When we asked a student what she could do to understand the topic of a unit, she answered, “Go through the book and try to understand the definitions.” When we asked this same student a similar question after she completed our unit on area, she responded: One good way to understand area is by relating it outside your math class ... think of other ways it is needed in your personal life.... By doing the problems different ways and changing the shapes around, you'll be able to use the formulas no matter what the shape is.
Why the difference in these two responses? Is it a different subject? A different teacher? The difference, we claim, is in the teaching.
Attempting to put into practice the four principles of the Teaching for Understanding framework, developed at the Harvard Graduate School of Education (see “Putting Understanding Up Front,” p. 4), we worked with a team of four teachers at two high schools to learn more about what teaching for understanding looks like in the classroom.
What Should I Teach?
In applying the four principles, we first had to ask: Were the topics of teachers' preexisting units generative? In other words, did the topics contain a rich array of genuinely meaningful connections to students' lives? Were the topics central to an understanding of the discipline?
In answering these questions, we expanded, connected, and extended the typical subject matter of units into generative topics. In considering whether the topic “the external anatomy of fish” was a generative one, for example, Steve Roderick, a science teacher at Lincoln-Sudbury Regional High School in Sudbury, Massachusetts, recognized a different topic that he considered more meaningful to his students: “The delicate balance between the environment and survival, where the most minor alterations could result in drastic consequences—even affecting our own lives.”
When Phil James, a history teacher at Lincoln-Sudbury, considered whether his initial topic, “The Industrial Revolution,” was generative in nature, he began rethinking his curriculum. He recognized his implicit desire for the students to not only become familiar with the facts and figures of that period, but also to understand the dramatic impact of the Industrial Revolution on society. James eventually recast his initial topic into the evocative question: “In what sense was the Industrial Revolution progress?”
This reconsideration of topics resulted in better student understanding. Roderick's students came away from their exploration of fish anatomy having also explored the balance of the ecosystem and their personal responsibility to the environment. James's students developed both an understanding of the impact of the Industrial Revolution on the lives of people during that period and the many ways the developments of that era had altered their own lives.
What Should Students Understand?
With a fundamental reconsideration of a unit's topic came a necessary rethinking of each teacher's goals for understanding. For a unit on surface area, Bill Kendall, a math teacher at Braintree High School in Braintree, Massachusetts, began with the topic of formulas for computing the surface area of particular shapes. In considering the generativity of this topic, however, Kendall found a potentially more fruitful endeavor: the application of surface area formulas to real-world projects.
With this reinvention of the unit's topic, Kendall had to revisit his primary goal for the unit. Originally, he had wanted his students to be able to correctly recall and apply the right formulas to specific shapes. In reframing the topic, Kendall now believed students needed to (1) understand formulas and their application in real-world problems, and (2) use what they already knew to make sense of what they didn't know.
Reacting to this second goal, Roderick remarked, “But that's what education is all about!” Kendall agreed, but the difference, he said, was that he now overtly shared this goal with his students. He had never before approached his teaching in this way. Instead, he had too often succumbed to the demands of coverage and the lack of student interest, making straightforward presentations of formulas and requiring the rote practice of algorithms.
Because they knew the goals, Kendall's students became more involved in their projects and felt more at ease in exploring and assessing their achievement of the goals. Students did not have to interpret reasons for what they were doing or try to figure out what Kendall wanted them to be able to do.
Understanding Takes Time
In putting theory into practice, we gradually recognized perhaps the greatest challenge in teaching for understanding: it takes time to engage students in performances of understanding.
Kendall, for example, created a long-term project in which students designed a dance floor with several different shapes of a particular area. In designing these shapes, students created formulas for the surface area of new shapes out of the formulas for familiar shapes. Students then collaboratively designed their dance floors, presented their designs to classmates, and judged their mathematical explanations against a variation of Vermont's Portfolio Assessment Criteria for mathematical reasoning.
The need for this kind of long-term, sustained inquiry was evident in all the other units the teachers redesigned as well. In a unit on global warming, Roderick had groups of students research and build an argument for the stance of one of the competing constituencies—economic, social, or environmental—addressing the issue. Roderick then played the role of President Clinton, having groups of students present their cases.
Giving students opportunities to actively argue, inquire, and articulate their understandings requires time. Yet the teachers felt the extended time was worth it. In Kendall's words: It is more time-consuming ... [but the other day, when I said we were going to miss a math class], students actually said, “Oh, but I like this.” More important, giving students more time invites them to get involved in the kind of thinking that will foster the building of their own understandings.
How Do I Know They Understand?
In assessing student work from earlier units, we quickly recognized how useful it would have been to have articulated our criteria and standards for assessing student understanding before rather than after the unit. These criteria and exemplars became powerful guides as we thought about what we wanted students to be able to do.
For example, when we assessed students' understanding of the relationship of fish anatomy to the environment, we decided that we wanted students to be able to extend, elaborate, and create novel connections. With this in mind, we designed the global warming unit to build on the concept of feedback, highlighting the role of ongoing assessment. To overcome the pitfalls of typical feedback, which can be open-ended, vague, and dependent on the personal discernment of reviewers, we judged work against clearly articulated public criteria.
By making his criteria for good work a central focus of the unit, Roderick transformed the global warming unit. Dee Gould, a science teacher who teamed with Roderick, noted students' continual use of the explicitly presented criteria: Having the standards of assessment from the very beginning of the unit to the very end was important.... When you write an essay question, you say, “Be sure that you're clear, and you answer all parts of the question,” and you get an answer that is half of that. This time, they really were looking at each [criterion]. They went right down them. They said, “Look at this one,” and they'd talk about it.... They were much more focused.
Why Am I Doing This?
By putting the four principles of the framework into practice, we developed a better grasp of what it means to teach for understanding. Our topic was generative: What does it mean to understand, and how do we teach for understanding? The goal was clear: What does it look like when we put the Teaching for Understanding principles into practice? The performances of understanding were self-evident: We put teaching for understanding center stage. And we practiced ongoing assessment, constantly asking ourselves, Is this teaching for understanding, and if so, how?
This is not to say we always practiced the framework principles as fully as we would have liked. With the typical demands of schooling—curriculum coverage, little time for reflection, and school calendars—we found ourselves shortchanging the potential for continuous self- and peer-assessment, for the sustained engagement in rich performances of understanding, and for focusing on generative topics that could be expanded across the entire curriculum.
What we learned, nonetheless, is that the framework helped us put understanding “up front.” The teachers with whom we worked are good teachers—devoted to the construction of their students' understanding, to actively engaging students in activities and assessments that press them to understand. None of them teaches strictly for rote memorization. None of them typically engages students in exercises divorced from making their own connections. And none of them ever gives assessments that are exclusively multiple choice or fill-in-the-blank.
Yet, by applying the framework, we all deepened our knowledge of teaching for understanding. Roderick recognized the need for larger, overarching themes and stronger, more valued goals of understanding in his curriculum. Kendall came to once again appreciate the ability of students to teach themselves to think mathematically and to apply mathematical understandings to real-world problems. And James came to appreciate the power of having students understand past political and social issues by bridging them to meaningful contexts in the present.
As one of our research collaborators said: Where to start? Be up front about what you want to happen in your classroom. Let students know that their education is not something that you, the teacher, do to them. Don't just tell them. Share with them that you are going to ask them to do certain things, and tell them to ask you why they are doing them if they don't understand why. If you can't answer why, then there's a problem. Putting the four principles of the framework into practice makes us ask once again, “Why am I doing this?” And we should resolve to share that answer with our students.
