How To Teach Math To Preschoolers

While some people groan that they will never use the math skills from school in real life, studies show quite the opposite. Starting children on the right track by teaching them math concepts in preschool is extremely beneficial. If kids learn math skills early on, they will be more likely to succeed in elementary school as well as later in life. Here are some important math skills to teach preschoolers.

Number sense

The first important math skill that a child should develop early on is number sense. Number sense is a term used to describe the basics of learning about numbers. This involves the child learning to count forwards and backwards. This will allow the child to learn the relationship between numbers later on. Number sense is an important skill that more preschool educators should implement so children are equipped with this knowledge before they reach kindergarten.

Learning numbers through pictures or representation

Children are able to build relationships between numbers and represented items because children are naturally visual. Using representation is a great way to make the use of mathematics real to a child’s mind. Early childhood education should involve representing numbers with pictures, items, or even family members. This will allow children to make connection between math skills and the real world. If they do not make this connection, they could be confused about the information provided in a classroom.

Adding and subtracting

Typically, concepts are taught before skills. This prevents many preschool teachers and parents from teaching young children addition and subtraction. But there is a way to teach young children the basics of adding and subtracting before they reach elementary school. These basic skills can be used in many typical childhood interactions. For example, if you are sharing cookies, you are subtracting from the total number in order to make sure that each person has the same number of cookies. If teachers focus on adding and subtracting, the students are likely to have a stronger foundation in mathematical skills that they can use in the future.

Geometry

In order to teach children geometry, you can use a number of supplies found in the classroom. One helpful exercise is to prompt students to name the shapes of blocks and other objects. You can also let students learn about geometry through creative activities. For example, you can allow the students to arrange colored blocks into various patterns and shapes. You can then ask them to name the shapes they have created. Geometry can even be brought into “arts and crafts” time. Have students create objects, such as shapes cut out of paper that then can use later in math lessons and activities.

Measurement

If you want to teach children about measurement, one great way to do this is to weigh and measure various objects on a “science table”. This will get the students enthusiastic not only about math, but about science as well. You can also use some of the students’ favorite manipulatives, such as toy cars or plastic animals, to teach them measurement. You can ask questions like “What are there more of, cats or dogs?” You can also play guessing games that have to do with measurement. For example, saying “I’m thinking of a number greater than 4 but less than 7” is a fantastic way to teach children about measurement.

There are a number of simple ways that you can teach preschoolers about math so that you can apply it to the world around them. Learning math skills from a young age makes a huge difference in students’ lives and leads them to success.

What It Takes To Be A Successful Tutor

Mathematics & Tutoring With David Seff

Effective tutoring is all about producing results. From a tutor’s perspective, it’s about witnessing your student achieve proposed academic goals and make way for more, in-depth learning. From a student’s point of view it’s about feeling confident with newly learned information and putting it to good use. As simple as this may all sound, effective learning is not something that happens overnight. Effective tutoring should be a direct reflection of a solid tutor-student relationship that’s been built over time, where both parties have a reached a point where they each feel fulfilled by their progress and excited for the academic challenges that are bound to come their way. Here are three key points for tutors to become more effective:

Prepare

You can never be prepared enough, but there more you are the better. Because tutoring can take place outside of a formal educational environment, it is very important that you shape your own. Make sure to set a schedule with your students, preferably a consistent schedule ahead of time. Establishing a learning rhythm will be key for your students’ academic improvement. Also, try to carve out what each session will be about. Of course there will always be questions that may arise not pertaining to the day’s lesson, so it’s good to leave some extra room for that. But for the most part, focusing each session on either a specific chapter, theme, etc. will also help establish a beneficial learning rhythm for your students.

Know Your Student

The best way to build a good rapport with your students is taking the time to know them. Every student is different. Figuring out your student’s personality will help you understand their learning process and how you should angle your sessions with them. For example, does your student have a sense of humor? Do they need your undivided attention? Are they shy? Does the student feel confident about the subject matter you’re going over? The more empathetic you can be, the more the student will feel comfortable around you and will allow himself or herself to continue to learn from you.

Set Expectations

It is very important that both you and your student know your roles and responsibilities in a tutoring session. Tutoring is not the same as teaching. The tutor is not there to teach or re-teach something that the student has already learned. Instead, tutoring should be more of a well-balanced exchange of questions and information. In other words, tutors are there to guide the students to improve their studying skills, while the students are there to discover things through independent questioning and develop self-confidence.

Peano’s Postulates

The Natural Numbers,  = {1, 2, 3, …}, can be built from five basic axioms, as follows:

1.  There is a natural number 1; that is, 1 ∈ .

2.  If n ∈ , then there is another number called the successor of n and written as S(n) that is also a natural number; that is:  n ∈   S(n) ∈ .

3.  Two different numbers have different successors; that is:  m  n  S(m)  S(n).

4.  Every natural number but 1 is the successor of some (other) number; that is:

n ∈ , n  1 m ∈  S(m) = n.

5.  The Principle of Mathematical Induction:  All natural numbers are formed in this way—either 1, or a successor.  More formally,

If there is a subset S of natural numbers such that 1 ∈ S and n ∈ S  S(n) ∈ S, then S is the set of Natural Numbers, that is S = .

Define addition:  n + 1 = S(n).

Can prove addition follows commutative and associative laws.

Define subtraction to be S(n) – 1 = n.

Algebraic Structures — Field

Mathematics does not deal so much in numbers, but in “mathematical structures.”  These are basically abstract concepts that fit together according to certain rules or patterns.

One abstract concept that is very important is that called a Set.  A set in mathematics is a very abstract concept, but unless we are involved in higher mathematics, a set is something very similar to what we call a “collection” in everyday English.  Since we are not involved in higher math in this course, we will consider a set to be a collection of some objects (often numbers, but it could be a collection of other things, such as functions, triangles, etc.).  Many books will define a set to be a collection.  This definition is not correct, so as a matter of honesty, we point to that it is not correct.  However, the differences between a set and a collection are so subtle and abstract, we need get involved, but will consider a set to be a collection.

Another important concept is that of an operation—it means a process where some members of a set are combined together to get something new, like mixing flour and water to get dough.  In mathematics, in dealing with numbers, the most common operations are addition, subtraction, multiplication, and division.  There are others, of course, but we need not be concerned with them.  In dealing with functions, the most important operations are there are these four and a FIFTH operation that is important, called composition.  We studied it in class–f◦g means f(g(x)).

An operation is called binary, if, by definition we must always combine exactly two things together.  All the above operations are binary.  Note we cannot combine three numbers with addition together, for example, we cannot add 2+3+4 all at once.  We must first add only two numbers together, get a temporary answer, and then we add the third number to this temporary answer.  This means the operation is binary—we can only deal with two at a time, although we can keep adding more numbers to our answer after we get it.

An operation is called unique, if, unlike tossing a pair of dice, for example, combining the same two inputs always produces the same output.

Rules for addition:

1. The sum of any two real numbers a and b is a real number written a+b

This is called the Law of Closure for addition of Real numbers.

2. For all real numbers a, b, c:  (a+b)+c=a+(b+c)

This is called the Associative Law for addition of Real numbers.

3. There is a one unique real number called zero, written 0, such that a+0 = a and 0+a = a  for all real a.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems.)   That is to say, that there is one and only one real number that can be used for I to make the following two equations come true:  For any Real number, a, a+I = a and I+a = a  for all real a.  (And of course, I is 0).

This is called the Additive Identity Law for addition of Real numbers.

4. For each real number a there is a real number called –a such that a+(-a)=0 and

    (-a)+a = 0.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems.)

This is called the Inverse Law for addition of Real numbers.

5. a+b=b+a

This is called the Commutative Law for the addition of Real numbers.

Rules for multiplication:

1. The product of any two real numbers is a real number written ab or ab

2. For all real numbers a, b, c:  (ab) c=a (bc)

3. There is a real number called one, written 1, such that a1=a and 1a=a for all real a.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems).

4. For each real number a0,  there is a real number called 1/a such that
     a (1/a)=1 and (1/a) a= 1.   1/a is also written as a-1.  (Note both parts are needed because Rule 5 does not necessarily hold in all systems.)

5. ab=ba

Rule 11 connecting addition and multiplication:

a(b+c)=ab+ac

Rule 1 is called the Law of Closure.

Rule 2 is called the Associative Law.

Rule 3 is called the Identity Law.

Rule 4 is called the Inverse Law.

Rule 5 is called the Commutative Law.

Rule 11 is called the Distributive Law.  (Adding columns and rows—switching order.)

Note:

1.  Rule 2 does not make sense unless Rule 1 is first established.

2.  Rule 4 does not make sense unless Rule 3 is first established.

3.  If a system has just one binary operation and the first four laws, it is called a “Group.”  Since Rule 5 is not necessarily true in an arbitrary group, it is necessary to state the identity and inverse laws on both sides.

4.  If a system has just one binary operation and the first five laws, it is called a Commutative Group or an Abelian Group.

5.  If a system has rule 2 and 5, then the General Associative Law is true, which says you can change the order and the parentheses any way you want.

6.  If a system has two binary operations and all 11 laws, then it is called a field.

You should be able to state these rules in the abstract—If S is a set, and if a, b, and c are any arbitrary members of S, and if * is a unique binary operation defined for the members of S, then (commutative law, for example, would be a*b = b*a.

Now some results or consequences of these rules:

1.  If a+c=a+b then b=c.  (This will be true in any group that has + as an operation.)

2.  If ab=ac, then b=c, providing a0  (This will be true in any group that has multiplication as an operation.)

3.  These laws enable us, if we are in a field,  to solve any equation of the following forms:

a+ x=b,

x+a=b,

ax=b (provided a0),

xa=b (provided a0), and

ax + b = c

4.  a0=0  for any Real number a.

5.  ab = 0 means a = 0 or b = o or both.

6.  The zero principle:  In a field, ab = 0if and only ifa = 0 or b = 0 or both.

This is a consequence of #4 and #5 above.  We needed all 11 laws (except for the commutative laws) to prove #3 and #4, so we see the zero principle will always hold in a field, but not necessarily in a group.

5 – 7

7.  (a)  b = (ab)  (Henceforth, we will omit the  for multiply, or use a dot instead.)

All the above represent the basic algebraic properties of a field.  There are many fields besides the real numbers.  Some are finite  and some are infinite, such as , , and –and there are many other examples of both finite and infinite fields not mentioned.  Any system that is a group will look a lot like our number system, , the real numbers.  However, any field will look even more like .

To characterize the Real Numbers mathematically, we need to say it is an infinite field with the following additional properties—these additional properties, however, are generally not considered “algebraic” properties, but “geometric” properties:

It is infinite, ordered, Archimedian, and complete.  These three properties are explained briefly below, but you are not responsible for them.

Ordered means given any two different numbers, then one is greater than the other (or equivalently, one is smaller than the other).

Archimedian means that given any small number, s, regardless of how small, and a big number, b, regardless of how big, then it is possible to add enough s’s together so that eventually the sum  s + s + … + s > b.  In other words there is a number n, so that ns > b.  For example, is s is 1/100 and b is 1,000,000 we can keep adding 1/100 + 1/100 etc. until eventually the sum is bigger than 1,000,000.  Equivalently, if n is any number larger than 100,000,000 then ns > b.

 

Russel’s Paradox

What is Russell’s Paradox?  What is the solution to it?

Background (you will need to know to understand the answer):

Russell’s Paradox arises when sets get “too big” and sets contain sets.  It is possible for sets to contain sets, but not all such combinations are acceptable.  We will now explain how the idea of a “set containing a set” can be problematic.

To start, first notice that many sets, even if they are infinite in size, can generally be expressed in English sentences containing a few words.  For example, “The set of all even numbers” is a sentence describing an infinite set—the even numbers—and the description should be very clear to anyone, even someone in the second grade.  Yet this description only takes six words!  (Side point:  In general mathematicians like to be very terse, and the brevity of mathematical statements can confuse people who are not used to paying attention to the detail and exact meaning of every word—an absolute necessity in mathematics.)  In spite of the tendency towards terseness, there are some sets that might take many words to describe, for example, consider this set:  “Let K = the set of all people who (1) were born in New York, (2) have red hair, (3) moved to California after age 10, (4) drive a Ford, (5) did not graduate college, (6) like to follow football games on television, (7) like Big Macs, (8) own a computer, etc. “  If we were to list 100 different conditions, even using terse wording, the description would certainly be over 100 words long.  Of course, it could be there is no person on earth who meets all 100 conditions, especially if some of them are rare, such as (over 6 feet 5 finches tall).  While that one condition might be met by some people, the number of people meeting this condition might be relatively small, so it could be there is no one who meets all 100 conditions.  If this is the case, then K, the set described is the empty set, but we would not know that without a survey unless we created a ridiculous condition, such as the person must be over 10 feet tall.  In short, this set could very well not be the empty set, but the description could be very long—over 100 words.

Now let H be the set of all sets that can be described in an English sentence containing less than 100 words.  By definition, the elements of H are themselves sets.  Clearly, the set K described above is not inside of H, because its description is too long.  However, all other sets mentioned above are sets that are inside of H—as well as many other sets, such as the set of real numbers, the set of rational numbers, the set of all people on earth, etc.  These sets are infinite, but their descriptions are short, so they are all inside of H.  In fact, the set H is also inside of H because the description of H above contained 17 words (which is certainly less than 100).  Thus we have something like the following H ={H, N, Z, Q, S, T, ….}  H is in fact a member of H!  This concept is counter-intuitive, because if we think a set as a collection, a collection cannot obviously contain itself.  So while it is nice to think of a set as a collection, the analogy breaks down when we deal with more complicated examples, as is so in this case.

The Answer as it Should Appear on a Test:

Now, here is Russell’s Paradox.  All the sets we have above, or “standard” sets, such as whole numbers, natural numbers, integers, etc. we will call “ordinary” sets.  However, any set, such as H, which contains itself, we will call an “extra-ordinary” set.

Let Obe the set of all ordinary sets.  Question:  Is O  ordinary or extra-ordinary?  We will find we have a problem (contradiction) or paradox, either way we look at it.  If O  is ordinary, then since O  contains all ordinary sets (by its very definition) then it must contain itself (since it is ordinary)—and by containing itself it would then become extra-ordinary—a contradiction!  Again, if O is extra-ordinary, then O  must contain itself, by the definition of the word “extra-ordinary.”  But this fact contradicts the definition of O , since O is supposed to contain only ordinary sets, not an extra-ordinary ones!  So as a result, we cannot decide whether O is ordinary or extra-ordinary.

What is the solution to Russell’s Paradox?

The solution to this paradox, is to redefine or rethink what we mean by sets.  They are not simply any collections, but they must be limited so they do not grow too big.  There are several ways to achieve this re-definition—but most are too sophisticated to explain here.  We will explain, briefly (not in detail) the simplest method here.

We start by defining some set, which we call U, the universal set.  U could be the real numbers, natural numbers, or any other set you want it to be.  However, we cannot just create “new” sets at random.  They must be formed by certain rules—for example, by taking elements from U, by taking subsets from U, or by taking unions or intersections, or complements of these sets.  So while we can make an infinite number of sets, and we can even create a set of subsets, we can’t create “higher levels” of sets containing sets, so we stay clear of this paradox.

Questions:  Let S be the set of all sets.  Does S exist?  

Answer:  No.  It is too big.  The solution to Russell’s Paradox sets limits as to how big a set can get—and S violates those limits.  We need to have some “upper” set such as the real numbers, and all other sets are subsets of this upper set or universal set.

 

Response To The Mathematical Association Of America: The Importance Of Mathematics

The Mathematical Association of America is considering removing calculus from the high school curriculum. The Association asked to hear from the community at large on whether not people agreed with this decision. In response, many proper mathematicians provided a variety of well-thought answers — both supporting the decisions and cautioning against deleting calculus from high school curriculum.

I, as typical, have a completely independent view on the potential decision, which I would like to share below. 

There are a lot of things wrong with the way we teach math that need to be fixed.  I hope to write a book on it some day.  In short, those who truly understand are few and feel the curriculum is MUCH too slow.  Those who do not understand are lost on basic concepts in the first or second grade and then nothing makes sense, and anything taught later, regardless of how slow, seems too fast.  The students just parrot ideas without really understanding them—so if they memorize certain motions, as in a dance, they can get great grades but not really know anything.  

Adding or deleting any course, such as calculus will not solve anything. Instead we need to revise the entire system from the bottom up.  For those few who do understand, all of math through the sixth grade can easily be taught in one or two years, 7th and 8th grade math can easily be taught in one year (say the third grade), and in the fourth grade we should start teaching what we mistakenly call high school math.  For these students, we should not teach calculus in high school, but in the sixth grade.  For those who do not understand math well, we should never teach calculus.  But our goal is to make everyone understand.  

Using a method I developed to teach addition and subtraction of signed numbers, one of my students who was a teacher taught it in kindergarten–that’s right kindergarten!  And EVERY student could add and subtract signed numbers without any errors.  All problems given were solved 100% correctly by 100% of the students after just one week of instruction. Yet signed numbers are generally taught in junior high school for months, and even then, most students do not understand the concepts well.  The solution to our math problems educationally is NOT in changing the curriculum but in making RADICAL changes in the way math is presented, starting in kindergarten. 

 

Basic Definitions and Concepts

Philosophically and educationally there are a number of ways to introduce and teach mathematical ideas.  Most mathematicians, however, prefer to use Sets as the basic structure in math, and build all ideas from there, using sets to define everything.

At first, mathematicians used to look at sets as just collections of things, but Bertrand Russell, the famous British philosopher and logician, showed that viewing sets as collections is a bit naïve—if a set gets “too big” it is possible that contradictions will result.  Before we discuss what the problem is, we will use this simple or “naïve” view of sets and consider a set just to be a collection of things.

SETS

Here are some important ideas about sets:

1.  In describing sets we often use the word “all” or “only.”  Even if we use only one of those two words, or neither of them, we ALWAYS mean both.  For example, if we talk about the set of all positive even numbers less than 10 we mean the set {2, 4, 6, 8}.  (Notice, we generally put the members or elements of a set between two curly-cue brackets, and we put commas between the different members.)  This set, {2, 4, 6, 8}, is both ALL the positive even numbers less than 10 and ONLY the even numbers less than 10.   If we only say “the set of positive even numbers less than 10” and omit the word “all” we still mean this set.  If we say “the set of only positive even numbers less than 10,” we still mean this set.

2.   Repeating an item in a set, or changing the order does not change the set.   For example, the sets {2, 2, 4, 4, 4, 6, 8} and {8, 4, 6, 2} and {2, 8, 4, 6, 8} are all equal to each other and the set mentioned in the previous paragraph.

3.   There does not have to be any connection between items in the set or uniformity.  For example, we can form a set that consists of all red Cadillac’s, Mexican immigrants, and stars that are less than 100 light years from the sun.

4.  The things inside a set are called members or elements of the set.  And there is a special symbol to so indicate:  For example, we can write 2∈{2, 4, 6, 8} to mean that “two is a member of the set of all positive numbers less than 10.”   We can also indicate something is not a member of the set by using the same symbol with a line through it.  For example, 3∉ {2, 4, 6, 8}.

5.  We can also talk about one set being inside another set—that is to say, if every member of one set is inside another set, we use this symbol:⊂ .  For example,

{2, 8}⊂ {2, 4, 6, 8}.  This may be read as “The set {2, 8}is contained in (or inside of) the set {2, 4, 6, 8}.”  The set {2, 8} is called a subset of the set {2, 4, 6, 8}.  Therefore, we can also read the above expression as “The set {2, 8}is a subset of the set {2, 4, 6, 8}.”

6.  There are two types of subsets—proper and improper.  If someone starts with a set and “removes” some elements and “keeps” some elements, then the result is a proper subset.  For example, if the starting set is {2, 4, 6, 8} and one “keeps” 2 and 8 and “removes” 4 and 6, then what we have left is a proper subset, namely, {2, 8}.  It is possible, though, to “keep” everything and “remove” nothing.  In this case the “subset” is the full set,

{2, 4, 6, 8}.  This full set, when thought of as a subset is called an improper subset.  Similarly, one can “keep” nothing and “remove” everything, in which case the subset, is empty or has nothing in it.  It is written as {} (empty brackets) or as .  It is also called an improper subset, and called “the empty set.”  If we want set A to be a subset of set B, and we allow that A could be all of B or the empty set—that is A could be a proper or improper subset–we write A  B.  If we do not want to allow the possibility that A could be all of B (but we still allow the possibility that A could be the empty set) we write

A  B.  If we want A to be a subset of B but we do not want to allow the possibility that A could be empty, there is no special symbol for that—we just add a sentence that “A is not empty.”

7.  There is a difference whether we write 2∈{2, 4, 6, 8} or {2}⊂{2, 4, 6, 8}, even though they basically mean the same thing.  2 without any set brackets is just the number 2, and it is an element of the set.  {2} is an entire set.  You can look at is as being the number 2 put inside a basket or fancy package.   This mathematical statement “{2}⊂{2, 4, 6, 8}” says everything in the “basket” on the left is also inside the basket on the right—even though the basket on the left has only one item.  However, this statement “2∈{2, 4, 6, 8}” is not talking about two baskets.  It talks about one basket only.  It merely says the number 2 is a member of, or an item in, the “basket” on the right.

8.  A set can contain a set, but if we are not careful, we can get into trouble (Russell’s Paradox).  Here is an example of how a set can contain a set, without getting into trouble.

If we call S = {2, 4, 6, 8}, we can create a set T = {S, 1, 2, 3, 4}.  How many members are there in T?  There are 5.  One of them is a “basket” which contains 4 items—namely the set S.  But T only has 5 items in it, not 9, and not 7.  Note that we can write

{2, 4, 6, 8} ∈ T just like we can write {1}∈ T or {2}∈ T, because the set S is a member, not a subset.  Note that because 2 is in both S and T we can write 2∈ S and we can also write 2∈T (because 2 is also a member of T—a fact totally independent of whether or not S is also a member of T).  Similarly with 4 which is both a member of S and of T.  However, with 6, we can write 6 ∈ S, but we cannot also write 6∈T, because 6 is not a member of T.  Now we can also write 2∈ S ∈ T, which means “the element, 2, is a member of set S, and the set S is a member of the set T.”  Note that this last statement does not say whether or not 2 is in T.  It merely says that 2 is in S, and that S is in T.  Similarly, we can write 6 ∈ S ∈ T, which means that 6 is in S and that S is in T.  It does not say whether 6 is in T or not.  Be sure to spend the time necessary to understand carefully and exactly each of the examples just mentioned in this paragraph before going on.  Do not skim, and do not just read it.  Understand it slowly and carefully in detail.

9.  Now we will explain how the idea of a “set containing a set” can be problematic.

To start, first notice that many sets, even if they are infinite in size, can generally be expressed in English sentences containing a few words.  For example, “The set of all even numbers” is a sentence describing an infinite set—the even numbers—and the description should be very clear to anyone, even someone in the second grade.  Yet this description only takes six words!  (Side point:  In general mathematicians like to be very terse, and the brevity of mathematical statements can confuse people who are not used to paying attention to the detail and exact meaning of every word—an absolute necessity in mathematics.)  In spite of the tendency towards terseness, there are some sets that might take many words to describe, for example, consider this set:  “Let K = the set of all people who (1) were born in New York, (2) have red hair, (3) moved to California after age 10, (4) drive a Ford, (5) did not graduate college, (6) like to follow football games on television, (7) like Big Macs, (8) own a computer, etc. “  If I were to list 100 different conditions, even using terse wording, the description would certainly be over 100 words long.  Of course, it could be there is no person on earth who meets all 100 conditions, especially if some of them are rare, such as (over 6 feet 5 finches tall).  While that one condition might be met by some people, the number of people meeting this condition might be relatively small, so it could be there is no one who meets all 100 conditions.  If this is the case, then K, the set described is the empty set, but we would not know that without a survey unless we created a ridiculous condition, such as the person must be over 10 feet tall.  In short, this set could very well not be the empty set, but the description could be very long—over 100 words.

Now let H be the set of all sets that can be described in an English sentence containing less than 100 words.  By definition, the elements of H are themselves sets.  Clearly, the set K described above is not inside of H, because its description is too long.  However, all other sets mentioned above are sets that are inside of H—as well as many other sets, such as the set of real numbers, the set of rational numbers, the set of all people on earth, etc.  These sets are infinite, but their descriptions are short, so they are all inside of H.  In fact, the set H is also inside of H because the description of H above contained 17 words (which is certainly less than 100).  Thus we have something like the following H ={H, N, Z, Q, S, T, ….}  H is in fact a member of H!  This concept is counter-intuitive, because if we think a set as a collection, a collection cannot obviously contain itself.  So while it is nice to think of a set as a collection, the analogy breaks down when we deal with more complicated examples, as is so in this case.

Now, here is Russell’s Paradox.  All the sets we have above, or “standard” sets, such as whole numbers, natural numbers, integers, etc. we will call “ordinary” sets.  However, any set, such as H, which contains itself, we will call an “extra-ordinary” set.

Let Obe the set of all ordinary sets.  Question:  Is O  ordinary or extra-ordinary?  We will find we have a problem (contradiction) or paradox, either way we look at it.  If O  is ordinary, then since O  contains all ordinary sets (by its very definition) then it must contain itself (since it is ordinary)—and by containing itself it would then become extra-ordinary—a contradiction!  Again, if O is extra-ordinary, then O  must contain itself, by the definition of the word “extra-ordinary.” But this fact contradicts the definition of O , since O is supposed to contain only ordinary sets, not an extra-ordinary ones! So as a result, we cannot decide whether O is ordinary or extra-ordinary.

The solution to this paradox, is to redefine or rethink what we mean by sets.  They are not simply any collections, but they must be limited so they do not grow too big.  There are several ways to achieve this re-definition—but most are too sophisticated to explain here.  We will explain, briefly (not in detail) the simplest method here.

We start by defining some set, which we call U, the universal set.  U could be the real numbers, natural numbers, or any other set you want it to be.  However, we cannot just create “new” sets at random.  They must be formed by certain rules—for example, by taking elements from U, by taking subsets from U, or by taking unions or intersections, or complements of these sets.  So while we can make an infinite number of sets, and we can even create a set of subsets, we can’t create “higher levels” of sets containing sets, so we stay clear of this paradox.

Abstract Thinking

What is abstract thinking, and why is it important in mathematics?   Name at least three advantages.

Let us start with an example:  If there are two rooms, room A and room B that are connected by a door, and a person starts in room A, but changes room every time a bell rings, then what room will the person be in if the bell rings three times?  256 times?  Most people get the answer to the first question by moving their finger three times and correctly conclude the person will be in room B.  Sometimes a person does not move a finger but mentally pictures the person going back and forth.  While this mental picturing is an example of abstract thinking, it only represents one level of abstraction that corresponds closely with a physical operation (going back and forth).  Other levels of abstraction exist.  Most people do not attempt to answer the question of “256 times” by going back and forth physically or mentally.  They rely on an additional layer of abstraction.  They realize that an odd number of rings of the bell puts the person in room B, and an even number of rings puts the person in room A.  Abstract thinking is an objective mental process whereby concepts rather than motion or physical objects or activity (or a mental picture thereof) are used to come to a conclusion.  There are many additional layers of abstraction used by mathematicians.

Abstract thinking is generally quicker than more concrete methods.  It also creates a deeper level of understanding.  Hence, it enables one to understand the essence of problem and from the principles involved, thereby, solve other problems similar to the original.  In addition, use of abstract thinking helps to “objectify” a problem and helps eliminate red-lining.

 

Red-Lining

We mentioned in class an article I had seen in Scientific American magazine many years ago.  A psychologist wanted to test how mathematicians think, so he devised the following experiment which was conducted by a graduate assistant.  People, one at a time, were brought in a gym to do an experiment.  These people were volunteers and included both professional mathematicians and non-mathematicians.  The gym was, let us say, 25 feet long.  About 5 feet from one wall was a red line going across the width of the gym floor, and on the opposite wall was a large nail.  The assistant conducting the experiment held five rings, such as those used in a carnival “ring-toss” game.  He told the volunteer (one at a time) that he/she was to play a game whereby the goal was to get the rings on the nail as quickly as possible.  The assistant then said “I will show you how it might be done,” and, standing behind the red-line, proceeded to toss the rings, one at a time across the gym floor, trying to get them on the nail.  The first time maybe only one or two landed on the nail.  The assistant then ran across the gym floor, picked up the remaining rings, and ran back to the other end, stood behind the line, and again tried to toss the rings on the nail.  Each time some rings missed, he would run to the other end, pick them up, return to the initial place behind the red line, and continue to toss the rings until all were on the nail.  The assistant further explained that the score of the volunteer was based upon the time.  He said explicitly that the volunteer could use any method he or she wished to use, as long as it was understood the object of the game was to get all five rings on the nail as quickly as possible.  He then asked if there were any questions.

When the volunteers had each performed the task and the results of the experiment were analyzed, the experimenter discovered a very clear division between the mathematicians and non-mathematicians.  All the mathematicians had times less than 10 seconds, and all the non-mathematicians had times over two minutes!.  Why?  Because the mathematicians took an entirely different approach to the problem.  While all the non-mathematicians had mimicked the assistant and tried to toss the rings onto the nail, the mathematicians heeded the instructions very carefully—in which it was said any method could be used as long as he got the rings on the nail as quickly as possible—and they simply ran to the nail and deposited all the rings onto it directly without throwing them.

In short, the non-mathematicians were deceived by the red line and the method used by the assistant.  It had no meaning with regards to the game.  They falsely assumed it did and thereby made it much harder.  I have termed the coin “red-lining” to refer to this false type of mental processing whereby a person makes a problem harder than it really is by attaching incorrect ideas to the problem based upon previous associations.

Often when we have to make a decision, we will use past experiences or associations to color the situation with which we are current dealing.  In fact, we may add to, subtract from, or alter the facts of, the case in front of us based upon these associations.  The type of associative thinking that enables us to change the facts is called “red-lining.”  Red-lining may be useful in many “human” or interpersonal situations.  For example, one may have a feeling in walking down a street at night that a certain character down the block is dangerous or to be avoided, and accordingly, one may cross to the other side of the street.  Objectively, there may be no evidence that there is something wrong, but intuitively, one may feel something is wrong and chose to be careful based upon this intuitive feeling rather than on purely objective evidence.  This decision is certainly a good precaution in this case even if there is a lack of objective evidence to support this decision.  In the world of mathematics, however, the thinking we use is to be based upon purely objective standards only.

The experiment mentioned in class about getting the rings on the nail is a classical example of red-lining.  All the mathematicians, and only the mathematicians, crossed the red-line on the floor (by running) to hang the rings on the nail.  The non-mathematicians, literally saw the red line on the floor, and saw how the experimenter stood behind the red-line and tried to toss the rings onto the nail.  Although the experimenter said nothing about the red-line, and therefore, the red-line played no real role in the problem, by association from past experiences and the experimenter’s method, most people “added” the red-line to the problem thinking (in error) that it was forbidden to cross the red line, and therefore made it more difficult to solve.

Another example is the nine-dot problem.  Students were challenged to connect nine dots placed at the centers of the cells in a tic-tac-toe board by drawing four straight lines, without taking the pencil off the paper and without retracing any lines.  Most people mentally draw a “red line” (square) around the nine dots and cannot solve the problem because they do not draw lines that go past the border of this square.  In truth, though, the problem set no conditions about such limitations, and can easily be done if the lines go beyond this square.  Symbolically, this problem may be said to typify the need to “think outside the box.”  Avoiding redlining often means thinking outside the box.  Using abstract thinking is NOT the same as avoiding redlining, but it can be useful in helping one remove the non-essential elements from the problem (such as a red line) and think outside the box.

Similarly, another challenge given to the class was to make four equilateral triangles out of six match sticks, with each side of each triangle being exactly one match stick (not less or more).  This problem cannot be done in two dimensions, but can be done easily by using three dimensions (by building a pyramid with a triangular base, called a tetrahedron).  The red-lining here is the unnecessary restriction to two dimensions.

Of interest here is that I have often found many students may present alleged solutions by using half-match sticks as the length of a side.  It has often happened that when I point out that they must use a full match stick, they deny that I had so said when I first gave the problem.  This denial has taken place so often, that I actually tape-recorded once the lesson wherein I assigned this problem.  I clearly stated that condition no less than five times.  Yet, at the next class, many students—in fact, almost, the entire class—were very adamant that I had not stated that condition.  I then played back the tape.  Most students were nonchalant about it.  They said sort of neutrally that they had made a mistake.  I found it amazing.  First, they did not hear something said five times.  Secondly, they were very adamant and even upset that I had claimed I had so stated, and some even swore I had not stated this condition.  Thirdly, it did not seem to bother them very much that they had made such a mistake.  I know I would feel upset if I had made such a serious error that I was adamant about something that had been said five times, and I had totally missed it.  Part of being a good student in math—or just a good student—is to pay attention.  Mathematics is more detailed than most other subjects, and missing just one idea can be critical.  It is important to get all the ideas exactly straight.  One should neither omit one idea (by accident or not paying attention), and one should not add an idea, based upon past experiences or other reasons.  Extreme accuracy in communication—both in speaking and in listening–is an important skill that must be mastered before one can be successful in a math course.

Summarizing, most people use mental processing they call thinking, but mathematicians do not consider to be true thinking because true mathematical thinking is totally objective, whereas the mental processing used by most people involves subjective elements, the addition of which can actually make a problem appear to be harder than it really is.