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• W40256640 abstract "Mathematical modeling can perhaps be best defined as “the process of scientific inquiry” for mathematics. This is obviously a comfortable mode for teachers of science, but is rarely seen in the mathematics classrooms of today. This paper explores the possibilities of using interactive web pages to help facilitate an understanding of practical applications based mathematics. Because the scientific process is emphasized as the general operating framework, situations where students can hypothesize and experiment, and create data tables are most valuable. Special emphasis is placed on the fact that students and teachers both need to re-conceptualize effective mathematics instruction in order to really embrace a modeling approach. Introduction: An important aspect of the continually changing reform movement in secondary level mathematics is that teachers are able to absorb and integrate what they have learned from both the classroom dynamics and from new research. It is perhaps most important that teachers of mathematics continue to grow with respect to the pedagogical techniques that have the greatest classroom potential. Although finding these techniques requires a great deal of effort, good teachers would certainly agree that the resources they bring to bear on behalf of their students set a foundation of success or failure for those students from both a competence standpoint and from a motivational standpoint as well. The reform efforts of the past decade have resulted in a mass of professional documents, curriculum standards, and reports, all of which are intended to strengthen a teacher’s profile of techniques. Yet with all of the various forms of assistance, the mathematics classrooms of the 21st century will probably look very similar to those that have been so common for the past 50 years. The fact that we know so much more now than we did 50 years ago, at least from a scientific standpoint, has appeared to have very little impact on what is taught or how things are taught in the secondary math classroom. Agreed, technology has brought flavor to the mathematics classroom, but the textbooks along with their very familiar format still seem to be the preferred method of instruction. Although there are instructional perks to this classroom format, the fact that students aren’t internalizing the information would suggest that other formats merit exploration. Instructional activities using a mathematical modeling approach have proven to be both effective and engaging for students. Additionally, some of the most valuable curriculum-application considerations in today’s mathematics classrooms can be revisited in the context of an interactive web based format that preempts the “what do we need this for” question. The mathematical modeling approach to instruction is indeed a “front heavy” technique for teachers, but allows for the kind of valuable exploration in mathematics that has been absent to date. Classroom Considerations for Mathematical Modeling: Mathematical modeling can perhaps be best defined as “the process of scientific inquiry” for mathematics. This is obviously a comfortable mode for teachers of science, but is rarely seen in mathematics classrooms. Students engaging in mathematical modeling activities would spend the majority of time experimenting in applied physical situations in an attempt to find patterns and consistencies in sets of data. Data sets could already exist in a number of different forms, or they could be collected as part of a classroom activity. Part of the impetus for mathematical modeling activities in the classroom is to help students understand that mathematics is not a discipline where complex solutions to problems are innately obvious or solvable in a matter of just a few minutes. In fact, any good mathematical modeling activity should be appropriately vague so that the students don’t get the impression that the activity is just another textbook assignment. The teacher designing the activity has the difficult task of articulating the problem in such a way as to provide clarity without being too prescriptive. This is done to emphasize that mathematical modeling is a process of continual refinement and modification. In most cases, this process of refinement serves two distinct tasks. First, the refinements are intended to create a working model that is more efficient, faster, or more accurate in some way than any previous model. Secondly, refinement and modification are natural processes of building any axiomatic system of notation. Students in essence build their own mathematical system of notation and in turn, greater mathematical understanding. Some instructional considerations related to the use of mathematical modeling activities in the classroom are as follows: 1. Students have some control over how they approach a problem. This is not typically the case with most textbook problems. 2. Good modeling activities are adaptable to many different ability levels. 3. Good modeling activities are easily scalable to different grade levels. 4. Problem solving and mathematical modeling are different processes. Problem solving typically acts as a process oriented approach whereby students find a specific solution to a specific problem. Mathematical modeling is an experimental approach where a problem is solved and continually refined over time in order to be more efficient, faster, or more accurate. Problem solving in many cases has a solution that is either correct or incorrect. Mathematical modeling is a process where few answers are incorrect, they just require continual revision. 5. Mathematical modeling focuses primarily on the “general case.” Students must at least generally understand the concept of a variable, which is why modeling activities below the fifth grade are difficult for teachers to construct. 6. Mathematical modeling activities are difficult to assess. An elegant solution may be an approach that works in a way that appears to be coincidental, but a student can justify why. Another solution to the same problem may utilize some specific procedure from the textbook, yet the student has no understanding of why they chose that method nor why it works. The premise of the mathematical modeling concept is not that the traditional courses in the curriculum need to be replaced, but rather accented in the appropriate spots to better emphasize the practical use of the concepts we do teach. Because mathematical models can take on many forms, the processes by which problems are approached are numerous and varied. Some of the more basic modeling structures lend themselves very well to established secondary level curriculum (i.e. numerical tables and patterns, graphs, systems of equations, etc.). Others may be more algorithm-based problems that require a computer or graphing calculator as an extension. Although no one set of rules is inherent to all mathematical modeling activities, the following set of steps can act as general guidelines for students engaged in mathematical modeling activities: 1. Identify what the problem and resulting model should look like 2. Establish the factors that affect the outcome 3. Define which of the factors are parameters and which are outcome variables 4. Establish a relationship between the parameters and the variables to derive a formula or alternately defined model or algorithm 5. Test the model with known values from previously collected data 6. Refine the model for accuracy and efficiency Using Web Pages for Modeling Activities: Although modeling activities in the mathematics classroom don’t have to be technology driven, the interactive nature of Java applications on many web pages can provide a physical context which students can use to test conjectures and build generalizations. Because the scientific process is emphasized as the general operating framework, situations where students can hypothesize, experiment, and create data tables are most valuable. Well designed web pages using Java allow for the kind of interactive experiments needed for success without the hassle of setting up a physical lab situation. The following example illustrates a possible modeling problem that could be practically used on the web: Problem: Suppose we wanted to find the time of day without using a clock. In ancient times, sundials were used for this purpose, and were fairly accurate. The first step in finding the time without using a clock is to use the relative movement of the sun and earth to predict how shadows might fall at different times of the day. Assuming that the meridian line (or noontime mark) has been established and the gnomon has been angled, we must find a way to mark the hour lines on the dial plate. Create a mathematical model that uses the angle of the sun on the style (top of the gnomon that creates the shadow) to mark hour lines on the dial plate of the sundial. Using angle as the base angle of the gnomon, and angle t as the angle of the arc the sun passes through in a given time frame, we should be able to calculate angle h by using the length of the resulting shadow. This is illustrated in figure 1. Figure 1: Shadow used to mark the dial plate" @default.
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• W40256640 title "Using Web Pages to Teach Mathematical Modeling: Some Ideas and Suggestions" @default.
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