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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:
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To focus on important mathematics;
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To focus on mathematics learning, as a foundation for teaching mathematics;
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To focus on evidence-based judgment; and
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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):
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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:
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Organizing data in ways useful for uncovering patterns and the rules that define
the patterns;
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Noticing a rule at work, and trying to predict how it works;
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Looking for repeating chunks in data that reveal how a pattern works; and
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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:
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"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.
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What surprises you or interests you about how students evolved and refined
their ideas?
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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:
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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.
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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.
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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.
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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:
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Using evidence to make valid inferences about student understanding;
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Reaching consensus about quality when looking at mathematics tasks;
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Framing questions and structuring tasks so that what is important and intended
is elicited;
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Deciding to make tasks involving important mathematics accessible to the
broadest range of students;
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Determining what are reasonable student answers to a problem when there is no
one "correct" answer; and
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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:
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How do they begin to describe the situation?
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What ways of communicating do they use-pictures, tables, formulas, etc.?
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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:
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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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