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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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