by Cynthia Ballheim
for the Editorial Panel
In a previous issue, Dialogues posed questions dealing with integrated mathematics. The first were directed to teachers of prekindergarten to grade 8:
- To what extent do you integrate mathematics with other content? How?
- To what extent do you integrate such mathematics content as geometry and arithmetic? Give examples.
We heard from more than 100 readers, and their responses indicate that integration "outside" and "within" mathematics occurs regularly. Content areas most often studied with mathematics include science, social studies, technology, art, health, fitness, and even religion. Most respondents blended geometry and arithmetic all the time, blurring any distinction between those subjects. For many, the goal was to integrate every lesson with either content areas or real-life situations. Written projects, class gardens, class quilts, mathematics poetry, mathematics cartoons, logic questions, career investigations, and amusement-park physics are a few of the connections that readers use to engage their students. Only 3 percent of them believed that integration between content areas was not a good thing. The following are some of their responses:
I integrate mathematics whenever a problem presents itself in language arts—for example, comparing size, dates, or ages—and regularly in science—for example, graphing, coordinates of latitude and longitude, and classifying into sets and subsets.
Grade 5 Teacher, Pennsylvania
I consult with other subject teachers about graphs, maps, the metric system, and formulas. I often use their subject as examples for problems.
Barbara J. Clemens
Grade 7 Teacher, South Carolina
We do not integrate nearly enough. We work with other disciplines to try to cover some topics that students will work with in those classes. But that is not enough!
Brian G. Fay
Grades 7–10 Teacher, Rhode Island
I just completed a student workshop on Frank Lloyd Wright designs to integrate with the Mars Millennium Project that my classroom is working on. A solid-geometry hunt was done in our classrooms, and students graphed the results of how many shapes we had. We are also working on a schoolwide "census" project for our school.
Grade 2 Teacher, Pennsylvania
Geometry is integrated into daily calendar routines; reinforced in many literature-related activities; integrated into such science activities as discussing observations of living or nonliving objects; and used in such arithmetic concepts as counting, comparing, decomposing, combining, and so on.
Constance B. Naylor
Kindergarten Teacher, Ohio
In science and in health class, we use fractions and decimals as we study weather and use a thermometer. In social studies, we use time lines and the stock market as we discuss losses (negatives) and gains (positives).
Kindergarten–Grade 7 Teacher, New Jersey
Integration examples include reading books and such special activities throughout the year as origami, pentominoes, and tangrams. Many "teachable moments" also occur, such as finding averages on tests and measuring for science projects and bulletin boards.
Grade 5 Teacher, Georgia
The advantage of integrating mathematics is answering the question that all mathematics students have: "When will I ever use this?"
Grades 6–8 Teacher, Minnesota
Integrated mathematics is a holistic approach to learning—one in which students can see firsthand the relevance of their academic lives.
Phyllis L. Walder
Grades 5–6 Teacher, Maine
Our high school audience was asked to discuss both the obstacles and the advantages to integrating mathematics. Perhaps the most surprising findings were that 10 percent of our readers considered the classroom teacher to be the most significant obstacle to integration, whereas only one of our readers saw no value at all in integrating mathematics. Common obstacles mentioned include the lack of available textbooks and support materials, the master schedule, college placement tests, parental preferences for a more traditional approach, administrative attitudes, and the basic inertia of our school systems. The advantages focused on an answer to that age-old question: "When are we ever going to have to use this?" as well as enhanced opportunities for visual and tactile learners, longevity, and an emphasis on education as a whole. The following is a sample of what our readers said:
Some of the obstacles to integrating mathematics content are teachers who do not want to give up doing what they are familiar with: they are accustomed to blaming students for lack of achievement and never think that the problem is what or how they are teaching. Related to this is teachers' limited understanding of what mathematics is.
Grades 11–12 Teacher, New Jersey
Obstacles to integrating mathematics content include curricula written by the authors of textbooks and not by classroom teachers who are teaching.
William J. Hewitt
Grades 9–12 Teacher, New Jersey
Both within and between disciplines, there remains far too much isolation and territoriality. Although specialization helped us learn details about our disciplines, it has caused us to forget how to communicate with one another.
Grades 6–12 Teacher, Georgia
Teachers who have always done it a certain way and refuse to change are obstacles to integrating mathematics content. They are out there, and they need to retire!
Nancy L. Hetrick
Grades 9–12 Teacher, North Carolina
Students get exposed to a little of this and a little of that without understanding or practice. I think that students should master algebra and geometry; then they have the necessary skills to do meaningful integration.
Grades 9–12 Teacher, Maryland
Lack of common preparation time and lack of planning or meeting time for discussion of "inter-" and "intra-" department needs is an obstacle to integrating mathematics content.
Grades 9–12 Teacher, New Jersey
Not being open to the idea is the main obstacle to integrating mathematics content. The teacher, students, and parents must all realize that with one lesson, ideas and concepts from many other lessons will be pulled together.
Constance T. Duzza
Grades 9–12 Teacher, Pennsylvania
Obstacles to integrating mathematics content include not knowing how mathematics is used in other fields and teachers with narrow or minimal mathematics background.
Grades 9–12 Teacher, Wisconsin
Students transferring into or out of a school with integrated mathematics may have difficulty with mathematics placement in the new school: this situation presents an obstacle to integrating mathematics content.
Mary B. Rhodes
Grades 11–12 Teacher, Pennsylvania
The integration of mathematics tends to deliver a defocused content filled with distractions. The textbook is hard to use as a reference because the topics tend to be scattered.
Daniel E. Sealey
Grades 7–12 Teacher, Michigan
I have taught integrated mathematics. I do not like it. A topic is presented at the beginning of a chapter. Then another. Students forget the first topic.
Grade 9 Teacher, California
An obstacle to integrating mathematics content is that the master schedule does not allow for mixing classes.
Grades 10–12 Teacher, Ohio
Integrating mathematics has no advantages that I have seen, unless it is to dilute the content to the point where more students now know less than ever about mathematics.
Grades 9–12 Teacher, California
Integrating mathematics enables the practitioner to view mathematics as a harmonious and spectacular mosaic.
Dr. C. Sethumadhavan
Mathematics Department Chair, South Carolina