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Tom Romberg

Researchers and educators have learned a lot about the teaching and learning of mathematics in schools during the past 30+ years, much of it from the work of the emerging mathematics education research community. However, UW-Madison education professor emeritus Tom Romberg says researchers and educators still have lots to learn about the teaching and learning of mathematics in the “messy” social environment of school classrooms.

For research to be productive and useful in any discipline, it must be conducted within a research community. Mathematics education is a relatively young academic field, and research on the teaching and learning of mathematics is even younger. A research community in mathematics education has only gradually begun to emerge in the past half century, and the process is still ongoing.

Reviewing mathematics education research, Romberg identifies 12 findings that distinguish what is known today from what was known in the 1960s. The first five come from research conducted by the National Center for Improving Student Learning and Achievement in Mathematics and Science on student learning with understanding, located at WCER.

1. Educators have underestimated students’ capability to learn mathematics with understanding. Given the opportunity to explore a domain using a set of structured activities, all students can learn important mathematics with understanding.
2. Learning the concepts and skills in a mathematical domain requires that students be engaged in a rich set of structured activities over time. Specifically, students need an opportunity to investigate problems that encourage mathematization. By that is meant problems
   
   
   • that are subject to measurement and quantification;
   • that embody quantifiable change and variation;
   • that involve specifiable uncertainty; and
   • that involve our place in space and the spatial features of the world we inhabit and construct.

   In addition, problems should encourage the use of languages for expressing, communicating, reasoning, computing, abstracting, generalizing, and formalizing. Such problems require systematic forms of reasoning and argument to help establish the certainty, generality, consistency, and reliability of students’ mathematical assertions.

3. Learning with understanding involves more than being able to produce correct answers to routine problems. Mathematics should be viewed as a human activity that reflects the work of mathematicians—finding out why given techniques work, inventing new techniques, and justifying assertions. Learning with understanding occurs when it becomes the focus of instruction, when students are given time to discover relationships and learn to use their knowledge, and when they reflect about their thinking and express their ideas. Doing mathematics cannot be viewed as a mechanical performance or an activity that solely involves following predetermined rules.
4. Modeling and argumentation are important aspects of mathematics instruction that foster learning with understanding. Modeling offers a way to represent phenomena in the world by means of a system of theoretically specified objects and relations. Modeling is critical in developing understanding in a domain. In classrooms, it is important to consider modeling as a cycle comprising model construction, exploration, and revision. Additionally, as students make conjectures, they need to learn to justify them. Thus, argumentation and standards of evidence, with an emphasis on promoting students’ skills for generalization in mathematics, are critical.
5. Student learning should be seen as a product of situated involvement in a classroom culture. Learning with understanding is a product of interactions over time with teachers and other students in a classroom environment that encourages and values exploration of problem situations, modeling, argumentation, and the like. The very nature of mathematics is defined communally, making participation by all not only a fundamental civil right, but also a critical prerequisite to the continued vitality of mathematics in the nation.

Romberg also points to four general findings from research on teaching. The reform approach to teaching represents a substantial departure from most teachers’ prior experience, established beliefs, and present practice.

6. Teacher knowledge of student thinking is critical. Teachers need to listen and hear what students are saying as they conjecture and build arguments. Teachers also need to judge the quality of students’ justifications and explanations in examining student work.
7. Teachers must understand the structure of mathematical domains. Knowledge of the network of relationships in a domain is critical when making decisions about student understanding and the sequence of instruction.
8. Rather than just cover the content in a textbook, teachers need to base instruction on the needs of their students. This finding follows directly from the previous two. If teachers know the level of their students’ thinking, and understand how it fits within the structure of the domain of interest, then they can design appropriate instruction.
9. Professional development cannot be done well in isolation. Professionalism is the key to quality classroom instruction, but it can only be achieved if teachers join together to collaboratively undertake professional assistance.

Finally, one of the key aspects of teaching has always been monitoring students’ progress. Mathematics teachers have traditionally done this by giving curriculum-based quizzes and tests, scoring and counting the number of correct answers on each, and periodically summarizing student performance in a letter grade. Unfortunately, this standard approach to assessment is not consistent with what educators now know about how students learn mathematics.

10. External tests have an effect on instruction in that teachers take classroom time to prepare students to take the test. However, such tests are not often well tailored to classroom instruction, nor are the results useful for monitoring growth over time.
11. Curriculum-based quizzes and tests tend to include items that are very similar to exercises in daily lessons and that reflect reproduction, definitions, or computations. Such instruments rarely contain items that require students to relate concepts or solve nonroutine problems.
12. Most mathematics teachers are aware that they acquire considerable informal evidence about their students, yet they rarely use such evidence in judging student progress. In fact, current data show that most mathematics teachers could benefit from professional development designed to help them learn how to make good use of their informal assessments.”

Schools are social and political organizations that operate within a coherent set of traditions. Changing such organizations involves more than just making the changes educators believe need to be made in mathematics classrooms; it involves understanding and dealing with the partisan political and ideological perspectives on schooling that permeate our society.

Researchers and educators who hope to ground school procedures in the findings of research rather than in politics and ideology are likely to face either of two arguments: that the research is based on grossly unrealistic reductions of complex phenomena, or that it involves conflicts of value that cannot be resolved by evidence. In fact, Romberg believes it is naïve to believe that research findings can curtail partisan prejudices about schooling. Nevertheless, research is increasingly providing insights, understanding, and new approaches that lead to instruction that more effectively promotes student achievement.

For more information contact Romberg at tromberg@facstaff.wisc.edu.

Funding for research conducted by the National Center for Improving Student Learning and Achievement in Mathematics and Science (NCISLA) is provided by the U.S. Department of Education's Office of Educational Research and Improvement.

Thomas A. Romberg is Sears Roebuck Foundation-Bascom Professor of Education at the University of Wisconsin-Madison. His research has analyzed young children’s learning of initial mathematical concepts, methods of evaluating students and programs, and the integration of research on teaching, curriculum, and student thinking.

Romberg was a leader in the development of the National Council of Teachers
of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics
(1989). He is former director of WCER’s National Center for Improving Student Learning
and Achievement in Mathematics and Science. He offers the following observations
looking back over a career spanning more than 30 years.