Math II (Несколько текстов для зачёта), страница 11

Qualitative Understanding through Rate Problems

IN CONJUNCTION WITH THEIR QUANTITATIVE study of curved graphs, the students solidified their understanding of slope with qualitative study of rate problems involving distance-time graphs. The club members were presented with sketches of distance-time graphs (see fig. 4) and asked to think of a video character moving along a line on the video screen, with the distance variable measuring how far the character is from the leftmost part of the screen. For distance-time graphs, the slope represents the ratio of the change in distance to the change in time, or the velocity of the character.

Beneath each distance-time graph, the students were asked to make a graph of velocity versus time. For example, in figure 4, the character moves forward at a constant speed (A to B) and then slows down (B to C) until it eventually sits motionless for a while (C to D). The character then moves back with increasing speed (D to E), meaning increasingly negative velocity, until reaching a constant speed coming back (E to F). Although a verbal description of these graphs can be a bit cumbersome to read, making the velocity-time graphs was fairly straightforward for the students. Indeed, in my seventh- and eighth-grade classes I have presented the students with such distance-time graphs and asked them to walk in the front of the classroom according to the graph. The students have an intuitive understanding of how to translate a distance-time graph into walking speed, realizing, for example, that a curve in a graph means that one is slowing down or speeding up.

Connecting Qualitative Understanding with Slope Patterns

A NATURAL COMPARISON OF THE STUDENTS' quantitative and qualitative studies created a final activity for the project. The club members were asked to graph some simple polynomial equations using the computer, then make a qualitative prediction of the corresponding slope graph, just as they had done in the previous rate problems. The students next checked their predictions against a computer-generated slope graph, made by entering the slope equation deduced by the patterns that they had found in table 1.

An example is shown in figure 5a with the computer-generated graph of the equation y = x[sup 2] - 4x. Having performed many slope calculations through zooming, the students were comfortable enough to make qualitative predictions about the slope at various points along the curve. For example, zooming in would give a negative slope at the point (-1, 5), zero slope at the point (2, -4), and positive slope at the point (5, 5). Indeed, the students were able to predict qualitatively how slope would vary throughout the graph: going from left to right, the slope starts off very negative and keeps increasing to become very positive, passing through 0 at x = 2. The qualitative study in the previous rate problems helped the students step back and see the variation of slope throughout an entire graph.

The qualitative predictions were then confirmed by an exact calculation of the slope using the patterns of table 1, that is, m = nx[sup n-1] to deduce the actual slope equation. Knowing that the x[sup 2] term has a slope of 2x and that the 4x term has a slope of 4, the students deduced a slope equation of m = 2x - 4, whose computer-generated graph is shown in figure 5b. Comparing the slope of the resulting graph with the qualitative predictions, one indeed sees the slope increasing from left to right, passing through 0 at x = 2.

Conclusion

THE ZOOMING TECHNOLOGY OF GRAPHING CALculators and some computer software allows a concrete and visual setting for middle schoolers not only to be introduced to the idea of slope in a curved graph but also to determine these slopes using familiar two-point calculations. The exploration described here would, in its entirety, certainly be an advanced topic at the middle school level. Portions of the exploration, however, could fit comfortably in any middle school study of graphing where zooming technology is available. A good starting point would be two-point calculations of slope along a single curved graph, such as y = x[sup 2] with the analogy of a skateboard ramp in mind, or any other nonlinear equation representing a real-world situation.

TABLE 1: Slope Pattern Discovered by Students

ORIGINAL SLOPE

EQUATION EQUATION

y = 1 m = 0

y = x m = 1

y = x[sup 2] m = 2x

y = x[sup 3] m = 3x[sup 2]

y = x[sup 4] m = 4x[sup 3]

y = x[sup n] m = nx[sup n-1]

THE MATH POEM: INCORPORATING MATHEMATICAL TERMS IN POETRY

Source: Mathematics Teacher, May2001, Vol. 94 Issue 5, p342, 6p

Author(s): Keller, Rod; Davidson, Doris

Mathematical imagination and imagery, closely linked, provide the vision that allows us to see the hidden but exquisite structure below the surface.
--ROBERT OSSERMAN, Poetry of the Universe

It began simply enough, with a conversation between two high school teachers standing beside a copying machine:

Mathematics teacher. Have you ever thought about having your students write a math poem?

English teacher. No, but I like the idea.

Mathematics teacher. What would you think of doing something together--letting the students get credit in your class and in mine? We do share a lot of the same students. It might be fun for us and for them.

English teacher. I agree.

Mathematics teacher. We would be having them put together two subjects that they usually do not think of as having any connection. Sometimes it seems like interdisciplinary work just means English and history or mathematics and science.

English teacher. I know what you mean.

Mathematics teacher. But how would we do it?

English teacher. I do not know, right off, but I am reminded of this great poem by Stanley Kunitz, "The Science of the Night."(n1) It has all these terms from astrophysics, but it is a love poem. I do not want the students to write math poems about mathematics. The results would be predictable.

Mathematics teacher. Poems about how hard and boring mathematics is.

English teacher. I am afraid so. Same thing would happen with a poem about any class. But mathematics is a way to look at, well, almost anything. That should be the point.

Mathematics teacher. Then let's think about it and come up with something.

When we discussed the project a few days later, we articulated our objectives more clearly. We wanted students to apply their knowledge of mathematics to another field and give evidence that at least some of what they had learned had become a useful and easily accessible part of their general experience. In English, we wanted a method that would help young people avoid bland and hackneyed ideas; write fresh, clever, and memorable poems; and gain more of the skills and confidence necessary to approach new topics in new ways.

We formulated an assignment that helped us meet these objectives. Various factors inherent in our situation--including school climate, courses taught, and time of year--influenced our thinking. We teach at Lebanon High School (current enrollment, 790) in Lebanon, New Hampshire. Our school's administrators treat teachers with respect, grant them a great deal of independence, and encourage new approaches to classroom instruction, so we did not need to seek special permission for a project linking mathematics and English or expect anything from the administration other than support.

About 60 percent of our graduating seniors go to four-year colleges, with another 10 percent going to two-year schools or into the military. Most of our students want to succeed and know that they have to put forth effort to accomplish their goals; like many teenagers, however, they often exhibit a feisty independence and a keen desire to avoid routine assignments. They might be skeptical about the kind of project that we had in mind, but they would probably be intrigued by a new challenge.

The project involved two mathematics classes that had mostly ninth-grade students: transition mathematics, the focus of which was basic geometry integrated with arithmetic, algebra, and problem solving; and math topics 2 (honors), which covered such topics as inductive and deductive reasoning and proof, probability, statistics, matrices, coordinate geometry, and quadratic and cubic equations. The four ninth-grade English classes that also participated were general English courses that included instruction in such areas as vocabulary, grammar, and spelling; public speaking; classic and contemporary literature; composition; and analytical thinking.

The idea for a math poem came near the end of the school year. Although we felt rushed, the timing actually worked to our advantage. Our students were accustomed to unusual and creative assignments. They had almost a year's worth of mathematics topics and concepts to draw from, and the students who had this English teacher were about to begin the poetry unit.

We based the assignment on mathematics terms that had been taught during that particular school year, and we chose from a long list the words that seemed most appropriate for the assignment. Careful logic was not always used in selecting the words. Indeed, the English-teacher half of the team, who was only vaguely familiar with some terms and completely ignorant of others, simply liked the way some of them sounded. The students were expected to incorporate a certain number of words from the list in their poems. A minimum length would encourage students to develop their idea in some depth. We tried to offer easy, difficult, and ambiguous terms in a flexible mix that was long enough for students to find the right words to describe and explore their subject but narrow enough to force creative selectivity and the use of important terms learned during the school year. Because some advanced students knew all the words on the list and because many of them would be getting credit in both English and mathematics, we required them to use four more terms than students who were taking other mathematics courses.

We emphasized one other requirement--to write about anything other than mathematics class itself. Although we recognized the importance of students' feelings, we believed that writing about the class might lead to a trite description of it as difficult, boring, and purposeless--even when students actually enjoyed it. In contrast to our approach, Peggy A. House and Nancy S. Desmond, editors of the anthology of poems and stories Mathematics Write Now! (1994), have inspired students to write about mathematics itself with wit and intelligence.

We distributed the assignment sheets. The one shown in figure la was for the more advanced students, and the one in figure lb was for the less advanced classes. We allowed students several days to complete their first drafts.

The results were diverse, delightful, and thought-provoking. Students wrote about a wide range of topics, from amorous encounters to loneliness to shooting a basketball. The styles were as varied as the topics. Some poems barely reached the minimum number of lines; others were several pages long. Most used free verse, but students probably used as many styles and patterns of free verse as the number of poems without rhyme. The moods were many. We enjoyed--but were not particularly surprised by--the humorous pieces. We had not expected so many sincere, thoughtful, deeply personal poems.

The students have revised the sample poems included in this article. Thus, the poems do not necessarily have the number of mathematics terms or lines of poetry specified in the assignment.

When given the chance to write a poem--any kind of poem--young people often become reflective. In this situation, attempting to include the mathematics terms encouraged a healthy distance from, and verbal control over, emotions that might have otherwise run in deep, but narrow and predictable, patterns.

Although the phrasing was sometimes awkward and although the use of the mathematics terms was occasionally forced, the language was often fresh and fascinating. When one student, in a poem called "My World," wrote, "There is no need / to coordinate everything / Things don't need to add up / and balance the equation," terms usually associated with precision helped profess a desire for imprecision, as if a freely roaming imagination is itself a kind of calculation. In another poem, the terms helped a student compress the immensity of sadness into a small and vulnerable shape:

"I wish I knew where I am, to stop my endless rotation around the globe. The variation of the sky seems no longer of any importance, as if it was transformed into a box."

Even the well-worn use of box to suggest loneliness, conformity, and entrapment seemed justified somehow in the context of geometrical terms.

The completed poems were also instructive. Many students desperate for adjectives employed such clumsy expressions as parabola-shaped and cylinder-shaped. Suddenly obvious to us were the grace and efficiency of parabolic and cylindrical. We saw the importance of emphasizing the different forms of mathematics terms as they were introduced during the school year and requiring students to use them in writing. The next year, we added the following instructions to our assignment sheet: "You may use the words in whatever form you wish. (For example, you may use 'parabolic' instead of 'parabola' and 'matrices' instead of 'matrix.')"