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Find more Kappan articles in the Why? Why? Why?: Future Teachers Discover Mathematical Depth In mathematics, it is not just the how, the procedures for solving problems, that is important, but the why, the underlying concepts, Perla Myers explains. When teachers regain the childlike wonder that leads them to seek the why of math, they will be better able to help their students learn.
EVERY parent knows well the most common question asked by a young child: "Why? Why? Why?" Answering such questions is one of the joys -- and at the same time, one of the biggest challenges -- of parenthood. How much detail should you include, without being dishonest or providing more information than a child wants or needs? When is it better to set up a situation in which a child can explore and discover his or her own answers? How can you answer in a way that encourages more questions and helps build a path toward lifelong curiosity and learning? Across the country, institutions of higher education offer a course in mathematics concepts for preservice elementary teachers. One goal of such a course is to help future teachers get back to that childlike stance of asking "Why? Why? Why?" Do they need to know the answers? Yes, but the questions themselves are just as important. Adults are generally less inclined to ask good questions than are children, and this is especially true in mathematics. So returning to the questioning mind of a child is an essential first step in preparing to become a mathematics teacher. Each teacher affects the attitudes of hundreds of children. Teachers can nurture or negate the innate curiosity young children bring with them into the elementary school classroom. Ultimately, teachers will help determine whether children see mathematics as a subject to be feared or revered. But in order to nurture curiosity among their young charges, teachers need to know that they should be asking questions, they need to know what questions to ask, and, above all, they need to want to ask the questions. Questions are crucial to understanding the depth of mathematics and getting a glimpse of its beauty. Depth is a measure of a subject's intensity and fullness. In mathematics, depth is not necessarily determined by the level of difficulty, for mathematics is more complex than complicated. Depth in mathematics is about the underlying connections between concepts, the generalization of ideas, the power of seemingly simple results, and the realization that mathematical facts make sense. For example, everything in elementary mathematics has a logical explanation. These connections and generalizations take our understanding of mathematics -- and of the world around us -- in new directions. To get a better sense of mathematical depth, let's begin with some questions about simple arithmetic -- addition and multiplication. Many youngsters (and their teachers) place addition and multiplication in the same category. After all, multiplication is simply repeated addition, right? Many people argue that if you know how to add, you know how to multiply. But is that true? Let's assume that we have two, two-digit numbers that we intend to add together. How do we go about it? We all know that is it correct to add the units digits and then add the tens digits. Why should the algorithm for multiplication be any different? Why not multiply the units digits and then multiply the tens digits? What happens to those products? Should we add them? Multiply them? Are there other algorithms that might work? Why? And to return to our original question: Is multiplication really just repeated addition? Before you answer, what about the square root of 3 x the square root of 2?2 The ability to pursue such inquiry and the disposition to want to do so are particularly important qualities for teachers if they are to help their students come to understand mathematical concepts in any depth. Teachers also need to become adept at creating a range of activities to achieve their goals. Since students grasp concepts in different ways and at different rates, helping individual students requires that teachers know several different paths to the same end, are able to construct easier problems as well as more challenging extensions, and have the ability to encourage students to ask their own questions. Teachers also need to anticipate student responses and evaluate them for correctness.3 All of these skills can be improved by knowing how to ask and by expecting good questions. Here is an example of the way some mathematical questions and concepts may be linked with others via questioning, and this is a true story. A few months ago we had our backyard redone and the whole lawn resodded. Unfortunately, the contractor made a big mistake in his calculation of the grassy area, and approximately one-eighth of our lawn remained without sod for several weeks. The following questions came to mind: What is the actual area? How did the contractor make his estimate? How could he make such a mistake? How do we find area? For many people, the immediate response to a question involving area is "length times width." But that applies only to very specific shapes. The contractor certainly could not have used only this formula to determine how much sod would be needed to cover our irregularly shaped lawn. How do we find the area of irregular shapes? How many different strategies could we use? Could we use a scale model of the lawn to find the area? Could we figure out the area of the lawn using string? Do we know any other formulas that we can throw at this problem? Why is the area of a rectangle length times width? What is the area of a parallelogram? A trapezoid? A pentagon? Is there an area formula that applies to all pentagons? Why do the area formulas that we learned in school work, anyway? Why is the area of a circle p times the square of its radius? We all learned about area in elementary school. My teacher introduced some of the formulas when I was in fourth grade. Consider the following questions from the previous paragraph and try answering them before reading on. • Could we figure out the area of the lawn using string? That is, encircle the lawn using a long string, then create a square with the string, and simply find its area? Would this process result in the correct area? • Could we use a scale model of the lawn to find the area? How would we use it? For example, if we drew a scale model of the lawn such that the lengths in the picture were 1/k of the lengths in the lawn, could we multiply by k to find the area of the lawn? These ideas are very clever! However, as tempting as these strategies would be, the answers to the questions are not all yes. Consider, for example, a 2 cm x 4 cm rectangle. This rectangle has an area of 8 square cm and a perimeter of 12 cm. If we create a longer, skinnier rectangle, say, 1 cm x 5 cm, the perimeter is still 12 cm, while the area changes to 5 square cm. In fact, any given perimeter can enclose an infinite number of regions of different areas. Take a piece of string and tie the two ends together. How small can you get the enclosed area to be? How large? What is the shape of the largest possible region that can be enclosed by the string? For the second question, imagine two squares, a 1 cm x 1 cm square and a 5 cm x 5 cm square. (Do units matter?) The lengths in the second square are five times as large as the lengths in the first square. Yet the area of the second square is 25 times the area of the first square, not five times as large. The ratio of the areas of two shapes that are similar is not the same as the ratio of two corresponding lengths. These facts are elementary, yet many of us were not exposed to asking these types of questions and so may make certain assumptions about their answers that are not correct. Our intuition often fails us. Note that in the case of the scale model, we can actually find the area of the lawn if we know the area of the scale model and multiply it by k2. However, in the case of the string, it is not possible to find the area of the lawn just by knowing its perimeter.4 This is depth! It's like pulling a string on a sweater and having the whole sweater unravel. As we pull, we unravel the barrier that hides the root of the issue and so leads us to new issues. A single "thread" connects many different mathematical ideas that create a whole picture. Our teachers need this ability. They must feel comfortable with asking questions of themselves, and they must encourage their classes to do likewise. At the university level, we must prepare these teachers for such experiences. Then they will pass their insights on to their pupils. Let's pose a few more seemingly obvious questions. Note that many, many apparently unrelated questions are actually connected to one another. The number of questions might seem overwhelming, but if we approach them one at a time we will find that the answer to each individual question is not complicated. For example, the following questions have a thread running through them. Let's unravel the sweater. Why are there 360º around a circle?5 Why is the sum of the measures of the interior angles of a triangle 180º? What is the sum of the measures of the interior angles of a square? Will this sum remain the same if we enlarge the square? What happens if we distort the square? What is the sum of the measures of the interior angles of a quadrilateral? Of a pentagon? Of a hexagon? Of a polygon with n sides? Why? Does it matter if the polygon is convex or concave? Why is it that we see only certain shapes of tiles in our kitchens and bathrooms? How many different shapes of tiles can there be and why? Which shapes can cover our kitchen floor without leaving gaps or overlapping? That is, which shapes tessellate a plane? Why? How many regular polygons tessellate a plane? How do the answers to these questions apply to the art of M. C. Escher? What does the measure of an angle have to do with a soccer ball? If an infinite number of regular polygons exist, why are there only five regular polyhedra (the equivalent of a regular polygon in three dimensions)? How are all these questions related to one another? Again, this is depth. The ability to ask relatively simple questions leads us in wonderfully interesting and interrelated directions. Our teachers must have the ability to engage in this kind of questioning so that our children will have the opportunity they deserve to learn how to think in this way. Such explorations will increase teachers' and students' appreciation and understanding of math. Knowing how to ask questions when we encounter a mathematical concept and exploring where those questions lead will introduce students to the depth -- and, yes, the beauty -- of mathematics. From a very young age, we are taught to appreciate the beauty of literature, of art, of music, of poetry. There are classes in music appreciation and art appreciation (perhaps not enough these days), and we go to concert halls to enjoy symphonies and to art museums to gaze at masterpieces and marvel at the creativity of our species. Unfortunately, mathematics is not viewed in this way. It is widely seen as a necessity, a gatekeeper for achieving career and educational success in many fields. And it is. In a technologically advancing society, a good mathematical foundation is becoming crucial for a large number of careers in science, engineering, medicine, technology, and even business.6 However, the applications of mathematics in those fields, as well as mathematics, in general, constitute much more than just skills. In the words of Richard Feynman, the Nobel laureate in physics for 1965, "To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. . . . If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in."7 Mathematics is more than just the steps needed to solve a problem, more than a set of isolated rules and procedures. After all, if mathematics were just a set of rules and regulations, would we mathematicians waste our time exploring it? Would we be so enthralled by it? Would we find it so fascinating? It is the depth and the beauty of mathematics that attract so many of us. But it can be difficult to get a glimpse of the beauty of mathematics. Many of us were trained in rote mathematics without the opportunity to appreciate the underlying richness. Yes, of course, there are rules. And there are procedures. But these rules and procedures make sense. They are interrelated, and deep beneath them there is a web that connects them conceptually. It is the simple elegance of the structure underneath that makes mathematics beautiful. Reforms in school mathematics are challenging the ways in which math has traditionally been taught. One of the goals of these reforms is to understand the processes, to know more than just algorithms. Mathematics requires more work as it gets more abstract, but understanding can be attained. As one of my students said in a letter to future students: "Get ready for an intense math experience. You are going to take everything you know about math and bring it to the next level. . . . I now feel a lot more confident in my knowledge of math because I have the background knowledge and the proofs of everything I've learned. I know the why and not just the how of the math I do." Our future teachers are very motivated to do anything that will benefit their future students. We can help them become comfortable asking questions that explore the depth of mathematics and make its beauty visible. Then they will be more likely to spend the time required to understand the concepts, and, once they understand the concepts, the procedures will fall into place. When teachers develop their inquiry abilities, they become more motivated to persevere when stuck on a problem, for they can ask the questions needed to break the problem down into bite-size pieces or to approach the problem in a different way. They may be surprised at how much they actually understand if they continue to try and fight the instinct to give up, even after experiencing frustration. Then our teachers will be better able to help their own students acquire these same skills and dispositions. Let the music of mathematics resonate within us! Let us seek its depth, its beauty. Let our future teachers recall their own wonder and curiosity and their tenacity in searching for answers. Then they and their future students will ask not just "How?" in mathematics but "Why? Why? Why?" 1. G. H. Hardy, A Mathematician's Apology (Cambridge: Cambridge University Press, 1940). 2. Liping Ma gives some nice examples of depth in elementary mathematics in Knowing and Teaching Elementary Mathematics (Mahwah, N.J.: Erlbaum, 1999). 3. Deborah Ball and her colleagues have published many articles about specialized mathematical knowledge for teaching. 4. There are some laser-guided measuring tools used by craftsmen that can calculate the area of a region by measuring its perimeters. However, such tools take into consideration the shape of the region. 5. Actually, this question is not a mathematical one. It was a choice made by the Babylonians and is related to the fact that the period of rotation of the Earth around the Sun is 365 days. Some "mathematical facts" are historical accidents, arbitrary choices made in the past, while others are necessary and logical consequences of definitions and axioms. It is crucial for teachers to understand this distinction and to help children do so as well. 6. "Math Will Rock Your World," Business Week, 23 January 2006, pp. 54-62. 7. Richard Feynman, The Character of Physical Law (Cambridge, Mass.: MIT Press, 1965.) PERLA MYERS is an assistant professor in the Department of Mathematics and Computer Science at the University of San Diego. |
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