Reasoning With Mathematics
Lesson 7 --- Look at This!
Most humans use their eyes almost constantly, at least during the time they're awake. And, in
a way, they use their eyes their visual sense even while sleeping, for dreams almost always
involve seeing things.
This constant use of sight leads impresses people and leads to comments like these:
Humans are highly visual creatures
Most people use vision more than any other sense to collect and process information
Humans communicate visually in all kinds of ways
Most people use vision to recognize what's around them
English and most other languages use everyday words and phrases that refer to vision, both literal seeing and thinking visually. You can probably add to this list of common terms:
"I can't see that," meaning something like I disagree or I don't understand.
"Do I have to draw a picture for you?" meaning something like Do I have to give you a simpler explanation?
"We need to look at the big picture," to mean approximately We should think about the context.
"Let's look at the next problem," to mean Let's go on the next problem and think about it
"We need a new vision," to mean We need a new statement of what we're trying to do and why it's important
And then there's this very old and widely used cliché: A picture is worth a thousand words.
Is the Cliché Accurate?
Is a picture worth a thousand words? As with so many clichés, this one is sometimes accurate
and fitting. There are times when a picture works wonderfully, especially if we think of
"pictures" to include drawings, sketches, photos, charts, graphs, animations, film, video, signs
and other visual renderings. These pictures bombard us every day.
To take one obvious example, you could look up triangle in a dictionary and find a definition
in words, such as a geometrical figure having three angles and three sides; any three-sided or
three-cornered figure, area, object, part, etc. After reading it, you may or may not have a
good picture in your mind.
But if you have examples to look at, you probably get along much better.
{graphic}
Most of us probably learn what triangles are from looking at them time after time, long before
we ever read or hear a definition in words. Later we learn words, numbers, and other symbols
that are useful for thinking about triangles such as right triangle, 900, acute, and more.
Eventually we can use triangle patterns to visualize a problem and solve it.
An Example in Front of Your Eyes
If you're at a computer now, you have a highly visual display right in front of you at least if
you're using a Macintosh or Windows operating system. Both use what's called a GUI
(pronounced "gooey"), an acronym for graphical user interface. GUIs set in front of you a
whole bunch of little pictures called icons, that you can point to and click on. The little
pictures stand for hundreds of commands, thousands of lines of instructions, and eventually
millions of 0s and 1s of binary code.
The GUI lets you imagine that your screen is a desktop where you have tools. Maybe not
literally a desktop or literally tools, but like a desktop, like tools. And the icons themselves
are simple pictures that remind you what they are and what they do: a scissors to cut things, a
check mark to run a spellchecker, a loudspeaker to adjust sound, a downward pointing arrow
to scroll that direction, an X to close, and so on and on.
Many computer users, especially younger people, have never used anything except a GUI like Mac or Windows. Those who have used something else, say old Apples or MS DOS machines, find the point and click systems are much easier to learn and use. In fact, many people say that Apple made it possible for nonspecialists to use computers when the Macintosh desktop that is, its GUI was developed. It works partly because it's so visual and so easy.
More Mathematical-Looking Examples
For many people, numbers make things look more mathematical. And although mathematics
doesn't always include numbers, everyday math usually does involve numbers and quantities.
| MAR | 1.5 |
| APR | -1.2 |
| MAY | 4.1 |
| JUN | -0.2 |
| JUL | 1.4 |
| AUG | -4.5 |
| SEP | 4.5 |
| OCT | 0.5 |
| NOV | -1.7 |
| DEC | -1.7 |
| JAN | 3.2 |
| FEB | 1.3 |
| MAR | -2.2 |
| APR | 1.4 |
Numbers and visualization go well together in all kinds of situations. Look at the list at the
left, which reports changes in orders for durable goods things like cars, refrigerators,
washing machines, and other products expected to last for three years or more. (This report
comes out once a month and it's one of several that some investors watch as signs of what's
going on in the U.S. economy.)
The numbers report how many percentage points those orders have gone up or down in one
month. So the first line is reporting that in March orders went up 1.5% compared to the
month before.
The numbers certainly give you information. But if what you want includes the pattern of
month-to-month changes, this graph might help a lot.
{graphic}
The two versions, the table with numbers and the graph, contain the same information but
don't communicate it the same way. In the graph, for instance, it's hard to figure out exactly
what the percentage is. (Is the percentage for the first month 1.5, or 1.6, or 1.7? Is there some
percentage change in June, and if so, how much?) In the list, it's hard to get a picture of how
often or much the percentages go back and forth. (Is the trend for a full year up or down or
unchanged?)
As with many cases having the numbers, and the graph, and words works best which makes a kind of multimedia package. The challenge, of course, is designing the package so that all elements work together effectively and honestly.
Commonly Used Graphs
Simple graphs pop up all around us, just about every day. A television news report might
show us the ups and downs of the stock market; a newspaper sports page might show how the
number of home runs hit in the major leagues has changed during the past ten years; a
magazine article might have a graph summarizing how tax money is spent.
WWW sites store vast numbers of basic graphs, too. At just one site, Statistics Canada, you
can see examples of three very common types of graphs.
One is a line graph [ link to: http://www.statcan.ca/english/Products/Farmfacts/ff9571.gif also stored on math downloads diskette A]
showing how the population of Canadians living on farms changed between 1931 and 1991.
graph
Line graphs often get used to show how something has changed over time, in this case
population living on Canadian farms. You can probably imagine how a line graph could be
used to show changes in the number of home runs, for instance.
Line graphs can give a good picture of ups and downs, of course, but also can say a lot about
when things were changing most dramatically. If you went back to the line graph above, you
could see a steep drop between 1956 and 1961.
A bar graph [ link to: http://www.statcan.ca/english/Faq/Glance/Tables/demo10.gif -- also
see math downloads diskette A] from Statistics Canada gives a picture of immigrant arrivals
at different times between 1960 and 1994.
graph
This graph also connects or relates numbers and time. So, you might ask, why not use a line
graph? In this case, a line graph would be misleading. The first four bars, starting at the top,
use numbers for 1960, 1970, 1980, and 1990 ten years apart. The last four bars use numbers
for 1991, 1992, 1993, and 1994 one year apart.
Even the bar graph is misleading if you don't notice the different intervals used for years, but a
line graph would be worse.
If you look at this bar graph or others, you can probably see a way to turn it into a line graph.
Do you?
A pie chart [ link to: http://www.statcan.ca/english/Faq/Glance/Tables/demo6.gif ] at the
same site summarizes the ethnic origins of Canadians as of 1991. Although the chart doesn't
say so, the numbers probably came from census information collected in 1990.
graph
As the name says, these graphs look like a pie and the size of each piece does the illustrating.
In the chart from Statistics Canada, the whole pie represents the nation's population; the piece
labeled British represents the 28% of the population who say they're British.
Pie charts are used to show the fraction, percentage, or decimal parts of something. (And as
you know fractions, percentages, and decimal parts are interchangeable equivalents. One-fourth, 25%, and .25 are equivalent, for instance.)
Spreadsheets: Automated Ways to Get a Good Look
Charts and graphs like the ones you just looked at can be made quickly and easily using
spreadsheet applications on a computer. (You probably have one on your computer.
Microsoft Works, which many Mindquest students use, includes a spreadsheet; so does
Microsoft Office, which has the Excel application. Claris, Corel, FoxPro all have
spreadsheets, too. Next to wordprocessors, spreadsheets probably are the most common
application for computers.
Spreadsheets crunch numbers; that is, they can do many different calculations, from basic
arithmetic to relatively complicated statistics. But they also display numbers including the
results of their calculations. What spreadsheets do to display numbers and results fits into this
course.
The person using a spreadsheet creates categories and enters numbers; those numbers and
categories are the raw material for a spreadsheet. Here's a small spreadsheet for a person's
monthly income and expenses:
Three big categories appear in the top row: income, expenses, and balance. The income and
expenses categories are broken down. The balance category is the difference between income
and expenses: how much is left over or, at least, not accounted for. Actually, the spreadsheet
has been told to 1) add up all the numbers in the income column and put the result in the cell
named B-5 (that's where you see 1240 now), 2) add up all the numbers in the expenses
column and put the result in cell E-14 (where 1220 is now), subtract the number in E-14 from
the number in B-5 and put the result in cell G-2 (where 20 is now). The applications does this
at electronic speed, which is blazing fast. Spreadsheets are very, very fast and very, very
accurate.
The sheet itself is visual, although not very fancy. Just seeing information spread out
(probably where the name spreadsheet came from) helps most people see what it is and what
it means.
With only a few short steps, an application can make a chart from the data on a sheet. Here's
one graph a bar graph obviously showing the data from the sheet above:
graph
To begin with, a spreadsheet application gives users two potentially useful visual tools to
display categories and numbers: the sheet itself and charts based on a sheet.
In addition, spreadsheets can change almost instantly when different categories or
numbers go into them, producing new sheets and charts. They're powerful tools for
some kinds of "what if" thinking.
Look at some what ifs about the income and expenses above: What if the person loses her
part-time job, then quits smoking, and disconnects her phone? What's the end-of-month
balance according to the spreadsheet?
Here's the new spreadsheet, which a computer application will spit out immediately after the
new numbers are entered:
According to this, she'll be $100 short under this what if. You can study the new sheet for a
moment, see where the differences, and see the famous bottomline (although in this case it's
the right hand column that shows the result).
And what if she gets another part-time job that pays $180, gets her phone connected again,
and stays away from cigarettes. Here's a spreadsheet with those changes:
The bottomline, that is the end-of-month balance, from the three sheets you just looked at
could be put into a graph, too. In the first month the balance was $20, the second month was
a negative $100, and the third month was $40. Here's one graph of these numbers:
graph
What if can go on as long as the user can think up new numbers, new categories, or new
calculations. If you want to, try using the spreadsheet application on your computer to run
some what ifs based on the example used here. You could add some new categories for either
or both income and expenses. You could change the numbers for categories already there, for
example raising or lowering the rent.
What If and If, Then The Same Principle
The What If idea used in this section is the same as the If, then strategy you've read about
in other parts of this course and in other Mindquest courses. It's a strategy that people use
over and over, including in their reasoning about a whole range of problems or questions.
Thinking "What if it rains today? Should I take an umbrella with me?" is very much like
thinking "If it's likely to rain today, then I should I should take an umbrella."
Visualizing helps people use their what if and if, then thinking patterns.
Looking Ahead to Different Visualizations
The charts and graphs in this section are simple. Usually, they deal with two things at a time:
how income and expenses balance out, how many people came to Canada as immigrants over
a period of years, how orders for durable goods change from one month to the next.
The charts and graphs seen here use two dimensions: horizontal and vertical. They are flat, in
other words, which makes it easy to produce them on paper, on a computer screen, or any
other flat surface.
Their simplicity can be a huge advantage in some ways. They can communicate quickly and
clearly, partly because they're simple.
Simplicity can be a limit, too. Many of the questions people have involve multiple factors,
not just one, or two, or three. So it's no surprise that people have developed more complex
ways to visualize what they think about.