__The Purpose of This Site__

This site is intended for educators who teach mathematics and are interested in integrating common technologies into their daily instruction. Our target audience includes intermediate and middle-grade teachers (particularly those who teach remedial math classes) and secondary special educators. While much of this site focuses on mathematics, there are a number of lessons and activities that are intended to blend mathematics with writing and make use of mathematical reasoning in other content areas such as social studies.

The idea of "technology integration" is being used frequently in
education. In this site, you will find a considerable range of
technologies that augment mathematical understanding. For example,
you will find two areas or *strands* in this site that
incorporate spreadsheets into a range of lessons. There are other
lessons that describe how to use calculators, word processors, and
presentation software as part of the problem-solving process or as a
tool for completing a project. Microsoft® Office or Microsoft
Works are the application tools that we suggest you use when a
spreadsheet, word processor, or presentational program such as
Powerpoint™ is needed.

But this site also contains an entire strand on how to use math journals (i.e., simple paper and pencil "technologies") as a complement to daily instruction. The range of common technologies described in this site-from calculators and paper and pencil to spreadsheets and word processors-is intentional. We believe that we serve our students best when we show them how to use the best tool in a given context.

The range of technologies in this site allows you, the educator, to use the lessons and materials in a way in which you feel most comfortable. Depending upon your background, interest, and the needs of your students, you can select lessons or focus on an entire strand and gradually use other materials over time. We feel that this orientation will be particularly important if you are not familiar with using spreadsheets. If this is the case, do not be intimidated. Our experience has been that students generally find Microsoft® Office or Works programs intuitive and easy to use. Again, learning to use a spreadsheet program such as Excel can be done gradually.

Three basic assumptions have guided the materials and lessons that we have developed for this site. First, we believe that there is a growing need to integrate technologies into everyday instruction. Not only are technologies becoming increasingly common and inexpensive, but these technologies help educators redefine the instructional content of their daily lessons. Graphing calculators are a striking example of how a common and relatively inexpensive technology has profoundly changed secondary mathematics.

We believe that there are other common and relatively inexpensive
technologies that are *underused* in education. Like the
graphing calculator, students can use these tools in a "cognitive
partnership" as they complete tasks. These technologies help extend
understanding, and, over time, they can become a natural part of
learning. For example, sometimes students need to model a problem,
particularly if they have collected data. A spreadsheet is an
incredibly useful tool for modeling the problem and analyzing the
data.

A second assumption that guides our work has to do with the
profound need to move students beyond rote learning. This is
particularly true in the area of mathematics, and highly influential
documents such as the National Council of Teachers of Mathematics
1989 *Curriculum and Evaluation Standards* consistently speak to
the need for instruction to be more rigorous and more conceptual. The
*Standards* also advocate increased use of technology in
instruction.

The interplay between mathematics and technology also appears in
national reports such as the Secretary's Commission on Achieving
Necessary Skills (SCANS) Report, *What Work Requires of
Schools*. This report forcefully argues that mathematical
knowledge today and in the future will have to be more conceptual
(hence, more sophisticated) in order for students to find employment
that pays more than the minimum wage. Ironically, common and
inexpensive technologies like the ones used in so many of our lessons
are one key reason why students need greater conceptual
understanding. These technologies have taken over the hand
computations that for so many American students in the 20th century
was the central goal of mathematics instruction.

A third and final assumption is rooted in the nature of student failure in secondary mathematics. For too long, we have viewed students who have had learning difficulties in mathematics as lacking basic skills. The logical solution to this perceived problem has been to work on the basic or prerequisite skills first, even if this means endless drills on computational procedures such as multi-digit subtraction or learning how to divide fractions by inverting and multiplying. While these skills have some value, they must be viewed in the context of what they mean to the students we teach. We must also acknowledge that in so many cases, there is little time left in the day for most secondary remedial or special education students to study mathematics. Learning to do complex long division problems by hand is likely to have little personal or social value to an eighth grader who is only taking mathematics because he or she is required to do so (and plans to get out of mathematics classes as soon as possible).

In light of the limited time that so many students have left to
study mathematics, the materials in this site attempt to make the
subject more meaningful. In addition to using technologies as a way
to compensate for computational methods that have not been mastered,
many of our lessons draw the connection between mathematics and their
application in everyday contexts. Furthermore, the problems are
sufficiently challenging that they cannot be solved quickly. On some
occasions, students will need to clarify the problem, collect data,
and *then analyze it* before coming to any conclusions. This
kind of problem solving resembles what is expected of students in the
world of work.

__What You Will Find in This
Site__

Currently, there are three main *strands* in this site. We
intend to expand the number of strands in the future. The first
strand is *Math Concepts*, and it covers a range of math topics
commonly taught at the late elementary and middle school level. The
lessons typically incorporate the use of spreadsheets and/or
calculators. These lessons are designed to extend conceptual
understanding and allow students to apply their knowledge in everyday
contexts. Our interviews with intermediate and secondary teachers
indicate that they struggle the most in developing this kind of
application or extension activities.

Thus, it is important to understand that these materials *do not
comprise a complete program of instruction*. They are not
day-to-day lessons that you can or should use continuously. Rather,
they are meant to extend a concept such as multiplying with fractions
or show how ratios can play a role in data collection and analysis.
These materials are not a substitute for a solid, conceptually-based
approach to mathematics, one that is consistent with the 1989 NCTM
Standards. We suggest that you consider programs such as *Connected
Mathematics* (Lappan, Fitzgerald, Friel, Fey, & Phillips,
1996) or *Math in the Mind's Eye* (Bennett, Maier, & Nelson,
1994) as the foundation for your daily instruction.

The second strand that you will find in this site is *Integrated
Lessons*. This strand provides a detailed framework for
constructing lengthy problems that integrate math problem solving and
written expression. The strand provides a model for initially
developing a problem and working with students to clarify its key
dimensions. Students then collect data on the problem and analyze the
data using a spreadsheet. The final step is to develop a written
report that summarizes the findings and makes specific
recommendations. The exercises in this strand are generally much
longer than the lessons in the *Math Concepts* strand. The
*Integrated Lessons* strand contains two examples of how
students work a problem-solving exercise from beginning to end.

The final strand is *Journaling*. This strand provides a
developmental model for incorporating writing into a math class. The
strand includes specific suggestions for managing journals,
developing prompts for writing, and providing students with feedback
on their writing. In addition, the site includes two sample lessons
for introducing students to important ideas related to writing about
their mathematical thinking.

__How to Use This Site__

As mentioned earlier, we have intentionally presented you, the
educator, with a range of options because we realize that not every
teacher has the same student needs, instructional interests, and
background or access to technology. For teachers who want to
gradually begin using technologies such as spreadsheets or
calculators, we recommend that you find a lesson in the *Math
Concepts* strand that matches the topics that you teach and try
them one at time.

If you have longer blocks of time and want your students to engage
in an extended problem-solving activity, we suggest that you look at
the framework provided in the *Integrated Lessons* strand. For
teachers who may not have consistent access to a computer lab or
spreadsheets but want to develop conceptual understanding, we
recommend the *Journaling* strand.

The majority of the lessons in the Teaching for Understanding strands contain FILES TO USE documents. These documents-either word processing or spreadsheet files-are to be used with a particular lesson. In order to preserve cross-platform compatability between Microsoft® Windows™ (3.1 or Windows 95 and 98) and Macintosh™ operating systems, all of these files are in one compressed file below. Please select either the Windows™ or Macintosh™ FILES TO USE below and download it. This compressed file will have to be decompressed by using a program like Aladdin Expander™ for Microsoft® Windows™ or Stuffit Expander™ for the Macintosh™ operating system. Please consult someone knowledgeable in computers if this procedure does not make sense to you.

Once these files are decompressed, you will find the word processing documents to be in Microsoft® Word 5.0 format. The spreadsheets are in Microsoft® Excel 5.0 format. You should be able to open these documents on either Macintosh™ or Windows™-based computers. If you have any problems doing this, open Microsoft® Word or Excel first, and then open the FILES TO USE document required for a particular lesson.