How does it happen that children inherit certain characteristics from their parents? What determines exactly what characteristics each child will inherit? Why is
it, in some cases, that two brown-eyed parents have all brown-eyed children, and in other cases two brown-eyed parents have some brown-eyed children and
some blue-eyed children?
    These can be very complicated questions, and they require the entire science of genetics for a complete answer. However, some of the simpler principals of
genetics can be grasped through the application of statistical techniques and mathematical models to life situations. For example, Gregor Mendel first discovered
and understood the laws of heredity by gathering and organizing with painstaking care a mass of data on pea plants. Mendel followed up the statement of his
laws by comparing his data to a probability model. After studying the relationship between his data and the model, he was able to develop a plausible explanation
of how characteristics are passed from parents to offspring. Although Mendel's experiment took place in the middle of the nineteenth century, the results of those
experiments form the basis of modern-day science of genetics.
    Most curriculum experts and learning theorists seem to agree that the application of mathematics to real life situations helps the students master skills and understand concepts. Mendel's experiments concerning inherited characteristics in plants can be drawn on to provide applications in junior high school mathematics and science classes in two categories:

                                                           1.    Basic statistical techniques of gathering and organizing data
                                                           2.    The use of probability models to explain observed patterns.
 
 

    Gregor Mendel, an Austrian monk and teacher, was interested in characteristics inherited by offspring from their parents. In particular, he was intrigued by his observation that when two unlike plants were crossbred, a characteristic of one of the parents would sometimes disappear entirely in the first generation of offspring, only to reappear in some of the plants of the second generation. For example, he had crossed a tall pea plant with a dwarf pea plant, and the seeds from this crossing had produced plants that were all tall. The characteristic of dwarfness had disappeared completely in the first-generation offspring, he found that some of the second-generation offspring were tall and some were dwarf.
    Mendel began to wonder whether one of the opposing characteristics from two different types of parents would always disappear in the first generation of offspring after a hybrid crossing and whether the missing characteristic would always reappear in the second generation. If so, would the missing characteristic always reappear in approximately the same proportion in the second generation? He designed an experiment that would allow him to gather data that might resolve these mysteries. He was well equipped to study these problems of heredity because he was trained in both biology and mathematics. Mendel's work was the first in history to link heredity and mathematics [5].
    Mendel used garden peas in his initial experiments because there were several varieties available with easily distinguishable characteristics. He identified seven pairs of traits that could be clearly described, observed, and tabulated. For example, he chose one type of pea plant that was short and bushy and paired that type with another that was tall and climbing. One kind had yellow seeds; another had green seeds. One variety had smooth seeds and another had wrinkled seeds.
    Mendel first obtained seeds from plants exhibiting the seven pairs of characteristics he was interested in studying. Before he began the experiments that would produce the data he needed, he had to make sure his seeds would breed true. That is, seeds from tall plants must produce only tall plants, seeds from dwarf plants must produce only dwarf plants, and so on for all of the seven pairs of traits he planned to study. To make sure of this, Mendel planted his seeds one year, took seeds from those plants that bred true, and planted these seeds the second year. When his plants bred true the second year, he was confident that he now had pure seeds, and so he began his experiments.
    In the first year of his experiment, Mendel planted his fourteen varieties in different plots. The separate plots allowed him to study each pair of characteristics one pair at a time. All other characteristics of the plants in a particular plot could be disregarded, making possible a clear and simple tabulation of results [127, p. 122].
    The plants in the first year of the experiment were called the parent generation. Before the plants in the parent generation could self-pollinate, they were cross-pollinated artificially to produce hybrid pea seeds. For example, the pollen of a flower from a dwarf plant was applied to the stigma of a flower from a tall plant [127. p. 127]. Hybrid seeds from these crossings were collected and carefully labeled in relation to the type of crossing from which they resulted.
    Now Mendel's curiosity was about to be satisfied. He sowed the hybrid seeds from the parents generation in the second year of his experiment so that he could see what the first generation of offspring would look like. He would soon be able to state, under controlled conditions, what plants resulting from hybrid seeds could be expected to look like. For example, if a tall plant is crossed with a dwarf plant, would the first generation offspring be expected to be tall like one parent, or short like the other parent, or perhaps of medium height somewhere between the two parents? Because of his previous observations, Mendel conjectured that all plants resulting from the crossing of pure tall with pure dwarf would be tall. His conjecture was proved correct by his experimental results. In every one of the seven pairs of traits he was studying, one trait disappeared entirely in the first generation of plants after the crossing. It appeared that one member of each pair of contrasting traits had overpowered the other. This led Mendel to conclude that for each of the seven pairs of traits, one characteristic could be said to be dominant over the other [127, p. 133].
    Mendel's plan next was to determine what would happen if the first-generation plants were allowed to self-pollinate. He wondered if the resulting seeds would produce second-generation plants with exactly the same characteristics as the first-generation plants. For example, would seeds from first-generation tall plants always produce second-generation plants that were also tall? It might seem logical to assume that seeds from a tall plant would always produce tall plants, but Mendel's experimental results proved otherwise. He kept careful records on which seeds came from which first-generation plants, and then sowed these seeds  in the third year of his experiment. This is the step by which Mendel made history. He found that some of the second-generation plants were tall and some were short. None was in between. Furthermore, he found that the ratio of tall plants to short plants was approximately three to one in the second-generation. He found similar results for all seven of the pairs of traits he was studying.  Table 24.1 gives, as an illustration, his results for three of the seven pairs of traits.
 
 

    Table 24.1 shows how Mendel used large numbers of plants and organized his information carefully in order to get an accurate description of the results. His mathematical training enabled him to analyze the results and to develop a theory to explain what happened. Mendel did not know of the existence of genes, of course, but he responded that there must be some factors present in the mixed plants of the first generation that transmitted the missing trait to some of the plants of the second generation. He conjectured that one of these factors (genes, as they are called today) must come from the male parent and the other from the female parents and that these factors must combine in some way at the moment of fertilization to determine the characteristics of the offspring.
    This reasoning by Mendel illustrates the potential value of the basic statistical techniques of carefully gathering and organizing data. Using these methods, he was able to perceive patterns accurately and to state these patterns as laws of heredity with a fair amount of confidence. He next turned to some mathematical modeling to develop a possible explanation of the observed patterns.
 
 

    After tabulating the results of his experiments, Mendel noted that the dominant characteristic appeared in the second-generation plants about three times as often as the other characteristic. This ratio was approximately the same for each of the seven pairs of characteristics he studied. From his conjecture that each trait is determined by a pair of factors, Mendel concluded that the two factors within a particular plant must have been obtained from the parents by a chance paring at the moment or fertilization. By comparing his results to a probability model, Mendel was able to show how his observed three-to-one ratio could be expected to occur. It is remarkable that Mendel was able to use a probability model to develop a possible explanation of inherited characteristics even though very little was known in his day about what occurs physically within the sex cells when an egg is fertilized by a sperm cell.
    Mendel's thinking can be studied by using modern symbolism for genes. A pair of genes that determine height can be represented by TT in a pure tall plant and ss in a pure short plant, in which each letter stands for a single gene. Suppose a pure tall plant (TT) is crossed with a pure short plant (ss). Since each parent contributes only one gene for height to the offspring and since both genes in the pure tall parent are alike, the gene from the pure tall parent must be a T. This situation is somewhat like flipping a two-headed coin. Whichever way the coin falls, the result must be a head. Similarly, the pure short parent can contribute only an s gene to the offspring.
    The result of crossing a pure tall plant with a pure short plant is similar to tossing a two-headed coin and a two-tailed coin at the same time. The possible results are summarized in table 24.2. The model shows that no matter how the coins land,, there is always a head on one and a tail on the other.
    The possible results of crossing a pure tall plant with a pure short plant are represented in table 24.3. The structure of table 24.3 is exactly the same as that of table 24.2. The model shows that each pair of genes in the first-generation offspring is made up of a T gene from a tall parent and an s gene from the short parent. Plants with this type of genetic makeup are called hybrid. Mendel's observation was that all his hybrid plants of the first generation were tall. He concluded that this observation could be explained by assuming that tallness is dominant over shortness.  For clarity in interpreting the symbols, capital letters are used for the dominant gene and lowercase letters for the others.  The model in table 24.3 shows that all the hybrid plants of the first generation, resulting from crossing two pure plants, would be tall.  This corresponds with Mendel's actual results.
 
 

TABLE 24.2 POSSIBLE RESULTS OF TOSSING A TWO-HEADED COIN  AND A TWO-TAILED COIN AT THE SAME TIME Second Coin t t First h ht ht Coin h ht ht

TABLE 24.3 POSSIBLE RESULTS OF CROSSING A PURE TALL  PLANT WITH A PURE SHORT PLANT Short Plant          (ss) s s Tall T Ts Ts Plant   (TT) T Ts Ts

Now it is appropriate to examine the possible results of crossing two hybrid tall plants.  Table 24.3 shows the genetic stricture of each offspring of a pure tall plant and a pure short plant to be Ts.  Table 24.4 shows the possible results of crossing a Ts with another Ts.  The model provided by table 24.4 shows that three of every four plants in the second generation can be expected to be tall.  Also, the shortness characteristic, which disappeared entirely in the first generation, can be expected to reappear in the second generation at the rate of one in every four plants.  These expected results correspond closely to the results actually found by Mendel in his experiments.
 
 

    The male can contribute either a T gene or an s gene to the offspring, and the female can also contribute either a T gene or an s gene. Chance determines which combinations are made. Table 24.4 shows the four possible combinations, and all combinations are equally likely. However, three of the possibilities contain T genes, and since T is dominant over s, the probability of getting a tall plant in the second generation is 3/4, and the probability of getting a short plant is 1/4.
    Table 24.4 is similar in structure to a table showing possible results of tossing two coins. The only difference between the gene model and the coin model is that the idea of dominance is missing in the latter.
 
 

    Many human characteristics, such as eye color, hair color, sex, ear lobe type, and tongue type, are inherited [95, pp. 44-45]. Some of these characteristics can be illustrated and studied with relatively simple probability models. One such characteristic is tongue rolling. Some people can roll their tongues into a U-shape, and others cannot. Rolling is dominant over nonrolling. In table 24.5, R stands for a rolling gene and n stands for a nonrolling gene. Each offspring from this union has a probability of 1/2 of being a roller and each offspring has a probability of 1/2 of being a nonroller.
 
 

    Just as it cannot be told in advance whether a flipped coin will land heads or tails, neither can it be determined with certainty whether the next child a couple has will be a boy or a girl. However, it can be safely assumed that of all babies born during a specific period of time, about half will be boys and about half girls. Actually, a few more boys are born than girls, but the figures are close enough so that a one-to-one ratio can be assumed for most practical purposes.
    Units called genes are responsible for transmitting characteristics from parents to offspring, and genes are located on bodies called chromosomes. Chromosomes are located in cells that form the structure of living things.
    Every cell in the human body, except for the sex cells, contains twenty-three pairs of chromosomes. The sex cells have twenty-three single chromosomes instead of twenty-three pairs of chromosomes. Then, when fertilizations occurs, a male cell joins a female cell, and the twenty-three single chromosomes in the male cell join the twenty-three single chromosomes in the female cell to form twenty-three pairs of chromosomes. As a result, the cells of the offspring then have twenty-three pairs  of chromosomes each, just as the cells of the parent did.
    One of the twenty-three pairs of chromosomes in each cell is a pair of sex chromosomes. the female has a pair of identical, rod-shaped sex chromosomes called X chromosomes. The corresponding pair of chromosomes in the cell of the male are different. One member of the pair is an X chromosome like that in the female, but its mate, instead of being rod-shaped, is bent like a hook. Its shape is similar to a Y, and so it is called a Y chromosome.
    Since the sex cell of the female (the egg) contains only single chromosomes instead of pairs of chromosomes, the sex chromosome found in each egg would always be a single X chromosome. The male sex cell (or sperm), however, could contain either a single X chromosome or a single Y chromosome. About half the sperm cells contain an X chromosome, and the other half contain a Y chromosome. The sex of an individual is normally determined at the time of fertilization [82, p. 216].
    If the sex chromosomes of the female are denoted by XX and the sex chromosomes of the male by XY, a model showing the possibilities for the sex of offspring can be shown in table 24.5. Sex, like the tounge-rolling trait, is determined by chance. If a sperm with an X chromosome happens to be the one that fertilizes the egg, the sex of the offspring will be female. If a sperm with a Y chromosome happens to be the one that fertilizes the egg, the sex of the offspring will be male. The model in table 24.6 shows that half the children born can be expected to be boys and half can be expected to be girls.
 
 

    Some people cannot distinguish certain colors or combinations of colors. Some types of color blindness are red, green, red-green, blue-yellow, and complete color blindness. Only about 1 percent of woman are color-blind, but somewhere between 5 and 8 percent of men are color-blind [55,p.486]. It has been observed that a color-blind father may have a daughter with normal vision, and then this daughter may have a son whom the trait of color blindness reappears after having been missing for a generation. Inherited characteristics that behave in this way are called sex-linked characteristics.
    The occurrence of sex-linked characteristics can be explained by assuming that X chromosomes contain genes that transmit certain characteristics and that Y chromosomes do not contain genes that transmit these characteristics. These genes will express themselves in male offspring even though they are not dominant because Y chromosomes, which males get and females do not get, contain no genes to prevent the expression of these sex-linked traits [81, p. 187]. A female would be color-blind only if the gene for color blindness is carried by both of her X chromosomes.
    Models can be constructed to depict probabilities of color blindness occurring in offspring. In these models, X denotes a chromosome with a normal gene for color vision, and X' denotes a chromosome with a gene for color blindness. Normal color vision is dominant over color blindness. The following combinations are possible in regard to color blindness:

        XX: Normal female
        XX': Carrier female
        X'X': Color-blind female
        XY: Normal male
        X'Y: Color-blind male

    Since there are three combinations relating to color blindness that could exist in females and two combinations that could exist in males, there are 3 X 2 = 6 possible mating combinations. Tables 24.7 and 24.8 show two of these six possibilities. It is possible to conclude from table 24.7 that none of the offspring from this union would be color-blind. All the female offspring, however, would be carriers of the gene for color blindness.
    Table 24.8 shows that half the males born of a normal male and a female carrier can be expected to be color-blind, and half the females can be expected to be carriers of the gene for color blindness. Reflection and the construction of other tables would show that female offspring could be color-blind only if the father is color-blind and the mother is either color-blind or a carrier of the gene for color-blindness.
 
 
 

TABLE 24.7 POSSIBLE COMBINATIONS OF OFFSPRING RESULTING  FROM A COLOR-BLIND MALE AND A NORMAL FEMALE Normal Female    (XX) X X Color-blind X' XX' XX' Male   (X'Y) Y XY XY

TABLE 24.8 POSSIBLE COMBINATIONS OF OFFSPRING RESULTING  FROM A NORMAL MALE AND A CARRIER FEMALE Carrier Female    (XX') X X' Normal X XX XX' Male   (XY) Y XY X'Y