Interactive simulation of projectile motion 

Contexts in source publication

Context 1

... teachers with experience of (i) how a teacher can take a problem from curricula materials, and extend it by using a spreadsheet; (ii) what does it mean to use technology appropriately, (iii) how a teacher can engage students in guessing, conjecturing, and debating; (iv) how a teacher can use a spreadsheet with children to explore in depth a "realistic" situation (content-bounded problem); (v) how can a teacher make students experience the need for justification and proving of emerging conjectures; and (vii) how technology can present mathematics as a dynamic discipline and subsequently create an interest toward mathematics. During the five-year BA/MT secondary mathematics teacher education program at the University of Virginia, preservice teachers are required to take a two-semester secondary mathematics methods course. Within this course, preservice teachers are engaged in a variety of different activities and discussions that address such topics as problem solving, the National Council of Teachers of Mathematics Standards (1989), constructivism, assessment, curriculum and instruction in middle and high school mathematics, and effective use of technology. The Curry Center for Technology and Teacher Education is currently funded to develop materials to help pre-service and in-service secondary mathematics and social studies teachers learn to incorporate technology into their teaching. The second author and other members of the mathematics group at the University of Virginia are currently developing activities that give teachers a variety of experiences using graphing calculators, spreadsheets (Microsoft Excel), The Geometer's Sketchpad, MicroWorlds, java applets, internet resources, and specific software programs such as Green Globs. These activities are then integrated throughout the aforementioned secondary mathematics methods course. The second author has developed spreadsheet activities that are designed to introduce teachers to several different uses of a spreadsheet for exploring mathematical concepts with middle and high school students. The teachers experience the various capabilities of the spreadsheet by: 1) analyzing data numerically and graphically, including importing data from the internet, 2) exploring recursive patterns numerically and graphically, 3) creating interactive simulations such as probability events and projectile motion (figure 2), and 4) creating templates for use as instructional tools to investigate problems such as maximizing area of a rectangle given a fixed perimeter and the effect of coefficients on the graph of general forms of equations. Many of the activities investigate mathematical concepts that arise from real world contexts (e.g., analyzing natality data, exploring the relationship between smoking and lung cancer, investigating projectile motion) and are connected with topics in the social studies and science curriculum. All activities are designed to enhance the teachers’, and subsequently their students’, understanding of mathematical concepts by using several mathematical representations (numerical, graphical, symbolic, verbal) and taking advantage of spreadsheet capabilities that allow the learner to extend beyond or significantly enhance what could be done using paper-and-pencil. Using spreadsheets to teach the same mathematical topics, in fundamentally the same ways, that could be taught without technology does not strengthen students’ learning of mathematics and belies the usefulness of spreadsheets. Thus, the activities take advantage of the available statistical analysis tools, the ability to link cells to create dynamic calculations and graphs, and the control forms (e.g., sliders) that can promote interactive and open-ended exploration of mathematical concepts. By using a spreadsheet to investigate mathematical concepts and problems, the teachers gain experience using a variety of functions and features of the spreadsheet. The teachers are also engaged in classroom discourse about mathematics as well as how and when to integrate spreadsheets in middle and high school mathematics curricula. The approach used by the second author to prepare teachers to use spreadsheets as an instructional tool has multiple purposes. First and foremost, the activities are designed to enhance and extend teachers’ knowledge about mathematics and to promote a problem solving, exploratory approach to doing mathematics that take advantage of the capabilities of a spreadsheet. Secondly, the teachers learn features of a spreadsheet within the context of solving mathematical problems and thus gain experience and knowledge of appropriate uses of a spreadsheet for teaching middle and secondary mathematics curriculum. A third purpose is to engage teachers in discussions on effective and appropriate uses of spreadsheets, and technology in general. As a final purpose, throughout the two semester course, the teachers create lessons and/or spreadsheet templates (such as the one shown in figure 2) that utilize features of the spreadsheet to teach mathematical concepts in such a way that significantly extends what is possible with paper-and-pencil. The use of technology in mathematics teaching should support and facilitate conceptual development, exploration, reasoning and problem solving, as described by the NCTM (1989; 1991). Furthermore, both authors believe that teachers who learn how to use technology (e.g., spreadsheets) while using it to explore mathematical topics are more likely to see its potential benefits and use it in their subsequent teaching. Although the contextual setting for the courses taught are quite different, the purposeful use of a spreadsheet as a cognitive tool for learning mathematical concepts is similar. Various instances of using an electronic spreadsheet program in mathematics education suggest that incorporation of this software into teacher education courses and subsequently into K-12 school settings can significantly affect school mathematics instruction. The authors argue that spreadsheets may be construed as a new generation of school-based educational software which use is not limited by financial constraints and commercial availability. Indeed, spreadsheets are commonly available in most schools as part of packages such as Microsoft Office, ClarisWorks, and Microsoft Works and even considered by many school administrators as one of the basic components of computer literacy. Nevertheless, there is a gap between using a spreadsheet as an accounting program and as a tool for conceptual development and educative growth. Therefore teachers' proficiency in the use of a spreadsheet as a cognitive tool becomes a crucial factor in advancing its use in the ...

Context 2

... understanding of underlying relationships that structure the quantitative (problematic) situation. This approach reflected an intrinsic nature of any mathematical concept that arises from an intuitive observation in order to be utilized further in attending more and more abstract structures. Through the use of a spreadsheet as a medium for developing such structures the teachers were given an opportunity to experience the discovery and re-invention of mathematical ideas through computer-mediated activities, appreciate meaningful links among different concepts through exploiting their multiple representations - iconic, numeric, and graphic-in a spreadsheet environment. The first author also taught a technology-enhanced 3 credit hours course for elementary pre-service teachers at the State University of New York, Potsdam that incorporated a spreadsheet from another perspective. In many instances, a didactical focus of using a spreadsheet as a cognitive tool was to challenge a conventional belief held by elementary pre-teachers that there is one and only one way of acting, representing, and comprehending. In other words, such focus promoted the diversity of thinking among teacher education undergraduates. More specifically, consider the following problem from pre K-2 Core Curriculum in Mathematics (New York State Education Department, 1998): A pet store owner sold only birds and cats. One legs. How many cats and birds might there have been? This mathematically rich problem can be presented in a variety of ways one of which is a spreadsheet modeling. A computational approach to the problem can help preservice elementary teachers develop a cognitive bridge between arithmetic and algebra. In addition, the spreadsheet modeling of the above "realistic" situation may help the teachers to comprehend a mathematical model provided by the Diophantine equation 2B+4C=L, where variables B, C, and L stand for the number of birds, cats and legs, respectively. Figure 1 presents both in numeric in graphic forms the results of modeling of the problem for different totals of legs counted. In particular, the case of 18 legs (numbers in a dark part) brings about 5 different solutions (within a specified range of the total of pets) whereas 70 legs allow for 2 solutions only (within the same range). The use of a spreadsheet makes it possible to easily explore numeric patterns and their graphic representations related to some “monsters” with 3, 5, 7, etc. legs. To conclude this section, note that the focus of the above computer-mediated activities was to provide pre-service elementary teachers with experience of (i) how a teacher can take a problem from curricula materials, and extend it by using a spreadsheet; (ii) what does it mean to use technology appropriately, (iii) how a teacher can engage students in guessing, conjecturing, and debating; (iv) how a teacher can use a spreadsheet with children to explore in depth a "realistic" situation (content-bounded problem); (v) how can a teacher make students experience the need for justification and proving of emerging conjectures; and (vii) how technology can present mathematics as a dynamic discipline and subsequently create an interest toward mathematics. During the five-year BA/MT secondary mathematics teacher education program at the University of Virginia, preservice teachers are required to take a two-semester secondary mathematics methods course. Within this course, preservice teachers are engaged in a variety of different activities and discussions that address such topics as problem solving, the National Council of Teachers of Mathematics Standards (1989), constructivism, assessment, curriculum and instruction in middle and high school mathematics, and effective use of technology. The Curry Center for Technology and Teacher Education is currently funded to develop materials to help pre-service and in-service secondary mathematics and social studies teachers learn to incorporate technology into their teaching. The second author and other members of the mathematics group at the University of Virginia are currently developing activities that give teachers a variety of experiences using graphing calculators, spreadsheets (Microsoft Excel), The Geometer's Sketchpad, MicroWorlds, java applets, internet resources, and specific software programs such as Green Globs. These activities are then integrated throughout the aforementioned secondary mathematics methods course. The second author has developed spreadsheet activities that are designed to introduce teachers to several different uses of a spreadsheet for exploring mathematical concepts with middle and high school students. The teachers experience the various capabilities of the spreadsheet by: 1) analyzing data numerically and graphically, including importing data from the internet, 2) exploring recursive patterns numerically and graphically, 3) creating interactive simulations such as probability events and projectile motion (figure 2), and 4) creating templates for use as instructional tools to investigate problems such as maximizing area of a rectangle given a fixed perimeter and the effect of coefficients on the graph of general forms of equations. Many of the activities investigate mathematical concepts that arise from real world contexts (e.g., analyzing natality data, exploring the relationship between smoking and lung cancer, investigating projectile motion) and are connected with topics in the social studies and science curriculum. All activities are designed to enhance the teachers’, and subsequently their students’, understanding of mathematical concepts by using several mathematical representations (numerical, graphical, symbolic, verbal) and taking advantage of spreadsheet capabilities that allow the learner to extend beyond or significantly enhance what could be done using paper-and-pencil. Using spreadsheets to teach the same mathematical topics, in fundamentally the same ways, that could be taught without technology does not strengthen students’ learning of mathematics and belies the usefulness of spreadsheets. Thus, the activities take advantage of the available statistical analysis tools, the ability to link cells to create dynamic calculations and graphs, and the control forms (e.g., sliders) that can promote interactive and open-ended exploration of mathematical concepts. By using a spreadsheet to investigate mathematical concepts and problems, the teachers gain experience using a variety of functions and features of the spreadsheet. The teachers are also engaged in classroom discourse about mathematics as well as how and when to integrate spreadsheets in middle and high school mathematics curricula. The approach used by the second author to prepare teachers to use spreadsheets as an instructional tool has multiple purposes. First and foremost, the activities are designed to enhance and extend teachers’ knowledge about mathematics and to promote a problem solving, exploratory approach to doing mathematics that take advantage of the capabilities of a spreadsheet. Secondly, the teachers learn features of a spreadsheet within the context of solving mathematical problems and thus gain experience and knowledge of appropriate uses of a spreadsheet for teaching middle and secondary mathematics curriculum. A third purpose is to engage teachers in discussions on effective and appropriate uses of spreadsheets, and technology in general. As a final purpose, throughout the two semester course, the teachers create lessons and/or spreadsheet templates (such as the one shown in figure 2) that utilize features of the spreadsheet to teach mathematical concepts in such a way that significantly extends what is possible with paper-and-pencil. The use of technology in mathematics teaching should support and facilitate conceptual development, exploration, reasoning and problem solving, as described by the NCTM (1989; 1991). Furthermore, both authors believe that teachers who learn how to use technology (e.g., spreadsheets) while using it to explore mathematical topics are more likely to see its potential benefits and use it in their subsequent teaching. Although the contextual setting for the courses taught are quite different, the purposeful use of a spreadsheet as a cognitive tool for learning mathematical concepts is similar. Various instances of using an electronic spreadsheet program in mathematics education suggest that incorporation of this software into teacher education courses and subsequently into K-12 school settings can significantly affect school mathematics instruction. The authors argue that spreadsheets may be construed as a new generation of school-based educational software which use is not limited by financial constraints and commercial availability. Indeed, spreadsheets are commonly available in most schools as part of packages such as Microsoft Office, ClarisWorks, and Microsoft Works and even considered by many school administrators as one of the basic components of computer literacy. Nevertheless, there is a gap between using a spreadsheet as an accounting program and as a tool for conceptual development and educative growth. Therefore teachers' proficiency in the use of a spreadsheet as a cognitive tool becomes a crucial factor in advancing its use in the ...