An Illustration of the Rediscovery Approach to Science Teaching



     Instructional time is inevitably short.  So, in this less than perfect world, science teachers often feel that their primary task is to enlarge their students' scientific store of knowledge as rapidly as we can.  Many of them are familiar with the scientific method, with the history of science, with the fluidity, open‑endedness, excitement, and uncertainty that surround any frontier of the natural sciences, and might be occasionally troubled by this pedagogical approach.  But given its popularity, its obvious advantages, and its fairly successful record, they feel constrained to embrace it.


     Many students spend much time in the laboratory, acquiring hands‑on experience in natural science.  They thereby, we go on reassuring ourselves, climb along that very path of self‑discovery which the typical textbook so studiously bypasses.  Sadly, however, lab exercises in both high schools and colleges resemble, at their worse, cooking classes.  At their near‑best, they foster appreciation for the procedures and methods other scientists employed in making their discoveries.  In either case, they deny learners the rewards of travelling the same path along which the discovery was originally made.  We do so despite the insistence of many philosophers of science, psychologists, and educators, that the process through which discoveries are made is just as important as the discoveries themselves.


     There is much to be said for the present system.  There is also much to be said against an educational approach that relies solely on self‑discovery.  For one thing, practitioners of the latter will know a great deal about a few things but, in this information age, they would not possess a broad enough background to understand the world around them, let alone teach or practice natural science.  For another, some topics simply don't lend themselves well to this approach‑‑How can you self‑discover, for instance, the cultural anthropology of nineteenth century Eskimos?


     In my science classes, I try to strike a compromise between the two.  For the most part I rely on the typical textbook approach.  This enables students to derive the considerable benefits which this standard approach affords.  But, depending on the topic, I often combine this approach with one or two self‑discovery exercises.  These occasional detours allow students to capture a few aspects of science which are largely overlooked by the textbook approach.  Students are delighted to find out that some scientific problems resemble the puzzles they sometimes encounter in their day‑to‑day lives.  They gain a greater appreciation for the scientific method and for the convoluted nature of scientific progress.  They come to see that science is not a collection of dry facts, but an intellectually exciting enterprise.  A few students may acquire greater confidence in their own intellectual powers.  Such learning experience may lead an occasional student to join the motley crew that makes up the world's community of natural scientists.  Most students rarely articulate these and other advantages in formal class evaluations.  They do, however, often say that they enjoyed the one or two self‑discovery exercises they carried out and that they hoped to see more of them in future science classes.


 Obviously, the type of self‑discovery approach, and the relative weight it receives in any given class, depends, among other things, on the topic, on the instructor's background and educational philosophy, and on students' backgrounds and aptitudes.  To illustrate this second approach, I should like to reprint here one of the handouts I give liberal arts students in my introductory biology classes.  I chose Mendel's principle of segregation because it makes for a good story, because it is familiar to most readers of this journal, and because of my own professional background.  Needless to say, the same basic approach can be adopted in many other areas of the natural and social sciences. 


     This exercise was written for students totally unfamiliar with probability and genetics, hence its length.  Hopefully, besides its illustrative value, it might be useful to high school and college science instructors whose students' background is similar to mine and who wish to integrate a self‑discovery segment in their introductory biology or general science classes.  When greater familiarity and facility with the hard sciences and mathematics can be assumed, correspondingly shorter and more challenging versions of this exercise will be required.



                            LAW OF SEGREGATION




     Science begins with simple questions.  "Why is the sky blue?"  Why do blue‑eyed parents have only blue‑eyed children?"  "Why do pea plants only give rise to peas, not to cows or roses?"  Some scientific discoveries are made, routinely, by all of us.  For instance, sometime during your early childhood you looked at a mirror and figured out, all by yourself, that you were actually facing your own reflection.  Similarly, discoveries, even great discoveries, made by professional scientists do not seem to require much more than a healthy dose of curiosity, open‑mindedness, and imagination.  Columbus opened a new world for the old.  With his improved telescope, Galileo could see Jupiter's moons.  Though we admire the technical ingenuity, courage to challenge dogmas, or curiosity about the world which such discoveries require, we understand how the likes of Galileo and Columbus could come around to making them.  With a bit more time, brains, and imagination, we too, we feel, could have discovered America or Jupiter's moons.  It took genius to discover them, but only "ordinary" genius.  There is, however, another kind of genius which has an almost magical quality to it.  We might, or might not, understand such work, but we are astounded by the psychological process that gave rise to it.  How in heaven's name, we wonder, did it occur to a "magical" genius to carry out his/her work?  And how did this person manage to interpret it in such a radical new way?     


     One such magical genius was Gregor Mendel, perhaps one of the greatest scientists of the nineteenth century.  His experiments not only revolutionized biology, but left historians of science wondering:  What mental processes led to their conception?  How did Mendel have the intellectual courage to traverse new territories for so many years, totally alone?  What made it possible for him to come up with his beautiful deductions and with a radically new theory of heredity?  How did he come to see, for the first time in the history of biology, that simple numerical ratios provided an important key to unlocking one of nature secrets.


     Mendel lived in central Europe, in a region which now belongs to Czechoslovakia but that, during his lifetime, belonged to Austria.  He spent most of his adult life as a monk in an Augustinian monastery.  As a farmer's son in class‑conscious Europe, the monastic life, in his words, spared him the "perpetual anxiety about a means of livelihood."  It also gave him free time to tinker with peas. 


     He carried out his experiments with the common edible pea in his small garden plot in the monastery.  They were begun in 1856 and completed some eight years later.  In 1865, he described his experiments in two lectures he gave at a regional scientific conference.  In the first lecture he described his observations and experimental results.  In the second, which was given one month later, he boldly explained them. 


     The forty or so scientists which made up his audience listened to him politely enough, but no one asked a single question.  Most likely, they didn't understand what he was talking about.  Later, he also carried on a correspondence with one of the greatest biologists of the time, who also failed to appreciate Mendel's work.  In 1886, his results were published in a scientific journal.  He might as well dropped a leaded spoon into the ocean; the paper was ignored by the scientific community.  At times, he must entertained doubt about his work, but not always:  "My time will come," he reportedly told a friend.

      His time came, though later than he might have expected.  His emergence from obscurity began in 1900‑‑sixteen years after his death.  In the West his work was generally accepted by the scientific community by the 1920s.  In the Communist block, it was not fully established until the 1960s.


     Conceptually, Mendel's experiments were not all that difficult.  His scientific contemporaries lacked, for one thing, open‑mindedness.  Moreover, they might have been unable to accept, deep down, the proposition that an obscure monk, a farmer's son, and a person who was discouraged from completing his university education because he "lacked insight," could have something important to say.  At any rate, in your case all this is not a problem.  You probably accept history's judgment that he was a great and insightful person and that geniuses can spring from impoverished, plebian backgrounds.  What you do have in common with Mendel's scientific contemporaries is, most likely, ignorance of elementary statistics.  The next section will give you the necessary background information.


Probability:  Random Events


     1.  Let us first examine the results of single random events.  Suppose you toss a coin once.  What are the chances of getting heads (H)?  ____  Tails (T)?  ____


     2.  What are the chances that your next grandchild will be a boy? ____  A girl? ____


     3.  Now toss a coin 30 times and record the number of heads (Hs) and tails (Ts) in the space below.  (Make sure that the process is indeed random.  Flip the coin a few times in your hand before each toss; examine it only after it lands on the floor).

                         H ___       T ___



     Record class results below:


     4.  Most likely, some of your classmates' observations are not in line with the expected 50/50 distribution?   Why? 


     5.  Are class totals more or less in line with the expected distribution?  Why?


     Mark Twain‑‑a contemporary of Gregor Mendel‑‑observed that there are "lies, damn lies, and statistics."  In a sense, he was right.  If you tossed a coin just once, and got H, then you didn't get the 1 in 2 distribution statistical theory seems to demand, but a 1 in 1 distribution.  Similarly, although each participant flipped a coin 30 times, the data of some diverged considerably from the expected 1 in 2 distribution.  Assuming a minimum of 10 students in your class, however, data for the entire class almost certainly approached the theoretical expectation of equal odds.  These commonsense observations suggest two generalizations:


     ‑‑Actual distributions of random events rarely match theoretical expectations.  A coin that has been randomly tossed 400 times will almost never yield 200 Ts and 200 Hs.  As a rule, something like 195 Ts and 205 Hs is all we can expect.

      ‑‑The larger the number of trials, the closer the fit between observed and theoretical values.  Thus, if you toss a coin just once, theory demands a 1 in 2 chance of H or T, in real life you would always observe just one event.  In 30 trials, you very well might get a 20/10 distribution.  But in 3000 trials, you will almost never get a 2000/1000 distribution.


     Until artificial offspring sex selection becomes a reality, the distribution of boys and girls obeys similar rules.  The coin flipper in this case is nature.  To put it in a somewhat simplified form, as far as a child's sex is concerned, a woman's anatomy plays a fairly passive role.  Semen contains two populations of sperm in roughly equal numbers.  One type, when united with a woman's egg, gives rise to a girl (G).  The other, when united with the same egg, gives rise to a boy (B).  For any given conception, chance determines which type of sperm will fertilize the egg.


     6.  What are a couple's chances of having a girl?     


     7.  Assuming that either type of sperm is as likely as the other to fertilize an egg (reality is a bit more complicated), what is the expected distribution of boys and girls in a small village of 100 souls?  ___         In a city of two million?  ___


Probability:  Two Independent Events

     Imagine that you simultaneously tossed a nickel and a dime.  Each coin alone, we know already, has an equal chance of yielding H or T, but what can we expect when the 2 are flipped at the same time?  Obviously, the nickel can be H or T, and so can the dime.  Our problem is putting the two together to determine the ypes of events and their respective probabilities.  

     For any given throw, the nickel has a 1 in 2 chance of yielding H.  Assuming that the nickel yielded an H, the dime is equally likely to yield H or T.  So the chances of [nickel:  H; dime: H] is one‑half of that one‑half, or one‑quarter.  Similarly, there is a one‑quarter chance of [nickel: H, dime T]. 


     The nickel also has a one‑half chance of yielding T, leading again to 2 equally probable events; [Nickel: T, dime H]; and [Nickel: T;, dime: T].  Looking then at the simultaneous toss of two coins, we have four equally probable events:














     Similar probabilities prevail in a large population of two‑child families: 








     In other words, in a population of 100,000 such families, one expects some 25,000 with 2 boys, 25,000 with an older girl and a younger boy, 25,000 with an older boy and a younger girl, and 25,000 with two girls.


     These examples suggest the following generalization:


     ‑‑The probability that two independent events will occur together equals the product of their individual probabilities.  Thus, when you randomly toss two coins, the two events are independent:  the outcome of one has no bearing on the outcome of the other.  Since the probability of each is 1/2, their combined probability is 1/2 X 1/2 or 1/4.


     You are ready now to work out a few more puzzles.


     8.  In this population of 100, 000, what are the chances of having a boy and a girl, irrespective of their order of birth? ___


     9.  What are the chances for two girls? ___


     10.  Let's divide these 100,000 two‑child families into 2 categories.  The first consists of families with 2 girls.  The second consists of all other two‑children families.  How many families would belong to the first group (the one with 2 girls)? _____   How many to the second (all the other combinations)?   If you divide the second number by the first, the expected ratio between them would  be ___ to 1.


     11.  Imagine that a philantropist came up with a $1000 prize to any family in your community that meets the following conditions:  it has 2, and only 2, natural born daughters.  Imagine that all qualified families applied for the bonus, and that after being checked for accuracy, 100,000 received the prize.  The philantropist now comes up with an even more curious announcement:  He will give a $1,000,000 prize to the first person who would tell him, and then properly explain his deductions, the approximate number of two‑child families in this community who do not have two girls.  How would you go about getting this prize?


     Imagine yourself a banker.  To save money, you devised an ingenius scheme of monitoring the vigilance of your bank's night watchman.  At the opposite ends of the building, there are 2 light switches.  Every hour during his 10‑hour shift, the guard must approach one switch and flip a coin.  If it's Heads, he is to turn the switch on, if it's Tails, he is to turn it off.  He then must approach the second switch and repeat the same procedure.  If both switches are on, the light is on.  If either one is off, or if both are off, the light is off.  You have installed a special machine that tells you, for every hour, whether the light has been on or off.  Four months and 1000 hours later, you examine the lighting record to determine whether the guard should receive a raise or be fired.  You observe that during that time, the light has been on 570 hours and off 430 hours. 


     12.  What is your theoretical expecations for the number of hours the lights should have been on and off?  Why?  Should the guard be promoted or sacked? 




     Obtain 2 vials from your instructor, and place in each one red and one white toothpick.*  [Available as the "Genetic Concept Biokit' from Carolina Biological Supply Company]  As you can see, there is at the bottom of each a hole through which, at any given time, the head of one, and only one, toothpick can pass when the vial is shaken.  So shaking a vial leads to the segregation of the two toothpicks‑‑they separate from each other, one remains wholly inside, the other is protruding out of the vial.


     You need to ascertain now that the two events‑‑Red and White‑‑are equally probable by shaking one vial 30 times. 


     13.  If a red toothpick is just as likely to penetrate the hole as the white one, what is your theoretical expectation for these 30 trials?


     Please record actual observations below:


     Your data:          Red ___      White ___


     Class Data:         Red ___      White ___


     Now let's examine the situation when the two vials are simultaneously shaken:

      15.  What are your theoretical expectations for the total number of trials carried out by the class?















     16.  What were your actual observations?



Total number of trials: ___                                                                                                                                    










17.   Record below the observed total for the entire class:                 














     Let's recapitulate this last experiment.  You held 2 vials.  By shaking both at the same time, either a red or a white toothpick popped through the hole:  the two segregated from each other.  We expect in this case 4 equally probable 2‑toothpicks permutations.  From this we can calculate any given ratio.  

     18.  For instance, out of 100 trials, how many are expected to be a White/White combination? ___  


    How many are expected to be either Red/White or Red/Red? ___


    What is the ratio between them [Red/White + Red/Red] : [White/White]? ___ to 1.




Mendel's Observations


     One wonders what might have happened had Mendel given three lectures on the subject, instead of two, with the first paper containing the elementary lesson in statistics you have just struggled through.  But let's leave these idle speculations to historians, and see if we can make heads or tails of Mendel's observations.


     In his day, gardeners could obtain all kinds of true‑breeding pea varieties from commercial seed houses.  For example, one variety was guaranteed to give only tall plants (6 ft or so); another, only short plants (about 1 ft in height).  If you crossed one tall plant to itself or to another tall plant, collected the resultant seeds some three months later, planted them, and observed the height of this second‑generation of plants, all would be tall.  Similarly, only short plants would result from a cross between true‑breeding short peas.  This would continue generation after genaration.  One stock would only give rise to tall plants; the other only to short plants. (Incidentally, in humans, blue‑eyedness is more or less true‑breeding‑‑without knowing any of the blue‑eyed couples you are familiar with, a geneticist can be fairly certain that their children are blue‑eyed too).


     Mendel then crossed the two varieties.


     19.  Before we continue, can you guess the height of the offspring?   The cross Short X Short, we have seen, yields short plants.  Tall X Tall yields only Tall.  But how tall would the descendants of a Tall X Short cross be?


     The surprising answer is that, in this case, all the first generation plants were tall.


     Mendel then crossed these new tall plants (the offspring of the Tall X Short cross) to themselves.  Can you guess the outcome?


     The actual results from this cross were:  787 plants among the second generation ("grandchildren" of the original true‑breeding tall/true‑breeding short "couple") were tall, and 277 were short. Note that the short characteristic which disappeared from sight in the first generation, reappeared in the second.  This means that although the factor which caused short stature was temporarily out of sight, it was still there.  Note also that the ratio between tall and short plant was 787/277, or 2.84 to 1 (approximately 3 to 1).


     Mendel obtained similar results for many other characteristics, suggesting that a general rule is at work here.  For instance, in Mendel's time order seeds with either smooth or wrinkled surfaces.  Again, Mendel established that seither type was true‑breeding.  When plants arising from smooth seeds where crossed to plants arising from wrinkled seeds., all their offspring produced smooth seeds.  In the next generation, 5,474 of the plants produced produced smooth seeds and 1,850 wrinkled seeds, yielding a 2.96 to 1 ratio (an almost exact 3 to 1 ratio).


     These are some of the observations Mendel recounted to his audience on that cold February evening of 1865, in the little Moravian town of Brno.  A month later he reappeared before them and put forward an elegant explanation for his results. 


     20.  Imagine you found yourself in Mendel's shoes.  What would you tell your audience? 


     Mendel deductions are by no means confined to peas.  Owing to the close affinity between them, living creatures often obey the same rules.  When crossed to flies of the same strain, red‑eyed individuals produce only red‑eyed offspring while brown‑eyed individuals produce only brown‑eyed offspring.  In one experiment, all the progeny of a cross between a red‑eyed male and a brown‑eyed female had red eyes.  When these sisters and brothers were mated to each other, they produced 308 red‑eyed and 93 brown‑eyed offspring.  Please use the insights you gained from this exercise to explain thes observations.







     Carlson, E. A.  The Gene:  A Critical History (1966).


     Fristrom, James W. and Spieth, Philip T.  Principles of Genetics (1980).


     Iltis, Hugo.  Gregor Mendel and his Work (1943).  Reprinted in:  Shapley, H. et al. (eds)  A Treasury of Science (1958)


     Kolata, Gina.  Remembering a "magical genius."  Science, June 19, 1987, pp. 1519‑1521.


     Mendel, Gregor.  Experiments in plant‑hybridization.  In:  Peters, James A. (ed)  Classic Papers in Genetics (1959).


     Strickberger, Monroe,  W.  Genetics (2nd edition; 1976).