Somewhere in a New Jersey elementary school:

The students in Mrs. Chaplain's fifth grade class eagerly return from recess, excited by the prospect of working on another of her famous Chaplain's Challenges. Mrs. Chaplain regularly uses a Challenge with her math class, and today she has promised the children that the problem would be a great one. She believes that all of her students will be up to the Challenge and expects that it will engage them in an exploration and discussion of the relationship between area and perimeter.

Suppose you had 64 meters of fencing with which you were going to build a pen for your large pet dog. What are some of the different pens you could build if you used all of the fencing? Which pen would have the most play space? Which would give the most running space? What would be the best pen? ^*

As the students file into the classroom, they stand, scattered around the room, reading the Challenge from the board even before finding their seats. Then they begin to ask questions about it. What is fencing? Wouldn't the pen with the most play space be the same as the one with the most running space? What shapes are allowed? Mrs. Chaplain answers some of these questions directly (she has brought a sample of fencing to class so she could show the students what it looks like), but, for the most part, she tells the students that they can discuss their questions in their regular working groups.

The groups begin their exploration by discussing how to organize their efforts. One of the first questions to arise is what kinds of tools would help solve the problem. The students have used a variety of materials to deal with Chaplain's Challenges, and they have often found that the groups that fashioned the best models of the problem situations were the ones that found it easiest to find solutions. Today, one group decides to use the 10-by-10 geoboards, figuring that they can quickly make a lot of different "pens" out of rubber bands if they let the space between nails equal four meters. Another group decides to get some graph paper on which to draw their pens, because that gives them a lot of flexibility. Still another group, reluctant to be limited to rectangular shapes and work spaces, thinks that the geometry construction software loaded on the computers in the back of the room would let them draw a variety of shapes and even help them measure various characteristics of the shapes. One last group, striving for realism, decides to use a loop of string sixty-four inches long. As the session progresses, the groups of students make many sample pens with whatever materials they have chosen to use. Some groups switch materials as they perceive other materials to be less restrictive than the ones they are using. Keeping the perimeter a constant 64 meters, they measure the areas of the pens using some of the strategies they developed the week before. Mrs. Chaplain circulates around the room, paying careful attention to the contributions of individual students, making notes to herself about two particular children, one who seems to be having difficulty with the concept of area, and another who is doing a nice job of leading her group to a solution.

Gradually, the work becomes more symbolic and verbal and less concrete. The students begin to make tables to record the dimensions and descriptions of their pens and to look for some kind of pattern, because they have learned from experience that this frequently leads to insights.

One group follows the teacher's suggestion and enters their table of values for rectangular pens into a computer, generating a broken-line graph of the length of the pen versus its area.

Toward the end of the class, the students become comfortable with their discoveries. Mrs. Chaplain reflects again on how glad she is that the faculty decided to organize the school schedule in such a way as to allow for these extended class sessions. When she sees how involved and active the students are, how they try to persuade each other to follow one path or another, how their verbalizations either cement their own understandings or provide opportunities for others to point out flaws in their thinking, she realizes that only with this kind of time and this kind of effort can she do an adequate job of teaching mathematics.

The summary discussion at the end of the session allows the students an opportunity to see what their classmates have done and to evaluate their own group's results. Mrs. Chaplain learns that everyone in the class understands that if you hold the perimeter constant, you can create figures with a whole range of areas. Moreover, she feels that a majority of the class also has come to the generalization that the more compact a figure is, the greater its area, and the more stretched out it is, the smaller its area.

But the students still have very different answers to the question, What would be the best pen? That fits her plans perfectly. For homework, Mrs. Chaplain asks each student to design the pen that he or she thinks is best, draw a diagram of it, label its dimensions and its area, and write a paragraph about why that particular pen would be best for the dog. Mrs. Chaplain plans to move on from this activity to others where the students concentrate on more efficient strategies for finding the areas of some of the non-rectangular shapes they explored in this Challenge.

Somewhere in a New Jersey high school:

Ms. Diego's algebra class and Mr. Browning's physical science class are jointly investigating radioactive decay. The two teachers, with the support of the school administrators, have worked out a schedule that enables their classes to meet together this month to explore some of the mathematical aspects of the physical sciences. Both teachers regularly incorporate some content from the other's discipline in class activities, but this month was specially planned to be a kind of celebration of the relationship between the two areas. By the end of the month, they expect that the students will really appreciate the role that mathematics plays in the sciences, and the problems that are presented by the sciences that call for innovative mathematical solutions.

The classes are average. Nearly every student in the high school takes these two classes at some point during their stay and, over the past few years, because of exciting real-world problems like the one on which they are working this week, the classes have become two of the most popular in the school.

Monday's class begins with a presentation by Mr. Browning about the process of carbon dating. He describes the problem that archaeologists faced in the 1940s with respect to determining the age of a fossil. They knew that all living things contained a predictable amount of radioactive carbon that began to diminish as soon as the organism died. If they could measure the amount that remained in some discovered fossil and if they knew the rate at which the carbon "decayed," they could figure out the age of the object. An American chemist named Willard Libby developed a technique that allowed them to do so. Ms. Diego explains that the classes will spend the next few days exploring the concept of radioactive decay and, toward the end of the week, they will be able to solve some of the same kinds of problems solved by those archaeologists.

On Tuesday, working at stations created by the teachers, the students begin to explore both the mathematical and scientific aspects of radioactive decay. Working in groups, the students use sets of 50 dice to simulate collections of radioactive nuclei. Each roll of the collection of dice represents the passage of one day. Any time a die lands with a "1" showing, it "spontaneously decays" and is taken out of the collection. The students plot the number of radioactive nuclei left versus the number of days passed in an effort to determine the half-life of the element - the amount of time it takes for half of the element to decay. Because the experiment is relatively well controlled, each group working on the task produces a graph that effectively illustrates the decay, but, because the process is also a truly random process, each group's results are slightly different from those of other groups.

On Wednesday, in a very different kind of activity, students use graphing calculators in a guided activity to discover properties of exponential functions, and the effects on the graphs of various changes to the parameters in the functions. Working from a worksheet prepared by the teachers, they start with the general form of an exponential function, y = ab^x. Using the values a = 1 and b = 2, they input the equation into the calculators and study the resulting graph. Then, they systematically change the values of a and b to discover what each change does to the graph. They are directed by the worksheet to pay particular attention to the effect of changing b to a value between zero and one, because graphs of that type will be especially important for their work with radioactive decay. The culminating problem on the worksheet is a challenge to try to find the values of a and b that produce a graph that looks like the ones that resulted from the experiment with the dice. The students enjoy the problem and use their calculators to quickly check and refine their solutions, zeroing in on the critical numbers. There is a lot of discussion about why those numbers might be the correct ones.

On Thursday, the students discuss a reading that was assigned for homework the night before, focusing on carbon dating and addressing some of the mathematical processes used to determine the age of fossils. This discussion is led by the two teachers, who have brought in some fossilized samples to better acquaint the students with the kind of materials they read about. Ms. Diego then leads a session to develop the computational procedures for solving the carbon dating problems using exponential functions. The students will be given some homework problems of this type and will spend tomorrow's class discussing those problems and wrapping up the unit.

The teachers are very pleased with what the classes have accomplished. The active involvement with a hands-on experiment simulating decay, the symbolic manipulations and graph explorations made possible by the graphing calculator, and the study of a particular scientific application of the mathematics have been very productive. By working together as a team, the teachers have been able to relate the different aspects of the phenomenon to each other. The students have learned a great deal of both mathematics and science and have seen how strongly they are linked.