Mathematics Education
Peter Appelbaum,
Excerpt from Critical Thinking and Learning”: An Encyclopedia
for Parents and Teachers, Joe Kincheloe and Danny Weil (eds). Greenwood
Press. 2004
Teachers
of mathematics have been searching for ways to describe and enact critical
thinking in their classrooms for a very long time. On the one hand, mathematics
itself is often held up as the model of a discipline based on rational thought,
clear, concise language, and attention to the assumptions and decision-making
techniques that are used to draw conclusions. In the nineteenth century, a view
of the mind as a muscle that could be trained and strengthened in particular
skills (this has come to be called “faculty psychology”) led many people to
justify a central place for mathematics in the school curriculum simply based
on the belief that mathematics would train the mind in clear, logical thinking.
To this day, employers often hire mathematics majors fresh out of college under
the presumption that they have been trained to “think.” On the other hand,
teachers of mathematics have always been disappointed in the critical thinking
that their students demonstrate. And there have been many research studies that
point to a dismal chance of any sort of transfer of learning of critical
thinking from the mathematics classroom into other realms of intellectual
effort. Indeed, research has failed to document any consistent transfer of what
might be called “critical thinking skills” from even one branch of mathematical
inquiry to another.
Back in
1938, Harold Fawcett was asked to publish his work with geometry students as
the yearbook of the National Council of Teachers of Mathematics. This book
introduced the idea that students could learn mathematics through
experiences of critical thinking. This was a big leap from older ideas that
promoted specific ideas about how to “teach” the skills of critical thinking. Fawcett
wrote that there had never been a time in the history of education when the
development of critical and reflective thought was not thought of as an
important outcome of school; but he thought that this particular outcome had
assumed increasing importance in the 1930s, and, further, that this importance
held strong implications for the nature of the school curriculum itself.
The
point Fawcett made was that it was pointless to try to teach critical thinking
skills (e.g., comparing, contrasting, conjecturing, inducing, generalizing,
specializing, classifying, categorizing, deducing, visualizing, sequencing,
ordering, predicting, validating, proving, relating, analyzing, evaluating, and
patterning; see O’Daffer & Thomquist 1993). Better would be to take
advantage of the critical thinking skills that students bring with them to
school mathematics, in order to learn the mathematics. His goals included the
following ways that students could demonstrate that they were, in fact,
thinking critically, as they participated in the experiences of the classroom.
By:
1. Selecting the
significant words and phrases in any statement that is important, and asking
that they be carefully defined.
2. Requiring evidence to support conclusions
they are pressed to accept.
3. Analyzing that evidence and distinguishing
fact from assumption.
4. Recognizing stated and unstated assumptions
essential to the conclusion.
5. Evaluating these assumptions, accepting
some and rejecting others.
6. Evaluating the argument, accepting or
rejecting the conclusion.
7. Constantly reexamining
the assumptions that are behind their beliefs and actions. (Fawcett, pp. 11-12,
paraphrased)
Years later, in 1989, the
National Council echoed this earlier call for critical thinking in its Curriculum
and Evaluation Standards. Calling
for classrooms that place critical thinking at the heart of instruction, the
new Standards stated clearly that a pervasive emphasis on reasoning
would be an essential aspect of all mathematical activity.
Presumably,
more than a half century after Fawcett’s lovely yearbook, thanks to years of
research and accumulated teacher-lore, the Council had a clear set of ideas for
how to accomplish this. Perhaps more interesting is the lack of any direct
attention to “critical thinking” in the more recent Principles and Standards,
published in 2000. Critical thinking is still present in the goals, but it has
been subsumed by more holistic notions of what it means to teach, do, and
understand mathematics. For example, in a discussion of communication in
any K-12 mathematics classroom, we are urged to design experiences that enable
students to:
How
similar these ideas are to those promoted by Fawcett so many years earlier! We
can see that little has changed in the mainstream ways that people tend to
define critical thinking in the context of mathematics education. Yet careful
attention to the details in the Standards does reveal increased
sophistication in what the Council means by these goals. For example, there is
increased interest in the idea that students must truly understand the
strategies and mathematical thinking of others in the classroom community.
Students are expected to search for the strengths and weaknesses of each and
every strategy offered. It is no longer good enough to reach an answer to a
problem that was posed. Now, students are cajoled into communicating their own
ideas well, and to demand the same communication from others. A shift
has occurred from listing skills to be learned toward attributes of classrooms
that promote critical thinking as part of the experience of that classroom.
One way
in which teachers can create such a classroom is by designing good ways for
students to communicate with each other. More specifically, it is now
recognized that reflection and communication are intertwined processes in
mathematics learning. The Council recommends explicit attention and planning by
teachers, so that communication for the purposes of reflection can become a
natural part of mathematics learning. And the Council further urges that
teachers focus on the building of a classroom community in which students feel
free to express ideas; teachers are asked to look for evidence that the
students understand that contributions to this community include the skills of
listening, and developing an interest in the thoughts of others. It seems that
the Council is asking teachers to spend more time on developing mathematics as
an evolving literacy, rather than as a set of conventions and techniques to be
mastered. Rather than rush to formal mathematical language, mathematics
teachers should recognize mathematics as a group experience that requires
reading, writing, listening, speaking, and the use of various modes of representation.
Students
who have opportunities, encouragement, and support for speaking, writing,
reading, and listening in mathematics classes reap dual benefits: they
communicate to learn mathematics, and they learn to communicate mathematically.
(NCTM 2000, p. 60)
Nevertheless,
from Fawcett in 1938 to the Council in 2000, we can identify a strand of
undeveloped theory: students are supposed to be communicating freely in some
form of ideal democratic environment. They not only have freedom of speech, but
indeed are guaranteed an audience from the other members of their community. In
this perfect democracy, students know that when they talk, people listen to
them. And they grow to understand that it is this sort of communication that
leads mutually to richer understanding of the material and increased
sophistication in talking about this mathematics. What is undeveloped, however,
is a critical consideration of the context in which this perfect democracy is
supposed to take place: the classroom is embedded in a society that determines
a wide range of ways that students do not come to the same table with equal
opportunities and resources at their disposal. Political theorists have noted
that ideal speech communities require as much attention to these contextual
realities and to responses to them as they require the generation of ways to
organize democratic participation.
For the
last century teachers of mathematics have been figuring out how to drop the
teaching of critical thinking in favor of establishing environments that allow
for the critical thinking that is possible through discussion and interaction.
This has meant abandoning long-cherished notions about what mathematics
classrooms look like and what the products of such classrooms should be.
Instead of timed tests with the number correct circled at the top, students now
bring home portfolios of material amassed over time, or complex reports on
open-ended investigations. Yet in the new millennium, teachers of mathematics
are now beginning to realize that this can only go so far. They are likely to still
be disappointed in the ways some of their students participate or
contribute. The response to this current malaise, growing out of the more
nuanced political awareness, is a movement called Critical Mathematics
Education.
Critical
Mathematics Education demands a critical perspective on both mathematics and
the teaching/learning of mathematics. In doing so, it takes one step further in
questioning our assumptions about what critical thinking could mean and what
democratic participation should mean. As Ole Skovsmose (1994) describes a
critical mathematics classroom, the students (and teachers) are attributed a
“critical competence.” A century ago, we moved from teaching critical thinking
skills to using the skills that students bring with them. We accepted that
students, as human beings, are critical thinkers, and would display
these skills if the classroom allowed such behavior. It seemed that we were not
seeing critical thinking simply because we were preventing it from happening;
through years of school, students were unwittingly “trained” not to
think critically in order to succeed in school mathematics. So we found ways to
lessen this “dumbing down of thinking through school experiences.” Now we
understand human beings more richly as exhibiting a critical competence,
and because of this realization, we recognize that decisive and prescribing
roles must be abandoned in favor of all participants having control of the
educational process. In this process, instead of merely forming a classroom
community for discussion, Skovsmose suggests that the students and teachers
together must establish a “critical distance.” What he means with this term is
that seemingly objective and value-free principles for the structure of the
curriculum are put into a new perspective, in which such principles are
revealed as value-loaded, necessitating critical consideration of contents and
other subject-matter aspects as part of the educational process itself.
Keitel,
Klotzman and Skovsmose (1993) together offer a new way for teachers to think
about the mathematics that is being taught. New ideas for lessons and units emerge when teachers describe
mathematics as a technology with the potential to work for democratic goals,
and when they make a distinction between different types of knowledge based on
the object of the knowledge. The first level of mathematical work, they write,
presumes a true-false ideology and corresponds to much of what we witness in
current school curricula. The second level directs students and teachers to ask
about right method: are there other algorithms? Which are valued for our need?
The third level emphasizes the appropriateness and reliability of the
mathematics for its context. This level raises the particularly technological
aspect of mathematics by investigating specifically the relationship between
means and ends. The fourth level requires participants to interrogate the
appropriateness of formalizing the problem for solution; a mathematical/technological
approach is not always wise and participants would consider this issue as a
form of reflective mathematics. On the fifth level, a critical mathematics
education studies the implications of pursuing special formal means; it asks
how particular algorithms affect our perceptions of (a part of) reality, and
how we conceive mathematical tools when we use them universally. Thus the role
of mathematics in society becomes a component of reflective mathematical
knowledge. Finally, the sixth level examines reflective thinking itself as an
evaluative process, comparing levels 1 and 2 as essential mathematical tools,
levels 3 and 4 as the relationship between means and ends, and level 5 as the
global impact of using formal techniques. On this final level, reflective
evaluation as a process is noted as a tool itself and as such becomes an object
of reflection. When teachers and students plan their classroom experiences by
making sure that all of these levels are represented in the group’s activities,
it is more likely that students, and teachers, can be attributed the critical
competence that we envision as a more general goal of mathematics education.
In formulating a democratic, critical mathematics
education, it is also essential that teachers grapple with the serious
multicultural indictments of mathematics as a tool of post-colonial and
imperial authority. What we once accepted as pure, wholesome truth is now
understood as culturally specific and tied to particular interests. Philip
Davis and Reuben Hersh (1987) and David Berliner (2000), for instance,
have described some aspects of mathematics as a tool in accomplishing a fantasy
of control over human experience. They
use the examples of math-military connections, math-business connections, and
others.
Critical
mathematics educators ask why students, in general, do not see mathematics as helping them to interpret events in their
lives, or gain control over human experience. They search for ways to help
students appreciate the marvelous qualities of mathematics without adopting its
historic roots in militarism and other fantasies of control over human
experience. Arthur Powell and Marilyn
Frankenstein (1997) have
collected valuable essays in ethnomathematics and the ethnomathematical
responses that educators can make to contemporary mathematics curricula.
Ethnomathematics makes it clear that mathematics and mathematical reasoning are
cultural constructions. This raises the
challenge to embrace the global variety of cultures of mathematical activity
and to confront the politics that would be unleashed by such attention in a
typical North American school. That is, ethnomathematics demands most clearly
that critical thinking in a mathematics classroom is a seriously political act.
One
important direction for critical mathematics education is in the examination of
the authority to phrase the questions for discussion. Who sets the agenda in a
critical thinking classroom? Stephen Brown and Marion Walter (1999) lay out a
variety of powerful ways to rethink mathematics investigations through The Art of Problem Posing, and in doing
so they give us a number of ideas for enabling students both to “talk back” to
mathematics and to use their problem solving and problem posing experiences to
learn about themselves as problem solvers and posers. In the process, they help
us to frame yet another dilemma for future research in mathematics education:
Is it always more democratic if students pose the problem? The kinds of
questions that are possible, and the ways that we expect to phrase them, are to
be examined by a critical thinker. Susan Gerofsky (2001) has recently noted
that the questions themselves reveal more about our fantasies and desires than
about the mathematics involved. Critical mathematics education has much to gain
from her analysis of mathematics problems as examples of literary genre.
And finally, it
becomes crucial to examine the discourses of mathematics and mathematics
education in and out of school and popular culture (Appelbaum 1995). Critical thinking in mathematics education
asks how and why the split between popular culture and school mathematics is
evident in mathematical discourse, and why such a strange dichotomy must be
resolved between mathematics as a “commodity” and as a “cultural resource.”
Mathematics is a commodity in our consumer culture because it has been turned
into “stuff” that people collect (knowledge) in order to spend later (on the
job market, to get into college, etc.). But it is also a cultural resource in
that it is a world of metaphors and ways of making meaning through which people
can interpret their world and describe it in new ways. Critical mathematics
educators recognize the role of mathematics as a commodity in our society; but
they search for ways to effectively emphasize the meaning-making aspects of
mathematics as part of the variety of cultures. In doing so, they make it
possible for mathematics to be a resource for political action.
The
history of critical thinking in mathematics is a story of expanding contexts.
Early reformers recognized that training in skills could not lead to the
behaviors they associated with someone who is a critical thinker. Mathematics
education has adopted the model of enculturation into a community of critical
thinkers. By participating in a democratic community of inquiry, it is
imagined, students are allowed to demonstrate the critical thinking skills they
posses as human beings, and to refine and examine these skills in meaningful
situations. Current efforts recognize the limitations of mathematical
enculturation as inadequately addressing the politics of this enculturation.
Critical mathematics educators us the term “critical competence” to subsume
earlier notions of critical thinking skills and propensities. A politically
concerned examination of the specific processes of participation and the role
of mathematics in supporting a democratic society enhances the likelihood of
critical thinking in mathematics.
Suggested
Brown, Stephen I. (2001). Reconstructing school
mathematics: Problems with problems and the real world. NY: Peter Lang.
Glazer, Evan (2001). Using internet primary sources to
teach critical thinking skills in mathematics.
Skovsmose, Ole (2000). Aporism and critical mathematical
education. For the Learning of Mathematics. 20 (1): 2-8.
Brown, Stephen and Walter,
Davis, Philip, and Hersh, Reuben (1986). Descartes' dream: The world according to
mathematics.
Fawcett, Harold (1938/1995). The nature of proof (NCTM 1938 Yearbook).
Gerofsky, Susan (2001).
Genre analysis as a way of understanding pedagogy n mathematics education.
In John Weaver, Marla Morris, and Peter Appelbaum (eds.) (Post) modern
science (education): propositions and alternative paths: 147-176. NY: Peter
Lang.
Keitel, Christine, Klotzmann, Ernst, and Skovsmose, Ole (1993).
Beyond the tunnel vision: Analyzing the relationship between mathematics,
society and technology. In Christine Keitel and Kenneth Ruthven (eds.), Learning
from computers: mathematics education and technology, 243-279. NY:
Springer-Verlag.
National Council of Teachers of Mathematics (NCTM)
(1989). Curriculum and evaluation standards for school mathematics
National Council of Teachers of Mathematics (NCTM)
(2000). Principles and standards for school mathematics.
O’Daffer,
Phares G. and Bruce Thomquist (1993). Critical thinking, mathematical
reasoning, and proof. In Patricia S. Wilson (ed.), Research Ideas for the
Classroom: High School Mathematics , NY: Macmillan/NCTM.
Powell, Arthur, and Frankenstein, Marilyn (1997) Ethnomathematics: Challenging eurocentrism
in mathematics education.
Skovsmose, Ole (1994). Toward a philosophy of critical mathematics education.