Jason K. Sanchez's CU

Educational Setting

Currently, I teach at Harrison Middle School, which is located in Albuquerque’s south valley area along the Rio Grande. Harrison Middle School has just been removed from the “schools in need of improvement” list by the New Mexico State Board of Education. So, there has been a concerted effort on the part of its teachers of mathematics to find effective ways of teaching mathematics to its students. The majority of Harrison’s students are comprised of the lower socio-economic echelons with, approximately, eighty-five percent eligible for the free or reduced lunch program. Although I write this curriculum unit with my current, 7th grade, mathematics students in mind, it may be used at higher levels, with some minor adaptations.

Context and Background

“This Isn’t Math”

When the administration decided to move Harrison Middle School from a “traditional” math program to what has been termed a “contextual” program, many students began asking, “Why don’t we do real math anymore?” leaving me in the position of justifying the reasons behind this change. Three years have since passed and still there are debates in our math department regarding what an education in mathematics should be and whether we should use rote work methods or contextual ones, etc.... Ultimately, the answer to this question lies in our perception of what mathematics is. Neither the rote-work approach nor the newer contextual approaches appear to be justified in claiming a monopoly on what mathematics is, or what a mathematics education should be. It is my conviction that we must find a balance between all aspects of mathematics in the classroom. This may prove to be difficult, however, since the time that teachers spend with their students is limited. Therefore, our methods of doing this need to be efficient, yet effective.

Furthermore, other rifts exist in mathematics education. Both the traditional and contextual sides of mathematics teaching are function or application oriented. That is, they deal with the tangible aspects of mathematics. They ignore the fact that students are not encouraged to spend time in mathematical thought. Students are not required to ask themselves “what is math?” One “cut and dried” definition of mathematics says that it is “the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically... Mathematical procedures, operations or properties.” (Random 825) A more detailed definition is that “Mathematics is a global activity which has developed into a worldwide language, with a particular kind of logical structure; it contains a body of knowledge relating to number and space, and prescribes a set of methods for reaching conclusions about the physical world. And it is an intellectual activity which calls for both intuition and imagination in reaching conclusions.” (Joseph 77) Our own definitions of what mathematics is will color the perceptions of our students. We must then be sure to facilitate a well balanced approach to mathematics that will foster perceptions of mathematics that are meaningful to our students. “The definition of mathematics changes. Each generation and each thoughtful mathematician within a generation formulates a definition according to his lights.” (Davis 8) We do not allow students to do this in public education, we simply want them to manipulate symbols and arrive at answers, then apply this skill in solving some “real life” problem and we fail in this. Perhaps students will gain more ownership of their mathematical experience if we give them the opportunity to arrive at conclusions like those of some philosophers who concluded that the type of mathematics we teach in the classroom were “mere shadow manifestations of the real mathematics. That math exists eternally and independently of this actualized universe” (9). Perhaps, they would arrive at their own more meaningful definitions of mathematics.

Such debates exist because of the “secularization” of mathematics. That is, the divorce of mathematical symbolism, procedure and application from mathematical thought, philosophy, religion and history. For so long we have tried to teach our students how to work problems and how to make some applicable use of them. Yet, we have denied them the opportunity of appreciating how they might interpret the world around them, philosophically, intellectually, spiritually and historically in the light of mathematics. It is not my intent to address Mathematicism, which is the use of rigorous mathematic methods in illucidating philosophy. However, a good understanding of mathematics and, specifically, probability can be extremely beneficial in infusing meaning and understanding to all aspects of living.

So, now that I have said all of this, let me make it clear that my intent is to teach a unit on probability, which will include historical, religious and philosophical implications of chance, probability and risk, as well as games of chance. In order to gain a “big picture” of what mathematics is, because it is the “parent of probability,” I will also engage in some general history of mathematics, as well as some general philosophical and religious implications. Let us do this with the conviction that knowing about math history in general will make better teachers of probability and that the student of mathematics will gain greatly by having an instructor that is well versed in mathematics history, philosophy, religion and applications. Once I have reached the discovery, or systematization, of the mathematics of probability in the 15th and 16th centuries AD, I will cease to discuss general math history in the interest of keeping this narrative from becoming interminable.

Before proceeding with this history, it might be advantageous to give brief definitions of chance, probability and risk. (They will be discussed in more detail later on.) Chance can be described as the unpredictability, randomness, un-willed-ness, or uncertainty of events. Probability is the measure of an event’s uncertainty and risk is a measure of an unfavorable outcome.

Prehistory of Mathematics

It is credible to assume that the earliest use of mathematics consisted of humans enumerating or counting as survival efforts dictated. (To this day counting sheep comforts us to sleep.) Dutch philosopher and mathematician L. E. J Brouwer regarded “the starting point for mathematics to be the intuitive understanding of the generation of the sequence of (positive) integers.” According to Brouwer, this understanding proceeds for a “basic inner awareness of time.” (New 570) However, once the immediate needs of survival were met and society advanced, mathematics evolved as well.

Archaeologists have uncovered artifacts providing clues to the nature of prehistoric mathematical accomplishments. One such artifact, the Ishango bone, which is estimated to be about 25,000 years old was discovered in Zaire. The Ishango bone, which has notches carved into it, is believed to have been used as a six month lunar calendar. (Joseph 76, Powell 56) Another early mathematical artifact found was a Peruvian Quipu made up of woven knots. The knots contain the record of a population census. Different sizes of knots represent different magnitudes of numbers. “In a predominantly pastoral or simple agricultural society , such ingenious devices were invented to satisfy the main mathematical requirements...But as societies evolved, mathematical demands became more varied and sophisticated.…” Leading, eventually, to place value notation, devices such as the abacus (Joseph 76), and more recently, calculators and computers. (Surprisingly, many mathematicians today choose not to use computers at all!)

Did early humans perceive that some things happened more often than others and that some things happened at random, or by chance? Many argue that early humans viewed everything as deterministic; that is, decreed or ordained by God or the gods. It is logical to assume that, as with most things in life, there was probably a continuum or spectrum of varying perceptions or beliefs. At any rate, “archaeologists have found evidence of games of chance on prehistoric digs, showing that gaming and gambling have been a major pastime for different peoples since the dawn of civilization.” Would societies that did not have some concept of chance or probability engage in games of chance? In fact, “given...other great mathematical discoveries (many of which predated the more often quoted European works) and the propensity of people to gamble, one would expect the mathematics of chance to have been one of the earliest developed” (Math 1). However, this was not the case.

Gambling

The advent of gambling occurred early in the history of civilization. Whether it arose from leisurely game playing or religious activity is unknown. The ancestor of the die was developed prior to 1400 BC. The term used for these primitive die is the “astralagus.” Astralagi seemed to be present in more than one civilization. Astralagi were made from the ankle bones of a sheep and had two rounded sides, so that, unlike modern dice, they had only four possible ways for any particular die to land. Eventually, the rounded sides were ground down, creating a six sided, or cubic, figure. The current method of arranging the dots on each face was in place by 1400 BC. Other games of chance were also played before 3500 BC. (Lightner 623)

The Mesopotamians

Much, but not nearly enough, is known about the mathematics of the civilizations that flourished between the Tigris and Euphrates rivers, in the Mesopotamian valley. These early civilizations consisted of the Sumerians, Akkadians, Babylonians, Assyrians and Persians. Our knowledge about the mathematics of these civilizations comes from their writings upon clay tablets. (New 576)

There have been no records found in which the Sumerians or Babylonians addressed the issue of chance or probability. Considering that they were both highly religious societies, it is reasonable to assume that they were quite deterministic; that is, chance was not seen as important, since the gods controlled things. Some of the mathematics that we do know about were developed and used by the Sumerians and the Babylonians.

The Sumerians (4,000 - 3,000 BC) developed two counting systems. One was a base ten system, which was then converted into a base 60 system. The base 60 (sexagesimal) system was used for measuring time and areas of circles. We inherited our method of dividing 1 hour into 60 minutes and each minute into 60 seconds from the Sumerians. (World 27) There are also other aspects of Sumerian mathematics that we still retain, such as division of circles into degrees, seconds, etc. Why the Sumerians settled upon a base 60 numeration system remains “obscure.” A mathematical benefit of using a base 60 system would be that 60 has many divisors (2, 3, 4, 5, 6...) (New 576). Some enlightenment may come when one considers that the Sumerians considered the numbers 7, 12 and 60 sacred. Since mathematics, in ancient times, was closely knit with philosophy and religion, it comes as no surprise that the Sumerians would integrate religion into mathematics and mathematics into religion.

Sumerian numbering systems could work in an additive manner, as follows: I = 1, < = 10. Therefore <<<< II = 42. These same symbols could be used in a place value system for larger numbers. For example, “I” could equal 60, making I<< = 80. “I” could also be used to represent any power of 60.

The Babylonians (2,000 - 1,000 BC) attained a relatively high level of mathematical competence. Like the Sumerians, they used a base 60 number system. They were able to predict eclipses of the sun and moon and devised the first system of weights. (World 44) Babylonian mathematics was not only practical (survival oriented), but somewhat academic as well. The Babylonians considered mathematics as “worthy of study in its own right, not just a tool.” (New 576,577) The number system developed by the Babylonians used place value, with a place value holder that functioned as zero. How exactly they used this “zero” is not clear. However, that they were able to arrive at a concept that is not “intuitive” testifies to their competence. That is, their numbering system was much more sophisticated than that used by the Roman (and Europeans into the middle ages). They also developed computational methods, solved linear and quadratic equations and worked with “Pythagorean triples.” They were familiar with the relationship of the two sides of a right triangle and its hypotenuse “more than a thousand years before the Greeks…”. They developed methodology for working with square roots and had calculated the square root of two to an equivalent of 1.414213 long before Pythagoras was forced to lose sleep over irrationalities.

Historically, it has been taught that Babylonian mathematicians were prevented from creating general algebraic systems (generic formulas, etc.) by reason of never having developed algebraic symbols and that their mathematical problems were merely specific. However, this has been refuted by English mathematician and statistician, George G. Joseph. Joseph shows that the Babylonians did make use of two symbols as variables. They used ush (length) and sag (width) for the same purpose that we would use x and y. Among other things, the Babylonians also used forms of substitution. Joseph also argues that it is misleading “to suggest that because existing documentary evidence does not exhibit the deductive, axiomatic, logical inference characteristics of much modern mathematics, (that) these cultures did not have an idea of proof.” He demonstrates that “what the Babylonian method involved was the step-by-step application of the general formula...”. Instead of providing generic formulae, the Babylonians gave example after example of how these formulae are applied. This practice would be equivalent to formal “proof.” (Joseph 73, 74)

The Egyptians

“Board games involving chance were, perhaps, known in Egypt 3,000 years before Christ.” (Everitt 3) One such game called “‘hounds and jackals’ was played as early as 3500 B.C.” (Lightner 623) However, given that the Egyptians believed that multitudes of gods controlled every aspect of life, the development of probability theory was not likely. Egyptian mathematicians, of course, understood that certain actions produced desired outcomes, especially, in architectural planning. This, however, might be seen as a “common sense” idea of probability.

The Egyptian mathematical community was comprised of (4,000 - 1,000 BC, and arguably later) scribes, since the scribes were the only literate group. (New 577) The scribes were trained in hieroglyphics, which means “priestly writing.” (Columbia 390) They also recorded the practical applications of mathematics. Many have argued that Egyptian mathematics was “on the whole, elementary and profoundly practical in its orientation.” (New 575) That Egyptian math dealt only with counting, surveying and architecture. However, there are many indications that this may not have been the case. In general, Egyptian counting systems were additive like that of the Romans. For example, �� = 23. The scribes, however, did have a system for abbreviating numbers. The Egyptians also utilized multiplication and division tables, such as;

1	28
2	56
4	12
8	224
16	448	...

To multiply 11 x 28, one would choose numbers in the first column that add up to 11. Then add up the corresponding numbers in the second column to get the answer. In this case 1 + 2 + 8 = 11 and 28 + 56 + 224 = 308, which is the answer. The Egyptians also represented fractions differently. For example, 6/10 would be expressed as � + 1/10. They tended to utilize fractions with 1 in the numerator.

Each year the Egyptians had to redefine property boundaries after the flooding of the Nile River, which they considered a god. In their spare time they may have calculated the answer to questions, such as this one from the Golenishchev Papyrus:

Find the total of “7 houses, 7 cats per house, seven mice per cat, 7 ears of wheat per mouse and 7 hekat of grain per ear.” The result is 19,607. (New 577)

An interesting word problem that was used in training scribes was recorded on the Rhind Papyrus (located in the British Museum). It reads:

You are the clever scribe... a ramp is to be built, 730 cubits long, 55 cubits wide, with 120 compartments-it is 60 cubits high, 30 cubits in the middle... and the generals and the scribes turn to you and say, “you are a clever scribe, your name is famous. Is there any thing you don’t know? Answer us, how many bricks are needed?” Let each compartment be 30 cubits by 7.

The Egyptians utilized mathematics in the construction of their sacred monuments, such as the great pyramids which so astonish tourists today. Though the architectural products of Egyptian mathematics are magnificent, Egyptian mathematicians were not credited, until recently, with brilliance by modern mathematicians. Currently, the nature of Egyptian mathematics is being re-evaluated. The ancient Greeks, however, seemed to hold Egyptian mathematicians in high esteem. Thales, Plato and Herodotus credit the Egyptians with providing a solid basis for Greek mathematics (in teaching and application). Indeed, not only is it now conjectured that the Egyptians did impact Greek mathematics prior to the 4th century BC (577-579), but that “if the Greeks were the founders of rigorous mathematical demonstration” (and not the Egyptians, or someone else) that “they would not have failed to boast about such an accomplishment.” (Powell 54, 55) However, the “ancient Greek writers...claimed Egyptian mathematics and astronomy were superior to their own.” (55)

The Hebrews (Chance, Numbers and Pi)

The Hebrew scriptures, which have influenced western civilization more than any other writings, declare that God brought order to primeval chaos by the operations of dividing and multiplying - and by the narrowing down of an infinite number of possibilities. If probability is a function of a defined set of possibilities, then God may be viewed as the one who defines the sets of possibilities, producing probable outcomes. Indeed, God demands of Job, “Gird up now thy loins like a man, for I will demand of thee, and answer thou me. Where wast thou when I laid the foundation of the earth? Declare, if thou hast understanding. Who hath laid the measures thereof, if thou knowest? or who hath stretched the line upon it?” (Job 38:3-5)

The Arabics also created a “systematic and consistent language of calculation which came to be known by its Arabic name ‘algebra’.” In AD 825, Mohammed ibn-Musa al-Khwarizmi wrote an “algebra book” in which he describes “restoration” and “reduction.” That is moving negative terms from one side of an equation to the other and adding like terms, respectively. These are “the two main operations in solving an equation.” The term “algorithm” is also of Arabic origin. It is a corruption of the name “al-Khwarizmi.” (Joseph 70)

Although Arabic scholars wrote books on algebra (and mathematics in general), pulmonary circulation, light refraction, gravity, the scientific method and even evolution (Joseph 78), there are apparently no books regarding probability theory. Perhaps, this is due to the Islamic idea of kismet or fate. Islam teaches that one must fatalistically submit to the will of Allah. Calculating probabilities or odds would seem to run counter to such beliefs.

Why No Probability Theory

With such a long history of gambling, it might be supposed that elements of probability theory would have been developed. “Yet no direct link between gambling and mathematics seems to have been observed.” (Lightner 624)

Why probability was not developed as a distinct mathematical discipline may have been primarily due to societal factors. In some societies, the mathematical capability was not in place. Others had a high level of mathematical competence, but viewed chance as a leisure activity. In societies with a high deterministic view, chance was downplayed, or seen as divine will. The more a civilization integrated religious or philosophical thought into mathematics, the less its people may have needed to rely or focus upon chance, thus reducing the probability of probability being viewed as a distinct discipline in mathematics. Reasons that probability theory did not exist until relatively modern times are given in more detail below: (Davis 21-23)

Previously, there existed an obsession among humans “with determinism and personal fatalism. Once the wheels of the cosmos had been set in motion, everything subsequent was determined and hence potentially knowable.” However, the predetermined could be divined by the casting of lots, card reading, sacrificing and looking at the entrails of animals, etc. “The consultation of a random element was often thought to reveal the will of God.”

A second reason that probability theory did not exist was that people believed that God spoke via randomization, therefore, coming up with a theory of how God did this might have been considered sacrilegious.

A third reason for the late advent of probability theory is that people need to have access to data or “numerous easily understood empirical examples,” in order to calculate probabilities. Hence, the importance of John Graunt’s work in statistics as described in the following section.

Fourth, it is proposed by some that “science develops according to economic needs,” and that there was no need for the science of probability until insurances and annuities became a part of modern life. Currently, probability theory is used in a host of disciplines, from biological experimentation to physical applications, as well as in the day to day operations and thinking of the billions of humans on earth today.

The last reason that probability theory was not developed until the last three hundred years, or so, was that mathematics, in general, was not yet capable of supporting probability theory. “Techniques of arithmetic were still primitive, and calculus, which is necessary for the description of probability distributions, was not available till the middle of the seventeenth century.”

Probability Gets a Chance

Because Christian condemnation of gambling has never succeeded in completely stamping out the practice, gaming has flourished during various times throughout European history. Dice were common gambling tools of the middle ages (cards were not invented until the 14th century). Galileo recognized that “the six faces of a fair die were equally likely to be the result of a throw.” However that was about the extent of his musings. (Everitt 10, 11)

In 1545, an Italian mathematician (who liked to gamble), named Cardano, wrote the first theoretical work on probability. His work was ignored for about 100 years until probability gained interest. (Miller 1) We know of at least one person who applied some form of probability in earning a living. Abraham DeMoivre, a persecuted religious exile of France, placed himself “at the beck and call of gamblers” to calculate odds and was “paid small sums.” (Everitt 11)

In order to calculate probabilities one must have access to statistical data. That is, if you want to know what the likelihood of something happening is, you must have some idea of how often it occurred in the past. John Graunt (1620-1674) of England was the first person to collect data for such a purpose. He calculated what the risks of dying from plague were in certain years by collecting data on how many people died, what they died of, and so forth. (Everitt 12, 13)

In 1654, James Bernoulli (who had trouble deciding whether he wanted to be a theologian or a mathematician) took what little was known about chance “from a guide for wagering” and transformed it into a genuine theory. During this time insurance companies also became interested in probability theories as a way of determining the likelihood of catastrophes occurring.

There is a famous story or problem which is listed in most books on probability called “the problem of points.” In some of the versions, Blaise Pascal poses this conundrum to Pierre Fermat. In another, Pascal and Fermat are the characters of the problem and the drama takes place in a Parisian cafe.

Two players are interrupted in the midst of a game of chance, with the score uneven at that point. The player who is winning gets a message that a friend is dying. The carriage driver offers to take him if he leaves immediately. How should the stake be divided?

Pascal and Fermat continued to write back and forth regarding the solution to this question. Both men arrived at a correct solution to the conundrum in different ways. (Lightner 626) Their letters are considered the basis for probability as a distinct mathematical discipline.

Interestingly, Pascal, whose “life focused alternately on religion and philosophy and on mathematics,” reasoned in this way: “We know neither the existence nor the nature of God...Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.” (Lightner 626)

Calculating Probability

As stated earlier, probability is a measure of the uncertainty (certainty) or chances of an event occurring. By convention, probability is measured on a scale of zero to one. Zero being used to describe the impossible and one to describe absolute certainty. Probability may be expressed in fractional, decimal, or percent form. If we wish to describe the likelihood of any one event occurring out of a possibility of three events, we say that the probability is 1/3, or 1 chance in 3, or about 33%. Often gamblers use the term “odds” instead of “probability.” Both terms are synonymous. If a gambler wishes to express that the probability of a certain horse winning a race is 1/10 or 10%, he or she might say that the odds are 9 to 1 for (or against as the case may be). (Everitt 14, 15)

There are different approaches that may be taken in calculating probability. In the relative frequency approach we divide the frequency that an event occurs by the number of possible events. That is, if 25 out of 100 students received A’s on an exam, the relative frequency is 25/100, or 0.25. (14, 15)

The second is the classical approach. We obtain a measure of likelihood by dividing the outcome in question by the number of a set of possibilities. For example, if we wish to calculate the probability of throwing a 3 on a die, we desire one outcome out of a possibility of 6 (6 sides on one die), so we divide 1 by 6, or 1/6. (15)

When we have no real way of assessing frequencies or possibilities, we can form a subjective opinion, or hunch, arriving at what is termed a subjective probability. For example, if I want to know how many dogs have nightmares, I will have to form some subjective opinion, based upon other information or perhaps upon very little information. (15)

We may then take these probabilities and use them in determining special cases. Such as, when we want to know the probability of more than one event occurring when one precludes the other. These are called mutually exclusive events. The formula for mutually exclusive events involves adding the probabilities of however many mutually exclusive events there are. That is, Pr(A or B) = Pr(A) + Pr (B) where A is one event and B is another. (16)

Let’s say we want to know the probability of two, or more, events, that are not mutually exclusive of each other, happening at the same time. In this case, the probability is simply a product of the probabilities of each event. That is, Pr(A and B) = Pr(A) x Pr(B), where A is one event and B is another. (17)

Conditional Probability

In 1761, when Thomas Bayes, an English clergyman-mathematician died, an essay was found among his papers in which the following theorem (more or less) was found.

P(A�B) = P(B�A) x P(A) / P(B)

Bayes Theorem suggests a procedure for the appropriate way to combine evidence. It also “provides the means of writing term of the conditional probability of event B given that A has occurred and the unconditional probabilities of events A and B.” (Everitt 99, 100)

For example, given that 85% of house cats eat goldfish and that 15% don’t [P(A)=0.85], and that a witness, who is right 80% of the time [P(B�A) = 0.8], swears your cat is innocent, what is the probability that your house cat, is in fact innocent [P(A�B)]?

To make this decision, using Bayes Theorem, we need to calculate the “unconditional probability” that the witness says the feline is innocent [P(B)]. P(B) is given by the sum of the probabilities of the following two events. The probability that the cat is innocent and the witness is correct is 0.8 x 0.15 = 0.12. The probability that the cat is guilty and the witness is wrong is 0.85 x 0.20 = 0.17. The sum is, therefore, P(B) = 0.12 + 0.17 = 0.29.

Now we plug in all of the terms: P(A�B) = 0.8 x 0.15 / 0.29 = 0.41.

Despite the witness’ report (another cat no doubt) your cat is more likely to be guilty! (Everitt 99-101, modification)

Calculating Risk

Risk is defined as “exposure to the chance of injury or loss...a dangerous chance...the degree of probability of such loss.” In order to determine risk, we must calculate and compare probabilities. One way of doing this is to divide the number of unfavorable occurrences by the number of possible unfavorable occurrences. For example, if 10 out of 50 cats have fleas, then we say 50 / 10 = 5 or 1 out of 5 cats is flea ridden. Or, if 56,000 auto fatalities occur each year out of 224,000,000 Americans, we can calculate 56,000/224,000,000 = 0.00025 or 1 out of 4,000.

Stochastic Implications Today

Let us use the word stochastic to mean chaotic, chancy or random. That is, “it refers...to the utilization of those methods of ...statistics and probability which are intended to reduce the chaos of the single unpredictable event of a less wild and more predictable pattern.” (Davis 18) This pattern, may in turn, then be used in decision making. Probability theory is indispensable when it comes to making certain decisions. Any one event cannot be predicted; however, by averaging many events, one can make fairly accurate decisions about things.

People everywhere in modern society are convinced that they have some sort of idea as to the likelihood of certain events (whether this is true or not). We often hear such phrases as “the odds of ‘such and such’ happening are...” or people rhetorically ask “what are the odds of...?” Yet, how many are actually able to calculate what the odds referred to are? Hence, the host of bad decisions that people make from day to day. Insurance agencies, polling firms, Mendelian geneticists, epidemiologists, and others “educated” in probability theory may have a workable, if not expert, understanding of how probability works. However, the vast majority of society simply tends to “take their word for it” or base its odds upon hunches or by taking the common sense approach. Who then is actually responsible for the decisions that society makes? Especially when “the whole world is indeed stochastized and is becoming more so as each day passes.” (Davis 20) The student, who has no understanding of probability theory is ill equipped for society. “The stochastization of the world so permeates our thinking and our behavior that it can be said to be one of the characteristic features of modern life.” “Probability is a net that supports us and a cage that confines us.” (Davis 31)

These hunches and common sense approaches may actually be built upon to create a better understanding of probability. Pierre Simon de Laplace (1749-1827), who was the first to apply probability theory to astronomy and who made valuable contributions to probability theory (continuous probability) wrote, “The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which oft-times they are unable to account....It teaches us to avoid the illusions which often mislead us....there is no science more worthy of our contemplations nor a more useful one of admission to our system of public education.” (Moritz 340)

Implementation

It is my intent that students will develop a “holistic” approach to teaching math. Students will research, discuss and write reflectively about the origins of mathematical concepts, symbols, etc., as well as mathematical thought. This will be in addition to practical application in problems, work or games. Specifically, students will focus upon probability and risk situations. Students will research the origins of probability, chance and risk, then discuss or write reflectively upon their philosophical and religious history. Students will also explore the implications of probability, chance and risk in their world.

Lesson Plans

Lesson plans consist of a series of activities that may be used as seems appropriate to the instructor. I have listed them in the order that seems logical to me. However, the order or activities may be changed as the need arises.

Activity One

Materials/Resources: Students will need access to library research materials and the Internet

Students will conduct research upon an historical personage or general mathematics topic. Students will write a brief report (1-2 pages) on the topic they have chosen in which they narrate the history and address any societal changes or views that were affected.

Activity Two

Materials/Resources: Dictionaries and enough newspapers or magazines for each student to peruse.

Students will become familiar with the history and vocabulary of probability. Students will gain an awareness of stochastics in the world around them and create a list of risky events.

Students will become familiar with the following terms by first defining them, then writing a story in which they use them appropriately or by using them correctly in sentences. They are: Average, complementary, chance, chaos, data, likely/likelihood, mean, median, mode, mutually exclusive, possible, probable/probability, random, risk, statistics.

Activity Three

Materials/Resources: Enough newspapers and/or magazines for each student to peruse.

Students will complete a newspaper or magazine search to locate articles in which some, or all, of the terms from activity two are used, and summarize/ explain what is conveyed or described in the articles, etc. They will then discuss or write a reflective paragraph on how the articles in question affected their opinions about the subjects of the articles and how this might affect any decisions that are made by people who read the articles.

Activity Four

Materials/Resources: Enough newspapers or magazines for each student to peruse.

Students will locate and cut out graphs from newspaper or magazine articles, in which statistical information is presented, and use these to answer the questions: “Which event in each graph is more likely to occur?” “Which event has less chance of occurring?”

Activity Five

Materials/Resources: Internet access for each student.

Students will conduct an on-line search for information on the history of mathematics and probability, respectively. They will then organize and summarize the information collected into a written or pictorial report.

Activity Six

Materials/Resources: One die per student, one sheet of graph paper per student.

Students will use one die to make an indefinite number of tosses (say 100 or more) and record the outcome of each toss. They may then add up the number of times each result occurred and present it in two or more graphical forms. Students will be asked to explain their results. Teacher should facilitate discussion that leads to an understanding that the six faces on a fair die each have an equal chance of happening. Teacher should describe how this is expressed as 1 out of 6 or probability = 1/6, etc. That is, the probability of one event occurring out of a possible set of six. Explain how this may be expressed as a fraction, decimal or percent. Have students use all three methods of expressing probabilities.

If appropriate, have students practice or review converting between fractions, decimals and percents.

Have students “brain-storm” and discuss other real-life situations in which assessing simple probabilities may be helpful.

The teacher will then facilitate a discussion on whether it is possible for a die to land on two sides. The obvious answer is that it would be impossible. Explain to students that we can calculate the probability that one or the other will occur, however. These outcomes are termed mutually exclusive and could be determined in the following way: [P(A or B) = P(A) + P(B)]. Therefore, the probability of a die landing with either a 2 or a 4 face up are P(2 and 4) = P(2) + P(4) = 1/6 + 1/6 = 2/6 or 1/3 or .33 or 33%.

Activity Seven

Materials/Resources: Two dice per student and one sheet of graph paper per student.

Students will make an indefinite number of die tosses using two dice. Students will record the results of each toss. Have students add up the number of tosses in which one event occurred. For example, the number of tosses in which one die only landed on a “3." This would correspond to the frequency method of determining probability. Show students how to calculate the probability of rolling one event (“3") using the classical method of determining probability. [P(3) = 1/12].

Have students sum the number of times that both dice landed on “3" (frequency). Have students calculate the probability of the same event (rolling two 3's) occurring on each die on the same toss. [P(“3") = 1/6 x 1/6 = 1/36]

Activity Eight

Materials/Resources: Two number cubes or dice per pair of students.

The following is an adaptation of the Number Cube Game. Have students work in pairs. First they can make intuitive predictions on which sum will occur most often when they toss two number cubes or dice. The students may each roll the cubes or dice 20 or more times and record the results. Students will then be asked to present their results graphically. Using this graph, students will be asked to predict the outcome of a second round of the game. Ask students to calculate the probability of the sum which occurs most often by creating a table diagram of the possibilities. (APS 1)

Activity Nine

Materials/Resources: Six tongue depressors or Popsicle sticks per student, access to permanent colored markers, examples of decorations.

Students will create their own version of Native American game sticks and use them to play. Students will use colored marking pens and tongue depressors to decorate one side of each stick with intricate designs of their own (alternatively, they may employ traditional Native American designs, see Regents citation) Allow approximately 30 - 40 minutes for students to decorate their sticks. Additional time will have to be given for students to actually play the game, etc. Have students write their names of the “clear” side of each stick.

Explain that the object of the game is for one team or person (when played in pairs) to take all of ten counters that are placed in the middle of the game area. Post, explain and demonstrate the game rules. Students take turns dropping the game sticks. The person, whose turn it is, holds up the game sticks vertically, in one hand, then to let them drop onto the a hard flat surface (game area). If all 6 game sticks land with the decorated sides facing up that person takes three of the counters. If all six sticks land with decorated sides facing down (clear sides facing up) the person, whose toss it was, takes two counters. If the game sticks are split evenly, that is, three decorated sides facing up and three decorated sides facing down, the person who tossed them takes only one counter. If the sticks land in any other configuration, the person who tossed takes nothing. Ask students to record results of tosses in subsequent games and to determine how the game sticks tend to fall. Ask students to draw (individually, in groups, or as a class) the possibilities, that may occur, in a logical sequence. Ask students to describe, orally or in writing, “what probability has to do with game sticks.”

Ask students to determine how many ways just one game stick can land. What is the probability of the decorated side landing face up? [P(1) = �]. How many ways might two game sticks might land? What is the probability of just one decorated side landing face up? [P(1)= � x �], and so forth. (Regents 63-78, adapted) This activity and the next allow for comparison of frequency and classical ways of calculating simple probability.

Activity Ten

Materials/Resources: Two pennies per student pair.

Have students, in pairs, decide what the probability of a coin toss will be, either intuitively, or by reasoning as follows: the desired event (heads) over the number of possibilities (2).

Have one of the students in the pair toss a coin (penny, nickel...) twenty times, while the other records the results of each toss. Is the actual number of heads/tails consistent with the intuitive probability determined above? Each person in the pair should alternate tossing and recording until the total number of tosses is at least 100. The students should determine whether the results are getting closer to the intuitive probability. That is, do 50% of the tosses result in “heads” and 50% in “tails?” This activity should help students understand that a larger sample size, in this case, may provide more accurate results. Ask them “does sample size matter?”

Alternatively, many coins may be placed “on edge,” on a flat table. (You will need many coins per pair.) Then have students shake, or nudge, the table so that the coins fall on a face. Does this give a 50/50 result? Why or why not?

Students may also explore the frequencies and calculated probabilities of two coins landing face up, face down, or split. This information may be tabulated and compared to probability calculations derived using the classical method. [P(of heads using 1 coin) = �, P(2 head/2 coins) = � x � = 1/4, and so on] Demonstrate how this information may be displayed using a tree diagram. Such as:

		First toss		Second toss
			[	heads ( � )
		heads ( � )	[	tails ( � )
coin	[
		tails ( � )	[	heads ( � )
			[	tails ( � )

Activity Eleven

Materials/Resources: Two sheets of paper per student, 1 sheet graph paper per student.

Students will investigate how the length of a paper airplane affects the distance it will fly. Students will plan an experiment in which they gather data, present it using a histogram, bar graph or scatter plot. Students will make and evaluate predictions or conjectures, about what the optimal length is, based on the data/graphical representation.

Activity Twelve

Materials/Resources: Copies of the list below and calculators.

Students will investigate and assess the risk of certain disasters, or unfavorable outcomes, occurring. Have students study the following figures (Slovic 181-216):

Source of Risk	Annual Actual Deaths
Smoking	150,000
Alcohol	100,000
Motor Vehicles	50,000
Handguns	17,000
Electric Power	14,000
Motorcycles	3,000
Swimming	3,000
Railroads	1,950
Bicycles	1,000
Nuclear Power	100

Have students discuss which of the above items appear to pose great threat or risk. Have students calculate risk, assuming a population of 224,000,000 Americans. (Obviously, these odds will only apply within the United States.) For example the risk of dying from the effects of nuclear power would be 100/224,000,000 = 0.0000004 a very small risk. Have students investigate and determine other risks that may be relevant to them at school.

Have students write about something they fear and explore what risks are involved in a journal or letter.

Activity 13

Materials/Resources per student pair: 2 dice, 1 game board or grid sheet (12 x 7 grids), 12 beans or other icons

Students will play “race” in which runners, horses, or other “icons” are numbered 1 through 12. They will use two dice. At the start of the race the 12 icons are lined up along the first column of the grid sheet. An icon “moves towards the finish line when the sum of the two dice rolled equals its number…Students make predictions about which …will finish first.” Some sums will occur more often than others and one not at all (“1”). Afterwards the students will be asked to use what they know about mathematics and probability theory to explain the outcomes. (Cuomo 45-51)

Activity - Math Science Extension

Students can be given a brief, working description of the Punnett Square used in Mendelian genetics and then be asked to answer the following question: “What is the probability of male versus female children being born?” Given more information, students may investigate the probability inherent in the determination of eye color, etc.”

Assessment

It is my intent to present the students with the following situations in a practical assessment to be given after the completion of the activities.

Given a situation in which there are a set number of possibilities, students will be able to describe the probability of any one event taking place. [P(A) = event/set of possibilities]

Students will be able to explain, or demonstrate using a die, the concept of mutual exclusivity and how this might be calculated [P(A or B) = P(A) + P(B)].

Students will be able to explain, or demonstrate using a pair of die, the concept of complementary outcomes. Students will demonstrate how this may be calculated. [P(A and B...) = P(A) x P(B)...]

Students will collect data, present it graphically, and determine probabilities via frequency of occurrence. Students will compare this frequency probability with one that is calculated (theoretical).

Standards Addressed

The following Albuquerque Public Schools Mathematics Content and Performance Standards for grade 7 are addressed in this curriculum unit.

Strand I: Global Mathematical Process

Content Standards

The student understands and uses mathematical processes.

Performance Standard

15. Recognizes and applies mathematics in contexts outside the mathematics course.

Strand II: Number Sense and Operations

Content Standard

The student demonstrates number sense through experiences with meaningful mathematical problems that focus on number meaning, number relationships...relative effects of operations and multiple representations to communicate sound mathematical thinking.

Performance Standards

10. Explains the relationship that can be expressed as ratios of part to whole (e.g., five red apples out of a total of eight apples, expressed as 5/8).

12. Explains relationships that can be expressed as proportions or percents (e.g., � = 50%).

Strand IV: Data Analysis, Statistics, and Probability

Content Standard

The student identifies patterns and special features of data and events of chance through experiences with meaningful mathematical problems that focus on comparing, predicting, representing data, and making decisions to communicate mathematical understanding.

Performance Standards

1. Applies counting principles to determine sample space.

2. Determines simple probability in experimental and theoretical situations.

3. Determines probability of dependent and independent events in experimental and theoretical situations.

4. Explains and uses appropriate terminology to describe complementary and mutually exclusive events.

Documentation

Works Cited

Albuquerque Public Schools K-12 Mathematics Content and Performance Standards. Albuquerque: Albuquerque Public School System. 2001 Draft.

Albuquerque Public Schools (APS) Research, Development and Accountability. Task Bank:
Performance-Based Mathematics Assessments. Albuquerque: APS. 2000.

Berkeley University. In All Probability, Investigations in Probability and Statistics Teachers Guide. Berkeley, CA: LHS GEMS. 1993: 67).

Cuomo, Celia. In All Probability: Investigations in Probability and Statistics Teacher’s, Guide, Grades 3-6. Berkeley: LHS GEMS. 1993.

Davis, Philip J. and Reuben Hersh. The Mathematical Experience. Boston: First Mariner Books, 1998.

---. Descartes Dream. Boston: Harcourt, Brace and Jovanovich, 1986.

Everitt, Brian S. Chance Rules. New York: Springer-Verlag, 1999.

Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries: Development and History. New York: W. H. Freeman and Company, 1997.

Jeffrey, Grant R. JESUS: The Great Debate. Toronto: Frontier Research Publications, Inc., 1999.

Joseph, George G. “Foundations of Eurocentrism in Mathematics.” Ethnomathematics. Albany: State Univ. New York Press, 1997.

Lightner, James E. “A Brief Look at the History of Probability and Statistics.” National Council of Teachers of Mathematics (NCTM) Publication. November 1991: 623-630.

Moritz, Robert Edouard. On Mathematics and Mathematicians. New York: Dover Publications. 1958.

Regents of the University of California. In All Probability. CA: LHS GEMS, 1993.

Slovic, P., Fischhoff, B. and Lichtenstein, S. In Societal Risk Assessment: How Safe is Safe Enough. New York: Plenum Publishing Corporation, 1980

The New Encyclopaedia Britannica. “The Foundation of Mathematics.” Chicago: Macropaedia, 1997