# Mathematics

## What is GIS?

Geographic Information Systems (GIS) allows us to view, understand, question, interpret, and visualize data in many ways that reveal relationships, patterns, and trends in the form of maps, globes, reports, and charts. A GIS helps you answer questions and solve problems by looking at your data in a way that is quickly understood and easily shared.

## Why Use GIS in Mathematics Education?

GIS provides an excellent way to teach mathematical concepts and skills throughout all STEM disciplines. The value of visualizing numbers is affirmed throughout the US Principles and Standards for School Mathematics, designed by the National Council of Teachers of Mathematics (NCTM). Representing numbers, understanding patterns, relationships, and function, 2-D and 3-D geometric and spatial relationships, probability, statistics, change, models, measurements, problem solving, reasoning, connections, and communications are critical concepts. Every one of these can be explored using GIS tools and methods. Comparing graphs and maps of birth and death rates over time and region, analyzing the response of a stream to a recent storm through a real-time hydrograph, and creating cross-sections of terrain are three common activities in geography instruction, easily done in a GIS environment. All of them—and thousands more geographic activities—involve analyzing numbers. One might say that GIS is visualizing numbers, since its basis is representing numbers as cells, points, lines, or polygons on a map.

NCTM’s curricular “focal points” also connect well with GIS. A focal point must pass three rigorous tests: Is it mathematically important, both in mathematics and for use in applications in and outside of school? Does it “fit” with what is known about learning mathematics? Does it connect logically with the mathematics in earlier and later grade levels? When we connect latitude and longitude to the Cartesian coordinate system, when we measure area, shape, size, and distance in different map projections, when we compare geometric to exponential growth rates of agricultural output, even when we explain the Earth’s shape, rotation, and revolution, we are applying geographic and mathematical concepts and can use GIS to teach it.

The use of GIS brings math to life by making math visual. Think of common problems such as: “Where and when will these trains cross paths? One departs Point A at 6:00am and heads toward Point B at 70kph, while the other departs Point B at 6:30am bound for Point A at 60kph.” Where and when will they cross paths?” GIS allows the anchoring of these and other problems in the real world: Points A and B could be Cheyenne and Casper, and students can determine and plot the course and passing point at a real point on a GIS map layer.

Solving these and other math problems in a GIS environment allows students to grapple with biodiversity, crime, natural hazards, climate, energy, water, and other relevant real-world issues of the 21st Century. Students often do not feel that what they are learning is relevant to what they will be doing after they get out of school. Hundreds of jobs in geospatial technologies—not just surveyors and remote sensing analysts—require analytical, statistical, and computational skills that are learned in mathematics.

These connections between spatial analysis and mathematics have deep historical roots: Willis Ernest Johnson wrote Mathematical Geography back in 1908, and Eratosthenes connected geography to mathematics in measuring the Earth’s circumference over 2,200 years ago!

## How To Use GIS in Mathematics Education

Books

Mapping Our World

Mapping Our World from ESRI feature software, lessons, data, and assessments—everything you need to get started exploring math within the spatial framework and GIS. The lessons cover from local to global scale and include the analysis of climate, natural hazards, population, and other data.

Fun With GPS

Millions of people love their global positioning system (GPS) devices but do not know how they can use this amazing technology to do much more than simply calculate a location. The dozens of illustrated examples in Fun with GPS show how GPS devices can be attached to just about anything or anyone. Each example includes maps and photographs that will inspire hobbyists, airline passengers, athletes, animal lovers, drivers, teachers, and students to track and map their activities. This book also discusses ways to incorporate GPS technology in the classroom to reinforce science, math, and geography curricula, and provide groundwork for project-based learning.

GIS Tutorial

This book provides effective GIS training in an easy-to-follow format. By combining ArcGIS® tutorials with self-study exercises intended to gradually build upon basic skills, the GIS Tutorial is fully adaptable to individual needs, as well as a classroom setting. This book features two additional chapters that utilize ArcGIS® 3D Analyst™ and ArcGIS® Spatial Analyst applications.

Spatial Mathematics

What fundamental mathematics principles provide the foundation for geography and spatial analysis with GIS? Spatial mathematics and analysis, two different approaches to scholarship, yield different results and employ different tools. This book explores both approaches to looking at real world issues that have mathematics as a critical, but often unseen, component. Readers learn the mathematics required to consider the broad problem at hand, rather than learning mathematics according to the determination of a (perhaps) artificial curriculum. This format motivates readers to explore diverse realms in the worlds for geography and mathematics and in their interfaces. It not only provides fundamental principles and theory, but also practical applications with web maps, GIS tools, and with mobile GPS-enabled devices in the field.

Other Lessons

The following lessons in the ArcLessons library illustrate the diversity in scale, content, and skills possible through the use of GIS in mathematics. Each is packaged with rich data sets so that you can begin using them right away.

Measuring the Circumference of the Earth with GPS

With a GPS receiver, find out how you and your students can accurately measure the circumference of the Earth to within 1% of the accepted value. In this lesson, students start by learning about Eratosthenes, who calculated the circumference 2,500 years ago. They then go into the field to calculate the circumference in a variety of different ways, and also consider mass and volume. This lesson incorporates scale, measurement, field work, coordinate systems, and brings together mathematics, geography, and physics.

Geocaching: Quest for the Boulder Missing Map

This activity was established to illustrate the types of questions that could be posed in setting up and running a geocaching course. This course was set up and run for the annual International Map Trade Association conference in Boulder Colorado USA, but the ideas presented could be adapted for any local area. The course includes 21 questions. Mission (Fictitious): Find the missing map made by Arab cartographer Al-Idrisi (1099-1166) entitled "The Gardens of Humanity and the Amusement of the Soul."

A Century of Population Change: USA 1900-2000

Students and researchers will use maps, databases, graphs, and layouts along with a series of detailed spatial data and analyses to study the historical and modern-day reasons for the spatial pattern of population change by county.

Travel Time To Major Cities

Have you ever wanted to get away from it all? In this activity, you will explore the concepts of remoteness and access. Analyze global travel times to major cities in a spatial context within a Geographic Information Systems (GIS) environment.

The 10 Most Populous Cities: Spatial Statistics

Dig deeper into investigating the 10 most populous cities in the world for the past 2000 years, using GIS as your primary investigative tool in this lesson. You will pay special attention to the weighted mean centers of population over time and the directional distributions during your investigation.

Analyzing Water Use With GIS

In this lesson, you will understand (1) how to access and format data from the USA County database from the US Census Bureau within a GIS environment; (2) how to analyze these data, specifically about water and by extension, other variables—using GIS and spatial statistics techniques, including regression analysis, hot spot, scatterplot, and others.

Investigating Extreme Temperatures in the USA

In this activity, you will investigate temperature extremes in the USA for the month of January 2009. You will use GIS for your primary investigative tool and begin with 30 questions. What effect does proximity to coastlines, altitude, or latitude have on daily high and low temperature? Shapefile version.

On the Road Again: Transportation Analysis of the USA

This 70-question lesson and spatial data are to be used with ArcGIS software by ESRI to study the spatial pattern of transportation--roads, waterways, railroads, and airports, in the USA. Rich data on vehicle volume and more from the National Transportation Atlas Dataset (NTAD).

Crops of the World: Examining 5 Crops Using GIS

Investigate the spatial pattern of 5 crops around the world--corn (maize), soybeans, cotton, wheat, and rice, using GIS as the investigative tool. Explore the relationship of crops to demographic variables and to ecoregions and biomes. Create new data and calculate the extent of biomes, and consider how different biomes and ecoregions support different crops.