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Understanding the pedagogical theories that
underlie school mathematics reform
Research shows that most mathematics
teachers, including prospective teachers, have strongly-held
beliefs about student and teacher’s roles, desirable
instructional approaches, students’ mathematical
knowledge, how students learn and the purposes of schools
(Thompson, 1992). These beliefs have mostly developed as a
result of the teachers’ own schooling. Although rarely
made explicit, the following views of knowledge, learning and
teaching lie behind what takes place in most traditional
classrooms:
• Knowledge is a body
of established facts and techniques that can be broken down and
transmitted to novices by experts (positivistic view of knowledge).
• Learning results
from acquiring isolated bits of information and skills through
listening, watching, memorizing and practicing (behaviorist view of learning).
• Teaching is the
direct transmission of knowledge from teacher to student; it
takes place as long as the teacher provides clear explanations
for the students to absorb (direct instruction view of teaching) (Borasi & Siegel, 1992, 2000).
In contrast, the teaching practices
recommended by the NCTM Standards (NCTM, 2000) and illustrated
in our classroom vignette are grounded in views of knowledge,
learning and teaching informed by a constructivist perspective
(e.g., Brooks & Brooks, 1999; Davis, Maher & Noddings,
1990; Fosnot, 1996). Although different interpretations of
constructivism exist, current school mathematics reform efforts
are generally characterized by the following constructivist
assumptions:
• Knowledge is
socially constructed through human activity, shaped by context
and purposes, and validated through a process of negotiation
within a community of practice. Thus, it is always tentative
rather than absolute. However, although knowledge is
provisional in this paradigm, it does not mean that
“anything goes.”
• Learning is a generative process of making meaning
that builds on personal knowledge and social interactions. This
process may be stimulated by perceived dissonance. Prior
knowledge, context and purpose play critical roles in the
shaping of learning situations.
• Teaching is
facilitating students’ learning by creating a learning
environment conducive to inquiry, setting up problem-solving
situations to stimulate both student interest and cognitive
dissonance about important mathematical ideas, and supporting
students’ attempts to solve problems and make sense of
mathematical concepts (Borasi & Siegel, 1992, 2000).
To fully appreciate the constructivist
pedagogical approach recommended in the NCTM Standards,
teachers need to identify and understand the non-traditional
theories of teaching and learning mathematics and the research
supporting such approaches.
Understanding students’ mathematical
thinking
One of the main challenges that the
teacher in our vignette experienced during her inquiry on area
was interpreting her students’ thinking and responding
appropriately, especially when students proposed new strategies
or formulas for computing area and explained how they got their
results. The teacher benefited considerably by having already
investigated a range of possible strategies and solutions to
the open-ended tasks she posed – although some of the
students’ strategies still took her by surprise! Indeed,
understanding students’ mathematical thinking is
especially critical in any constructivist approach if teachers
are to design instructional experiences that help students
build on their existing knowledge (Confrey, 1991).
Research on Cognitive Guided Instruction
(CGI) has provided both theoretical arguments and empirical
evidence claiming that mathematics teachers benefit from
knowing about their students’ prior knowledge and ways of
learning specific mathematical concepts (Carpenter &
Fennema, 1992; Fennema, Carpenter & Franke, 1997).
Knowledge of child-constructed procedures is a crucial
prerequisite for designing learning experiences that capitalize
on, rather than override, the informal mathematical knowledge
children bring to school. For example, many elementary teachers
are surprised to learn that children often develop their own
procedures for solving simple arithmetic problems before
they enter school. Knowing this fact can help teachers rethink
how arithmetic operations might be introduced.
Further empirical support for the value
of teachers’ knowing how students think comes from the Integrating Mathematics Assessment (IMA) project. This project focused on making
teachers aware of the key features of student thinking about
fractions. As a result, students made significant gains in
solving problems involving fractions (Gearhart, Saxe, &
Stipek, 1995).
While the results of studies like CGI and
IMA are compelling, it is reasonable to ask whether we
should expect teachers to acquire research-based knowledge
about student thinking in all the mathematical areas they will
teach, especially when most topics taught in secondary school
are not as well researched as basic arithmetic and rational
numbers. Rhine (1998) suggests that rather than trying to
create such a knowledge base among teachers, it may be more
important to foster a new attitude, one that values analyzing
student thinking as part of teachers’ everyday practice
and provides strategies to help them do so.
Learning to use effective teaching and
assessment strategies
One element that most distinguished the
inquiry on area in our classroom scenario was the extensive use
of teaching practices that are usually absent from traditional
mathematics instruction. These included, for example,
orchestrating group work using a variety of techniques, such as
the initial “think-pair-share” activity;
facilitating class discussions in which students shared results
and jointly constructed new knowledge; using effective
questioning techniques to synthesize key mathematical ideas;
and assessing students’ learning in multiple ways, such
as the performance assessment in which students created an area
formula for a star.
The pedagogical recommendations
articulated in the NCTM Standards (NCTM, 1991, 2000) call
specifically for teaching practices like these that are not
currently used by many mathematics teachers, especially at the
secondary level (for comprehensive lists of such practices, see
Koehler & Grouws [1992] and Borasi & Fonzi [in
preparation]). Non-traditional practices include not only
facilitating what goes on in the classroom as lessons develop
but also planning and assessing lessons effectively. Assessment
has received special attention recently (e.g., Bright &
Joyner, 1998; Lesh & Lamon, 1992; NCTM, 1995; Webb &
Coxford, 1993) because determining what students know is
necessary for teaching effectively within a constructivist
paradigm. It is also critical for documenting the outcomes of
reform efforts.
Learning to use novel teaching practices
appropriately is not easy. Research on how people learn complex
tasks may shed some light on what it takes teachers to adopt a
new teaching practice. For example, Collins and his colleagues
(1989) have suggested the following three-phase process for
learning a complex task:
1. Modeling – The learner observes and examines
how an expert engages in the task.
2. Scaffolded
practice – The learner
engages in the task himself/herself, but with the help of an
expert and/or of other supporting structures.
3. Independent
practice – The learner
engages in the task without support.
Clearly, using new teaching practices
effectively goes far beyond simply knowing they exist. While
mathematics teachers should learn about a variety of teaching
strategies to enrich their repertoire of resources, they should
also have the opportunity to personally experience these
practices in supported situations in order to evaluate fully
their pedagogical potential. It is also critical for teachers
to learn not only to use specific practices well but also to
appreciate their strengths and limitations so they can choose
practices most appropriate to an audience and to unique
instructional goals.
Becoming familiar with exemplary
instructional materials and resources
When reading about a well-designed, complex experience such as the inquiry on area described in our vignette, teachers might feel daunted by the prospect of creating similar lessons on their own. Fortunately, today’s mathematics teachers are not expected to always create innovative units on their own as they may take advantage of the many exemplary instructional materials informed by the NCTM Standards that have been produced in recent years. As argued by Russell, this by no means demeans the professionalism of teachers: Curriculum materials, when developed through careful, extended work with diverse students and teachers, when based on sound mathematics and on what we know about how people learn mathematics, are a tool that allows the teacher to do her best work with students... . It is not possible for most teachers to write a complete, coherent, mathematically sound curriculum. It is not insulting to teachers as professionals to admit this. (Russell, 1997, p. 248)
Exemplary instructional materials may
consist of replacement units, which are individual units
designed to replace parts of the traditional curriculum while
expanding the instructional goals and introducing some
effective teaching practices or of comprehensive curricula.
These consist of a sequence of units intended to totally
replace the current mathematics curriculum at either
elementary, middle or high school. Among the latter group, a
set of instructional materials consistent with the NCTM
Standards has been recently developed with support from the
National Science Foundation (NSF) (see Figure 7 for a complete
list of these comprehensive curricula and their websites’
addresses). Additional exemplary mathematics curricula have
been identified in a study by the U.S. Department of Education
(U.S. Department of Education’s Mathematics and Science
Education Expert Panel, 1999).
Continued
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CHAPTER 1 continued