Using GIS to Transform the Mathematical Landscape
Using GIS to Transform the Mathematical Landscape Bob Coulter, Missouri Botanical Garden Joseph Kerski, US Geological Survey
Abstract: GIS is increasingly being recognized as a valuable tool in science and social studies classes. Its power in advancing the mathematics curriculum through measurement and data analysis activities is all too often overlooked. Using the Principles and Standards for School Mathematics issued by the National Council of Teachers of Mathematics, the potential for GIS to advance student understanding in several domains of mathematics is assessed and illustrated. Teachers seeking to implement state and local curriculum standards and meet the ambitious agenda described in the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics face many challenges. In addition to striving toward a higher level of mathematical skills, a number of additional curriculum areas are now brought into play as we seek to develop a mathematically literate student body. Proficiency in calculations, equations, and basic geometry are no longer considered sufficient measures of a mathematically literate person. Instead, mathematical fluency now includes skills in areas such as representations, communication, and connections to other disciplines.
While this broadening and deepening of our conception of mathematical literacy is greatly needed, it poses a daunting task for teachers who seek to meet these new standards while they continue to meet a curriculum agenda already packed with traditional topics. Adjusting one’ s practice is never easy, but the magnitude and high stakes attached to this shift make it a formidable challenge. Clearly, given a finite number of hours in school, simply adding demands cannot succeed. Instead, new approaches must be taken which enable this broader set of goals to be met. If ever there was a time to think “ out of the box”in regard to the mathematics curriculum, this is it.
Compounding the challenge to build mathematical fluency is the persistent desire to use technology in educationally meaningful ways—not simply as an add-on, but as a learning
tool that is integral to students’growing understanding of mathematics. Educators and parents alike are becoming increasingly sophisticated consumers of educational technology. It is no longer sufficient to say “ Our school has a computer program”that is in fact neatly confined to a weekly computer lab time detached from the rest of the curriculum. To achieve meaningful integration requires a fundamental rethinking of the role of technology as one of several learning tools in the curriculum.
Fortunately, as part of the process of rethinking math curriculum and pedagogy, higherlevel use of technology may provide a vehicle by which the new goals can be met. In our work with hundreds of teachers over the past few years, we have found that thoughtful use of geographic information system (GIS) software has enabled classes to meet these higher academic challenges as they engage in sustained investigations of real-world data. These investigations—structured carefully to support the development of key skills— promote higher achievement and a more comprehensive understanding of the power of mathematics. As such, using GIS as an environment that supports mathematical investigations can be one of the keys to unlocking the curriculum gridlock imposed by increasing demands for content, skills, and applications.
Seeing GIS through a mathematical lens A geographic information system, or GIS, integrates data about the location of something and its attributes. For example, climate data is ultimately based on a location. Wherever your home town is, it can be located within a coordinate system (such as latitude and longitude). Knowing this, your town can be plotted on a map. By itself, this work with coordinate systems can be educationally valuable, but there are of course simpler ways to achieve this goal that do not require sophisticated computer software.
The power of GIS is in its unique ability to integrate this location (geographic) data with specific attributes. In the case of climate studies, local conditions will be significantly different in southern Arizona and northern Minnesota. The climate attributes that go with a given location—such as average monthly temperature and precipitation—give a quantitative measure to this difference. The average January temperature in Yuma,
2
Arizona (56.5 degrees) is 55.4 degrees warmer than the temperature for the same period in International Falls, Minnesota (1.0 degrees). More broadly, these patterns hold true across regions and continents. Educationally, there is a world of difference between the student who claims that it is warmer in Arizona than in Minnesota, and the student who is able to quantify the difference and interpret it as part of a regional pattern. The latter student is well on the way to becoming a mathematically literate citizen, both in skills and disposition. Students using GIS can build this capacity as they gain practice in analyzing the quantitative differences between variables on a map.
Another example can be found in the use of GIS to analyze changes in the American population over the course of the 20th century. Looking at the locations of the largest
3
counties in the United States, one can see a dramatic shift from the northeast and midwest to the south and southwest over the course of the century. This, of course, leads to interesting considerations of what factors would cause this change, and the different mathematical techniques needed to measure and document the change. On a national scale, a map showing the counties that have actually lost population over the course of the century (at a time when the overall U.S. population more than tripled from slightly over 76 million people to more than 281 million) presents a striking pattern. The shift from a rural, agricultural economy to an urban one is captured quite dramatically in this data. Qualitative descriptions from a social studies textbook take on new meaning with the addition of a mathematical perspective.
In either of these cases, the examples given are just a sampling of the mathematically rich investigations that are possible with GIS. Studies of a wide range of other phenomena, including changes in global demographics, severe weather patterns, and the migration patterns of birds and mammals lend themselves to similar analysis.
4
How can GIS promote mathematical understanding? In any of these investigations, students are applying fundamental mathematical skills as they analyze quantitative and spatial patterns in the data. Through this application, basic skills are developed through meaningful practice, and the development of a mathematical worldview is supported as students see how math is used in different contexts through investigations of real-world issues about the Earth.
In the climate and Census examples described above, several aspects of mathematics are brought to bear on the study. In fact, if these dimensions of mathematics were not employed, the study would be less rich and less meaningful for the students. Among the ten major strands in the Principles and Standards for School Mathematics, each has the potential to have a significant presence in a curriculum enhanced by GIS:
Number and operations lie at the core of most mathematical investigations undertaken in school. Without the basis of a strong understanding of number to build on (in terms of what the numbers represent and the number system being used—Celsius or Fahrenheit, for example) students’success will be limited. Having this understanding is only the beginning however, as students’capacity for effective use of this understanding of number will be developed considerably through its application in a range of contexts. Algebra skills are enhanced as students investigate and describe change, either from month to month in temperature and precipitation, or by decade with the Census data. This process also lends itself to investigations of algebraic concepts such as the slope of a line, as dramatic changes are represented by a steep slope, while relatively gradual changes have a more moderate slope. Geometry skills are likewise developed as students work with coordinate geometry in locating climate monitoring sites on their maps. Spatial analysis is enhanced as students investigate broad national patterns (such as population growth) in a data set, as well as more specific regional variations. For example, how many tornados strike in your state each month? Are their locations clustered, or spread out in an apparently random pattern?
5
6
Measurement is an integral part of an investigation as students consider time frames by comparing long-term data sets with current measurements. As part of this process, a student needs to have at least a working knowledge the data being investigated, including what the data set represents and how it was collected. In order for temperature data from multiple sites to be comparable, it needs to be collected using a standard protocol. Otherwise, data from one city collected at 4 pm may be quite different from data collected in another city at noon. Similarly, the Census data is collected using a standard protocol, but it is worth considering who is less likely to be counted, and what the significance of that undercounting may be. Teachers of more advanced students can capitalize on current issues relating to the census and whether statistical sampling should be substituted for a full enumeration. Data analysis and probability skills permeate GIS-enhanced investigations. The premise of these projects is that they are informed by real-world data. As these data sets are explored, a number of analysis skills are developed. These range from traditional concepts of central tendency (such as mean and median) to more advanced issues such as how to handle outliers within a data set. At times outliers can be quite interesting, while at other times they may simply represent measurement or reporting errors. More generally, as noted in the discussion of measurement, students need to become critical consumers of what the numbers in a data set actually represent. Throughout an investigation, as questions are formulated about a data set and appropriate methods of analysis are chosen, data literacy skills are bolstered. Problem solving is an integral part of any real-world project. Engagement with authentic phenomena naturally invites a problem-solving disposition. Analysis of earthquake data from around the world offers students opportunities to investigate their local risk and to see how mathematics is used to help earth scientists monitor geologic processes. Likewise, thoughtful investigations of population studies invite very concrete considerations of how rapid changes in population affect life in an area, as well as more abstract but equally interesting issues that arise, such as the need for reapportionment of representatives every ten years based on the Census data.
7
Reasoning and proof capacities are enhanced as students develop explanations and justifications for patterns they observe. By working with real data students see that generalizations such as “ It’ s colder in the north and warmer in the south”are at best only partially true. Through spatial analysis, more carefully reasoned perspectives are developed that are informed by the data. This data in turn can be used to establish proof for the conjectures students develop. Communication skills are intimately involved in mathematics as students develop and express mathematical arguments such as explanations for the patterns they observe in population changes, and evaluate arguments offered by others. Ultimately, careful use of language depends on and in turn reflects the quality and refinement of a person’ s mathematical thinking about the issue at hand. In turn, students need the ability to critically examine the data that is now contained routinely in advertising and news stories. Connections among mathematical ideas are fostered as students integrate numbers, graphs, tables, and maps as part of their investigations. Rather than being separate units (“ This week we’ re doing graphs in math class…” ), an array of mathematical tools and concepts are brought to bear on a problem, further supporting an integrated view of mathematics as a discipline. Likewise, an understanding of the connections between math and other disciplines is supported as students see how math enables a better understanding of science and social studies topics. Representation—Numeric, graphic, and spatial representations of a set of data each have particular strengths and limitations. As students become critical consumers of each of these, they are better able to clarify their thinking about the investigation at hand. Taking a longer-term perspective, practice in the skilled use of these representations enables a transfer of these skills to other contexts.
Why use GIS in the math classroom? As the examples above illustrate, GIS has the capacity to support skill development in many dimensions of mathematics. As such, it is widely applicable in the math classroom. In addition to the investigations described here, any topic for which there is a spatial (or
8
geographic) component is ripe for a GIS-enhanced investigation. The key to effective use of the software is you—the teacher—as a role model actively posing meaningful, mathematically rich questions and leading your students in investigating relevant data. In this environment, GIS becomes an essential tool as it is able to work with data tables, maps, and graphs of data in support of your explorations.
Ultimately, mathematical understanding as exemplified in the Principles and Standards for School Mathematics and other documents goes well beyond computational skills and manipulating equations. It is precisely this broader view of mathematics that GISenhanced investigations support. This is where schools need to evolve toward if math education is to improve. When students experience math as a useful and at times even elegant way of seeing the world—and not just as an endless series of exercises—they are much more likely to sustain a meaningful engagement with the discipline.
Author information
Robert Coulter, Ed.D. Senior Manager Missouri Botanical Garden PO Box 299 St. Louis, MO 63166-0299 (314) 577-0219 [email protected] Joseph J. Kerski, Ph.D. Geographer: Education/GIS US Geological Survey Building 810 - Entrance W-5 - Room 3000 Box 25046 - MS 507 Denver CO 80225-0046 Voice 303-202-4315 [email protected]
9
Abstract: GIS is increasingly being recognized as a valuable tool in science and social studies classes. Its power in advancing the mathematics curriculum through measurement and data analysis activities is all too often overlooked. Using the Principles and Standards for School Mathematics issued by the National Council of Teachers of Mathematics, the potential for GIS to advance student understanding in several domains of mathematics is assessed and illustrated. Teachers seeking to implement state and local curriculum standards and meet the ambitious agenda described in the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics face many challenges. In addition to striving toward a higher level of mathematical skills, a number of additional curriculum areas are now brought into play as we seek to develop a mathematically literate student body. Proficiency in calculations, equations, and basic geometry are no longer considered sufficient measures of a mathematically literate person. Instead, mathematical fluency now includes skills in areas such as representations, communication, and connections to other disciplines.
While this broadening and deepening of our conception of mathematical literacy is greatly needed, it poses a daunting task for teachers who seek to meet these new standards while they continue to meet a curriculum agenda already packed with traditional topics. Adjusting one’ s practice is never easy, but the magnitude and high stakes attached to this shift make it a formidable challenge. Clearly, given a finite number of hours in school, simply adding demands cannot succeed. Instead, new approaches must be taken which enable this broader set of goals to be met. If ever there was a time to think “ out of the box”in regard to the mathematics curriculum, this is it.
Compounding the challenge to build mathematical fluency is the persistent desire to use technology in educationally meaningful ways—not simply as an add-on, but as a learning
tool that is integral to students’growing understanding of mathematics. Educators and parents alike are becoming increasingly sophisticated consumers of educational technology. It is no longer sufficient to say “ Our school has a computer program”that is in fact neatly confined to a weekly computer lab time detached from the rest of the curriculum. To achieve meaningful integration requires a fundamental rethinking of the role of technology as one of several learning tools in the curriculum.
Fortunately, as part of the process of rethinking math curriculum and pedagogy, higherlevel use of technology may provide a vehicle by which the new goals can be met. In our work with hundreds of teachers over the past few years, we have found that thoughtful use of geographic information system (GIS) software has enabled classes to meet these higher academic challenges as they engage in sustained investigations of real-world data. These investigations—structured carefully to support the development of key skills— promote higher achievement and a more comprehensive understanding of the power of mathematics. As such, using GIS as an environment that supports mathematical investigations can be one of the keys to unlocking the curriculum gridlock imposed by increasing demands for content, skills, and applications.
Seeing GIS through a mathematical lens A geographic information system, or GIS, integrates data about the location of something and its attributes. For example, climate data is ultimately based on a location. Wherever your home town is, it can be located within a coordinate system (such as latitude and longitude). Knowing this, your town can be plotted on a map. By itself, this work with coordinate systems can be educationally valuable, but there are of course simpler ways to achieve this goal that do not require sophisticated computer software.
The power of GIS is in its unique ability to integrate this location (geographic) data with specific attributes. In the case of climate studies, local conditions will be significantly different in southern Arizona and northern Minnesota. The climate attributes that go with a given location—such as average monthly temperature and precipitation—give a quantitative measure to this difference. The average January temperature in Yuma,
2
Arizona (56.5 degrees) is 55.4 degrees warmer than the temperature for the same period in International Falls, Minnesota (1.0 degrees). More broadly, these patterns hold true across regions and continents. Educationally, there is a world of difference between the student who claims that it is warmer in Arizona than in Minnesota, and the student who is able to quantify the difference and interpret it as part of a regional pattern. The latter student is well on the way to becoming a mathematically literate citizen, both in skills and disposition. Students using GIS can build this capacity as they gain practice in analyzing the quantitative differences between variables on a map.
Another example can be found in the use of GIS to analyze changes in the American population over the course of the 20th century. Looking at the locations of the largest
3
counties in the United States, one can see a dramatic shift from the northeast and midwest to the south and southwest over the course of the century. This, of course, leads to interesting considerations of what factors would cause this change, and the different mathematical techniques needed to measure and document the change. On a national scale, a map showing the counties that have actually lost population over the course of the century (at a time when the overall U.S. population more than tripled from slightly over 76 million people to more than 281 million) presents a striking pattern. The shift from a rural, agricultural economy to an urban one is captured quite dramatically in this data. Qualitative descriptions from a social studies textbook take on new meaning with the addition of a mathematical perspective.
In either of these cases, the examples given are just a sampling of the mathematically rich investigations that are possible with GIS. Studies of a wide range of other phenomena, including changes in global demographics, severe weather patterns, and the migration patterns of birds and mammals lend themselves to similar analysis.
4
How can GIS promote mathematical understanding? In any of these investigations, students are applying fundamental mathematical skills as they analyze quantitative and spatial patterns in the data. Through this application, basic skills are developed through meaningful practice, and the development of a mathematical worldview is supported as students see how math is used in different contexts through investigations of real-world issues about the Earth.
In the climate and Census examples described above, several aspects of mathematics are brought to bear on the study. In fact, if these dimensions of mathematics were not employed, the study would be less rich and less meaningful for the students. Among the ten major strands in the Principles and Standards for School Mathematics, each has the potential to have a significant presence in a curriculum enhanced by GIS:
Number and operations lie at the core of most mathematical investigations undertaken in school. Without the basis of a strong understanding of number to build on (in terms of what the numbers represent and the number system being used—Celsius or Fahrenheit, for example) students’success will be limited. Having this understanding is only the beginning however, as students’capacity for effective use of this understanding of number will be developed considerably through its application in a range of contexts. Algebra skills are enhanced as students investigate and describe change, either from month to month in temperature and precipitation, or by decade with the Census data. This process also lends itself to investigations of algebraic concepts such as the slope of a line, as dramatic changes are represented by a steep slope, while relatively gradual changes have a more moderate slope. Geometry skills are likewise developed as students work with coordinate geometry in locating climate monitoring sites on their maps. Spatial analysis is enhanced as students investigate broad national patterns (such as population growth) in a data set, as well as more specific regional variations. For example, how many tornados strike in your state each month? Are their locations clustered, or spread out in an apparently random pattern?
5
6
Measurement is an integral part of an investigation as students consider time frames by comparing long-term data sets with current measurements. As part of this process, a student needs to have at least a working knowledge the data being investigated, including what the data set represents and how it was collected. In order for temperature data from multiple sites to be comparable, it needs to be collected using a standard protocol. Otherwise, data from one city collected at 4 pm may be quite different from data collected in another city at noon. Similarly, the Census data is collected using a standard protocol, but it is worth considering who is less likely to be counted, and what the significance of that undercounting may be. Teachers of more advanced students can capitalize on current issues relating to the census and whether statistical sampling should be substituted for a full enumeration. Data analysis and probability skills permeate GIS-enhanced investigations. The premise of these projects is that they are informed by real-world data. As these data sets are explored, a number of analysis skills are developed. These range from traditional concepts of central tendency (such as mean and median) to more advanced issues such as how to handle outliers within a data set. At times outliers can be quite interesting, while at other times they may simply represent measurement or reporting errors. More generally, as noted in the discussion of measurement, students need to become critical consumers of what the numbers in a data set actually represent. Throughout an investigation, as questions are formulated about a data set and appropriate methods of analysis are chosen, data literacy skills are bolstered. Problem solving is an integral part of any real-world project. Engagement with authentic phenomena naturally invites a problem-solving disposition. Analysis of earthquake data from around the world offers students opportunities to investigate their local risk and to see how mathematics is used to help earth scientists monitor geologic processes. Likewise, thoughtful investigations of population studies invite very concrete considerations of how rapid changes in population affect life in an area, as well as more abstract but equally interesting issues that arise, such as the need for reapportionment of representatives every ten years based on the Census data.
7
Reasoning and proof capacities are enhanced as students develop explanations and justifications for patterns they observe. By working with real data students see that generalizations such as “ It’ s colder in the north and warmer in the south”are at best only partially true. Through spatial analysis, more carefully reasoned perspectives are developed that are informed by the data. This data in turn can be used to establish proof for the conjectures students develop. Communication skills are intimately involved in mathematics as students develop and express mathematical arguments such as explanations for the patterns they observe in population changes, and evaluate arguments offered by others. Ultimately, careful use of language depends on and in turn reflects the quality and refinement of a person’ s mathematical thinking about the issue at hand. In turn, students need the ability to critically examine the data that is now contained routinely in advertising and news stories. Connections among mathematical ideas are fostered as students integrate numbers, graphs, tables, and maps as part of their investigations. Rather than being separate units (“ This week we’ re doing graphs in math class…” ), an array of mathematical tools and concepts are brought to bear on a problem, further supporting an integrated view of mathematics as a discipline. Likewise, an understanding of the connections between math and other disciplines is supported as students see how math enables a better understanding of science and social studies topics. Representation—Numeric, graphic, and spatial representations of a set of data each have particular strengths and limitations. As students become critical consumers of each of these, they are better able to clarify their thinking about the investigation at hand. Taking a longer-term perspective, practice in the skilled use of these representations enables a transfer of these skills to other contexts.
Why use GIS in the math classroom? As the examples above illustrate, GIS has the capacity to support skill development in many dimensions of mathematics. As such, it is widely applicable in the math classroom. In addition to the investigations described here, any topic for which there is a spatial (or
8
geographic) component is ripe for a GIS-enhanced investigation. The key to effective use of the software is you—the teacher—as a role model actively posing meaningful, mathematically rich questions and leading your students in investigating relevant data. In this environment, GIS becomes an essential tool as it is able to work with data tables, maps, and graphs of data in support of your explorations.
Ultimately, mathematical understanding as exemplified in the Principles and Standards for School Mathematics and other documents goes well beyond computational skills and manipulating equations. It is precisely this broader view of mathematics that GISenhanced investigations support. This is where schools need to evolve toward if math education is to improve. When students experience math as a useful and at times even elegant way of seeing the world—and not just as an endless series of exercises—they are much more likely to sustain a meaningful engagement with the discipline.
Author information
Robert Coulter, Ed.D. Senior Manager Missouri Botanical Garden PO Box 299 St. Louis, MO 63166-0299 (314) 577-0219 [email protected] Joseph J. Kerski, Ph.D. Geographer: Education/GIS US Geological Survey Building 810 - Entrance W-5 - Room 3000 Box 25046 - MS 507 Denver CO 80225-0046 Voice 303-202-4315 [email protected]
9
