example: Students are asked to discuss what numerical understandings they need
in order to understand the political and personal imp lications of the chart shown in
Table 1. These figures were compiled in time-and-motion studies conducted by
General Electric, and published in a 1960 handbook to provide office managers
with standards by which clerical labour should be organized.
Table 1. Estimated time for various operations (Braverman, 1974, p. 321)
Open and close Minutes
Open side drawer of standard desk 0.014
Open centre drawer 0.026
Close side drawer 0.015
Close centre drawer 0.027
Chair activity Minutes
Get up from chair 0.039
Sit down in chair 0.033
Turn in swivel chair 0.009
To understand more deeply how numbers underpin worker control, we discuss
historical examples, like that of William Henry Leffingwell who, in the early
1900’s:
… calculated that the placement of water fountains so that each clerk walked,
on the average, a mere hundred feet for a drink would cause the clerical
workers in one office to walk an aggregate of fifty thousand miles each year
just to drink an adequate amount of water, with a corresponding loss of time
for the employer. (This represents the walking time of a thousand clerks, each
of whom walked only a few hundred yards a day.) (Braverman, 1974, pp.
310-11).

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We also look at contemporary examples, including situations where “managers
kept computer spreadsheets monitoring employee use of the bathroom” and female
workers were told “to urinate into their clothes or face three days’ suspension for
unauthorized expeditions to the toilet” (Robins, 2002, D5).
According to the Bureau of Labor Statistics, in 1979, 25 percent of
employees in medium- to large-sized companies did not have paid rest breaks
during which they could go to the bathroom. By 1993, the last year for which
there are statistics, that number had jumped to 32 percent. . . Not until 1998
did the federal government, under pressure from the labor movement, even
maintain that employers had to grant employees an ill-defined “timely
access” to the bathroom. (Robin, 2002, p. D5)
Scharf (2003) gives another current example where employees' talk to customers is
“scripted” by management:
Fast-food drive-through window workers must greet customers almos t
instantly—often within three seconds from the time the car reaches the menu
board. Digital timers . . . measure how long it takes the worker to issue the
greeting, take the order, and process the payment. ... Former McDonald’s
CEO Jack Greenberg claimed that unit sales increase 1% for every six
seconds saved at the drive-through.
To look at such outrages even more deeply, we think about the theory behind
the “scientific managemen t” of workers. Braverman (1974) states that the idea is to
conceive of the worker as a general-purpose machine operated by management,
displacing labourers as the subjective element of the labour process and
transforming them into objects.
This mechanical exercise of human faculties according to motion types,
which are studied independently of the particular kind of work being done,
brings to life the Marxist conception of “abstract labor.”… The capitalist sees
labor not as a total human endeavor, but [abstracts it] from all its concrete
qualities in order to comprehend it as universal and endlessly repeated
motions, the sum of which, when merged with the other things that capital
buys – machines, materials, etc.–results in the production of a larger sum of
capital than that which was “invested” at the outset of the process. Labor in
the form of standardized motion patterns is labor used as an interchangeable
part . . . (pp. 180-2)
Understanding the Meanings that Numbers can Hide in Descriptions of the World
Example: Students are given data of employment status for various categories of
workers such as “employed part-time, want full-time work,” and “not employed,
want a job now, have not looked for work in the last year.” They are asked to
decide which categories should count as unemployed, which are in the labour

REAL REAL-LIFE WORD PROBLEMS
9
force, and to calculate their unemployment rate, and compare it with the rate
calculated using the categories counted by the government.
Discussion brings out that there is political struggle involved in deciding who
counts as unemployed. In 1994, the USA official definition gave an unemployment
rate of 5.1%, whereas considering additional categories of discouraged workers as
unemployed changes the rate to 9.3%. Further, there are other groups we could
count, such as the 2.5 million people who worked full-time, year round, in 1994
and earned below the official poverty line (Sklar, 1995). And there are many other
political decisions to make about how to count unemployment (Figure 1):
Figure 2. The counting of jobs
Even more sophisticated ways of making the unemployment data seem “low” are
also considered. For example, students need to use algebra to show why removing
an unemployed worker from the labour force makes the unemployment rate go
down.
Several years ago BLS changed the criteria for determining labor force status so
that the higher the job turnover, the lower the official unemployment rate. To
illustrate: workers who expected to begin a new job but who were not ye t
working were formally counted as unemployed. Now most are defined as out of
the work force and are no longer included in the unemployment statistics.
Moreover, the worker who left with the intention of seeking another job is
defined as out of the work force until that person applies for another job. To be
counted as unemployed requires an “active” effort to find work. . . . Neither the
worker who expects to begin a job nor the individual who just left the same job
are collecting a paycheck. Yet the official unemployment rate among them is
zero. The greater the job turnover, the more this situation is replicated, and the
larger the gap between the official and real unemployment rates. (Brill, 1999, p.
40).
This is a case in which the larger context of the politics behind the collection of
data is important to include. In 1994 the Bureau of Labor Statistics stopped issuing
its U-7 rate, a measure that included various categories of discouraged workers not

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counted for the official government unemployment rate, so now researchers will
not be able to determine “alternative” unemployment rates (Saunders, 1994). Then,
in 2003, the “mass layoff” statistics (layoffs putting 50 or more workers out of a
job) were dropped by the Bureau of Labor Statistics. Boothby (2003) reports that
in November 2002 there were 2,150 mass layoffs affecting 240,000 workers.
Finally, more extended discussion brings up a number of related
criticalmathematical issues such as the effects of racism and sexism resulting in
differential unemployment rates and incomes. In 1986, even with a college degree,
blacks had higher unemployment rates than whites (13.2% to 5.3%); blacks with a
college degree even had higher unemployment than whites with only a high school
diploma (13.2% to 10.1%). Folbre (1987, charts 4.7, 4.12) explains that:
One way to measure the combined effect is to multiply the median earnings
of the different groups by the percentage of the labor force of that group that
is employed. This provides an estimate of the typical earnings of a member of
the labor force. . . . individual black men working full-time in 1983 earned
75% of what white men earned, but a typical black man in the labor force
earned only 52% of what a typical white man earned – because the black man
was far more likely to be unemployed.
Of course, the broadest theoretical context examines why 100% employment
cannot exist in a capitalist economic system, because that would create a situation
in which the workers would have too much power to change the conditions of their
exploitation.
Understanding the Meanings that Numbers Cannot Convey in Descriptions of the
World
Example: Following this is an example of art encoding quantitative information.
The numbers are the data of our world—our wars; the art allows us to understand
the quantities in ways we could not understand from the numbers alone. As Toni
Morrison states: “Data is not wisdom, is not knowledge” (quoted in Caiani, 1996,
p. 3).
The famous memorial in Washington, D.C. by artist Maya Lin lists the names of
57,939 Americans killed during the Vietnam War. In “The other Vietnam
Memorial” (Museum of Contemporary Art in Chicago, IL), Chris Burden etched
3,000,000 names onto a Rolodex-type structure, standing on its end, that fills the
entire room in which it is displayed. The names represent the approximate number
of Vietnamese people killed during the US war on Vietnam. Since many of their
names are unknown, Burden created variations of 4000 names taken from
Vietnamese telephone books. Also, the museum notes comment that by using the
form of a common desktop object that functions to organize professional and social
contacts, Burden underlines the unrecognized loss of Vietnamese lives in US
memory.

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Understanding the Calculations
The real real-life mathematical word problems whose calculations are an integral
part of understanding a situation focus on:
• verifying/following the logic of an argument
• understanding how numerical descriptions originate
• using calculations to restate information
• using calculations to explain information
• using calculations to reveal the unstated information
The purpose underlying all the calculations is to understand better the information
and the arguments, and to be able to question the decisions that were involved in
choosing which numbers to use and which calculations to perform.
Verifying/Following the Logic of an Argument
Example: Students are asked to read an excerpt from “The One-Percent Solution”,
the letter to the editor criticizing one of the examples in the excerpt, discuss the
arguments made, and fill in all the details of the mathematical operations the writer
performed to back up his argument. (One of the nice pedagogical aspects of this
exercise is that the letter writer, correcting one mathematical mistake made by
Dollars & Sense, makes another mathematical error himself.)
The excerpt from “The One-Percent Solution” (Dollars & Sense, December
1989, p. 32) is as follows:
Both the sheer volume of numbers and, in an era of multi-trillion-dollar
national debts, their overwhelming magnitude obscure their true meaning. …
Divorced from both the decisions of the powerful and the effects on the
powerless, the numbers just numb. A percentage point here, a percentage
point there; what’s the difference?
As the following examples demonstrate, the difference can be dramatic
indeed.
• A one percentage point reduction in the poverty rate, from 13.1% to 12.1%,
would lift 2.4 million people above the poverty line, including over 600,000
children.
• A one percentage point decrease in the official unemployment rate, from
5.3% to 4.3%, would mean 1.2 million people working and an additional $75
billion in annual output.
The letter to the editor, titled "One percent is bigger than it looks" (Dollars &
Sense, April 1990, p. 22) reads as follows:
I have a comment on your “Economy in Numbers” (December 1989). You
state that a 1% decrease in the average number of hours worked each week

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would create the equivalent of 17,000 new full-time jobs. I think this estimate
is much too low.
Average weekly hours for production or non-supervisory workers in the
private sector in 1988 was 34.7. A 1% reduction would be 0.347 hours. Total
employment in 1988 was 116.7 million. Assuming this average weekly hours
estimate applies to all workers, a 1% reduction in hours would mean that the
existing work force would work a total of 40.49 million fewer hours per
week. If both labor productivity and total labor demand remained unchanged,
1.167 million additional workers working 34.7 hours per week would be
required, nearly 70 times the Dollars & Sense estimate!
Of course, productivity tends to increase when the length of the work is
reduced (workers might have a specific number of tasks they must perform in
a shorter time period) so there would be some reduction in labor demand.
Surely, however, it would not be as much as implied by your estimate.
In closing, I find Dollars & Sense an excellent publication. Keep up the good
work!
Andrew Sharpe, Head of Research, Canadian Labour Market and
Productivity Centre, Ottawa, Canada
Understanding how Numerical Descriptions Originate (Seeing how Raw Data are
Collected, Transformed, and Summarized into Numerical Descriptions of the
World)
Example: Students are asked to read the excerpt below so that they are thinking
about issues of how to teach and how people learn mathematics at the same time
that they are earning the mathematics. Then, they are asked to: describe the study’s
methodology (i.e., what procedures were followed in the study, what the “raw”
data consisted of, and how the raw data were transformed and summarized); re-
write the findings described by creating a chart; discuss which presentation of the
data is clearest, and why; list conclusions they can and cannot draw from the data;
and indicate what other information they would want in order to clarify the data or
strengthen and/or change their conclusions. The excerpt to be analysed is as
follows (Sklar, 1993, p. 53):
Sixty-six student teachers were told to teach a math concept to four pupils -
two White and two Black. All the pupils were of equal, average intelligence.
The student teachers were told that in each set of four, one White and one
Black student was intellectually gifted, the others were labelled as average.
The student teachers were monitored through a one-way mirror to see how
they reinforced their students' efforts. The “superior” White pupils received
two positive reinforcements for every negative one. The “average” White

REAL REAL-LIFE WORD PROBLEMS
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students received one positive reinforcement for every negative
reinforcement. The “average” Black student received 1.5 negative
reinforcements for each positive reinforcement, while the “superior” Black
students received one positive response for every 3.5 negative ones.
Using Calculations to Restate Information (Changing the Quantitative Form)
Example: Students study a letter I wrote (Frankenstein, 2002) responding to an
article by Howard Zinn (2002) in which he argues that the numerical descriptions
of the deaths from the US war on Afghanistan can obscure those horrors. To
dramatize my argument that numbers can illuminate the meaning of data and
deepen connections to our humanity, I conclude that the 12 million children who
die every year from hunger “are dying faster than we can speak their names.”
(Frankenstein, 2002, p. 23).
Using Calculations to Explain Information
Example: Students are asked what calculations to perform to understand how
declining block rate structures, like the one illustrated in the chart below, transfer
money from the poor to the rich, and to propose other kinds of payment structures.
The Rate Watcher's Guide (Morgan, 1980) details that a 1972 study conducted in
Michigan, for example, found that residents of a poor urban area in Detroit paid
66% more per unit of electricity than did wealthy residents of nearby Bloomfield
Hills. Researchers concluded that “approximately $10,000,000 every year leave the
city of Detroit to support the quantity discounts of suburban residents.”
7
kwh/month
0-299 kwh/m 300-799 kwh/mo 800-1400 kwh/ho
6
5
4
3
2
1
0

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Figure 3. Declining block rate structure
Using Calculations to State the Unstated Information
Example: Students learn about percents while analyzing the following political
poster in the context of the politics of language where people who constitute a
majority of the world’s population are referred to as “minorities.” Students also see
that numbers are “behind” many economic, political, and/or social issues even if
there are no numbers “visible” in the picture (Figure 3).
Figure 4. Los Angeles Hispanics and other recent immigrants are demanding their piece of
the pie (Guardian, 1978, Mario Torero, with Zapilote, Ro cky, El Lton, and Zade)
CONCLUSION: PEDAGOGICAL AND POLITICAL DIMENSIONS OF TEACHING
THROUGH REAL REAL-LIFE MATHEMATICAL WORD PROBLEMS
Pedagogical Dimensions
Following ABC’s 1983 airing of a film about The Day After a nuclear war, the
network presented a panel discussion, chaired by Ted Koppel, of mostly
conservative government officials and Carl Sagan, a liberal scientist. At one point,
Sagan refuted the then Secretary of State Schultz’s contention that the
Administration was already disarming, pointing out that “its current build-up calls
for an increase in the number of strategic warheads, from 9,000 to 14,000.” Koppel
turned to Sagan and said “… I must confess statistics leave my mind reeling and, I
suspect, everybody else’s too.” (Manoff, 1983, p. 589)
Certainly, students need enough mathematics so that their heads do not reel
from comparing the size of two numbers! As a prerequisite to accomplishing the
goal of a Freirean “reading of the world” using a criticalmathematical literacy,
students need confidence that they can learn enough mathematics to use as part of

REAL REAL-LIFE WORD PROBLEMS
15
understanding public and community service issues. Some of this confidence is
gained from analysis of mystifications about learning mathematics, such as “there
is only one correct method for solving a particular mathematical problem” and
“only some people have mathematical minds” (Frankenstein, 1984). Of course,
there is an emotional part of analyzing these aspects of mathematics learning.
When students realize that their teacher has confidence in them and expects, with
studying, that they will learn the mathematics, they can begin to let go of the
negative expectations many have internalized from past mathematics learning
experiences.
Some confidence is gained from analysis of the societal uses of mathematics
“anxiety.” For example, I think it is important to address issues of “the politics of
language” in order for students to understand how the label “mathematically
anxious” can have contradictory effects. Naming that situation can initially
reassure students that their feelings about mathematics are so common that
educators have a name for them. However, the label can also focus the problem
inward, “blaming the victims” and encouraging solutions directed solely at them.
The label can direct attention away from the broader social context of how their
learning got mystified, and what interests might be served by widespread
mathematics “anxiety” and avoidance.
Some confidence is gained from understanding the politics of knowledge that
have discounted some people’s knowledge and privileged others’ knowledge. For
example, I ask students to reflect on Freire’s (Freire & Macedo, 1987) insistence
that “the intellectual activity of those without power is always characterized as
non-intellectual.” (p.122) and on Marcuse’s (1964) argument that:
In this society, the rational rath er than the irrational becomes the most
effective mystification…For example, the scientific approach to the vexing
problem of mutual annihilation—the mathematics and calculations of kill an
over-kill, the measurement of spreading or not-quite-so-spreading fallout… is
mystifying to the extent to which it promotes (and even demands) behavior
which accepts the insanity. It thus counteracts a truly rational behavior—
namely, the refusal to go along, and the effort to do away with the conditions
which produce the insanity (pp. 189-190).
Once students are confident in their ability to learn mathematics, and motivated to
reason quantitatively about public and community issues, then the question is: How
much of the structure of mathematics must be demystified in order for students to
be able to use numerical data for demystifying the structure of society?
It is important for students to understand enough concepts behind the basic
algorithms to be able to use those rules comfortably in many different situations.
However, as Lange and Lange (1984) found, although mathematics education can
be empowering in a more general way, it is not necessarily the best approach in
working with people on specific empowerment issues. The piece-rate workers they
were organizing in the textile industry in the southern United States were
struggling with a pay system made intentionally obscure. The Langes' experience
was that teaching the concepts of ratios and fractions behind that rate system was

MARILYN FRANKENSTEIN
16
not the most effective way to empower the workers in their struggle for decent pay.
It was more empowering to create a slide-rule distributed by the union that did the
pay calculations for the workers, making the mathematicl problem disappear, so
that the workers could “focus on the social and economic relations underlying the
way they are treated and paid” (p. 14).
Brown and Dowling (no date) argue for a research-based mathematics
education—“an approach which centralizes the consideration of social inequalities
as a goal in itself, and which subordinates the mathematics as one of a number of
possible means.” In my context, my curriculum is loosely organized by a linear
thread of underlying mathematical concepts (i.e., the meaning of whole numbers,
then fractions, later percents, and so on). But, the lessons also involve non-linear
explorations of real real-life public and community issues and much
interdisciplinary content.
However, in thinking about what numeracy citizens need to solve real real-life
problems, I am not advocating getting rid of college preparatory mathematics. As
Powell and Brantlinger (2008) argue, teaching “traditional" mathematics with
understanding to students who have been marginalized from college or certain
professions is another form of criticalmathematics education appropriate to that
context. I would argue that all citizens need the criticalmathematics I am
describing, but it does not need to replace more “traditional” mathematics.
Political Dimensions
I suspected trouble when, at a 1981 National Council of Teachers of Mathematics
(NCTM) Conference, the president of the organization opened the meeting by
stating that Ronald Reagan’s election was great for mathematics teachers. But, I
did not suspect how outraged the teachers would be by the biases in my real-life
word problems. They did not accept my argument that no mathematical word
problems are neutral.
A few years after my NCTM audience was furious at my biased word problems,
the NCTM journal, The Mathematics Teacher, (March 1984, December 1984) was
running multi-page spreads advertising a US Navy slide show “Math and Science:
START NOW!” Toll-free phone numbers to arrange for a class presentation by a
Navy representative were included. They published one critical letter that focused
on the inappropriateness of the Navy starting recruiting drives in junior high school
and questioned why there were no ads from government groups “whose mandate is
more closely tied to social and environmental problems” (Milne, 1984). The editor
answered that the Navy paid for the ad and any government agency could do
likewise. He did not publish my strong critique that accepting an ad from the Navy
implied:
… a certain level of support—especially since the NCTM’s Executive
Director is quoted in the ad as saying “Without hesitation, we endorse the
project”!! In addition, your ad policy will be skewed towards those
governmental agencies with the largest advertising budgets—therefore, those
agencies, such as the military, which are favored by the current

REAL REAL-LIFE WORD PROBLEMS
17
administration, will also be favored by NCTM ad policy. Finally, we did pay
for the ad—not through our NCTM dues as you stated—but certainly,
through our tax dollars.
As discussed in Frankenstein and Powell (1994), the epistemology of Paulo Freire
is in direct opposition to the NCTM’s and other dominant educational institutions'
paradigm of positivism which views knowledge, though a product of human
consciousn ess, as neutral, value-free, objective, and completely separate from how
people use it. Learning, in this view, is the discovery of static facts and their
subsequent description and classification (Bredo & Feinberg, 1982). On the other
hand, Freire insists that knowledge is not static; that there is no dichotomy between
objectivity and subjectivity, or between reflection and action; and that knowledge
is not neutral.
For Freire, knowledge is continually created and re-created as people act and
reflect on the world. Knowledge, therefore, is not fixed permanently in the abstract
properties of objects, but is a process in which gaining existing knowledge and
producing new knowledge are “two moments in the same cycle” (Freire, 1982).
Embedded in this notion is the recognition that knowledge requires subjects;
objects to be known are necessary, but they are not sufficient.
Knowledge…necessitates the curious presence of subjects confronted with
the world. It requires their transforming action on reality. It demands a
constant searching (Freire, 1973, p. 101).
Knowledge, therefore, is a negotiated product emerging from the interaction of
human consciousness and reality; it is produced as we, individually and
collectively, search and try to make sense of our world.
Because of the unity between subjectivity and objectivity, people cannot
completely know particular aspects of the world—no knowledge is finished or
infallible. As humans change, so does the knowledge they produce. In connection
with this, Lerman (1989) theorizes that objective statements are publicly-shared
social constructions. At particular moments in history, communities of people
discuss, debate, revise, adopt, and challenge concepts and theories. Thus
knowledge, “objective” beyond the visions of individual subjects, does not have
the “transcendental existence” that positivists ascribe to it (Lerman, 1989, p. 219).
Through constant search and dialogue, we continually refine our understandings
and theories of reality and, in so doing, act more effectively.
Further, action and reflection are not separate moments of knowing. On the one
hand, reflection that is not ultimately accompanied by action to transform the world
is meaningless, alienating rhetoric. On the other hand, action that is not critically
analyzed cannot sustain progressive change. Without reflection, people cannot
learn from each other’s successes and mistakes; particular activities need to be
evaluated in relation to larger collective goals. Only through praxis—reflection
and action dialectically interacting to re-create our perception and description of
reality—can people become subjects in control of organizing their society.

MARILYN FRANKENSTEIN
18
This praxis is not neutral. Knowledge does not exist apart from how and why it
is used, and in whose interest. Even, for example, in the supposedly neutral
technical knowledge of how to cultivate potatoes, Freire asserts that:
… there is something which goes beyond the agricultural aspects of
cultivating potatoes.… We have not only…the methods of planting, but also
the question which has to do with the role of those who plant potatoes in the
process of producing, for what we plant potatoes, in favor of whom. And
something more. It is very important for the peasant…to think about the very
process of work—what does working mean? (Brown, 1978, p. 63).
In Freire’s view, people produce knowledge to humanize themselves. Overcoming
dehumanization involves resolving the fundamental contradiction of our epoch:
domination against liberation.
One final point: the real real-life context illuminated by the real real-life
mathematical word problems in my adult criticalmathematical literacy curriculum
are outrageously horrible. How can these topics be taught without discouraging
people and thereby stopping resistance? The context of my students’ lives is such
that many have been involved in our struggle to change this situ ation. And
different groups of us have experienced some victories. However, given the
resources of those in power to regroup, we wind up fighting the same battles over
and over and often initial victories are overturned or co-opted. Nevertheless, those
of us who are committed to the struggle for a just liberatory world keep fighting.
Audre Lorde (1988) reminds us in A Burst of Light that:
… hope [is] a living state that propels us, open-eyed and fearful, into all the
battles of our lives. And some of those battles we do not win. But some of
them we do. (p. 80)
If you can walk, you can dance … (Figure 5).

REAL REAL-LIFE WORD PROBLEMS
19
Figure 5. Yes we can!
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1 When Arthur Powell and John Volmink and I formed the Criticalmathematics Educators Group
(CmEG) in 1991, following a conference we organized in October 1990, we decided to use one word to
describe critical mathematics because of our hope that one day all mathematics education will be
critical. See Powell and Brantlinger (2008) for a perspective on various ways of interpreting
criticalmathematics education. In the future we intend to create a web site for the group, which will
include the archive of the five CmEG Newsletters we distributed between 1991 and 1997.
2 Related to the politics of language, The Progressive (2007, p.11) cites a Washington Post article
indicating that the United States Department of Agriculture will no longer use the word “hunger” to
describe people who cannot get enough food to eat; instead these people will be described in official
government documents as having “very low food security.”
3 Due to space limitations the examples are presented in an abbreviated form. I am developing these and
others into a collection of columns for various websites and newsletters. Since June 2008, they have
been appearing in Numeracy Briefing, edited by Europe Singh. For more information contact them at
numeracy@basicskillsbulletin.co.uk. If any reader is interested in syndicating these columns, free of
charge, contact me at marilyn.frankenstein@umb.edu
4 I use quotes around illegal to draw attention to who gets to make the laws that determine who is
“legal” and who is “illegal”.