For more than 30 years, these researchers have put the learning of mathematics under the microscope in search of ways to improve teaching and student understanding. They’ve found, for instance, that math instruction can be strengthened by tapping into children’s informal knowledge, by teaching them to use the same practices as mathematicians, and by engaging them in real-world problem-solving instead of rote drills on abstract skills.
By making math more accessible to students of all ages and abilities, they hope that more people will recognize mathematics as they do – as a language for thought.
Having established a solid foundation, the trail-blazing cohort led by Carpenter, Romberg, and Fennema in recent years has been passing the torch at UW-Madison to a new generation of scholars.
“The Mathematics Education area is in good hands,” says Eric Knuth, associate professor in the Department of Curriculum and Instruction, who leads a group that includes three assistant professors – Amy Ellis, Victoria Hand, and Edd Taylor. Adapting a phrase used by Sir Isaac Newton, Knuth adds, “We are continuing on the shoulders of giants.”
Like those giants, all four are engaged in research aimed at adding to the body of knowledge of how diverse populations of students learn and understand mathematics. Likewise, they are dedicated to equipping current and future teachers with the best practices, based on the latest knowledge, for supporting all students in their development of mathematical understanding and reasoning.
The path of giants
Tom Romberg describes mathematics as “a human activity involving the ability to represent quantitative and spatial relationships in a broad range of situations, express those relations using the language of mathematics, and use various techniques to carry out numerical procedures.” While humans have used mathematics for centuries to help make sense of the world, he explains, research on the teaching and learning of math is a relatively young discipline.
Romberg is widely recognized for playing an instrumental role in creating the mathematics education research community. Since the late 1960s, he has held numerous leadership posts, including the chairmanships of the Research Committee for the National Council of Teachers of Mathematics (NCTM), the Special Interest Group in Mathematics Education for the American Educational Research Association, and the North American branch of the International Group for the Psychology of Mathematics Education.
In the 1990s, he chaired the NCTM committee that produced the mathematics curriculum and assessment standards, marking the start of the standards movement in education. He notes, “These documents have had considerable impact throughout the world.”
While the accomplishments to date have been substantial, Romberg and his colleagues acknowledge that plenty of work remains. “While all instructional programs have a goal of teaching mathematics so that students ‘understand,’ there has been little evidence that the goal has been reached,” he says.
With evident pride, Romberg and Tom Carpenter describe the contributions of mathematics education research at UW-Madison.
“As a consequence of our program of research for over 30 years, we have developed a workable conception of how to characterize ‘student understanding’ and some ‘powerful practices’ that lead to such understanding,” Romberg explains. “The impact of these conceptions is reflected in our most recent work on teaching early algebra, the development of a middle-school curriculum (Mathematics in Context), and the creation of a classroom assessment system.”
“We have been instrumental in bringing together research on the development of students’ mathematical thinking and the research on classroom interactions and classroom processes,” adds Carpenter, whose credits include serving as editor of NCTM’s Journal for Research in Mathematics Education, the leading journal in the field. He also has been honored for his research publications by the NCTM and the American Educational Research Association.
“This has been a major development in research during the last 15 years,” he says, “and our faculty members have played major roles both in articulating the need and conceptual framework for the integration and in the specific research that was at the forefront of the changes.”
Carpenter also points to his collaboration with Fennema – who is especially known for her research on gender differences in learning mathematics – and others in the development of Cognitively Guided Instruction, a highly regarded professional development program. CGI prepares elementary school teachers to recognize and build on their students’ informal mathematical knowledge by providing a framework that teachers can use in making their own instructional decisions.
“I would consider the remarkable accomplishments of teachers I have worked with in CGI as one of the most significant and satisfying aspects of my career,” says Carpenter. “Elizabeth and I clearly cannot take credit for all they have accomplished, but my relations with them and whatever I contributed to them has been exceptionally rewarding.”
In her CGI research of children in grades 1-3, Fennema noted gender differences in the strategies boys and girls used to solve problems, although not in the results. Girls tended to use more concrete strategies like modeling and counting, while boys used more abstract strategies. Fennema says this study revealed that gender differences emerged earlier and were more complex than previously recognized.
Both Romberg and Carpenter have directed the National Center for Improving Student Learning and Achievement in Mathematics and Science (NCISLA), a decade-long (1995-2004), federally funded initiative based at the Wisconsin Center for Education Research (WCER). NCISLA involved researchers at six institutions collaborating with K-12 teachers to advance effective reform of mathematics and science.
The researchers found, for example, that children are capable of learning more complex ideas at earlier ages than traditionally thought, that teachers need more substantive professional development about student thinking and subject matter, and that standardized tests do not adequately assess students’ long-term growth of knowledge nor depth of understanding.
Carpenter, Romberg, and other NCISLA staff summarized the center’s work in Understanding Mathematics and Science Matters (Mahwah, N.J.: Lawrence Erlbaum, 2005), and created a multimedia product, Powerful Practices in Mathematics and Science (Madison, Wis.: NCISLA, 2004) for use by practitioners.
Beyond the research findings and publications, the Mathematics Education program can measure its enduring influence in terms of people. “One of our most significant contributions has been the outstanding graduates of our program who have made important contributions to mathematics education,” Carpenter notes.
Since 1980, UW-Madison has conferred 84 Ph.D.’s in mathematics education and has graduates on the faculties of many universities, including major state universities in California, Georgia, Texas, Illinois, Colorado, Arizona, Missouri, Delaware, Indiana, Pennsylvania, and Minnesota.
New faces, same focus
Effective mathematics instruction, explains Eric Knuth, involves three key components: understanding how children learn, preparing teachers who can tap into and build upon that knowledge, and having a curriculum that supports these efforts. Like the pioneers who preceded them, Knuth and his mathematics education colleagues are engaged in all three parts.
Like Carpenter and others, Knuth and Amy Ellis – who joined the faculty in 1999 and 2004, respectively – are interested in promoting the development of algebraic reasoning. Math researchers describe algebra – which introduces students to the use of symbolic representations – as the gatekeeper between the concrete calculations of arithmetic and higher levels of mathematics.
“A lack of success in algebra means losing opportunities for advanced studies,” Knuth explains. Ellis notes that algebra – which involves “the study of structures and systems generalized beyond specific computations and relations” – plays a vital role in access to college and careers in the sciences and engineering, which are associated with higher earning power.
They regard the development of algebraic reasoning as far too important to wait until eighth or ninth grade, when many students first encounter algebra. The seeds of algebraic reasoning need to be planted and nurtured in the elementary and middle school grades, they say.
“We want students to move beyond solving one problem,” Ellis says.
In studies funded by the National Science Foundation (NSF), Knuth and Ellis are looking at the development of key practices used by mathematicians and scientists –generalization, modeling, and proof/justification – that are often not emphasized by traditional instruction.
Algebra marks the first time that students are encouraged to generalize patterns, relations, and functions, says Ellis, adding “it’s fairly common for them to struggle with this.”
Ellis, whose work on generalization is funded by a three-year NSF Research on Learning and Education (RoLE) grant, describes generalization as “a sophisticated mathematical activity that involves extending the range of reasoning beyond one specific problem.”
She has found that the development of the abilities to make generalizations and to construct arguments to justify mathematical claims seem to go hand in hand. She also has seen that grounding abstract lessons in measurable situations enhances students’ abilities to generalize.
In a five-year, longitudinal study funded by an NSF Career grant, Knuth has been examining how middle school students acquire and develop their understanding of what constitutes evidence and justification and how such understandings can be refined and extended. Traditionally, students first encounter – and struggle with – justification and proofs in high school geometry.
Knuth and Ellis also have been working with Charles Kalish, professor of educational psychology, to study relationships between student reasoning inside and outside of math. Understanding how children develop their reasoning abilities, especially those related to mathematics, can lead to instructional practices that support and foster their development.
Knuth and other UW-Madison researchers have looked at such essential concepts as how elementary and middle school students understand the equal sign (=). They’ve found that, instead of recognizing that this symbol indicates a relationship – that one side is equivalent to the other – many children interpret it as something like “find the total,” “the answer comes next,” or “do something.”
NCISLA’s Powerful Practices video provides an example: Asked to fill in the blank on 8 + 4 = __ + 5, a fourth-grade class reaches a quick consensus that the correct answer is 12 (the sum of 8 and 4, ignoring the 5). Instead of correcting them, the teacher poses a series of number sentences that prompt the students to re-evaluate their understanding of the equal sign and, ultimately, recognize that the correct answer to the original question is 7.
“We need to provide these kinds of experiences for kids much earlier,” Knuth says.
Edd Taylor and Vicki Hand – who joined the faculty in 2004 and 2005, respectively – address how issues of diversity and equity affect the teaching and learning of mathematics – an area where Elizabeth Fennema and other UW-Madison faculty have made significant contributions. Both Taylor and Hand are involved with Diversity in Mathematics Education (DiME), the National Science Foundation Center for Learning and Teaching based at UW-Madison and led by Tom Carpenter.
DiME – a consortium consisting of UW-Madison, the University of California at Los Angeles, and the University of California at Berkeley, and school districts in Madison, Los Angeles, and Berkeley – is engaged in preparing a new generation of mathematics education scholars, creating professional development programs for teachers, and facilitating research on equity issues in mathematics education. More information about this project is available online at www.wcer.wisc.edu/dime/.
To the casual observer, the teaching and learning of mathematics might not seem like something that’s affected by ethnic and cultural diversity. Yet, Hand notes, “The notion that mathematics education is culture-free is problematic.”
The broader cultural and social context in which mathematics education takes place influences teachers’ perceptions of what productive and unproductive learning look like – for instance, what “counts” as a justification for students’ mathematical ideas. Hand says misalignments can occur when these cultural differences aren’t taken into account.
Hand has examined structural issues, such as the impact of tracking on opportunities for learning and students’ trajectories for higher education. She has noted that, for a variety of reasons, students of color more often end up in low-tracked classes. Often, these classrooms are less rigorous and put students on a trajectory that doesn’t prepare them for college, she explains. This perpetuates the achievement gap, and feeds the stereotypical view that students of color cannot do math. DiME researchers have found that tracking, even when eliminated as policy, might continue in practice.
Hand also considers broader issues – for example, how the inequitable distribution of high-quality teachers across urban and suburban schools affects students’ opportunities to learn – as she investigates the interplay of structure and student backgrounds.
In his research, Taylor looks beyond the conventional methods used by the mathematics education community at the informal ways children think about math and solve problems outside of school. For instance, he has studied the mathematical development of children who spend money at corner stores in low-income neighborhoods.
Taylor explains that students might solve problems more easily if linked to their everyday practices. For example, a traditional problem – e.g., 160 – 100 = __ – can be presented in a way that draws upon their understanding of money: “If you have $160 and I take away $100, how much do you have left?”
Making teachers more aware of cultural understanding and experiences outside of the classroom can help them create classroom environments that tap into how their students reason through mathematics, he explains. He plans to extend his investigation of math reasoning outside of school to religious organizations and such practices as tithing.
“We want teachers to honor more ways of doing math,” he says. “That’s just good mathematics.”
Influencing practice
“The research has to impact more than just the academic community,” says Knuth.
He and his colleagues underscored the importance of working directly with classroom teachers and connecting their research to the preparation of new teachers. The bridge between research and classroom instruction includes curriculum development and effective teacher education and professional development.
“The kind of research we do has us engaged in the local schools,” explains Ellis. In addition to advancing the research, this benefits the school community and helps teachers address current needs.
Knuth and Ellis design and run the preparation program for secondary mathematics teachers. As the program director, Knuth arranges field placements for pre-service teachers, oversees the teaching assistants who teach methods courses and/or provide field placement supervision, and communicates with cooperating teachers. Taylor directs the preparation program for elementary mathematics teachers.
All four teach also undergraduate courses. Knuth has a class on teaching mathematics with technology. Hand, Taylor, and Ellis have taught various methods courses. Ellis helps run a seminar for pre-service teachers in their final year, and Hand has co-taught a geometry content course within the Mathematics Department for pre-service teachers.
“We all work with graduate students, as well,” Ellis adds. “There is a core sequence of four graduate courses that our math-ed students take, and we all four teach these courses.”
And, all four work with in-service teachers.
Knuth has directed several professional development programs for secondary school mathematics teachers, ranging from a three-year program designed to help high school teachers learn to teach with technology to multi-year programs designed to help middle-school teachers foster students’ mathematical reasoning.
He and Ellis ran a professional development program for pre-service teachers and their cooperating teachers that was geared toward promoting the mentoring relationship. The program was funded by a small grant from the Calculus Consortium for Higher Education.
Through DiME, Hand and Taylor have been involved in creating professional development programs for local school districts to make teachers aware of the learning opportunities that they create. Hand conducts professional development for Madison teachers on equity in mathematics instruction.
Hand and UW-Madison graduate students work with math teachers in the Madison Metropolitan School District in a group designed as a venue for sharing knowledge. She says that efforts by teachers to improve mathematics education for all students in the district have made significant progress over the last three years in narrowing the achievement gap.
The Mathematics Education group also has collaborated with the Mathematics Department to improve the preparation of middle school teachers in both content and teaching diverse populations.
“Teachers need to be given more respect for work they do in their field,” Hand says. “It’s not just about knowing mathematics, but about knowing how to teach mathematics to diverse learners.”
“The research influences how we teach the teachers,” Ellis says. “The teacher’s role is critical in shaping student reasoning.”
“In the end,” Knuth says, “we want all students to learn to meaningfully engage in mathematical practices and to develop increasingly more sophisticated ways of engaging in those practices.”
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