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|
How to Read Mathematics
How to Read
Mathematics
by
Shai Simonson and Fernando Gouvea
Mathematics is “a language that
can neither
be read nor understood without initiation.” [1]
A reading protocol is a set of strategies
that a
reader must use in order to benefit fully from reading the text. Poetry
calls
for a different set of strategies than fiction, and fiction a different
set
than non-fiction. It would be ridiculous to read fiction and
ask oneself
what is the author's source for the assertion that the hero is blond
and
tanned; it would be wrong to read non-fiction and not ask such
a
question. This reading protocol extends to a viewing or listening
protocol in art and music. Indeed, much of the introductory course
material in
literature, music and art is spent teaching these protocols.
Mathematics has a reading protocol all its own,
and just as
we learn to read literature, we should learn to read mathematics. Students need to learn how to read
mathematics, in the same way they learn how to read a novel or a poem,
listen
to music, or view a painting. Ed
Rothstein’s book, Emblems of Mind, a fascinating book
emphasizing the
relationship between mathematics and music, touches implicitly on the
reading
protocols for mathematics.
When we read a novel we become absorbed in the plot and characters.
We try
to follow the various plot lines and how each affects the development
of the
characters. We make sure that the characters become real people to us,
both
those we admire and those we despise. We do not stop at every word, but
imagine
the words as brushstrokes in a painting. Even if we are not familiar
with a
particular word, we can still see the whole picture. We rarely stop to
think
about individual phrases and sentences. Instead, we let the novel sweep
us
along with its flow and carry us swiftly to the end. The experience is
rewarding, relaxing and thought provoking.
Novelists frequently describe characters by involving them in
well-chosen
anecdotes, rather than by describing them by well-chosen adjectives.
They
portray one aspect, then another, then the first again in a new light
and so
on, as the whole picture grows and comes more and more into focus. This
is the
way to communicate complex thoughts that defy precise definition.
Mathematical ideas are by nature precise and well defined, so that a
precise
description is possible in a very short space. Both a mathematics
article and a
novel are telling a story and developing complex ideas, but a math
article does
the job with a tiny fraction of the words and symbols of those used in
a
novel. The beauty in a novel is in the
aesthetic way it uses language to evoke emotions and present themes
which defy
precise definition. The beauty in a mathematics article is in the
elegant
efficient way it concisely describes precise ideas of great complexity.
What are the common mistakes people make in trying to read
mathematics? How
can these mistakes be corrected?
Don't
Miss the Big Picture
“Reading Mathematics is not at all a
linear
experience ...Understanding the text requires cross references,
scanning,
pausing and revisiting” [2]
Don’t assume that understanding each phrase, will enable you to
understand
the whole idea. This is like trying to
see a portrait painting by staring at each square inch of it from the
distance
of your nose. You will see the detail,
texture and color but miss the portrait completely.
A math article tells a story.
Try to see what the story is before you delve into the details.
You can
go in for a closer look once you have built a framework of
understanding. Do this just as you might
reread a novel.
Don't
be a Passive Reader
“A three-line proof of a subtle
theorem is the
distillation of years of activity.
Reading mathematics… involves a return to the thinking that went
into
the writing” [3]
Explore examples for patterns. Try special cases.
A math article usually tells only a small piece of a much larger and
longer
story. The author usually spends months
discovering things, and going down blind alleys. At
the end, he organizes it all into a story that covers up all
the mistakes (and related motivation), and presents the completed idea
in clean
neat flow. The way to really understand
the idea is to re-create what the author left out.
Read between the lines.
Mathematics says a lot with a little.
The reader must participate. At
every stage, he/she must decide whether or not the idea being presented
is
clear. Ask yourself these questions:
·
Why is this idea true?
·
Do I really believe it?
·
Could I convince someone else that it is
true?
·
Why didn't the author use a different
argument?
·
Do I have a better argument or method of
explaining the
idea?
·
Why didn't the author explain it the way
that I
understand it?
·
Is my way wrong?
·
Do I really get the idea?
·
Am I missing some subtlety?
·
Did this author miss a subtlety?
·
If I can't understand the point, perhaps I
can
understand a similar but simpler idea?
·
Which simpler idea?
·
Is it really necessary to understand this
idea?
·
Can I accept this point without
understanding the
details of why it is true?
·
Will my understanding of the whole story
suffer from
not understanding why the point is true?
Putting too little effort into this participation is like reading a
novel
without concentrating. After half an
hour, you wake up to realize the pages have turned, but you have been
daydreaming and don’t remember a thing you read.
Don't
Read Too Fast
Reading mathematics too quickly results in frustration.
A half hour of concentration in a novel
might net the average reader 20-60 pages with full comprehension,
depending on
the novel and the experience of the reader.
The same half hour in a math article buys you 0-10 lines
depending on
the article and how experienced you are at reading mathematics. There
is no
substitute for work and time. You can
speed up your math reading skill by practicing, but be careful. Like any skill, trying too much too fast can
set you back and kill your motivation.
Imagine trying to do an hour of high-energy aerobics if you have
not
worked out in two years. You may make
it through the first class, but you are not likely to come back. The frustration from seeing the experienced
class members effortlessly do twice as much as you, while you moan the
whole
next day from soreness, is too much to take.
For example, consider the following theorem from Levi Ben Gershon’s
manuscript
Maaseh Hoshev (The Art of Calculation), written in 1321.
“When you add consecutive numbers starting with 1, and the number of
numbers
you add is odd, the result is equal to the product of the middle number
among
them times the last number.” It is
natural for modern day mathematicians to write this as:
A reader should take as much time to unravel the two-inch version as
he
would to unravel the two-sentence version.
An example of Levi’s theorem is that 1 + 2 + 3 + 4 + 5 = 3×5.
Make
the Idea your Own
The best way to understand what you are reading is to make the idea
your
own. This means following the idea back to its origin, and
rediscovering it for
yourself. Mathematicians often say that to understand something you
must first read
it, then write it down in your own words, then teach it to someone else. Everyone has a different set of tools and a
different level of “chunking up” complicated ideas.
Make the idea fit in with your own perspective and experience.
"When
I use a word, it means just what I choose it to
mean" (Humpty Dumpty to Alice in Through the Looking Glass by
Lewis
Carroll)
“The
meaning is rarely
completely transparent, because every symbol or word already represents
an
extraordinary condensation of concept and reference” [4]
A well-written math text will be careful
to use a
word in one sense only, making a distinction, say, between
combination
and permutation (or arrangement). A strict mathematical
definition might imply that "yellow rabid dog" and "rabid yellow
dog" are different arrangements of words but the same combination of
words. Most English speakers would
disagree. This extreme precision is utterly foreign to most fiction and
poetry
writing, where using multiple words, synonyms, and varying descriptions
is de
rigueur.
A reader is expected to know that an absolute
value is not about some value that happens to be absolute, nor is a
function
about anything functional.
A particular notorious example is the
use of “It
follows easily that” and equivalent constructs. It means something like
this:
One can now check that the next
statement is
true with a certain amount of essentially mechanical, though perhaps
laborious,
checking. I, the author, could do it,
but it would use up a large amount of space and perhaps not accomplish
much,
since it'd be best for you to go ahead and do the computation to
clarify for
yourself what's going on here. I
promise that no new ideas are involved, though of course you might need
to
think a little in order to find just the right combination of good
ideas to
apply.
In other words, the construct, when used
correctly, is a signal to the reader that what's involved here is
perhaps
tedious and even difficult, but involves no deep insights.
The reader is then free to decide whether
the level of understanding he/she desires requires going through the
details or
warrants saying “Okay, I'll accept your word for it.”
Now, regardless of your opinion about
whether that
construct should be used in a particular situation, or whether authors
always
use it correctly, you should understand what it is supposed to mean. “It follows easily that” does not mean
if you
can't see this at once, you’re a dope,
neither does it mean
this
shouldn’t take more than two minutes,
but a person who doesn’t know the lingo
might
interpret the phrase in the wrong way, and feel frustrated. This
is apart
from the issue that one person's tedious task is another
person's
challenge, so the author must correctly judge the audience.
Know
Thyself
Texts are written with a specific
audience in
mind. Make sure that you are the intended audience, or be willing to do
what it
takes to become the intended audience.
T.S.Eliot’s
A Song for Simeon:
Lord, the Roman hyacinths
are blooming in bowls and
The winter sun creeps by the snow hills;
The stubborn season has made stand.
My life is light, waiting for the death
wind,
Like a feather on the back of my hand.
Dust in sunlight and memory in corners
Wait for the wind that chills towards
the dead
land.
For example, Eliot’s
poem pretty
much assumes that its readers are going to either know who Simeon was
or be
willing to find out. It also assumes
that its reader will be somewhat experienced in reading poetry and/or
is
willing to work to gain such experience.
He assumes that they will either know or investigate the
allusions here. This goes beyond knowledge
of things like
who Simeon was. For example, why are
the hyacinths “Roman?” Why is that important?
Elliot assumes that the reader will read
slowly
and pay attention to the images: he juxtaposes dust and memory, relates
old age
to winter, compares waiting for death with a feather on the back of the
hand,
etc. He assumes that the reader will
recognize this as poetry; in a way, he's assuming that the reader is
familiar
with a whole poetic tradition. The reader is supposed to notice that
alternate
lines rhyme, but that the others do not, and so on.
Most of all, he assumes that the reader
will read
not only with the mind, but also with his/her emotions and imagination,
allowing the images to summon up this old man, tired of life but
hanging on,
waiting expectantly for some crucial event, for something to happen.
Most math books are written with
assumptions about
the audience: that they know certain things, that they have a certain
level of “mathematical
maturity,” etc. Before you start to
read, make sure you know what the author expects you to know.
An
Example of Mathematical Writing
To allow an opportunity to experiment with the guidelines presented
here, I
am including a small piece of mathematics often called the birthday
paradox. The first part is a concise
mathematical article explaining the problem and solving it. The second is an imaginary Reader's attempt
to understand the article by using the appropriate reading protocol. This article’s topic is probability and is
accessible to a bright and motivated reader with no background at all.
The Birthday Paradox
A professor in a class of 30 random students offers to bet that
there are at
least two people in the class with the same birthday (month and day,
but not
necessarily year). Do you accept the
bet? What if there were fewer people in
the class? Would you bet then?
Assume that the birthdays of n people are uniformly
distributed among
365 days of the year (assume no leap years for simplicity). We prove that, the probability that at least
two of them have the same birthday (month and day) is equal to:
 .
What is the chance that among 30 random people in a room, there are
at least
two or more with the same birthday?
For n = 30, the probability of at least one matching
birthday is
about 71%. This means that with 30 people in your class, the professor
should
win the bet 71 times out of 100 in the long run. It turns out that with
23
people, she should win about 50% of the time.
Here is the proof: Let P(n) be the probability in question.
Let Q(n)
= 1 – P(n) be the probability that no two people
have a
common birthday. Now calculate Q(n) by calculating the number of
n
birthdays without any duplicates and divide by the total number of n
possible birthdays. Then solve for P(n).
The total number of n birthdays without duplicates is:
365 × 364 × 363 × ... ×
(365 – n +
1).
This is because there are 365 choices for the first birthday, 364
for the
next and so on for n birthdays. The total number of n birthdays
without
any restriction is just 365n because there are 365
choices
for each of n birthdays.
Therefore, Q(n) equals
 .
Solving for P(n) gives P(n) = 1 – Q(n)
and hence our result.
Our
Reader Attempts to Understand the Birthday Paradox
In this section, a naive Reader tries to make sense out of the last
few
paragraphs. The Reader’s part is a
metaphor for the Reader thinking out loud, and the Professional’s
comments
represent research on the Reader's part.
The appropriate protocols are centered and bold at various
points in the
narrative.
Be aware that my Reader may seem to catch on to things relatively
quickly. However, be assured that in
reality, a great deal of time passes between each of my Reader’s
comments, and
that I have left out many of the Reader’s remarks that explore dead-end
ideas. To experience what the Reader
experiences, requires much more than just reading through his/her
lines. Think
of his/her part as an outline for your own efforts.
Know Thyself
Reader (R): I don’t know anything about probability, can I
still make
it through?
Professional (P): Let’s give it a try. We may have to
backtrack a lot
at each step.
R: What does the phrase “30 random students” mean?
"When I use a word, it means
just
what I choose it to mean"
P: Good question. It doesn’t mean that we have 30 spacy or
scatter-brained people. It means we
should assume that the birthdays of these 30 people are independent of
one
another and that every birthday is equally likely for each person. The
author
writes this more technically a little further on: “Assume
that the birthdays of n people are uniformly distributed
among 365 days of the year.”
R: Ah, that’s what that means.
Isn't that obvious? Why bother saying that?
P: Yes it is kind of obvious, but the author is just setting
the
groundwork for later. Keep reading.
R: I don't understand that long formula, what’s n?
P: The author is solving the problem for any number of
people, not
just for 30. The author, from now on, is going to call the number of
people n.
R: I still don't get it. So what's the answer?
Don't Be a Passive Reader
- Try Some Examples
P: Well, if you want the answer for 30, just set n =
30.
R: Ok, but that looks complicated to compute.
Where’s my calculator? Let’s see:
365 × 364 × 363 × ... × 336. That’s
tedious, and the final exact value
won’t even fit on my calculator! It
reads:
2.1710301835085570660575334772481e+76
If I can’t even calculate the answer once I know the formula, how
can I
possibly understand where the formula comes from?
P: You are right that this answer is inexact, but if you
actually go
on and do the division, your answer won’t be too far off.
R: The whole thing makes me
uncomfortable. I would prefer to be
able to calculate it more exactly. Is
there another way to do the calculation?
P: How many terms in your
product? How many terms in the product on the bottom?
R: You mean 365 is the first term and 364 is the second? Then there are 30 terms. There are also 30
terms on the bottom, (30 copies of 365).
P: Can you calculate the answer now?
R: Oh, I see. I can pair up
each top term with each bottom term, and do 365/365 as the first term,
then
multiply by 364/365, and so on for 30 terms.
This way the product never gets too big for my calculator.
(After a few
minutes)... Okay, I got 0.29368, rounded to 5 places.
P: What does this number mean?
Don't Miss the Big Picture
R: I forgot what I was doing. Let’s see. I was calculating
the answer
for n = 30. The 0.29368 is everything except for subtracting
from
1. If I keep going I get 0.70632. Now
what does that mean?
P: Knowing more about probability would help, but this simply
means
that the chance that two or more out of the 30 people have the same
birthday is
70,632 out of 100,000 or about 71%.
R: That’s interesting. I wouldn’t have guessed that. You mean that in my class with 30 students,
there’s a pretty good chance that at least two students have the same
birthday?
P: Yes that’s right. You might want to take bets before you ask
everyone their birthday. Many people don’t think that a duplicate will
occur. That’s why some authors call
this the birthday paradox.
R: So that’s why I should read mathematics, to make a few
extra
bucks?
P: I see how that might give you some incentive, but I hope
the
mathematics also inspires you without the monetary prospects.
R: I wonder what the answer is for other values of n.
I will try some more calculations.
P: That’s a good idea. We can even make a picture out of all
your
calculations. We could plot a graph of the number of people versus the
chance
that a duplicate birthday occurs, but maybe this can be left for
another time.
R: Oh look, the author did some calculations for me. He says
that for
n = 30 the answer is about 71%;
that’s what I calculated too!
And, for n = 23 it’s about 50%.
Does that make sense? I guess it
does. The more people there are, the
greater the chance of a common birthday.
Hey, I am anticipating the author.
Pretty good. Okay, let’s go on.
P: Good, now you’re telling me when to continue.
Don’t Read Too Fast
R: It seems that we are up to the proof.
This must explain why that formula works. What's
this Q(n)? I guess that P
stands for probability but what does Q stand for?
P: The author is defining something new. He is using Q
just
because it’s the next letter after P, but Q(n) is also a
probability, and
closely related to P(n). It’s time to take a minute to
think. What
is Q(n) and why is it equal to 1 – P(n)?
R: Q(n) is the probability that no two people have the
same
birthday. Why does the author care about that?
Don’t we want the probability that at least two have the same
birthday?
P: Good point! The author doesn’t tell you this explicitly.
But
between the lines, you can infer that he has no clue how to calculate P(n)
directly. Instead, he introduces Q(n) which supposedly equals 1
– P(n). Presumably, the author
will proceed next to
tell us how to compute Q(n). By
the way, when you finish this article, you may want to deal with the
problem of
calculating P(n) directly. That’s
a perfect follow up to the ideas presented here.
R: First things first.
P: Ok. So once we know Q(n), then what?
R: Then we can get P(n).
Because if Q(n) = 1 – P(n), then P(n)
= 1 – Q(n). Fine, but
why is Q(n) = 1 – P(n)?
Does the author assume this is obvious?
P: Yes, he does, but what’s worse, he doesn’t even tell us
that it is
obvious. Here’s a rule of thumb: when
an author says clearly this is true or this is obvious,
then take
15 minutes to convince yourself it is true. If
an author doesn’t even bother to say this, but just implies it,
take a little longer.
R: How will I know when I should stop and think?
P: Just be honest with yourself. When in doubt, stop and
think. When
too tired, go watch television.
R: So why is Q(n) = 1 – P(n)?
P: Let’s imagine a special case. If the chance of getting two
or more
of the same birthdays is 1/3, then what's the chance of not getting two
or
more?
R: It’s 2/3, because the chance of something not happening is
the
opposite of the chance of it happening.
Make the Idea Your Own
P: Well, you should be careful when you say things like opposite,
but you are right. In fact, you have discovered one of the first rules
taught
in a course on probability. Namely,
that the probability that something will not occur is 1 minus the
probability
that it will occur. Now go on to the next paragraph.
R: It seems to be explaining why Q(n) is equal to the
formula
shown. I will never understand this.
P: The formula for Q(n) is tougher to understand and
the
author is counting on your diligence, persistence, and/or background
here to
get you through.
R: He seems to be counting all possibilities of something and
dividing by the total possibilities, whatever that means.
I have no idea why.
P: Maybe I can fill you in here on some background before you
try to
check out any more details. The
probability of the occurrence of a particular type of outcome is
defined in
mathematics to be: the total number of possible ways that type of
outcome can
occur divided by the total number of possible outcomes.
For example, the probability that you throw
a four when throwing a die is 1/6. Because there is one possible 4, and
there
are six possible outcomes. What's the probability you throw a four or a
three?
R: Well I guess 2/6 (or 1/3) because the total number of
outcomes is
still six but I have two possible outcomes that work.
P: Good. Here’s a harder example. What about the chance of
throwing a
sum of four when you roll two dice? There are three ways to get a four
(1-3,
2-2, 3-1) while the total number of possible outcomes is 36. That is
3/36 or 1/12. Look at this 6 by 6 table
below
and convince yourself.
1-1,
1-2, ..., 1-6
2-1, 2-2, ..., 2-6,
3-1, 3-2. ..., 3-6,
...
6-1, 6-2, ..., 6-6.
What about the probability of throwing a 7?
R: Wait. What
does 1-1 mean? Doesn’t that equal 0?
P: Sorry, my bad!
I was using the minus sign as a dash, just
to mean a pair of numbers, so 1-1 means a roll of one on each die -
snake
eyes.
R: Couldn’t you have come up
with a better notation?
P: Well maybe I could/should
have, but commas would look worse, a slash would look like division,
and
anything else might be just as confusing.
We aren’t going to publish this transcript anyway.
R: That’s a relief. Well, I know what you mean now. To answer your question, I can get a seven
in six ways via 1-6, 2-5, 3-4, 4-3, 5-2 or 6-1. The
total number of outcomes is still 36, so I get 6/36 or
1/6. That’s weird, why isn’t the chance
of rolling a 4 the same as rolling a 7?
P: Because not every sum is equally likely.
The situation would be very different if we were simply spinning
a wheel with the sums 2 through 12 listed in equally spaced intervals. In that case, each one of the 11 sums would
have probability 1/11.
R: Okay, now I am an expert. Is probability just about
counting?
P: In some sense, and sometimes, yes. But
counting things is not always so easy.
R: I see, let’s go on. By the
way, did the author really expect me to know all this?
My friend took Probability and Statistics
and I am not sure he knows all this stuff.
P: There’s a lot of information implied in a small bit of
mathematics. Yes, the author expected you to know all this, or to
discover it
yourself just as we have done. If I
hadn't been here, you would have had to ask yourself these questions
and answer
them by thinking, looking in a reference book, or consulting a friend.
R: So the chance that there are no two people with the same
birthday
is the number of possible sets of n birthdays without a
duplicate
divided by the total number of possible sets of n birthdays.
P: Excellent summary.
R: I don’t like using n, so let me use 30. Perhaps
the
generalization to n will be easy to see.
P: Great idea! It is often helpful to look at a special case
before
understanding the general case.
R: So how many sets of 30 birthdays are there total? I can’t do it. I guess I need to restrict my
view even more. Let’s pretend there are only two people.
P: Fine. Now you’re thinking like a mathematician. Let’s try n = 2. How
many sets of two birthdays are there
total?
R: I number the birthdays from 1 to 365 and forget about leap
years.
Then these are the total possibilities:
1-1,
1-2, 1-3, ... , 1-365,
2-1, 2-2, 2-3, ...
, 2-365,
...
365-1, 365-2,
365-3, ... ,
365-365.
P: When you write 1-1, do you mean 1-1 = 0, as in subtraction?
R: Stop teasing me. You
know exactly what I mean.
P: Yes I do, and nice
choice of notation I might add. Now how
many pairs of birthdays are there?
R: There are 365 × 365 total possibilities for two people.
P: And how many are there when there are no duplicate
birthdays?
R: I can’t use 1-1, or 2-2, or 3-3 or ... 365-365, so I get
1-2,
1-3, ... , 1-365,
2-1, 2-3, ... ,
2-365,
...
365-1, 365-2, ... ,
365-364
The total number here is 365 × 364 since each row now has 364 pairs
instead
of 365.
P: Good! You are going a little quickly here, but you’re 100%
right.
Can you generalize now to 30? What is the total number of possible sets
of 30
birthdays? Take a guess. You’re getting good at this.
R: Well if I had to guess, (it’s not really a guess, after
all, I
already know the formula), I would say that for 30 people you get 365 ×
365
×... × 365, 30 times, for the total number of possible sets of
birthdays.
P: Exactly. Mathematicians write 36530. And what is the number of possible sets of
30 birthdays without any duplicates?
R: I know the answer should be 365 × 364 × 363 × 362 × ... ×
336,
(that is, start at 365 and multiply by one less for 30 times), but I am
not
sure I really see why this is true. Perhaps I should have done the case
with
three people first?
P: Good. You probably need to go through more details more
slowly to
really see it, but at least you know where to head. Let’s quit for
today. The
whole picture is there for you. When you are rested and you have more
time, you
can come back and fill in that last bit of understanding.
R: Thanks a lot; it’s been an experience.
Later.
[1]
Emblems
of Mind, Edward Rothstein, Avon Books, page 15.
[2]
ibid, page
16.
[3]
ibid, page
38.
[4]
ibid, page 16.
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