CRESMET Researcher Working in Arizona Math Classrooms
After Mesquite High School mathematics teacher Chuck Weeks teaches a lesson that he and Dr. Thompson have designed, the two of them confer on analyzing how it went and planning improvements. Graduate student Sharon Lima records their conversation. |
Dr. Thompson has been working in the Mesquite classrooms, as well as with teachers from the Higley High School District and other Gilbert high schools, as the principal investigator of a project that the National Science Foundation is funding with $4.5 million over five years. In total, 42 teachers have participated in the project, known as Teachers Promoting Change Collaboratively, or TPC².
With two years left in the grant period, the TPC² team finds that its interventions are yielding some encouraging results:
- Among Ms. Bishop’s 22 students, for example, all of them average, non-honors students, 100 percent said that her class made them want to go on to take four years of high school mathematics courses.
- Fourteen of Ms. Bishop’s 22 Algebra I students enrolled in Mr. Weeks’ geometry class, the only other reformed course option Dr. Thompson is leading at the school (the remaining eight students wished to continue, but either moved or had scheduling conflicts).
- The four Mesquite teachers have recruited the rest of the school’s math teachers to join them in forming professional learning communities in which they study mathematics, develop lessons, and observe and help one another improve their classroom teaching.
Driven by his own 30 years of research and that of others in the field of mathematics education, Dr. Thompson and his team of ASU faculty and graduate students are investigating a new model of professional development for instructors who teach precalculus mathematics courses, such as algebra and geometry.
The researchers are trying to understand what combination of experiences and learning materials can support teachers in developing a more effective way of understanding and teaching mathematics.
The Trouble With Math in the U.S.
With American students consistently scoring near the bottom of those from developed countries in mathematical problem-solving, and too few students entering math and technical careers just as a huge wave of baby-boomers is retiring from industry and academia, the need for reforming teaching to raise achievement and to keep more students excited about math and science has been cited as a pressing national concern by political and business leaders and numerous reports.
In Dr. Thompson’s view, the research strongly indicates that the U.S. culture of mathematics teaching—relying heavily on memorizing formulas and giving scant attention to mathematical reasoning and understanding of concepts—is the fundamental reason for students doing poorly in mathematics and disliking it.
There are two principal elements in the model Dr. Thompson has been researching. Teachers are taking a new, CRESMET-developed graduate course in mathematics, Extended Analysis of Functions, in which they deeply explore the ideas of algebra leading to calculus. And they are joining together in school-based professional learning communities. The collaborations with Ms. Bishop and Mr. Weeks arose in Year 3 of the TPC² project. Dr. Thompson initiated them because he realized that teachers in the project were having trouble visualizing the methods he was proposing. As he coaches and videotapes the Mesquite classes of Ms. Bishop and Mr. Weeks, he is producing video examples, print and multimedia materials that other teachers will be able to analyze and learn from.
What Am I Saying? More Important, What Are My Students Thinking?
Mesquite High School mathematics teacher Shannon Bishop hosted Dr. Thompson in her Algebra I classroom four mornings a week last year. Together they are developing and piloting a new sequence of idea-driven, highly interactive algebra lessons. |
The CRESMET course on functions intensively coaches teachers to fix their attention in the classroom on what students are understanding—on tactics for getting students to express their thinking so that the teacher can nudge it toward clarity.
In other words, under Dr. Thompson’s tutelage, teachers go from being tellers of a tale (their “material”) to being keen observers and elicitors of what their audience (their students) are understanding of that tale.
With coaching from the professor, teachers learn, most importantly, how to pose carefully constructed cascades of probing and facilitating questions that help students to resolve misconceptions and move toward accurate understanding.
As Ms. Bishop says, rather than asking students, “What’s the solution? You teach them to articulate, How did you think in order to produce that solution?”
This “new way” of teaching is radically different from the mathematics instruction that most adults recall from high school and college.
In old school math, teachers stood at the board scratching out equations and describing each step of the operations used to solve them. (In truth, according to a number of research studies, this is how most American teachers still teach.)
Rarely did instructors ask questions such as, “Why does this work?” or “What does this mean?” The students’ learning consisted of mimicking the teacher until they had memorized the operational steps that yielded a solution. Then, for homework, they repeated the calculations several dozen times.
But as Chuck Weeks emphasized in a recent interview in his geometry classroom, that vision has little connection with how mathematics is used in real-life applications.
A former nuclear submarine officer in the U.S. Navy and employee in high-technology industries who came to teaching as a second career, Mr. Weeks tells his students that “nobody in the Navy or industry ever told me to ‘solve for x.’ They told me to solve problems.”
A “Geometry Book of Knowledge” of One’s Own
That distinction was on display recently in Mr. Weeks’ early-morning geometry class. Traditional geometry instruction emphasizes memorizing theorems and proving them mechanically, with little discussion of what the geometry represents, and what problems it can be used to solve, in real situations.
But Mr. Weeks’ students are building their own “Geometry Book of Knowledge” comprising theorems that they have proved and understood and demonstrated they can apply. When complete, it will appear on the class website.
On this occasion, the class opened with Mr. Weeks drawing on the whiteboard a map displaying the location of farmhouses on a grid of intersecting country roads. Working from that physical image, relying on their knowledge of other mathematics and ability to use tools such as compass and straightedge, little by little the students demonstrated why: Given Point B and Line AB the distance from P to AB is the length of the perpendicular segment from P to AB.
Far from being a dispenser of answers in the discussion, Mr. Weeks was an asker of pointed questions, and he directed students’ questions to other students or the class as a whole.
Dr. Thompson, whom the teachers and students at Mesquite call “Dr. T,” sat in a corner of the classroom, largely silent as Mr. Weeks led the class.
Now and again, raising his hand for Mr. Weeks to acknowledge him, Dr. Thompson took a Socratic role. He asked questions—probing questions that prompted the students to spot an error or oversight in their reasoning; facilitating questions that stimulated new insights and ideas.
Even in a summarizing moment of tallying what was learned as the class drew to a close, Mr. Weeks and Dr. Thompson elicited answers from the students by asking questions, rather than delivering a traditional speech repeating the lesson’s main points.
The Debrief
With both Ms. Bishop and Mr. Weeks, Dr. Thompson spent Saturday mornings working with the teachers to develop the upcoming week’s instruction, activities and instructional materials.
With Ms. Bishop, he has now designed curricular materials and lesson plans for a non-honors ninth grade course in Algebra I. The work with Mr. Weeks will result in a new set of materials and lessons for non-honors geometry.
Dr. Thompson began with non-honors classes to generate empirical evidence that the students are capable of handling more difficult concepts that their usual courses provide. But advanced students also need and profit from a more rigorous conceptual approach to math, according to him and other researchers.
The professor and the teachers also talked in depth about the objectives of each lesson and anticipated how it would unfold in the students’ discussions.
The TPC2 project has also bought each teacher an extra planning period to follow the TPC2 mathematics class.
In this extra planning period, Dr. Thompson and the teacher debrief the lesson just delivered. The professor interviews the teacher and together they identify what went well and what might have gone better.
Every meeting of the classes and the following debrief is videotaped by graduate student Carlos Castillo-Garsow, with field notes taken by graduate student Sharon Lima. These tapes and notes serve as data for the TPC² researchers. What the researchers learn from studying this evidence feeds continual improvements in the courses and PLCs, builds the knowledge base of the field, and is resulting in research papers and presentations that faculty and students are delivering in the U.S. and other countries.
Changing How Teachers Think About Math
“I was never satisfied with the traditional teaching of math,” said Mr. Weeks. “I didn’t like math in high school, and I learned a lot more mathematics in physics class than I did in math class.”
He is unimpressed with Arizona’s standardized mathematics assessment, the AIMS test, believing that it drives the kind of rote learning without understanding that alienated him as a student. “When I first saw AIMS,” he explained, “I thought, is that all they want us to do? Solve for x? Where’s the challenging problem solving in real-world contexts?
“With CRESMET, I am learning a new way of teaching that’s context-rich, and really based on understanding the reason for the mathematics, not just doing procedures.”
Ms. Bishop also “grew up in a meaningless curriculum. I didn’t know the whys. I never learned why somebody back in the day figured out that a procedure would work. I was a good math student because I was really good at mimicking the teachers when they worked the procedures. But I never knew why the procedures worked.”
She is critical, too, of her college preparation for teaching, feeling that it gave her just “a handful of activities and ideas for how to make math bearable for the kids.”
“My ASU career never challenged me,” she said, “None of my professors ever talked to me about what to say when kids ask why. I left thinking my job was to help students become better mimics [when working math procedures]. Never did I hear that my job is to help kids see why the mathematics works.”
Working Together in Learning Communities
Professional learning communities (PLCs) for teachers are in fashion in many reform campaigns being played out around the country.
CRESMET learning communities are distinctive in insisting that participants focus on the specific content teachers are teaching, as well as pedagogy for teaching it. The TPC² has used grant funds to buy out time each week for the teachers to participate in PLCs in their schools. The project has also invested heavily in training selected teachers to serve as PLC facilitators.
The researchers have found that achieving a meaningful PLC, in which teachers rigorously explore content and push one another to gain new knowledge and skills, is highly difficult.
Thus, Dr. Thompson and his team now recognize that there are two identifiable stages in the typical growth of a PLC. In Stage 1, project leaders conduct “reflecting on practice sessions.” In this stage, the teachers are learning how to explore mathematical concepts, examine and analyze evidence of what students are learning, and productively coach one another’s teaching. In Stage 2, a genuine, teacher-led PLC emerges, and the participants can chart measurable improvements in their instruction and their students’ learning.
Stage 1—Reflecting on Practice Sessions
Project leaders help teachers learn to focus on student thinking. Teachers gain—
- Understanding of concepts
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Knowledge of pedagogy for teaching those concepts
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Classroom skills for teaching those concepts
Stage 2—Professional Learning Communities (PLC)
Outsider-led sessions evolve into self-directed communities of teaching professionals. Teachers hold—
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Shared goals
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Common repertoire of methods & activities
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Commitment to their PLC community
One finding from Dr. Thompson’s research into the TPC² professional learning communities is that teachers who take a course together evolve more quickly as an effective group.
Ms. Bishop and Mr. Weeks are mentoring new PLCs at Mesquite, whose teachers have not had the opportunity to take the CRESMET functions course. “It’s more difficult for them,” she said. In the first group, “we all went in skeptical about the benefits of teaching math with a foundation of ideas rather than procedures, but we got to grow together and accept Pat’s ideas at our own pace. So we’re taking it slowly with the new group, just focusing now on really probing students’ thinking and reflecting on our own teaching practices.”
Both teachers say that the biggest revelation of this experience has been to realize that they were underestimating their students’ capabilities.
“Pat would suggest some activity,” said Ms. Bishop, “and I’d say, no way can freshmen do that! It’s too hard! But he’d insist, and sure enough, they’d do it. I didn’t have enough confidence in my students. And that’s been the biggest benefit to me of working with Pat, learning that these kids are capable of doing far harder work than I thought they could.”