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Thinking About Students' Thinking
George W. Bright
The University of North Carolina at Greensboro
In my view, one of the fundamental characteristics of "mathematics reform
pedagogy" is that teachers are encouraged to understand the mathematics
thinking of each of their students and then use that knowledge in instructional
planning. As compared to typical pedagogy of 25 years ago,
explanation/demonstration by the teacher is relatively less important and using
questions to probe students' thinking is relatively more important. One
implication of this shift is that teachers have the burden of making sense of
what students say in their explanations of their thinking. Making sense
requires understanding not only the relevant mathematics that might underlie an
explanation but also the trajectory of development of students' (i.e.,
novices') mathematical understanding. Understanding each student's
mathematical development becomes a critical focus of a teacher's attention.
Questioning
Teachers can get information about students' thinking in a variety of ways, for
example, questioning, observation, analysis of students' writing, tests, and
project work. Probably the most important of these is questioning, since that
is the most interactive of these techniques. Through a series of questions, a
teacher can accomplish one or more of several different goals. First, carefully
sequenced questions can lead students to a correct answer. Indeed, many
teachers are very good at getting students to say almost anything desired, by
"leading the student down the garden path." The down side of this,
though, is that it may not be clear to the student why those questions were
asked or what the final, correct answer means.
Second, questions can help a student process, or even self-correct, an answer.
Teachers regularly recognize when a mistake has been made in computing or
processing or choice of operation, etc. Asking a question about the student's
work may have the effect of "slowing the student down" in order to
re-analyze that work. That rethinking may be sufficient reflection for the
student to see the error and correct it.
Third, questions can help expose a student's thinking so that the teacher can
understand it. Sometimes teachers really don't "see" immediately how
a student is thinking or what mathematics a student is using to generate a
response. In that situation, questions can help reveal a student's thinking so
that the teacher can determine whether the thought process is acceptable. It is
this kind of questioning that is probably most useful for teachers to use as
they learn to understand students' thinking, but this is also the kind of
questioning that is least familiar to teachers. It is this kind of questioning
that teachers need the most help at learning how to do.
Questioning has to occur "in real time" during face-to-face
interactions; teachers have to create questions immediately and with very
little reflection time. Teachers, obviously, have more time outside of class
than they do in class, so an alternate, though perhaps not as effective,
technique for understanding students' thinking is analysis of students'
writing. Students might (a) record solutions to problems in a journal, (b)
respond to specific prompts posed by the teacher, (c) create reports of
projects, or (d) maintain portfolios of their work over time. In many ways, the
"easiest" way for teachers to get students to respond to prompts,
particularly in middle and high school classes, is to include those prompts on
a test. Probably the most frequently used of these prompts is "show your
work." The difficulty with this prompt is that teachers and students often
do not share a common perspective on what constitutes "work." It is
important that as part of instruction, teachers and students reach an agreement
on what should be written down so as to allow the teacher to understand the
thinking that lies behind the solution to a particular problem.
The usefulness of writing as a window into thinking is influenced by the
language and writing ability of the students. It is less useful in primary
grades than in high school, and it is less useful when students are expected to
write in their second language. We must consider, however, that mathematical
symbolism is itself a second language for most students. Effort must be put
forth, both by teachers in teaching and by students in learning, to master this
form of written communication.
Professional Development Focused on Understanding Students' Thinking
There are several professional development projects that have helped teachers
gain the skill and knowledge of processes for understanding students' thinking.
The one I am most familiar with is Cognitively
Guided Iinstruction (CGI), described in a joint publication of the
National Council of Teachers of Mathematics and Heinemann (Carpenter et
al.,1999). CGI is an approach to teaching mathematics in which knowledge of
children's thinking is central to instructional decision making. Teachers use
research-based knowledge about children's mathematical thinking to help them
learn specifics about individual students and then to adjust instruction to
match students' performance. Teachers learn to assess students' thinking and
then use that knowledge to plan instruction. Fosnot's (1996) principles of
learning as development, reflective abstraction, and dialogue seem to be
integral to the implementation of CGI by teachers.
Learning as development seems critical if teachers are going to help students
increase the level of sophistication of their thinking. Teachers need to have
both a vision of what level of sophistication they want students to attain and
a sense of where each student is along the path toward that vision. In
addition, teachers need to be thinking about how to organize instruction so
that students move in the direction of the desired goals.
Reflective abstraction is important as a mechanism by which teachers can improve
their instruction. It is also important for children, in the sense that the
children must reflect on their own mathematical development. Thus, teachers
have to reflect on their own work and to help students reflect on their
thinking. In grades K-3, there would seem to be a special obligation to help
students develop the skills necessary for reflection; for example, through
listening to their peers explain solutions, learning to ask questions to
clarify those presentations of solutions, and keeping written records of their
solutions that can form data for reflection.
Dialogue is especially important in CGI classrooms, but for dialogue (as opposed
to multiple monologues) to occur, all parties involved must listen to each
other. Within the implementation of CGI, teachers have special responsibilities
for listening. During early implementation of CGI, teachers tend to listen to
children's explanations with the expectation that they (the teachers) are
supposed to understand children's thinking simply on the basis of what the
children say; this type of listening can be called "passive
listening." In contrast, teachers who are especially effective at
implementing CGI seem to understand that making sense of children's thinking is
a joint responsibility; both the teacher and the children must contribute to
generating shared knowledge about that thinking. These teachers often engage
children in conversation about solution strategies so that both the teacher and
the children come to understand how what is being said reflects real thinking.
This type of listening can be called "active listening." A teacher's
development of either passive or active listening skills is consistent with
allowing students to develop personal meaning.
One other advantage of listening is that it provides time, both for the students
and the teacher. When a teacher chooses to listen, all of the children in the
class have time to think and to develop meaning. When listening is combined
with questioning, the time required increases further, since students need time
to respond to those questions. The teacher has the opportunity to use the
questions to focus students' attention on particular aspects of a solution, so
that their reflection on those aspects of thinking is more intense. Having
patience to allow students to internalize deep understanding of mathematics
concepts is an essential characteristic of effective teachers.
Recent curriculum projects have also attempted to help teachers understand
students' thinking; these materials can be an effective vehicle for
professional development. For example, in Investigations materials
(e.g., Kliman et al., 1996), there are (a) dialogue boxes, which illustrate how
children might respond to particular questions, (b) lists of questions that
teachers can use to probe students' thinking, and (c) suggestions for what to
look for during observations. In Connected Mathematics Project materials
(e.g., Lappan, et al., 1998), a launch/explore/summarize strategy is suggested
for the activities, and within the notes for each of these there are numerous
suggestions for ways to assess what students know and can do. These aids for
teachers are designed to help teachers use classroom interactions to deepen
their understanding of children's thinking, with the hope that teachers will
then use this information to make better instructional decisions. This process
is what has been called classroom assessment (Bright and Joyner, 1998).
The pay-off for helping teachers become better at classroom assessment is
better instructional planning, which in turn should lead to greater student
learning.
Closing Remarks
Much research has tried to help understand the trajectory of students'
thinking. The research underlying CGI, for example, indicates a consistent
pattern of development of students' thinking as revealed in their solution
strategies. When teachers understand this path of development, they can
interpret where a child's thinking is and can project ways of helping that
child move to a more sophisticated level of thinking. Making instructional
decisions based on a clear understanding of children's thinking seems to have
significant pay-off for improved learning of mathematics. Unfortunately, not
much seems to be known about how to recreate these effects across all levels of
mathematics instruction.
Professional development materials and programs should help teachers understand
more about students' thinking. In addition to the obvious pay-off for student
learning, teachers themselves also seem to be renewed and become more excited
about being successful teachers (Vacc, 1998). Such renewal is certainly a
worthy goal in itself, but the more important outcome is better student
learning that follows from teachers' renewed excitement about teaching.
References
Bright, G. W. and Joyner, J. M. (1998). Classroom Assessment in Mathematics: View
from a National Science Foundation Working Conference. Lanham, MD:
University Press of America.
Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., and Empson, S. B.
(1999). Children's Mathematics: Cognitively Guided Instruction.
Portsmouth, NH: Heinemann and Reston, VA: National Council of Teachers of
Mathematics.
Fosnot, C. T. (1996). Constructivism: A psychological theory of learning. In C.
T. Fosnot (Ed.), Constructivism: Theory, Perspectives, and Practice (pp.
8-33). New York, NY: Teachers College Press.
Kliman, M., Tierney, C., Russell, S. J., Murray, M., and Akers, J. (1996). Investigations
in Number, Data, and Space: Mathematical Thinking at Grade 5. Palo
Alto, CA: Dale Seymour.
Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., and Phillips, E. D.
(1998). Connected Mathematics: Growing, Growing, Growing: Exponential
Relationships. Palo Alto, CA: Dale Seymour.
Vacc, N. N. (1998). Becoming a teacher leader in mathematics education. Mathematics
Education Leadership, 2(3), 6-16.
For Further Information
Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C-P., and Loef, M.
(1989). Using knowledge of children's mathematics thinking in classroom
teaching: An experimental study. American Educational Research Journal,
26, 499-531.
Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., and
Empson, S. B. (1996). A longitudinal study of learning to use children's
thinking in mathematics instruction. Journal for Research in Mathematics
Education, 27, 404-434.
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