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Type R Mathematics Teachers
By Douglas A. Grouws and Matthew
S. Winsor*
Are you a Type R mathematics teacher? Think for a few minutes
about how you would teach an important lesson in your school’s mathematics
curriculum. For example, how would your lesson on adding fractions with
unlike denominators, say 1/2 + 1/3, flow? How would the lesson begin?
What would the centerpiece of the lesson be? Jot your thoughts down before
reading on.
This article is designed to help you reflect on your teaching
practice and to gain insight into your philosophy of how students learn.
We will examine several approaches to planning a lesson, each with a different
starting point and underlying perspectives on student learning. By thinking
about these different approaches and comparing them to your own, you would
develop a clearer picture of your own theory of student learning. Such
reflection on teaching is much more valuable than classifying teachers
by type, and so, we will not be defining a Type R teacher in this article,
but you could interpret R to mean “reflection.” We hope that
the title, however, has sparked your interest in reading the article and
that you are not disappointed in our decision not to define Type R teachers.
Please read on.
Teaching styles and beliefs vary widely among well-qualified
mathematics teachers (Grouws & Smith, 2000), and thus, how they plan
lessons also differs. In planning the fraction lesson mentioned above,
some teachers will focus their attention on identifying the prerequisite
concepts and skills that students need to understand and master the lesson
objective. For instance, these teachers believe it is important for students
to (1) understand that two fractions may have the same value (e.g., 1/2
and 3/6), (2) be able to add fractions with like denominators, and (3)
know basic terminology (e.g., numerator and denominator). Once these prerequisites
are determined, the initial phase of the lesson focuses on making sure
these building blocks for learning are in place. Other teachers may approach
this same lesson by first locating a familiar real-life setting that is
interesting to the students. For example, these teachers might use the
following problem to begin the lesson. If a student brings a bag of marbles
to school and gives 1/2 of the marbles to a friend and 1/3 of the marbles
to another friend, then what fraction of the bag of marbles has he given
away? This approach reflects a belief that familiar situations facilitate
student assimilation of new ideas and provide motivation to learn as well.
Still other teachers may approa ch the fraction lesson by
constructing a challenging mathematical task to begin the lesson. For
instance, students might be asked to find two fractions in lowest terms
with unlike denominators whose sum is 5/6. The emphasis of the lesson
that follows this plan is to have students work in small groups to solve
the problem, share their ideas and solution methods, and finally formulate
their own generalizations that lead to dealing efficiently with routine
situations such as 1/2 + 1/3. The central beliefs behind such a lesson
plan are that mathematics should play an early and leading role in every
lesson and that student interaction has great value in learning situations.
In summary, each of the lesson plans above highlights a value or values
held by the teacher making the plan. In the first plan, there is an emphasis
on an orderly development of ideas, and value is placed on the logic and
structure of mathematics. In the second plan, the emphasis is on using
real-life contexts to foster learning and increase motivation. The third
plan recognizes the potential of using challenging mathematical tasks
and verbal interaction. Do any of these approaches to planning lessons
match the way you would plan the fraction lesson? What values and beliefs
form the foundation for your approach to planning this lesson and other
lessons in general?
It is beyond the scope and intent of this article to debate
the merits of these approaches to planning lessons, although others have
taken strong positions in journal articles and in research literature.
We are happy to leave such debates to others, because we believe that
a good lesson plan should take careful account of the particulars of the
teaching situation; that is, the students, the teacher, the curriculum,
the culture, and the teaching environment. We obviously do not have this
information for your classroom, and to make suggestions without it would
be difficult at best.
We do hope, however, that this article has caused you to
think about how you plan lessons and that this, in turn, has helped you
gain insight into your beliefs about how students learn and what you value
in a teaching situation. Reflecting on our work and gaining insight from
it makes us better teachers (Senger, 1999).
References
Grouws, D. A., & Smith, M. S. (2000). Findings
from NAEP on the preparation and practices of mathematics teachers. In
E. A. Silver & P. Kenney, (Eds.), Results from the Seventh Mathematics
Assessment of the National Assessment of Educational Progress. Reston,
VA: National Council of Teachers of Mathematics.
Senger, E. S. (1999). Reflective reform in mathematics:
The recursive nature of teacher change. Educational Studies in Mathematics,
27(3), 199-221.
*Douglas
Grouws and Matthew Winsor are faculty members of University of Iowa. Dr.
Grouws was a Visiting Scholar with the Pacific Consortium at PREL in 1999.
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