What is 21st Century Mathematics?

Concepts, Not Computation

By Robert Tinker and George Collison

Twenty-first century secondary mathematics calls for a very different curriculum. The demands on workers and voters to make informed decisions require greater knowledge of concepts related to data, models, computers, and rates of change. But ubiquitous access to computers in schools, work, and home means that procedural and computational techniques are far less important, while learning more sophisticated mathematics concepts is possible. A transformation is required that results in greater emphasis on the many ways math helps us understand the world, and less on math for its own sake. We need to focus on concepts, not computation.

In the coming decade, developing and using realworld content will require new technology tools and new approaches to teaching and learning. It will also require new assessment methods and a commitment to teacher professional development.

New Content and Technology Tools

Educators must bring the ideas of mathematics to students so that they can recognize the power and potential of the deepest ideas of mathematics. New content includes a greater emphasis on a conceptual understanding of functional relationships, as in rates of change and accumulation. It also includes the use of modeling to develop and illustrate ideas. And it must demonstrate ways mathematics can support decision making, for example, understanding graphs and percentages could help a voter interpret a political candidate’s views on global warming or the budget deficit.

Over ten years of research by Jim Kaput and colleagues (more info) with a large number of middle school and secondary students has shown that students can grasp and apply calculus concepts without first mastering formal notation. Using SimCalc software, students can create graphs of data obtained from a real motion detector or from simulations of the motion of an elevator, a car, or other object to explore the relationship between function and its derivative and integral.

Research has shown that using visualization tools such as TinkerPlots and Fathom, elementary through high school students can achieve sophisticated levels of statistical understanding. Computer-based models such as StarLogo, NetLogo, and AgentSheets are all environments in which students can create a set of rules (e.g., evolution, global climate change, phases of matter, or schools of fish) and watch for emergent behavior. This type of learning provides students with a broad conceptual understanding needed by everyone, not just those students planning a mathematics or technical career.

The Seeing Math algebra interactives developed by the Concord Consortium support an approach to algebra using the function concept as a central theme. With traditional approaches that offer exercises requiring manipulation of symbols and equation solving, teachers and students miss many opportunities to make connections to real-world, practical mathematics. The function concept unifies later study in algebra and the study of change in calculus; introducing functions earlier aids student understanding of mathematics significantly.

The Seeing Math interactive “Piecewise Linear Grapher” in the MW Platform. This puts all instructions, challenges, help, and assessments in the same environment that can be easily edited and delivered to students.

New Professional Development

In order to understand and use new content and new technology tools, mathematics teachers need a new type of professional development that mirrors the student experience by being grounded in student work as well as a cognitive theoretical framework.

Seeing Math online case studies (available at Teachscape and PBS TeacherLine) integrate videos of students at work, expert commentary on student thinking, online interactives that target key mathematical ideas, and threaded discussions. Within the course design, teacher participants watch online video clips of stu dents working through the same or similar activities to those the participants are working on. Participant collaboration and discussion of the activities helps illuminate key elements of mathematics and mathematics teaching. In contrast to older designs of online professional development in which an expert moderator acts as a "sage on the stage," in Seeing Math courses expert knowledge is imparted in two ways: using videotaped expert commentary targeting specific ideas in the student work and participants' experience gained by working through the activities themselves. As a result, participants make connections related to their own learning of mathematics as well as important new connections among graphic, symbolic, and dynamic representations that are critical in order to teach algebra effectively. The important aspects of mathematics they learn through these case studies are not accessible through traditional methods or refresher algebra or calculus courses.

Another powerful professional development tool developed at the Concord Consortium is the VideoPaper Builder(TM), which enables teachers or administrators to author their own web-based video cases. Just as students need to learn how to learn, teachers need to sustain their own professional development.

New Assessment Tools

Different forms of assessment must accompany these curricular and instructional innovations. The goal of the assessment should be to inform students and teachers about the level of understanding achieved, and of the next necessary steps in instruction. Large-scale standardized assessments provide information about an aggregate of student performances; they are of little use in addressing individual student needs. Extensive research by Paul Black at the Open University has shown that ongoing formative assessment that guides teaching and learning brings about increased learning as well as increased self-esteem for students.

The Concord Consortium's MW Platform offers an example of a highly innovative design for ongoing formative assessment. MW Platform allows teachers to create their own stand-alone or web-based lessons using Seeing Math interactives, or other Java-based software. The highly innovative report feature of the MW Platform enables students to create a series of annotated screenshots with explanations of their work. The MW report includes annotated graphs and student commentary that can be sent to the teacher or other participants. With MW, teachers can set up lessons using any of the above applets and customize activities and questions to suit their students' local needs (for example, to change the context of the question or to meet specific local standards). The report feature enables students to capture dynamic "footprints" of their work and share them with other students and with a teacher for the purpose of formative assessment.

An Integrated Vision

This integration of new content, new tools, new professional development, and new methods of assessment is not just a dream realizable ten years down the road. The Concord Consortium has assembled and tested in classroom settings all of these components. Using software developed by the Concord Consortium and its collaborators, including an elegant case study professional development design and the formative assessment capacity of the MW Platform, high-quality mathematics will be accessible to all students.

Robert Tinker (bob@concord.org) is President of the Concord Consortium. George Collison (george@concord.org) is an Associate of the Concord Consortium and Senior Curriculum Author for the Seeing Math project.