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June 1, 2008
Vol. 65
No. 9

Differentiating Math Through Expeditions

How can we push students to think for themselves?

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How can we help students achieve what they need most—the ability to think for themselves and pursue their own learning? To facilitate this kind of learning, teachers need to rethink class design, instructional design, and curriculum design—and create structures that enable students to set their own pace for learning and assessment.
For several years while teaching math in Taiwan, I have successfully experimented with redesigning instruction and curriculum to help students take more initiative. I call my approach conscious differentiation.

The "Expeditions" Strategy

I use an organizing strategy I call expeditions. Expeditions are long-term investigations of a core concept that feature much student choice and student-driven assessment. Students choose not only their own learning tasks but also their own schedule of assessments. For example, as part of a three-month expedition on linear functions, students explored how linear functions operate in the real world, an exploration that culminated in a multimedia presentation displaying their knowledge.
I set up three different pathways for each expedition, each reflecting a different ability level. Each pathway leads to mastery of the same benchmarks, but I provide more or less scaffolding—and assignments at different levels of difficulty—depending on the level.
In designing this structure, I drew on the Expeditionary Learning model (Expeditionary Learning Outward Bound, 2001). Organizing by expeditions helps cluster information around core concepts, one of the key ways that experts organize their knowledge. This approach takes students past superficial answers like "I'm studying Chapter 10" to deeper answers like "I'm exploring nonlinear functions" when asked what they are learning in math class.

What the Work Looks Like

Expeditions can spur student learning in any content area; in my current position at Hsinchu International School in Taiwan, I use this strategy to teach science and humanities courses. But I initially developed it while teaching a yearlong Algebra I course at Taiwan's Taipei American School. In this algebra course, students, broken into ability groups, moved through three expeditions: Linear Functions, Systems of Equations, and Nonlinear Functions. During the Nonlinear Functions expedition, I gave each group of students one of three digital portfolios to complete, depending on the group's level. Each portfolio required eventual mastery of the same benchmarks but was tailored to a different ability level. The portfolio pages, which I designed using Geometer's Sketchpad software, contained instruction, practice problems, and complex tasks such as building mathematical models.
Each student group chose its own daily tasks and pace as it moved through the portfolio. For example, one of the high-level groups decided to start the Nonlinear Functions expedition by building the mathematical models described in the later pages of that group's portfolio. These were complex tasks requiring students to use geometry software to, for example, build a model to calculate the optimal location for a warehouse that should be as short a distance as possible from three separate stores. These students spent a week trying to figure out how to build these models and finally realized they didn't have the knowledge to do so. So they went back to the first five pages of the portfolio and worked through the exercises until they had the skills necessary to build the models. Once the group members felt they were ready, they returned to the mathematical models, successfully completed these learning tasks, and passed the related assessments. Then they went back to complete the pages in the middle of their portfolio.
el2008summer lokenindepend-w
Photo courtesy of Brent Loken
A group collaborates on chosen tasks as part of an expedition at Hsinchu International School.
This high-level group worked best by jumping back and forth through their portfolio and picking up the necessary information as needed. The students did not assign themselves any textbook problems or use the math book. They obtained all their information online or by asking other students, but they successfully completed their portfolio.
One of the mid-level groups doing the Nonlinear Functions expedition worked through its portfolio in a more systematic way. This group used class time to complete their portfolio task by task in sequence, take quizzes, and ask me questions. These students used their time at home to work on self-assigned homework problems from the textbook that would prepare them to do upcoming work in their portfolio. They assigned themselves homework every day. This group of students knew they did their best "figuring out" at home and their best "doing" at school.
These examples highlight the flexibility of expeditions. When students are given a chance to design their own learning experiences, they will design experiences that suit their learning needs.

How to Plan for Expeditions

Organizing expeditions—and the instructional year—around core concepts helps teachers carefully review standards and benchmarks and determine what they believe is worth teaching. To choose concepts, try using four filters developed by Grant Wiggins and Jay McTighe (2000) as part of their Understanding by Design framework:
  • To what extent does the idea, topic, or process represent a "big idea" with enduring value beyond the classroom?
  • To what extent does the idea, topic, or process reside at the heart of the discipline?
  • To what extent does the idea require deep and analytical uncoverage?
  • To what extent does the idea give students freedom to make choices and exercise responsibility?
In designing an expedition, I also consider which big ideas will incorporate the school's standards and benchmarks. My students now master as much or more math as they did before I introduced expeditions because they make deeper connections and retain information longer.

Setting Up Assessments

The next planning step is to work backward, choosing authentic summative assessments that will demonstrate students' mastery of the standards and benchmarks. For summative assessments, students initiate, for example, digital portfolios, original videos, and even pencil-and-paper tests.
Once I have selected summative assessments, I use formative assessment to continually measure each student's grasp of each objective or benchmark, typically through a combination of written and oral quizzes. Students receive immediate feedback on their understanding. Although I administer oral quizzes, students take written quizzes when they are ready and grade them themselves. A learner can take each quiz as many times as necessary to pass, and grades are not final until the end of the expedition.
el2008summer lokenoralquiz-w
Photo courtesy of Brent Loken
Mr. Loken gives a student group an oral quiz, at their request.
Giving students the chance for retakes allows them to master content when they are ready. Many students who have a difficult time grasping a concept initially can master it at a later date.
I believe that each student should have the opportunity to receive an A in every class. In many classrooms, teachers target a more difficult interpretation of mastery, which makes it impossible for some students to earn an A. In this circumstance, the top tier of students is challenged, but the majority may be confused. In other classrooms, teachers lean toward a lower interpretation of mastery; more students master the standards, but a contingent of learners remains unchallenged. My approach challenges each student appropriately.
I use a color-coded grading system adapted from Clymer and Wiliam (2007) that clearly shows students which objectives they have mastered and which objectives they need to work on. In some expeditions, students progress through the objectives sequentially. In others, they can master objectives in any order they wish.

Assigning Groups and Writing Contracts

My last planning step is to design three different pathways students can take for the expedition. Students take a pretest to assess their knowledge, attitudes, and skills about key concepts. They then divide into three ability levels, each identified with a color to avoid stigma. Students group themselves into clusters of two to three students within their color level. Such choices give students ownership; 92 percent of the Algebra I students reported that grouping by ability helped their learning.
Students all master the same objectives, but the required tasks vary depending on each pathway's level. For example, the Nonlinear Functions expedition included the objective, "Be able to use mathematical models to solve real-world problems." The red group's assignment was to construct one model reflecting a real-world problem, the blue group completed two such mathematical models, and the white group constructed four models. This enabled the higher-level group to explore elementary calculus as well as parametrics.
Students write formal contracts in which they carefully describe their group's project and outline the objectives they must master. Before students begin the expedition, they discuss and propose a grading criteria for the whole class. Typically, students propose that the content portion of their grade be worth 75 percent of the overall grade and that the process (how students process the information and work as a group) be worth 25 percent.
<P ID="Loken"><XREF TARGET="fig1">Figure 1</XREF> shows a sample student contract developed for the Algebra I course at Taipei American School. Writing contracts takes time, but it's worth the effort because the resulting clear understanding of expectations increases student motivation. I help each group revise its contract if necessary and sign the final version. This agreement requires the teacher to pay the students with a letter grade upon successful completion of the contract, placing responsibility for the grade into the students' hands.
Next students plan the actions they will take in their expedition, particularly in three key areas:
  • Their homework assignments, including the number of problems they will need to do to understand the material.
  • Timing of assessments. Students take each quiz when they choose and grade it on the honor system.
  • Their daily tasks. Students plan what they will accomplish each day in class and track their actual accomplishments on a calendar.

Student Videos and Other Creative Projects

Each expedition concludes with a different type of summative assessment project. For example, for the Linear Functions expedition, each group produced a 30-minute multimedia presentation demonstrating its knowledge of linear functions in the real world. I gave students a video camera and a motion detector and assigned them to discover how things move in the surrounding community. The video titled "Linear Is Everywhere" (see my Web site) is an example of the type of project students created to demonstrate their understanding of linear functions.
As the final assessment for the Nonlinear Functions expedition, students completed electronic portfolio templates showing the relevant mathematics, discussing how their work fulfilled the directions, and incorporating photographs or real-world examples of math. To see examples of student portfolios, videos, and assessments used with expeditions in various content areas, visit my Web site.

Additional Benefits

At Hsinchu International School, where I now teach, my colleagues and I use expeditions in many different content areas in the secondary grades. Although each expedition I have conducted or observed has unique content, and each discipline calls for a slightly different design, each one uses the general format and grading philosophy described here.
Besides learning content, students achieve other important goals through expeditions:
  • Becoming independent learners. During expeditions, I do not preteach concepts or lecture to the entire group. Students explore learning objectives independently, using any resources available. When students have trouble understanding content, they may request that I give a minilecture to a small group.
  • Working as a team. Because students work with the same team for the entire expedition, they have ample opportunity to learn how to work together. It's important to provide some support and not expect a group to function as a team automatically.
  • Learning to organize and manage time. Because students are progressing at their own pace and schedule, they need to manage their time effectively. Typically, students begin working before class starts because they no longer need to wait for direction. I record daily comments about each group's progress and make this sheet public so students can keep track of how they are doing.
  • Learning to handle freedom. Ninety-six percent of Algebra I students at Taipei American School said freedom and teacher trust were important for their learning. In time, students begin to value this freedom and trust to such a high degree that they will work hard for it. Cheating has become almost nonexistent in expeditions classrooms.
  • Learning to reflect on learning. Once students have worked through their expedition and fulfilled their contract, groups reflect on their performance and each student writes a convincing argument asking for the individual grade that student believes he or she has earned. Groups can propose what all members deserve for the process portion of the grade as a group, or each member can request an individual process grade. Students can defend themselves against any infraction that I noted on their daily progress sheet or any part of the contract they failed to fulfill. I hold a final conference with each group, and with each student if necessary, to discuss grades.

Evidence of Success

I have used expeditions at the International School of Islamabad, Pakistan; the Taipei American School; and currently at Hsinchu International School. Overwhelmingly, students at each of these schools have favored the learning environment created through expeditions. More than 85 percent of my students at Taipei American School reported that this learning environment helped them learn more than a traditional one, and 97 percent believed they had learned important life skills (for detailed student comments, see my blog). During 2007–08, the first year that teachers used expeditions at Hsinchu, average math grades rose between 5 and 10 percent compared to the two previous years. (The school has only existed for three years.)
The environment that expeditions create helps schools prepare students to not only pass examinations, but also to become the citizens our world needs. Expeditions provide fertile ground for differentiating instruction, curriculum, and assessment. I encourage readers to take the plunge and try using this approach.

Clymer, J. B., & Wiliam, D. (2007). Improving the way we grade science. Educational Leadership, 64(4), 36–42.

Expeditionary Learning Outward Bound. (2001). Evidence of success: Expeditionary learning in year eight. Garrison, NY: Author.

Wiggins, G., & McTighe, J. (2000). Understanding by design. Upper Saddle River, NJ: Prentice Hall.

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