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essay
Inviting Professional Growth:
The Case of the Algebraic Thinking Toolkit
Mark Driscoll
Education Development Center
Background
How do teachers change? Or, more to the point, how do teachers change
for the better and grow professionally? Like many of you, my most accessible
source of data on this question is my own experience.
Back in the 1970s, I was teaching in an alternative high school in St.
Louis. The school was designed with a particular teaching model in mind-that
offered by the clinical psychologist Carl Rogers in his writings. I needn't
say much here about the Rogerian approach. Suffice it to say that teaching
was seen as a process of guiding students through discovery, analysis,
critical thinking, and problem solving. All of which constituted a significant
departure from my own schooling, where I had been taught quite a bit of
mathematics through the well-established "I tell, you listen, then
practice" approach. That traditional approach, which had been so
good to me in my mathematics education, had failed the kids I now was
teaching-most of them school dropouts-and I wanted to develop a very different
way of tapping their thinking and problem solving.
Over the first couple of years, my teaching skills developed fairly well
to reflect the school's philosophy-e.g., my skills in listening, inquiring,
clarifying, and motivating. These skills served me and, I believe, my
students well. Then, one day an interaction in class with a student set
me off on a course of change that transformed my professional life.
I wrote about the incident a few years ago in a journal article (Driscoll,
1995):
Our students were drawn from the city's school dropout population,
and many hadn't been in a mathematics classroom for years. Luckily,
our classes were relatively small, which permitted me on this day to
take a reflective look at what Billy had done on a task. I had put five
decimals, all between zero and one, on the board, and asked the class
to rank them in order of number size-a list something like .06, .607,
.6, .6707, .067. Billy arrayed them in descending order of length,
longest to shortest. I asked: "Which is the number with the smallest
value, Billy?" He pointed without hesitation to .6707. "How
come?" I asked. This time, Billy thought a bit and seemed to be
looking at what he had done through the mists of school memory: "I
don't know. I sort of remember one of my teachers saying, 'The farther
out you go in a decimal, the smaller the number.'"
The remark stuck with me for days. Somehow, I couldn't dismiss it as
a mere memory aberration. Evidently, Billy had been listening several
years before when his teacher had provided what was intended to be a
helpful guide to the symbol system of decimals, but the meaning he had
attached-or, at least, was attaching now-was distorted from what was
intended. I now realized, in a way that was permanently imprinted on
my teaching values, that the mathematical meaning I intended and the
mathematical meaning derived by students were not necessarily the same.
(p.420)
The change this realization initiated was not quick, nor was it easy,
but it eventually made me a much better teacher. The interaction had left
me both confused and intrigued, and I struggled with the implications
of what had happened. Billy had provided me with evidence of something
significant, but my professional judgment was slow to grab hold of the
evidence. In the end, however, after analyzing it with colleagues, comparing
it to similar stories from their experience, conjecturing together about
learners constructing meaning, the evidence and its significance grew
clearer. I started to believe that teaching had to be rooted in an understanding
of learning. Carl Rogers had been on the right track-teaching is about
enabling learning-but, absent knowledge about how learning happens, the
Rogerian philosophy sits partly in the dark.
I was fortunate that my own professional development in the school supported
my shift in professional judgment. We met as a staff each Thursday afternoon
to view a videotape of a teacher's class, taped that week. Typically,
the teacher who was having the most difficulty volunteered to have his
or her class taped. To this day, I remain impressed with our nerve in
instituting such a risky and anxiety-producing process. And I remain grateful
because, through the risk and anxiety, I learned how to teach.
The tapes and the discussions they prompted were a particular blessing
after my conversation with Billy because my colleagues and I could look
for other evidence of mismatch between teacher's intended meaning and
student's taken meaning. Over the past two decades I have tried in various
ways to make this kind of transformative and supportive professional development
more common in the teaching profession. This essay is an account of the
latest effort, the design of a set of professional development material
called the Algebraic Thinking Toolkit (Driscoll, 2000).
Like its companion book, Fostering
Algebraic Thinking, (Driscoll, 1999) the Algebraic Thinking
Toolkit, was born in work with and by teachers in Milwaukee, in an
NSF project called the Linked Learning in Mathematics Project (LLMP).
Though drawn from the Milwaukee work, the toolkit also has roots in my
teaching experience in St. Louis, as reflected in several of the purposes
being infused into the materials. We want the materials, when they are
in the hands of teacher groups using them:
- To focus on important mathematics;
- To focus on mathematics learning, as a foundation for teaching mathematics;
- To focus on evidence-based judgment; and
- To establish structures/norms/expectations for similar evidence-based,
collegial professional development.
The middle two purposes are particularly challenging to design for. I
will spend the rest of this essay describing what is challenging and how
we try to design to meet the challenges.
A Focus on Mathematics Learning
To keep the focus on learning rather than only on teaching, we have employed
a structure and a conceptual framework. The structure provides
alternative perspectives on mathematics learning, in the context of a
set of mathematics problems (many of them in Fostering Algebraic Thinking)
which we believe represent important mathematics and which are offered
to toolkit users to explore and discuss with colleagues. In every session,
teachers are given a mathematics problem to explore. They discuss their
thinking about the mathematics with colleagues, then use the problem with
students, followed by analysis of student work on the problem at the next
session. This structure can be represented by a triangle:
The structure is not unique to the toolkit. We have used it before (see,
e.g., Kelemanik, et al.,
1997), and so have others. However, the structure suits our purpose especially
well, in that each mathematics problem is looked at from different perspectives:
one's own, colleagues', and students'. For most people engaged in the
structure, the multiple perspectives work to focus attention primarily
on the richness of the mathematics and the different lines of thinking
that can mine the richness. Concerns about teaching the material usually
take a back seat.
While algebra is the focus of the toolkit, the content of algebra is
not essential to the approach to professional development. Other materials,
with other content foci, aim to build better teaching on the foundation
of teachers' deeper understanding of learning. For example, the Number
and Operations, Part 1
& Part 2 materials
(Schifter et.al., 1997) do this in the domain of elementary-level mathematics.
The conceptual framework behind the toolkit is based on the notion that
successful learners of algebra develop certain habits of mind. The toolkit
features three of these habits of mind, the same as in Fostering Algebraic
Thinking: Building Rules to Represent Functions; Abstracting from
Computation; and Doing/Undoing. How do they work to keep attention on
learning rather than teaching? Primarily, we believe, because they cast
light on how a learner gets to learning outcomes, not just on the outcomes
themselves. For example, take one of the NCTM algebra learning expectations
for middle grades (from NCTM, 2000), and featured throughout the toolkit):
- Represent, analyze, and generalize a variety of patterns with tables,
graphs, words, and, when possible, symbolic rules
We posit certain aspects of the habit of mind Building Rules to Represent
Functions as relevant to achieving this expectation:
- Organizing data in ways useful for uncovering patterns and the
rules that define the patterns;
- Noticing a rule at work, and trying to predict how it works;
- Looking for repeating chunks in data that reveal how a pattern
works; and
- Describing the steps of a rule without using specific inputs.
We do this for each of the NCTM algebra learning expectations. When teachers
are engaged in discussing the different ways of approaching the toolkit
mathematics problems, we shape the discussion questions to emphasize the
presence or absence in the various approaches of the aspects of the habits
of mind. For example, here is a small-group discussion question after
teachers work on the mathematics problem called "Carnival Bears":
| "While working on the Carnival Bears problem,
most people use the habit of mind Building Rules to Represent Functions
as they try to organize and chunk the data or information involved
in the problem. How did group members use some of the features of
this habit of mind?" |
Similarly, when we ask the teachers to analyze classroom evidence-in
the following example, student interview data from work on "Carnival
Bears"-we try to shape analysis so that it emphasizes students' thinking,
and not just on whether they got the answer. We ask small groups to discuss:
- "Start by trying to reconstruct the students' thought
processes. Say aloud, and ultimately write down, a sequential
list of the ideas and thoughts that students had when working
on the problem, as evidenced by what students say to one another
in the listening interview transcript.
- What surprises you or interests you about how students evolved
and refined their ideas?
- How do they decide whether an idea, rule, or hypothesis is
correct
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Experiences in the Linked Learning in Mathematics project convinced us
that the focus on learning and student thinking can be quite powerful
for teachers. Consider the words of one teacher, in her written report
about her interview of three students as they worked on "Carnival
Bears." She refers to a transcript excerpt, which she included in
her paper:
"This excerpt is right after they did it with three bears on each
side. Now they had to do it for different numbers of bears. They had
moved the bears successfully and were now looking at the worksheet to
see what might happen for other numbers of times. I tried not prompting
them. I deliberately did not use the word 'pattern' because I wanted
to see how they'd progress if I didn't introduce this leading word.
I guess if they got stuck, I might have jumped in, but I didn't have
to."
Because she went on to reflect that she found herself listening more
to students' thinking processes in class as a result of the project experience-indeed,
so did other project participants-we began to see how an emphasis on evidence
can help teachers refine their judgments. And we began, in the context
of the Algebraic Thinking Toolkit, to design materials in order
to focus on evidence-based judgment.
A Focus on Evidence-Based Judgment
The toolkit comprises four modules, each concentrating on a different
kind of classroom evidence:
- Analyzing Written Student Work
In this module teachers analyze written examples of students' problem
solving. The student examples have been selected to illustrate a range
of mathematical thinking and a variety of problem-solving strategies
and algebraic thinking. Teachers use these same problems in their own
classrooms and collect work from their students to share and analyze
in seminar meetings.
- Listening to Students
This module features the small group interview as the source of information
about student thinking. Teachers first review written accounts, by teachers
from previous projects, of interviews of students working on mathematics
problems in small groups. They also analyze a videotape of three students
working alone on a mathematics problem. Teachers then conduct interviews
of their own students, making written transcripts of segments of the
interviews. These transcripts are the focus of seminar work.
- Questioning in the Classroom
This module emphasizes teacher-student interactions in the real-time
context of the classroom. Using videotapes provided in the toolkit,
teachers first practice observing and analyzing the questions asked
of students by the teachers on the tapes, and analyze videotaped debriefings
between classroom teachers and an observer. Seminar participants then
pair up and conduct observations and debriefings in each other's classrooms.
- Documenting Patterns of Student Thinking
In this module, the focus is on drawing up and analyzing a qualitative
summary of students' responses to individual items on written assessments.
Teachers work on aligning instruction and assessment, using the algebra
learning expectations from the NCTM Principles and Standards for
School Mathematics (2000).
A fundamental challenge confronts anyone who puts evidence at the center
of professional development: to design so that, when teachers discuss
the evidence they bring in, the discussions are based on the evidence
rather than just opinion, and that their analyses and discussions encourage
them to examine and refine their judgments. In the toolkit, we particularly
have in mind the following directions of refining judgment:
- Using evidence to make valid inferences about student understanding;
- Reaching consensus about quality when looking at mathematics tasks;
- Framing questions and structuring tasks so that what is important
and intended is elicited;
- Deciding to make tasks involving important mathematics accessible
to the broadest range of students;
- Determining what are reasonable student answers to a problem when
there is no one "correct" answer; and
- Determining appropriate instructional actions in light of conclusions
from student evidence.
Too often, the only "evidence" that matters is whether the
student got the correct answer. We want teachers to dig deeper. For example,
we provide them four examples of student work on a problem called "Crossing
the River." During their discussion of these pieces of student work,
our materials ask them:
"Describe differences in how the four students
represent the problem. For example:
- How do they begin to describe the situation?
- What ways of communicating do they use-pictures, tables, formulas,
etc.?
- How might their methods of representation affect the mathematical
ideas they uncover?"
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The third question requires the teachers to make judgments, but it follows
two questions that ask them to establish the evidence on which they build
their judgments. This kind of structured discussion can be quite powerful.
On occasion, with this particular discussion prompt, we have seen teachers
recognize how the use of tables typically leads students toward one kind
of algebraic rule building, while drawing pictures and diagrams invites
another kind. This observation serves to reshape judgments about the advantages
of multiple representations for both teaching and learning in mathematics.
Changing Norms of Professional Development
So far, I have covered how we attend to the first three goals for the
Algebraic Thinking Toolkit, but have not said much about the fourth
goal:
- To establish structures/norms/expectations for similar evidence-based,
collegial professional development.
Arguably, this is the most ambitious and most challenging of the four
goals, difficult to design for, but no less important in improving the
teaching and learning of mathematics in this country. We believe that
mathematics teachers will grow professionally and become more effective
to the extent that their typical experience in professional development
engages them with evidence from their practice and challenges them to
refine their judgments about mathematics, learning, and teaching. How
to accomplish this through a set of stand-alone materials, with minimal
expectations for the practitioners who facilitate the use of the materials
by colleagues, is an abiding challenge. Our materials support facilitation,
and they carefully scaffold discussion prompts, but the capacity of materials
to foster learning on their own merits is limited.
As a case in point: nothing can guarantee how teachers will use evidence
from their practice. For example, we cited above a teacher who used interview
evidence to inquire into students' thinking and to shift her judgments
about her instruction. In the same cohort was a teacher who appeared to
have used the interview opportunity to reinforce his existing beliefs:
"I wanted to see how efficiently the students could solve the
problem. My strategy, which shows up in this excerpt, was to question
and guide a bunch in the beginning to make sure the students saw the
complete picture and had a sense of totality about the problem; then
to sit back and watch them work. I am convinced that patterning and
generating rules are not intuitive approaches that students take, and
that you have to let them know it is expected of them."
Of course, the fact that the two teachers chose very different ways to
deal with their interview evidence merely points to a universal in adult
learning: people will differ in how they seek evidence, in how they interpret
evidence, and in how they use evidence. No professional development approach
can guarantee teacher learning outcomes. Nonetheless, we are optimistic
that, as evidence-based approaches to professional development spread-like
our Algebraic Thinking Toolkit, or Number and Operations,
Part 1 &
Part 2, or the "Lesson Study" approach described and recommended
in the recent book
The Teaching Gap (Stigler and Hiebert, 1999)-the norms and expectations
for professional development will shift, and greater numbers of teachers
will find that their practice is changing for the better.
References
Driscoll, M. (2000). Algebraic Thinking Toolkit: Fieldtest version.
Newton, MA:
Education Development Center.
Driscoll, M. (1999). Fostering Algebraic Thinking: A Guide for Teachers
in Grades 6-10. Portsmouth, NH: Heinemann.
Driscoll, Mark (1995). "The farther out you go…": Assessment
in the classroom. Mathematics Teacher, 88 (5), 420-425.
Driscoll, M., Moyer, J., and Zawojewski, J. (1998). Helping teachers
implement algebra for all in Milwaukee Public Schools. Mathematics
Education Leadership, 2, 3-12.
Kelemanik, G., Janssen, S., Miller, B., and Ransick, K. (1997). Structured
Exploration: New Perspectives on Mathematics Professional Development.
Newton, MA: Education Development Center.
National Council of Teachers of Mathematics (2000). Principles and
Standards for School Mathematics. Reston, VA: NCTM.
Schifter, D., Bastable, V. and Russell, S.J. (1997). Developing Mathematical
Ideas. Parsippany, NJ: Dale Seymour Publications.
Stigler, J.W. and Hiebert, J. (1999). The Teaching Gap. New York,
NY: The Free Press.
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