Reasoning With Mathematics
Lesson 8a Visualizing With Maps
The graphics used in the previous lesson line graphs, bar graphs, and pie charts are
relatively simple, which can be both an advantage or a disadvantage. Simple visualizations
can communicate quickly and forcefully. In fact, the very word graphic gets used to label
something that portrays information quickly and forcefully, as in the phrase a graphic
example or a graphic photo. That's their main and very important advantage.
Disadvantages start to appear when the amount of information increases or the kind of
information becomes complex. A simple chart or graph may no longer work so well.
What's Your Living Space Like?
Suppose you wanted to describe the apartment or house you live in and decide to use some
visual aids. If you wanted to (and thought it would help your audience) you could do a pie
chart showing the size of rooms, say in square feet of floor space. Or you could do a bar
graph showing the number of windows in each room. Or a line graph showing the number of
doors in each room.
This starts to sound pretty silly as ways of showing visualizing what your place looks like.
And most people would immediately think of a much better way. Something like this, for
instance, would be much better:
Insert Davidson II floor plan at http://www.sarp.com/homes/floorplans/david2fp.htm
or equivalent
From this floorplan the standard name for this kind of graphic you can visualize a great
deal about the house it portrays including:
How many rooms there are, what they're designed to be, and their floor size
How rooms are arranged within the house
How many bathrooms there are, where they're located, and which have a tub and which
have a shower
Where doors are located, which side they're hinged on, and which way they swing open
The floorplan tells more than these things, in fact, which you can figure out in just a few
moments of looking at it.
At the same time, plenty of information about this house isn't on the floorplan. You can't tell
how high the ceilings are, for instance, or the height of anything else for that matter. (You
might think of one or two ways to show that information if you wanted to or needed to. And
later in this lesson, we'll come back to this kind of problem.) You can't tell which rooms, if
any, have carpet on the floors or wallpaper on the walls. (Again, you might have ideas about
how to show these details.)
But for right now, focus on what is on the floorplan. It's packed with information about
size and shape, two things that mathematics deals with over and over again. In other
words, a floorplan is very mathematical and puts its math visually.
What's shown in the sample floorplan used here could be described in words, spoken or
written. But if you try saying or writing out what the floorplan shows you would quickly see
that in this case the picture is worth a thousand words or more.
Maps, Maps, Maps
A floorplan is a kind of map, one kind among many. Maps are among the most useful and
creative things human beings have developed. Imagine the different maps you might use in
one day:
Weather maps on television or in a newspaper that tell you what to expect today and
tomorrow's weather will be like
Shopping center maps that tell "you are here" and where to find the shop you want
Highway maps that help you decide which route to follow and estimate how long your trip
will take
Freeway maps that tell you where there are delays because of accidents or road
construction
News maps that show where a plane crashed or a child is lost or floods are running
Lake maps that show where to rent a boat, go swimming, or try fishing for walleyes
Sky maps that show where to look for a constellation at different times of year
Plat book maps that show the location, size, and shape of the lot on which your house or
apartment is built
Getting Our Eyes on Some Maps
Maps are some of the oldest forms of visualization known to humankind. An ancient one is
on a clay tablet found in the Middle East and identified as Sumerian and thought to be at least
4,500 years old. (The ancient civilization of Sumeria was in what is now the nation Iraq.)
Egyptian maps from around the 14th century B.C. show property lines along the Nile River
and were used to keep track of taxes. Maps from ancient Greece show that people long before
Columbus visualized the earth as a sphere.
Like so many human creations, mapmaking has gone along with the times. When papyrus
and then paper was developed, maps were made on them; when printing was developed,
maps were printed; after European explorers returned home with their revised navigation
charts and other information, old maps were redrawn and new ones made; when airplanes
made aerial photography possible, mapmakers used those photos to make or improve their
products. Satellites and space vehicles make it possible to use cameras and other sensing
equipment from enormous distances and covering wide areas, which serves cartographers
(the technical name for mapmakers) in many ways. Computers revolutionized
cartography; in fact, mapmaking is a very highly computerized field today.
Predictably, the WWW includes a lot about maps and mapmaking. Because of that you'll
be using WWW resources for most of what you do with this topic.
There Are Maps and There Are Other Maps
Maps portray something about a place, typically practical information of some kind. The
place may be large say the North American continent or the Eastern hemisphere. Or the
place may be small a very small town or one square block of a city.
The most familiar maps, the ones most of us think of first, show some place on earth: a city,
park, county, state, country, subcontinent, continent. Probably the first uses that come to
mind are to know where we are or to get from one spot to another in map language for
location or for navigation.
Imagine that you or someone else is planning to visit Salt Lake City in the state of Utah, on
the shore of the Great Salt Lake, and near the Wasatch Range of the Rocky Mountains. [ use
photo @ http://www.slc.k12.ut.us/ ] You can see the mountains in the background of the
photo of downtown Salt Lake City.
A road map of the Salt Lake City area would be handy of course. [ link to road map at
http://info.er.usgs.gov/education/teacher/what-do-maps-show/index.html#road ] It
shows you major and secondary highways, maybe a few major streets within towns and cities,
and other basic information as well: a little about physical features such as mountains, rivers,
and lakes; some political features such as city limits, county lines, or park boundaries; a few
related topics such as the location of airports, rest areas, campgrounds, universities, churches,
and so on.
This relief map [http://info.er.usgs.gov/education/teacher/what-do-maps-show/index.html#shaded ]
of an area just north of Salt Lake City portrays some of the same information as the road map
but emphasizes the shape of the area it shows: the ups and downs of hills, valleys, and
mountains; the turns and curves of rivers, lakes, and roads. This map is called a shaded relief
map because it uses color shading to portray shapes. It might help if you imagine you are
looking at the territory from high above on a sunny day and seeing the shadows on the
landscape. Relief maps, using shading or some other technique, often appear in geography or
other schoolbooks to help students visualize what some part of the world.
This topographic map [http://info.er.usgs.gov/education/teacher/what-do-maps-show/index.html#topo ]
of a small part of Salt Lake City also portrays the ups and downs of its territory but does it in
a different way. If you look at the right side of the map, you can see a series of lines curving
along, many of them almost parallel to another one. Each one, called a contour line, stands
for a given elevation, a height above some known reference point usually sea-level. One of
the lines on this map, for instance, will show where it is 4,500 feet above sea-level. To get an
idea of how that line is drawn, imagine that you had some way of walking along the ground
on a path that stayed at 4,500 feet above sea-level. That's the line on the map. There's another
line for 4,450 feet and another for 4,550 feet above sea-level. And so on.
Topographic maps, called top-o maps for short, give very detailed information about the
shape of a territory. With a few skills and some practice, people can use top-o maps to
recognize hills, valleys, dry and wet areas, and other landmarks that tell them where they are
and how to get to another point. So they're popular with people who spend time in relatively
remote places, people like hunters, fishers, hikers, resource explorers, rangers, and so on.
There is a topographic map for every square mile of the United States, a project that took
many years to complete. These maps are made by the United States Geologic Survey and can
be purchased from them or in stores.
Some Key Ideas About Maps And That You Already Know
Maps are a striking example of the key ideas you dealt with in the Mindquest course Learning
Strategies. To make a decent map, let alone an excellent one, a mapmaker will use the
principles introduced in that course.
Mapmakers use chunking. A map covers a defined territory, a chunk. There usually is an
honest-to-goodness, literal line that says, in effect, "I'm going to show you something about
the chunk inside this boundary." As mentioned earlier, the chunk may be large or small or in
between. But it is a chunk.
Mapmakers pay attention. They pay attention to whatever it is about the chunk they want to
portray. This paying attention shows in several ways including what details they collect to
make their maps. They pay attention to what to include and, sometimes, what to exclude.
Mapmakers elaborate. They search for effective ways to clarify and emphasize their main
purposes. So if they are making road maps, the mapmakers elaborate the roads; if they're
making relief maps, they elaborate shapes. Other information might be included but it won't
be elaborated as thoroughly. They also decide how to elaborate, partly on the basis of how the
maps will be used. For example, if doing a relief map they might consider shading, color
coding, contour lines, or sets of elevation numbers. Shading might work well for general
users, color coding for young children, contour lines for resource explorers, and numbers for
construction engineers.
Mapmaker make connections. Maps deals with many kinds of connections. First of all,
there's the basic connection between information and location, like saying "This is right here."
Or "You're looking for a rest area on this highway and you're driving this direction. Here it
is."
There are other connections, too. Many maps connect users to another map for adjoining area
or for another kind of information, sometimes directly and sometimes indirectly. If you look
at a road map of Montana you will probably see around its edges small areas of Canada, North
Dakota, South Dakota, Wyoming, and Idaho. Without necessarily saying so, the map refers
you to maps for those locations. That Montana road map might also tell you to "see inset map
on other side" for a more detailed view of a city such as Billings, Helena, or Missoula city
maps, in other words.
Like other materials found there, map sites on the WWW depend a lot on connections in the
form of links which create one connection after another.
Three Features That Most Maps Use
Scale
Maybe you or someone you know has made a model car, space vehicle, airplane, ship, or
similar thing from a kit sold at a hobby store. Obviously, models rarely are the same size as
the real thing; instead they're usually proportional to the real thing. So a model car might be
1/30th or 1/40th of the actual one; a model ship might be 1/100th or 1/200th the size of the real
one.
Scale is another word for this proportion between the model and the real object. So the
package for a kit might say something like "1/40th scale" or "scale 1:40." Both mean the same
thing: one unit of measurement on the model equals 40 units on the real thing. For instance,
one inch on the model is the same as 40 inches on the real thing.
Maps almost always are drawn to scale and tell you what scale they use. The map may tell
you in words such as "one inch equals ten miles." Or it may use a line segment that looks
something like this:
0 25 km 50 km
Explaining scale in words or with a line segment usually helps the map user get an idea of the
size of the territory in some standard unit of measure such as miles, kilometers, or feet.
Or the map may use numbers in a ratio such as "1 : 20,000," which is an actual proportion
statement saying that one unit of measure on the map equals 20,000 units of the real thing.
(Because maps so often deal with geographic places, users often say that a 1 : 20,000 scale
means that one unit of measure equals 20,000 on the ground.) This way of explaining scale
has one big advantage: it works no matter what unit of measure is used. One inch on the map
equals 20,000 inches on the ground; one half-centimeter on the map equals 20,000 half-centimeters on the ground; one pencil length on the map equals 20,000 on the ground.
Knowing the scale also helps users know how detailed the map is likely to be. A map with a
scale of 1 : 20,000 covers less territory than one with a scale of 1 : 100,000 (if both maps are
the same size piece of paper). And a map covering less territory can show more detailed
information.
Orientation
North is at the top of most maps, something most of us know from using maps over and over
again. We don't need to be told. This pattern probably is the reason people say they're going
"up north" or "down south." (Try saying "down north" or "up south" and notice how
awkward they seem.)
Orienting maps with north at the top is a convention, an unspoken agreement among map
makers and map users. It's been the pattern for hundreds of years and throughout the world,
even though common sense might say that in the southern hemisphere south should be at the
top.
Of course general maps can be drawn with any direction at the top. If you had to draw a map
showing someone how to reach your home, you might draw it with some main street or road
going up and down (or left to right) and not worry about compass directions.
Small areas like cities, parks, lakes, or campuses frequently are drawn so the map is square
with the boundaries which can mean the top of the map is not straight north. Such maps
sometimes use a symbol that looks like a compass needle to point out straight north.
Topographic maps, which are often used along with a compass, have to deal with an
important detail about how a compass works. A compass needle lines up with the magnetic
force lines of the earth and points to magnetic north, which is not exactly the same as true
north. Topo maps show how to correct for the difference between them with what's called
declination.
Using a correction factor, like declination, is pretty common. Take the height of a person, for
instance. Is height measured while the person is wearing shoes? Or in "stocking feet"? Well,
it might be more convenient to measure someone's height with shoes on, then subtract an inch
to account for shoes. The amount subtracted is the correction factor. Similar correction
factors get used in measuring many different things: time, distance, weight, cost, and more.
All the examples of orientation used here are for maps of something on earth, some part of the
earth's surface. Different orientation questions pop up for maps showing something other than
the earth's surface. How would you orient a map of an underground mine? Or a map of our
solar system?
Legends
Many maps have a kind of index on them explaining special symbols. The index, called a
legend, usually appears as a separate box or section next to the map.
Road maps typically have a long legend because the maps include a lot of information. They
show different kinds of roads and use symbols to distinguish among them: maybe thick green
lines for interstate freeways, thinner but still heavy red lines for main highways, skinny blue
lines for secondary highways, and brown lines for unpaved roads. (It's been very common for
maps to use thin blue lines for secondary highways, which usually have less traffic on them
and go through smaller communities. So much so that some people call those roads "blue
highways.")
A Quick Look at Some of These Features
You can see examples of scale, orientation, and legends on most maps that show some part of
the earth's surface including this map of the Burlington, Ontario transit system
[http://www.wchat.on.ca/cob/tt/westsysmap.html - Burlington transit system map ]
Scale is shown indirectly on this map with the names of major streets and highways, which
users are supposed to know. Orientation is shown with the compass symbol in the lower left
of the screen. There's a button to click that will take you to the legend explaining the
symbols.
Thematic or Data Maps
Maps get produced and used for purposes other than location and navigation. Put into the
ideas and words of this course, maps are used to visualize other kinds of information
besides where you are or how to get somewhere. People use maps to show information
about a place, where something happens, or how something has spread from one place to
another, for instance. These might be called thematic maps (because they're organized around
one theme) or data maps (because they visualize data a set of numbers).
Imagine someone who works for a company that makes salsa and who is responsible for
marketing the products. That person might be interested in knowing where in the United
States salsa sells well and where it sells poorly, in other words where there is already a strong
market for salsa and where the market is weak.
Here's something the marketing specialist might find interesting. It's some but not all of a
short feature published in The Atlantic Monthly magazine for May, 1997. This first part is the
text; what's not here right now is the accompanying map that's mentioned in the text.
Read the text first. After that, you can see the map.
MACROBUTTON HtmlDirect * MACROBUTTON
HtmlResImg 
MACROBUTTON HtmlResImg 
MACROBUTTON HtmlResImg
MACROBUTTON HtmlResImg  LIKE stir-fry
and bagels, salsa has gone mainstream. The growing
appetite for salsa and other spicy Mexican sauces -- salsa
recently surpassed catsup as America's best-selling
condiment, with annual sales of $700 million -- has
made an unusual migration in the United States, as seen
in this map of the nation's 212 television markets.
Whereas most culinary trends begin in the large coastal
metropolitan areas and creep inward to the heartland,
salsa has its roots among rural Latinos in the Southwest
and has spread north and east. This pattern bucks the
usual "proletarian drift," whereby first upscale urbanites
become enamored of a new product, such as Starbuck's
coffee, and then, in time, even the most isolated market
in Appalachia is selling caffè latte. |
| MACROBUTTON HtmlDirect *Demographic surveys
show that salsa consumers tend to be of two types:
downscale Latino families, for whom salsa (Spanish for
"sauce") is a staple, and upscale Anglo families, who can
afford to buy a condiment that is more expensive than
catsup and who appreciate salsa's low fat content.
As a group, salsa aficionados are more likely to be
married than single and rural than urban. Salsa is
probably more popular among families than among
singles because larger households do more cooking at
home. This is also a reason why salsa sales are not high
in the nation's big cities, where singles, who eat out
regularly, make up a large proportion of residents.
Another reason is that salsa is not particularly popular
with African-Americans, who tend to prefer sweeter,
tangier flavors.
Although the map does generally reflect the settlement
patterns of the Latino population in the United States,
there are several anomalies. For example, two centers of
Latino settlement, Miami and New York, have relatively
low rates of salsa consumption; salsa is not as popular
among Latinos of Cuban or Puerto Rican descent as
among those from Mexico or Central America.
-- Michael J. Weiss
MACROBUTTON HtmlDirect <HR noshade size=1>
Copyright © 1997 by The Atlantic Monthly Company. All rights reserved.
The Atlantic Monthly; May 1997; The Salsa Sectors; Volume 279, No. 5;
page 85.
|
The map [ http://www.TheAtlantic.com/atlantic/issues/97may/count.htm ]
The map literally shows some patterns and some of them can be taken in even with a quick
look. Among other things, it makes it possible for a user to get some information on every
place or part of the map. You could, for instance, ask yourself "What does the map say about
the place I live?" or "What does it say about the Miami, Florida area?" and get an answer right
away.
The map shows the general picture; the text elaborates and interprets the map. For instance,
the text tells that the map uses information from television markets in the United States; the
map shows where those markets are. The map shows that salsa sells better in rural areas than
in urban ones; the text explains why that might be so. The map shows that salsa is more
popular in areas populated by people from Mexico and Central America; the text explains the
significance of that pattern.
A marketing specialist could use this map to help make decisions about where to distribute
products, where to buy advertising, what kind of advertising to use, in what kinds of retail
stores to place products, and so on. Obviously, the decisions depend largely on what the
marketing specialist thinks the information means.
Plenty of Examples on the WWW
Without much trouble at all, you can find thematic maps of different kinds on the WWW as
well as in newspapers, on television, in magazines, and in books. Check out some of these if
you want to:
Some U.S. Census data appears on maps from the Office of Social and Economic Data
Analysis at University Extension in Missouri. For instance, you can see a map showing
Percent of Persons Age 25 and Over With Less Than a High School Diploma, 1990
[http://www.oseda.missouri.edu/graphics/us/plthsdip.gif]. Or Percent of Persons in
Poverty, 1994 [http://www.oseda.missouri.edu/graphics/us/inc/pctpov94.gif] On the
same server, you can you can find a collection of thematic maps
[http://www.oseda.missouri.edu/mapslib.html], many of them about the state of
Missouri.
Tourist maps provide information that someone might want when planning a trip or
vacation: where to find hotels, restaurants, local transportation, or entertainment, for
instance. On the WWW many of these are "clickable maps," where the user clicks on
some location or on a button for a kind of information. Then the server computer brings
up a more detailed view. To see an example, check out Caribbean-on-Line
[http://caribbean-on-line.com/]. For a bigger list of tourist information sites, try the
Virtual Tourist. [ http://www.city.net/]
Maps of traffic conditions are being put online in many urban areas. One of the first to go
online was for the Chicago area
[http://www.ai.eecs.uic.edu/GCM/CongestionMap.html ] and shows the predicted time
it will take to drive from one point to another on freeways and tollways. The predictions
are changed as new information comes in.
Weather maps of many kinds are on the WWW. Some show current conditions in a local
area or region; others specialize in weather information used by pilots or farmers. Still
others show forecasts for the next 24 hours or next five days, for instance. To see
examples of these and other weather-related maps, try the Weather Channel site and look
for the "Maps" button on the left side of the screen.
[http://www.weather.com/twc/homepage.twc]
You can pick and map out selected information about states at Ptolemy,
[http://ptolemy.gis.virginia.edu/gicdoc/dlg/dlg.html ] which uses data about interstate
highways, other major roads, national parks, national forests, and so on. The site gives
instructions on how to create your own map.
A special section in the University of Texas Libraries
[http://www.lib.utexas.edu/Libs/PCL/Map_collection/map_sites/map_sites.html]
technically, a specialized library is devoted entirely to maps and mapmaking
information. From its WWW page you can link to their collections and to other map-related WWW sites around the world.
Mental Maps or How This Connects to Reasoning
Maps, along with other tools like charts, graphs, diagrams, and illustrations, put information
in forms we can see. They visualize information so that we can visualize it in our minds.
Once in our minds, we can use these visualizations to think. That's why we use words like
seeing the picture, seeing the big picture, mapping it out, sketching out the problem, and
tracing out a solution as metaphors for some kinds of thinking.
Another common metaphor is mental maps and it suggests that people carry maps in their
minds. Sometimes these may be very much like maps on paper or on a monitor screen. To
explore this for a moment, try to call up the mental map you have of the place you live (or
some place you used to live). Think about the size, shape, and compass direction information
you have on your map. Someone who lives in the St. Paul - Minneapolis area of Minnesota
might have information like this on the mental map:
Minneapolis is west of St. Paul
The Mississippi River runs through parts of both cities
The Mississippi divides the two cities at some places
The town of West St. Paul is actually south of downtown St. Paul
Interstate Highway 494 goes east and west for a long stretch but also turns and runs north
and south
Mall of America is in Bloomington
This mental map could have been constructed by looking at a literal map of the area. Or it
could have been constructed by living in the area, listening to other people, and picking up
information in dozens of different ways. And obviously, the mental map could have been
made from life experience and looking at literal maps.
Both literal maps and mental maps have some important limits. Recognizing some of those
limits can make maps (and other visualizations) more useful or, at least, less misleading.
Take this case: Say you had only a standard map of the United States, say a map of U.S.
interstate highways, and you wanted to check out what point in the lower 48 states is farthest
north. On a flat map it looks as if that point would be in northeastern Maine. After all, that's
closer to the top edge of the map and north is at the top. But actually, the most northerly point
in the lower 48 is the "northwest angle" of Minnesota.
Or this case: Suppose you have a chance to fly to Europe and have a choice between going
first of all to either Rome, Italy or Helsinki, Finland. And suppose you don't really like flying
very much and want to spend the shortest time possible in the air. Which is closer to New
York, Rome or Helsinki? On a regular flat map, Rome looks closer because the route looks
close to straight. But, as you know or can guess now, Helsinki is closer as you can see on a
special map of the great circle air routes [http://www.finnair.fi/hgateway.htm]
A huge problem for maps, of course, is that they're flat and the earth is a sphere. And flat
maps in slightly more technical terms, two-dimensional maps cannot give an exact picture
of the earth, a three-dimensional object. Different distortions happen when the three-dimensional earth is put onto a flat map, depending on how the map is made. But on many
maps the extreme northern and southern areas are distorted the most, which makes it look like
Greenland is as big or bigger than all of South America when in fact South America is nearly
eight times bigger than Greenland.
The most accurate way to show the earth's size and shape is, of course, a globe a three
dimensional model of the earth. But the trade-offs are obvious: it's awkward to carry a globe,
particularly one large enough to read comfortably, whereas flat maps are very portable. Flat
maps also can be put into computer systems, through fax machines, onto photographic film,
and so on.
We often have to trade the accuracy or exactness of a globe for the convenience and
versatility of a map.
And often we have to trade our own direct personal observation for someone else's.
After all, we can't literally go everywhere and see everything that interests us. So we
learn from other people, whose experiences are communicated in all kinds of ways
including maps, charts, graphs, and other visualizations.
Thinking is improved if we recognize when these trade-offs are going on.
The Map is Not the Territory
That phrase, the map is not the territory, is a reminder that trade-offs have consequences.
Looking at a map of a place (or reading a book, watching a movie or television program,
listening to another person, and so on) simply is not the same as being there in person. Each
way of "experiencing" the place has its advantages and disadvantages. Knowing the
difference can make a difference.
[Concluding assignment: Write about a place you've never seen in person. Describe your
mental map of the place. Tell how you constructed the mental map. Speculate on its
accuracy. Etc. ]