MESOPTAMIAN MATHEMATICIANS

Who were the Mesopotamian mathematicians ? 
About a thousand miles east of the Nile lies the fertile valley of the Tigris and Euphrates, once known as Mesopotamia. During early history this land was the home of the Sumerians, Chaldeans, Assyrians and Babylonians. In some ways, their society was similar to Egypt's. Their mathematicians also belonged to the priestly caste. Unlike Egypt, Mesopotamia carried on extensive foreign trade with the people to the west in Lebanon, to the north in Asia Minor, to the east in India, and possibly even with China. What we know of their mathematics comes to us from the baked clay tablets on which they wrote. The Babylonians had advanced mathematical knowledge as far back as 2500 B.C. 
We know that they inherited from the Sumerians their cuneiform, or wedge-shape, writing and numerals. We are indebted to these people for several of our basic mathematical concepts and notations.

THE HARDY - WEINBERG LAW - MATHEMATICS AND GENETICS

This was explained by the Hardy–Weinberg law, an application of basic mathematics to genetics. It explains how, in the Mendelian theory of inheritance, a dominant gene does not take over completely and a recessive gene does not die out.
G.H. Hardy was an English mathematician who prided himself on the nonapplicability of mathematics. He was a great researcher in pure mathematics but is probably more widely known for this single contribution to genetics – which started life as a piece of mathematics on the back of an envelope done after a cricket match. Wilhelm Weinberg came from a very different background. A
general medical practitioner in Germany, he was a geneticist all his life. He discovered the law at the same time as Hardy, around 1908.
The law relates to a large population in which mating happens at random. There are no preferred pairings so that, for instance, blue-eyed people do not prefer to mate with blue-eyed people. After mating, the child receives one factor from each parent. For example, a hybrid genotype bB mating with a hybrid bB
can produce any one of bb, bB, BB, but a bb mating with a BB can only produce a hybrid bB. What is the probability of a b-factor being transmitted ? Counting the number of b-factors there are two b-factors for each bb genotype and one b factor for each bB genotype giving, as a proportion, a total of three b-factors out
of 10 (in our example of a population with 1:1:3 proportions of the three genotypes). The transmission probability of a b-factor being included in the genotype of a child is therefore 3/10 or 0.3. The transmission probability of a B factor being included is 7/10 or 0.7. The probability of the genotype bb being included in the next generation, for example, is therefore 0.3 × 0.3 = 0.09. The complete set of probabilities is summarized in the table.

The hybrid genotypes bB and Bb are identical so the probability of this occurring is 0.21 + 0.21 = 0.42. Expressed as percentages, the ratios of genotypes bb, bB and BB in the new generation are 9%, 42% and 49%. Because B is the dominant factor, 42% + 49% = 91% of the first generation will have brown eyes. Only an individual with genotype bb will display the observable
characteristics of the b factor, so only 9% of the population will have blue eyes.
The initial distribution of genotypes was 20%, 20% and 60% and in the new generation the distribution of genotypes is 9%, 42% and 49%. What happens next ? Let’s see what happens if a new generation is obtained from this one by random mating. 
The proportion of b-factors is 0.09 + ½ × 0.42 = 0.3, the proportion of B-factors is ½ × 0.42 + 0.49 = 0.7. These are identical to the
previous transmission probabilities of the factors b and B. The distribution of genotypes bb, bB and BB in the further generation is therefore the same as for the previous generation, and in particular the genotype bb which gives blue eyes does not die out but remains stable at 9% of the population. Successive proportions of genotypes during a sequence of random matings are therefore
20%, 20%, 60% → 9%, 42%, 49% → . . . → 9%, 42%, 49%
This is in accordance with the Hardy–Weinberg law: after one generation the genotype proportions remain constant from generation to generation, and the transmission probabilities are constant too.
Hardy’s argument
To see that the Hardy–Weinberg law works for any initial population, not just the 20%, 20% and 60% one that we selected in our example, we can do no better than refer to Hardy’s own argument which he wrote to the editor of the American journal Science in 1908.
Hardy begins with the initial distribution of genotypes bb, bB and BB as p, 2r and q and the transmission probabilities p + r and r + q. In our numerical example (of 20%, 20%, 60%), p = 0.2, 2r = 0.2 and q = 0.6. The transmission probabilities of the factors b and B are p + r = 0.2 + 0.1 = 0.3 and r + q = 0.1 + 0.6 = 0.7. 
What if there were a different initial distribution of the genotypes
bb, bB and BB and we started with, say, 10%, 60% and 30%? How would the Hardy–Weinberg law work in this case ? Here we would have p = 0.1, 2r = 0.6 an d q = 0.3 and the transmission Probabilities of the factors b and B are respectively p + r = 0.4 and r + q = 0.6. So the distribution of next generation of genotypes is 16%, 48% and 36%. Successive proportions of the genotypes
bb, bB, and BB after random matings are and the proportions settles down after one generation, as before, and the transmission probabilities of 0.4 and 0.6 remain constant. With these figures 16% of the population will have blue eyes and 48% + 36% = 84% will have brown eyes because B is dominant in the genotype bB.
10%, 60%, 30% → 16%, 48%, 36% → . . . → 16%, 48%, 36%
So the Hardy–Weinberg law implies that these proportions of genotypes bb, bB and BB will remain constant from generation to generation whatever the initial distribution of factors in the population. The dominant B gene does not take over and the proportions of genotypes are intrinsically stable.
Hardy stressed that his model was only approximate. Its simplicity and elegance depended on many assumptions which do not hold in real life. In the model the probability of gene mutation or changes in the genes themselves has been discounted, and the consequence of the transmission proportions being constant means it has nothing to say about evolution. In real life there is ‘genetic drift’ and the transmission probabilities of the factors do not stay constant. This
will cause variations in the overall proportions and new species will evolve.
The Hardy–Weinberg law drew together Mendel’s theory – the ‘quantum theory’ of genetics – and Darwinism and natural selection in an intrinsic way. It awaited the genius of R.A. Fisher to reconcile the Mendelian theory of inheritance with the continuous theory where characteristics evolve.
What was missing in the science of genetics until the 1950s was a physical understanding of the genetic material itself. Then there was a dramatic advance contributed by Francis Crick, James Watson, Maurice Wilkins and Rosalind Franklin. The medium was deoxyribonucleic acid or DNA. Mathematics is needed to model the famous double helix (or a pair of spirals wrapped around a
cylinder). The genes are located on segments of this double helix.
Mathematics is indispensable in studying genetics. From the basic geometry of the spirals of DNA and the potentially sophisticated Hardy–Weinberg law, mathematical models dealing with many characteristics (not just eye-colour) including male–female differences and also non-random mating have been developed. The science of genetics has also repaid the compliment to mathematics by suggesting new branches of abstract algebra of interest for their intriguing mathematical properties.

THE FANO PLANE

The Fano plane geometry was discovered at about the same time as Pick’s formula, but has nothing to do with measuring anything at all. Named after the Italian mathematician Gino Fano, who pioneered the study of finite geometry, the Fano plane is the simplest example of a ‘projective’ geometry. It has only seven
points and seven lines.

The seven points are labelled A, B, C, D, E, F and G. It is easy to pick out six of the seven lines but where is the seventh? The properties of the geometry and the way the diagram is constructed make it necessary to treat the seventh line as DFG – the circle passing through D, F and G. This is no problem since lines in
discrete geometry do not have to be ‘straight’ in the conventional sense.
This little geometry has many properties, for example:

• every pair of points determines one line passing through both,

• every pair of lines determines one point lying on both.

These two properties illustrate the remarkable duality which occurs in geometries of this kind. The second property is just the first with the words ‘point’ and ‘line’ swapped over, and likewise the first is just the second with the same swaps.
If, in any true statement, we swap the two words and make small adjustments to correct the language, we get another true statement. Projective geometry is very symmetrical. Not so Euclidean geometry. In Euclidean geometry there are parallel lines, that is pairs of lines which never meet. We can quite happily speak
of the concept of parallelism in Euclidean geometry. This is not possible in projective geometry. In projective geometry all pairs of lines meet in a point. For mathematicians this means Euclidean geometry is an inferior sort of geometry.

If we remove one line and its points from the Fano plane we are once more back in the realm of unsymmetrical Euclidean geometry and the existence of parallel lines. Suppose we remove the ‘circular’ line DFG to give a Euclidean diagram.
With one line fewer there are now six lines: AB, AC, AE, BC, BE and CE. There are now pairs of lines which are ‘parallel’, namely AB and CE, AC and BE, and BC and AE. Lines are parallel in this sense if they have no points in common – like the lines AB and CE.

The Fano plane occupies an iconic position in mathematics because of its connection to so many ideas and applications. It is one key to Thomas Kirkman’s schoolgirl problem . 
In the theory of designing experiments the Fano plane appears as a protean example, a Steiner Triple System (STS). Given a finite number of n objects an STS is a way of dividing them into blocks of three so that every pair taken from the n objects is in exactly one block. Given the seven objects A, B, C, D, E, F and G the blocks in the STS correspond to the lines of the Fano plane.

THE PICK'S THEOREM

The Austrian mathematician Georg Pick has two claims to fame. One is that he was a close friend of Albert Einstein and proved instrumental in bringing the young scientist to the German University in Prague in 1911. The other is that he wrote a short paper, published in 1899, on ‘reticular’ geometry. From a lifelong
work covering a wide range of topics he is remembered for the captivating Pick’s theorem – and what a theorem it is!

Pick’s theorem gives a means for computing the area enclosed by a manysided (or polygonal) shape formed by joining up points whose coordinates are whole numbers. This is pinball mathematics.
To find the area of the shape we shall have to count the number of points • on the boundary and the number of interior points o. In our example, the number of points on the boundary is b = 22 and the number of interior points is c = 7. This is all we need to use Pick’s theorem:

From this formula, the area is 22/2 + 7 – 1 = 17. The area is 17 square units.
It is as simple as that. Pick’s theorem can be applied to any shape which joins discrete points with whole number coordinates, the only condition being that the boundary does not cross itself.

THE LEIBNIZ HARMONIC TRIANGLE

The German polymath Gottfried Leibniz discovered a remarkable set of numbers in the form of a triangle. The Leibniz numbers have a symmetry relation about the vertical line. But unlike Pascal’s triangle, the number in one row is obtained by adding the two numbers below it. For example 1/30 + 1/20 = 1/12. To construct this triangle we can progress from the top and move from left
to right by subtraction: we know 1/12 and 1/30 and so 1/12 − 1/30 = 1/20, the number next to 1/30. You might have spotted that the outside diagonal is the famous harmonic series

but the second diagonal is what is known as the Leibnizian series

which by some clever manipulation turns out to equal n/(n + 1). Just as we did before, we can write these Leibnizian numbers as B(n,r) to stand for the nth number in the rth row. They are related to the ordinary Pascal numbers C(n,r) by the formula:

In the words of the old song, ‘the knee bone’s connected to the thigh bone,and the thigh bone’s connected to the hip bone’. So it is with Pascal’s triangle and its intimate connections with so many parts of mathematics – modern geometry, combinatorics and algebra to name but three. More than this it is an exemplar of the mathematical trade – the constant search for pattern and harmony which reinforces our understanding of the subject itself.

PUZZLES TO PUZZLE YOU

PUZZLE 1 Handshake Problem

I invite ten couples to a party at my house. I ask everyone present, including my wife, how many people they shook hands with. It turns out that everyone questioned — I didn’t question myself, of course — shook hands with a different number of people. If we assume that no one shook hands with his or her partner, how many people did my wife shake hands with? (I did not ask myself any questions.) 

PUZZLE 2 Strawberry Ice Cream 

I visited a math professor of mine for dinner at his home (well, not really but shh! it’s part of the problem!) who had pictures of his three daughters on his mantle. He had had pictures taken of the three girls when each was a particularly adorable age — the same age for all three, as it happens. Unfortunately, this made it impossible for me to determine which was the oldest. So I had to ask him. Being a math professor, however, he declined to answer directly, telling me only that the product of their current ages was 72. “However,” he added, “since that isn’t enough information to determine their ages, I’ll also tell you that the sum of their ages happens also to be the number of our street address.” 
(Of course, I understood that each daughter’s age was to be considered an integer for this puzzle.) 
I darted outside to check the number on his mailbox. I was daunted to discover that I still didn’t have enough information to determine their ages, and I returned to tell him so. “That is an astute observation,” he said, smiling. “So you’ll be glad to know that my oldest daughter prefers strawberry ice cream.” 
Finally! I knew their ages. Do you ?

PUZZLE 3 Dissection Dilemma

The top two figures show how each of two shapes can be divided into four parts, all exactly alike. Your task is to divide the blank square into five parts, all identical in size and shape

PUZZLE 4 100 Light Switches

I give you a row of 100 light switches, all in the off position. Starting from the left, I ask you to flip every switch. Again starting from the left, I ask you to flip every other switch — so flip the 2nd, the 4th, etc. Again starting from the left, please flip every third switch. And so on: every fourth, then every fifth, etc, until on the last pass you flip only “every hundredth switch,” which means only the rightmost switch. When we are finished, which light switches are in the on position, and which are in the off position ?

PUZZLE 5 George’s Ropes

 George has six ropes. He chooses two of the twelve loose ends at random (possibly from the same rope), and ties them together, leaving ten loose ends. He again chooses two loose ends at random and joins them, and so on, until there are no loose ends. Find, with proof, the expected value of the number of loops George ends up with.

PUZZLE 6  Are You Sure There’s No Typo ? Find the missing number in this sequence:

MATHS AND MINDSETS

We currently live in a society where it is acceptable to be math phobic. We have embraced a culture of anxiety, stress and avoidance of mathematics. There is safety in this cohort, as many students, teachers, administrators, parents and the general population are members. 

How did this happen to a subject that the ancient Greeks viewed as beautiful and the key to unlocking the mysteries of the world ? 

When and why did we develop such a closed mindset to learning mathematics ? 

What are the implications for the students in our schools ? 

As educators, we have the most powerful impact on student attitudes and achievement, and yet there is open admission at times of fearing mathematics. Many researchers are investigating this phenomenon. The consensus is that a closed or fixed mindset is an important element of the problem. 
Carol Dweck released a ground breaking book in 2006, Mindset: The New Psychology of Success. According to Dweck’s research, a person with a fixed mindset believes that they are either smart or not. People who have a fixed mindset believe they fail at tasks or make mistakes because they just aren’t smart enough. They may develop traits such as avoidance of challenges, problem solving evasion and lack of perseverance.
 In traditional mathematics classes, we have placed value on the fastest correct answer. This is a dangerous practice as it reinforces the concept of intelligence and the ability to learn as being fixed. Dweck’s research found that a fixed mindset is evident across all achievement ranges, with the highest proportion found in high achieving girls (Boaler, 2013).
 Educators and parents inadvertently contribute to this problem. 

How often have we heard, “You are so smart! Look at how fast you answered the question!” 

When a student fails at a task or isn’t quick enough, a student (or adult) views this as a failure and surmise that they are not smart enough. 
We often reinforce this mindset in the classroom by ability grouping in mathematics problem solving settings. Students are highly perceptive to the fact they are judged and sorted on their abilities. Study after study indicates that this practice is deeply damaging and creates a fixed mindset in students (Boaler, 2013).

In fact, the research shows, “mathematics classrooms influence to a high and regrettable degree, the confidence students have in their own intelligence. This is unfortunate both because math classrooms often treat children harshly, but also because we know there are many forms of intelligence and ways to be smart and math classrooms tend to value only one.” (Boaler, 2008)

The opposite of a fixed mindset is a growth mindset. People who have a growth mindset demonstrate the following characteristics: embrace challenges, persist when faced with setbacks, see effort as the key to mastery, embrace constructive criticism and find inspiration in the success of others (Dweck, 2006). 

Failure is seen as an opportunity for growth, to learn and improve. A growth mindset can be taught to students and encouraged by the types of tasks students are provided. 

If students are given math questions that are closed and require right and wrong answers, the message that is communicated to students is that only a correct answer is valued. Offering more open tasks allows students to see the possibilities of high achievement and the opportunity to improve (Boaler, 2013).

Why is mathematics such a polarizing subject ? 

Why is it acceptable to dislike math and be open about it? 

It is very rare that someone would admit to disliking reading or brag about not being good at it. This bravado is a deflection or a shield. If one states they are terrible at math, then they can’t be judged if asked to do it on demand. 

When splitting a bill in a restaurant, when figuring out how much tile to buy at the home improvement store and most unfortunately in schools. 
Many educators today remember math instruction as a time to learn algorithms and apply those procedures. Evaluation and assessment was based almost entirely on how well a student could recall and apply those procedures. 
Remember racing to the blackboard and waiting with chalk in your sweaty palm as the whole class watched and waited until you found out which times table you would be asked to record as fast as you could with the class watching? 
As Carol Dweck would say, we believed math was about how good or correct we were, not on what we learned and how much growth was exhibited. This message is still being heard today. 
When teachers participate in professional development, the idea is that teachers are always learning and repositioning their understanding of a topic based on what information their students provide them. This is a position of a growth mindset. 
The extensive learning by teachers  around problem-based learning in mathematics places the teacher as a co-learner along with the students. 
Often, even with in-depth preparation for a three-part lesson in mathematics teachers will be fascinated by a strategy a student used that they didn’t expect. 
When a teacher feels they hold all the knowledge and it is their job to impart it to students it takes them away from a growth mindset. They are less open to new learning and to seeing mistakes and constructive criticism as a key tool in teaching.