Connecting Concrete and Abstract
We mentioned in the introduction to the chapter that our definition of mathematics is changing. It is important for us to take just a few moments to consider the vital and foundational question, "What is mathematics?" Before reading any further, take a few moments and write down some of your thoughts in answer to this question. If possible, perhaps through the use of electronic communications,2 try to share your ideas with some of your colleagues and take a look at their answers to the question before reading further.
It would be helpful if you would think of the water cycle as an analogy to the way mathematics operates. (In the water cycle, liquid water evaporates into a vapor, or gaseous form; then the gaseous water condenses onto bits of dust in the atmosphere, and these drops of water become large enough to fall from the sky in the form of precipitation.) The stage of the water cycle at which water is in its vapor form is the closest to mathematics itself. Water vapor is invisible, and it is in the air around us. The closest thing to visible water vapor is a cloud, which is normally high above our heads and out of our reach. To many people, this is a perfect description of mathematics. (We hope to change that perception!) Mathematics is an abstraction. In a sense, mathematics itself is invisible. But where does it come from? In our analogy, where does water vapor in the air come from? It comes from the process of evaporation, which operates on liquid water on the face of the earth. Mathematics, in a similar way, is abstracted from real-world situations. For example, we look around us and we see many objects in nature that have an identical shape: the sun, the moon, the shape of ripples on the water when a pebble is dropped, and so on. From those observations we construct a mental picture of a shape that we name as a circle. The idea of a circle is just that: an idea that a person must construct in his or her mind. An idea is an abstraction, and this particular abstraction has its roots in things we have observed in the physical world. Now we have something mathematical: our mental representation of a circle. We can study characteristics of a circle. We can measure parts of it; we can compute relationships between various parts of it; we can describe it in such detail that someone could follow our description and draw exactly what we have in mind. We can even use the idea of a circle as a model to make things that help us accomplish goals. For instance, we can make a wheel in the shape of a circle and use it for transportation. When we take a mathematical idea and bring it back down to earth in the form of something we have made, this is like the part of the water cycle where water returns to the earth in the form of precipitation. This is the application of mathematics from the abstract to the concrete.
The original connection between the physical world and abstract mathematics is important to note. Nearly all the mathematics that humans have developed have been developed in response to some actual, physical-world problem(s) that we needed to solve. The connection between abstract mathematics and the concrete, physical world in the form of applications of mathematics is also important to note. We do not typically study mathematics for its own sake (although we could, and some people like to do so). Ordinarily, the purpose for learning mathematics is to use it to solve problems. The problems we solve with a particular mathematics concept could be the same problems that originally gave rise to the mathematics in the first place. More commonly, the mathematics we learn is something we use in a vast array of new problems. The person who can successfully connect abstract mathematical ideas with new applications in real-world settings is a person who has succeeded in learning mathematics in a meaningful way. This is our goal for all learners.
There is a two-way relationship between the real world and mathematics. Educators have a very special need to understand this relationship completely. Noneducators need to have a body of mathematical knowledge that they can apply to real-world problems. Noneducators do not need to be concerned with how they have derived that body of knowledge. Their body of mathematical knowledge should be something that they can simply take as a given. Educators, on the other hand, must not only understand how to apply mathematical knowledge to real-world problems, but they must also understand how it is that mathematical knowledge can grow from real-world experiences. In other words, the pathway from the "real" world to mathematical knowledge has special significance to teachers. This pathway is where mathematics educators dwell. This pathway is the arena within which mathematics educators perform. This pathway represents mathematical pedagogical content knowledge, and it is the central focus of this text.
Ball and Bass (2000) express this unusual perspective held by teachers in this way: "Because teachers must be able to work with content for students in its growing, not finished state, they must be able to do something perverse: work backward from mature and compressed understanding of content to unpack its constituent elements" (p. 98). Much of what we examine in this text is centered on this idea of working backward from mature and compressed understanding of content. We believe that this is an important element of coming to understand mathematics in the special way needed by teachers.
2A note to math coaches and professional development coordinators: We recommend establishing an online chat room in which teachers can discuss issues that come up throughout their course of study.
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