## 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.

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.

## Important Advice For SAT Preparation

The SAT is a nerve wracking experience almost every high school student. The Scholastic Aptitude Test plays an integral part in the college application process. The 170-question test, which takes about 4 hours to complete, is an exhausting experience compounded by the fact that it is administered on Saturday mornings. While the test can feel overwhelming, you can lessen your stress levels and improve your score by familiarizing yourself with the test beforehand and following these essential tips.

If You Love Books, You Will Love The SAT

Students that excel at the SAT are those who are well read and eager to look up words when they come across a word they can’t define. The test really favors verbally inclined students. The SAT directly tests vocabulary in its sentence-completion section.

When In Doubt, Leave It Blank

The test has a “guessing penalty” that punishes students who take a wild swing at a question that is beyond their intellectual reach by deducting points for an incorrect answer. So if you do not know the answer to a question, if is probably in your best interest to leave the answer blank. If you are able to narrow down the answer to two or three choices, however, guessing may be the wiser option because the odds of you getting the question right outweighs the penalty for guessing.

Know The Classes That Matter

The Math section is almost entirely comprised of information that you learned in Algebra I and Geometry. The multiple-choice writing questions test basic elements of grammar, which tends to go uncovered in high school English classes. It is important to review these math classes and brush up on your grammar before taking on the SAT.

Use Time Wisely

Do not waste too much time on a single question. If a question has you stumped then you should move on quickly and revisit the question if you have extra time after finishing the rest of the section. Those who do well on the SAT give each question its fair share of time.

There is an Alternative to the SAT, the ACT, which is accepted at all four-year schools that accept the ACT. The ACT is geared towards students that excel at math as opposed to verbal skills. The best way to determine which test is best for you is to take a practice version of each exam. While it may seem like a lot of work all for nothing, in reality, taking the time now to determine which test is best for you will set you up to get into the college of your choice.

If you would like to learn more tips or need more information to see if the SAT is right for you then check out this article. Some students are much better suited for the ACT, so it is important to understand the pros and cons of both before deciding what to take.

## The Parking Spaces Problem

At a local business, parking spaces are reserved for the top executives: CEO, president, vice president, secretary, and treasurer with the spaces lined up in that order. The parking lot guard can tell at a glance if the cars are parked correctly by looking at the color of the cars. The cars are white, orange, red, black, and blue, and the executives names are Andrew, Brett, Caitlyn, Danny, and Erin.

• The car in the first space is black.
• A blue car is parked between the black car and the orange car.
• The car in the last space is red.
• The secretary drives a white car.
• Andrew’’s car is parked next to Danny’s.
• Erin drives a orange car.
• Brett’s car is parked between Caitlyn’s and Erin’s.
• Danny’s car is parked in the last space.

Question 1: Who is the secretary?

1. Erin
2. Danny
3. Caitlyn
4. Brett
5. Andrew

Questions 2: Who is the CEO?

1. Andrew
2. Brett
3. Caitlyn
4. Danny
5. Erin

Question 3: What color is the vice president’s car?

1. Orange
2. White
3. Blue
4. Red
5. Black

1. E – Caitlyn cannot be the secretary, since she is the CEO, nor can Erin, because she drives a orange car (and the secretary drives a white car). Danny’s, the red car, is in the last space. Andrew is the secretary, because his car is parked next to Danny’s, which is where the secretary’s car is parked.

1. C – The CEO drives a black car and parks in the first space. Erin drives a orange car; Brett’s car is not in the first space; Danny’s is not in the first space, but the last. Andrew’s car is parked next to Danny’s, so Caitlyn is the CEO.

1. A – The vice president’s car cannot be black, because that is the CEO’s car, which is in the first space. Nor can it be red, because that is the treasurer’s car, which is in the last space, or white, because that is the secretary’s. The president’s car must be blue, because it is parked between a black car (in the first space) and a orange car, which must be the vice president’s.

## The Snake Problem

The Snake Problem

You are asked to open one (and only one) of two boxes, labeled “A” and “B,” and are given the following facts:

Fact 1:  Each box contains either a poisonous snake or a large sum of money.  If you open a box containing a snake, which is highly poisonous, it is likely to attack you and kill you instantly.  If you open a box containing money, you are entitled to keep the money.

Fact 2:  Each Box has a sign attached with a written statement on it.  Either both of these statements are true, or both are false.

Here are the statements:

Statement on Box A:  At least one of these boxes contains a million dollars.

Statement on Box B:  The other box has a poisonous snake in it.

Question:  Which box should you open?

To answer that question we need to do an analysis of the given information. The first thing is to list all possibilities.

Notes To Help Answer The Question

1.  This is a logic problem, and a logic problem may present difficulties of “self-reference.”

2.  Do not jump into a problem right away.  Be sure you first understand the problem, even if it takes a few moments.

3.  If a problem clearly give a small number of possibilities, one way that is sometimes good to solve the problem is to “step” through it, and see how each possibility works out.

4.  There are two weak spots in this analysis—and they are typical of weak spots in almost any problem on a test or in real life that requires logical analysis.  They shuld be carefully investigated—on a test by reviewing when you are all done, or in real What life by someone in your company or on your team, especially if there is high risk, such as expense or health involve.  What are they?

(A)  Is the logic valid?  Is there a flaw in the logic?

(B)  Even if the logic is valid, the conclusion may be incorrect if you do not have the facts.  Are the claims used in the analysis reliable and complete?

– Reliable means honest and accurate (can have one without other).

– Complete means you have all the relevant information needed.

To do so we must be sure we understand all of the given information correctly.  It should be noted Fact 1 may be interpreted in one of two ways.

Wrong way:  One box has a snake and the other box has the money—two boxes cannot contain the same thing.  This gives us two possibilities:
Possibility 1:  A has a snake, and B has money.
Possibility 2:  A has money, and B has a snake.

Correct Way:  Any one box can contain either a snake or money—these possibilities are independent, and there is no guarantee that a snake is in either box or that there is money in either box.  Thus, there are four possibilities, the two above plus these two:
Possibility 3:  A has a snake, and B has a snake.
Possibility 4:  A has money, and B has money.

Question:  How do you know that the second way (as opposed to the first) is the correct one?

Answer:  Two possible ways.  First, from the wording itself , “Each box…”  This phraseology does not mean there is exactly one box with snake and exactly one box with money.  It merely means each time there is either a snake or money—there is no relationship between the two.  The second way is from the statement on box A saying “At least” one box has money—this statement allows for the possibility both have money (or both have snakes.)

Summarizing the gives us the following chart:

Our second step involves an analysis of the information once we have listed all cases.  Let us ask which statements, A or B, will be true in each case.  We put a Blue T or F (for True or False) next to each possibility if, in that case, Statement A is true or false, and we put a Red T or F next to each possibility if, in that case, Statement B is true or false.  NOTE: Our goal is to co-ordinate the two statements (Fact 1 & 2) with the two statements (A & B).  Here is our result:

Thus, we see in the first case, and only in the first case, both Statements A & B are true. There is no other case where both are true or both are false, as required by Fact 1.  Therefore, it must be that Possibility 1 is the only viable possibility, so open Box B.

In real life, however, we should not open any box, because, as humans, we can never be sure that (1) we did not make any mistake in our analysis, and (2) the information given to us (namely Fact 1 and Fact 2) are really correct.  We should consider these to be claims that other humans (who are fallible) have made, and not accept them as absolute fact.