Descriptive Statement

Calculators, computers, manipulatives, and other mathematical tools need to be used by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper-and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Historically, people have developed and used manipulatives (such as fingers, base ten blocks, geoboards, and algebra tiles) and mathematical devices (such as protractors, coordinate systems, and calculators) to help them understand and develop mathematics. Students should explore both new and familiar concepts with calculators and computers, but should also become proficient in using technology as it is used by adults, that is, for assistance in solving real-world problems.

Meaning and Importance

Both mathematics and the way we do mathematics have changed dramatically in the last few decades. The presence of technology has made more mathematics accessible to us, has allowed us to solve mathematical problems never before solved, and has brought a new and extraordinarily higher level of proficiency with mathematical operations to all members of our society. At the same time, our resulting increased need to truly understand that mathematics has brought on a renewed interest in developing materials and approaches that provide concrete models and that can engage learners. Both of these new directions combine in this one standard which calls for the appropriate and effective use of tools and technology.

Manipulatives are concrete materials that are used for "modeling" or representing mathematical operations or concepts. In much the same way that children make and use models of race cars or the human skeleton so that they can study and learn about them, students can make and learn from models of two-digit numbers or multiplication. The difference between the two situations is that while the car and skeleton models are smaller versions of actual concrete things, the number models or multiplication models are concrete models of abstract concepts and operations.

When students use bundles of sticks and single sticks to represent tens and ones, or algebra tiles to represent polynomials, they are using manipulatives to "model" mathematical ideas. Technically, even when young children count on their fingers, they are concretely modeling numbers.

The mechanism by which concrete modeling aids children in constructing mathematical knowledge is still not completely understood. There is little doubt, however, that it does. There is a good deal of research which shows that the optimal presentation sequence for new mathematical content is concrete-pictorial-abstract. Activities with concrete materials should precede those which show pictured relationships and those should, in turn, precede formal work with symbols. Ultimately, students need to reach that final level of symbolic proficiency with many of the mathematical skills they master, but the meanings of those symbols must be firmly rooted in experiences with real objects. Otherwise, their performance of the symbolic operations will simply be rote repetitions of meaningless memorized procedures.

Calculators and computers provide still other benefits for students. In the 1994 Position Statement on The Use of Technology in the Learning and Teaching of Mathematics, the National Council of Teachers of Mathematics states:

Students are to learn how to use technology as a tool for processing information, visualizing and solving problems, exploring and testing conjectures, accessing data, and verifying their solutions.
. . . In a mathematics setting, technology must be an instructional tool that is integrated into daily teaching practices, including the assessment of what students know and are able to do.

The availability of technology requires that we re-evaluate our mathematics curricula. What we teach and how we teach it are now inextricably linked to these new tools. Their presence makes some traditional mathematics topics obsolete - we certainly do not still teach the square root extraction algorithm, but what about hours and hours of paper-and-pencil practice with the long division algorithm? How much is necessary? How much is adequate? How much will our students need that skill when they become adults? The presence of the technology also makes some traditional topics more important than ever - efficient calculator use requires a high level of estimation, place value, and mental math skills so that calculations can be quickly checked for reasonableness and accuracy. And the technology allows some topics to be dealt with that were never accessible to students previously - graphing calculators allow students to instantly see the graphs of complex functions that would have taken a whole class period to graph by hand, and computer-based geometry construction tools allow students to experiment with animation to see the effects of transformations of figures in three-dimensional space.

Stephen Willoughby, a former president of the National Council of Teachers of Mathematics, offers this analogy to those who might be reluctant to change the traditional curriculum:

When automobiles first appeared, there were undoubtedly many people who kept a spare horse in the garage lest the automobile fail, but very few of us still do today. Calculators, with and without batteries, have become so inexpensive and reliable that it is more efficient to keep an extra calculator handy than it is to learn to do well everything a calculator does better. Most of us no longer find the ability to shoe a horse or cinch a saddle to be essential skills. Is it not reasonable that in the near future we may feel the same way about multi-digit long division? (Willoughby, 1990, p. 62)

The New Jersey State Board of Education, recognizing the need for students to develop appropriate and integrated technology skills, requires the use of calculators in the state's mathematics assessment program at the eighth and eleventh grade levels. To be adequately prepared to use calculators on those assessments, it is critical that the students have ample opportunity to use them as a regular and natural part of their mathematics classes and other testing.

We are only just beginning to explore the ways in which technology can be helpful to mathematics learners, but already there are tremendous opportunities at all levels. Two additional tools that have not yet been mentioned in this Overview possibly provide the most futuristic vision of what mathematicsclassrooms might soon look like. Calculator-Based Laboratories (CBLs) allow measurement probes to be connected to hand-held calculators to collect and analyze scientific data such as light, distance, voltage, temperature, and so on. This direct observation, collection, and display of data allows the student to focus more on the hypothesis, interpretation, and analysis phases of experiments. And the World-Wide Web, or graphical Internet, holds untold amounts of data and information. From geographic and census data, to current information about almost any mathematical or scientific subject, to rich sources of mathematical problems for K-12 students, the Internet will greatly expand the research capability of both teachers and students.

Strategies and teaching approaches which utilize technology have been shown to improve student attitudes, problem solving ability, ability to visualize mathematics, and overall performance. Here, possibly more than anywhere else in this Framework, we need to be open to new ideas and receptive to new approaches.

K-12 Development and Emphases

Many specific suggestions for appropriate uses of calculators, computers, and manipulatives are given in the following pages and, indeed, throughout this Framework. The point to be made here, though, is that the frequent, well-integrated use of these tools at all levels is essential. Young children find the use of concrete materials to model problem situations very natural. Indeed they find such modeling more natural than the formal work they do with number sentences and equations. Older students will realize that the adults around them use calculators and computers all the time to solve mathematical problems and will be prepared to do the same. Perhaps more challenging, though, is the task of getting the "reverse" to happen as well, so that technology is also used with young children, and the older students' learning is enhanced through the use of concrete models. Such opportunities do exist, however, and new approaches and tools are being created all the time.

In summary, mathematical tools play an ever-more important role in today's mathematics. Students who will be expected to be knowledgeable users of such tools when they leave school must see those tools as a regular and routine part of "doing mathematics" in school.

References

National Council of Teachers of Mathematics. Position Statement on The Use of Technology in the Learning and Teaching of Mathematics. Reston, VA, 1994.

Willoughby, S. Mathematics Education for a Changing World. Alexandria, VA: Association for Supervision and Curriculum Development, 1990.

General References

Barnes, B., et al. Tales from the Electronic Frontier. San Francisco: WestEd Eisenhower Regional Consortium for Science and Mathematics Education (WERC), 1996.

Kinslow, J. Internet Jones. Philadelphia, PA: Research for Better Schools, 1996.