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March 1, 1993
Vol. 50
No. 6

Thinking Mathematics

Instructional Strategies
Traditional staff development is replete with inservice in cavernous halls, admonitions about bringing up test scores, lots of talking at teachers, and publishers' sessions on “how to use the teacher guide” for new textbook adoptions. Absent are interaction, sense-making, sufficient hands-on experiences related to students, a well-grounded understanding of rationale or continuum, and follow-up. In rare instances when rich hands-on activities are provided, they are often presented or received with no understanding of their place in a subject field. Thinking Mathematics seeks to change this approach. In the Thinking Mathematics collaboration, teachers have become equal partners with researchers to create staff development specifically aimed at mathematics instruction.

How it Works

Researchers at the Learning Research Development Center (LRDC) and five Visiting Practitioners have worked together to create research syntheses and training modules related to how children learn mathematics (Thinking Mathematics, Volume 1: Foundations and Thinking Mathematics, Volume 2: Extensions). The substance of this research has been conceptualized as Ten Principles to guide the teaching of mathematics.
With these materials, teachers are trained in eight-day summer sessions for two successive years through the American Federation of Teachers' Education Research and Dissemination (AFT ER&D) Network. Midway through each year, participants are reunited at a winter ER&D weekend workshop in Washington to share their fall experiences and receive additional training.
First-year workshops focus on the 10 principles of Thinking Mathematics. The principles are then combined with specific content research and applied to counting, addition, and subtraction. In the second year, multiplication, division, and initial proportional reasoning are addressed. Rather than offering a set of packaged activities and lessons, the goal is to change dispositions and understandings about math teaching. The kind of knowledge teachers receive enables them to apply the principles to their local curriculum, create activities uniquely relevant to their classrooms, and understand how other materials and activities fit with their goals.
Some 91 teachers who have been trained nationally are expected to implement what they have learned in their own classrooms and be trainers for other teachers in their home districts. As local teachers are trained, they meet regularly to reflect on their classroom experiences, offer mutual support, and share the excitement of what students are accomplishing. Representatives from the national project visit each site the year immediately following training to assist with local training, demonstrate lessons, and answer any questions about Thinking Mathematics. In all, some 500 teachers have been trained in the principles of the project.

Implementation Results

According to a recent survey of 151 teachers trained in Thinking Mathematics, 71 percent of them now use the 10 elements of the program about three times a week as opposed to twice a month before their training.
Students have also shown positive effects of the project. Researchers at the University of Pittsburgh compiled data from student scores on standardized tests for the group of pilot classes in 1990–91 (Hojnacki and Grover 1992). Although standardized tests are not a good measure of the goals of Thinking Mathematics (or NCTM Standards), the Thinking Mathematics students did at least as well as, if not better than, non-Thinking Mathematics' students on both Computation and Concepts and Applications measures on these tests. This result inspires optimism because scores are expected to drop when teachers try new curriculum methods because it takes time for teachers to become as proficient with the new skills as they were with the old.
The report also stated that there are multiple indications that student learning and attitudes were enhanced by teachers' participation in the Thinking Mathematics program. A positive attitude towards math was evident at all sites, including populations with low socioeconomic and high minority composition, learning disabled classes, and suburban settings. Teachers have reported that students like mathematics and they are confident in their abilities to do mathematical computations and problems. They are also willing to persevere to obtain solutions to math problems. These observations have been corroborated by an attitude survey administered to students themselves.
What has led to this success? A change in teaching that produces such results does not stand much chance of survival if care is not taken to construct a supportive environment. Thinking Mathematics is part of a program that recognized this need more than 10 years ago.
In the early 1980s, the American Federation of Teachers' (AFT) Educational Research and Dissemination Program (ER&D) won the first American Educational Research Association award ever presented for successfully connecting teachers to research. AFT knew it wasn't sufficient to simply print and distribute research. The ER&D process involves investigating the knowledge that is available, synthesizing it for teachers in understandable terms, talking about classroom implications with colleagues, using what is helpful, and having an opportunity to give feedback to the research community. This differs greatly from programs that deliver prescriptions for what teachers should do. At the core of the program is a belief that the effort must be non-threatening and non-judgmental; then teachers will take the research and use it according to their best professional judgment.
The original ER&D units focused on effective teaching research. Thinking Mathematics has moved the ER&D process into subject matter and discovered that even more supports are crucial. The content is more complex and must be taught daily. As they have implemented Thinking Mathematics, teachers have also recognized how inadequate traditional school structures and practices are for adequately supporting a new teaching paradigm. Teachers discovered they needed to assess differently, and some wanted to spend “book money” on manipulatives. Lessons didn't fit into neat time slots. Required planning and follow-up—and the energy to sustain them—demanded serious consideration about reconstructing a teacher's day and the perception of teacher work.
Teachers are not likely to venture into a new paradigm if they think the risk is too great. So, it is interesting to see what Thinking Mathematics teachers believe enabled them to change. Four influences stood out: the local union, national AFT support, colleagues, and the nationally trained leaders (Math Site Coordinators).
The union, which is often seen as an obstacle to change, led the way. In many instances, the union made essential connections and negotiated for time, stipends, or credit. It gained releases from textbook requirements, negotiated a procedure to suspend certain regulations, and was available in case the teacher evaluation process went awry. The union's willingness to enter into partnership with the district to improve learning added to every teacher's stature. The local union played a crucial role in establishing a non-threatening atmosphere for teachers willing to take the risks.
Principals and math departments also influence the process. A supportive principal made efforts easier. Work succeeded even where a principal was not particularly helpful but did nothing destructive. Yet an overtly hostile principal could thwart the change effort. On the whole, “math departments” were perceived as even less helpful than principals, although districts differed. The biggest obstacle teachers identified, however, was districts' use of standardized tests, a perception in line with recent studies about what drives curriculum.
  • adequate professional development, including a sufficient knowledge base and enough time;
  • support from the union;
  • a credible leadership team;
  • support from the school administration—philosophically and materially;
  • continuing support by those who understand the new practice(s);
  • regular opportunities for reflection and problem solving;
  • relief from the constraints of traditional evaluation and testing while new ways are learned;
  • hope that structural changes teachers begin to find necessary to sustain their efforts can indeed happen.
The power is here. Teachers have approached implementation of the Thinking Mathematics project with almost religious fervor. They say with conviction they will “never go back.” But whether they can sustain their heroic efforts will depend in part on the reality of the environment surrounding them. If support is adequate, they themselves can become agents of significant change.

Hojnacki, S. K., and B. W. Grover. (1992). “Thinking Mathematics: What's In It for the Students?” Paper presented at the annual meeting of the American Educational Research Association, San Francisco.

Alice J. Gill has been a contributor to Educational Leadership.

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