Pseudomath and Pseudoresearch:
Some Consequences of Mathematical Ignorance
A Talk Presented by UCSMP Director Zalman Usiskin at the Twelfth Annual UCSMP Secondary Conference, November 9-10, 1996.
The major change that has occurred in U.S. mathematics education in recent years is the realization that mathematics beyond arithmetic is important for everyone. With that change has come the expectation that everyone should learn some algebra, geometry, probability, and statistics, as well as other mathematics traditionally reserved only for more advanced students.
This fights a strong tradition- a tradition that mathematics is a subject for very good students and that other students should just learn whatever they can. In the past, when poorer-performing students would take algebra, it was usually an algebra based on skills, an algebra without word problems and without theory. But algebra without problems and theory is algebra with its life taken from it. It is form without substance. When poorer-performing students would take geometry, it was usually an "informal geometry," that is, a geometry with the proof taken out. But the sequence in most geometry courses is based on proof- that is why a theorem like the base angles of an isosceles triangle are congruent precedes the Pythagorean Theorem-and to take out proof from such a course is to take out its skeleton, leaving a blob of ideas. It is no surprise that students in such courses-who already were taking more mathematics than they thought they needed- wondered why they were taking them.
I don't think it is at all an overstatement to say that we in UCSMP helped initiate the movement that has convinced most people that algebra and geometry courses with applications and theory interwoven are better courses for all students, including the poorer-performing, for they give meaning and sense to the mathematics in a way that courses de- voted to skills cannot. We have been adamant from the start in believing that virtually all students can learn a significant amount of mathematics, and that poorer students should not be given different mathematics from better students.
We do differ from some others who share our belief in that we do not believe all students can do this at the same age. Some students are, for whatever reason, ready before others.
And because work is involved and because prior knowledge is essential in mathematics, not all students learn at the same pace, so that many schools have extended the length of UCSMP courses for their lowest-performing students.
This talk was originally motivated by my concern that some students, but particularly the lower-achieving students, are likely to be misled by the mathematics they find around them, by unscrupulous mathematics, by superstition, by charlatans. But the more I got into the topic, the more it seemed to me that what I was finding applied to all students and even to educated adults.
Beliefs Without Evidence
In mathematics, deduction from statements assumed true is our sole criterion for truth. Science allows, in addition to deduction, generalizations from consistent results from replicable experiments, what we know as the scientific method. To understand numerical results obtained by the scientific method, one needs statistics. In past generations, most students did not study proof in mathematics, and they took only one or two years of science in high school. Although today's students may be better-informed and better-prepared than any students in the past, the lack of knowledge by adults of mathematics beyond arithmetic, and of science beyond vocabulary and assorted facts, means that what constitutes evidence is not something that many students have ever dis- cussed either in school or at home.
In the absence of a knowledge of criteria to establish truth, phenomena are believed to occur for which there is no evidence. A Gallup poll of 1,236 adults in 1990 found that many people believe in things like ghosts and clairvoyance, even in the healing power of pyramids (see p. 8). In fact, more people believed in such paranormal phenomena in 1990 than in 1978, when a previous poll was taken.
Thus you are quite likely to have some students who believe in this sort of magic, and when you speak of the magic of mathematics, these students take it literally. You are likely to have some students who believe in hidden causes for random events, and when you speak of probability, they won't have a clue of what you are talking about-to them, events either occur or they don't. Someone might have just willed it to happen.
The Pythagoreans of ancient Greece viewed mathematics as mystical, and many people still speak of the mystery and magic of mathematics. There has even been a popular book: Mathematics: Magic and Mystery. But to me this approach has its drawbacks. If people believe that mathematics is magical, then it would follow that they do not need as much evidence to believe its true results or to be misled by false reasoning.
For this reason, we need to play up the wonder of mathematics and the way everything fits, and downplay anything magical. Mathematics is not magic; all of its results can be deduced. It is not a bunch of tricks; everything fits and can be verified even if not deduced. The similarity between mathematics and magic is only that there are results that some of us cannot explain. A major difference is that magicians believe it important to keep their tricks secret, while our job as teachers of mathematics is to work hard to spread knowledge of mathematical techniques. In centuries past, top European mathematicians often kept their knowledge secret because they challenged each other to solve problems for which there were prizes. Today, virtually the only mathematics kept secret is at the advanced levels: by people who are working on mathematics considered important for national security, or by researchers who are keeping their ideas secret because they want to be first to publish them.
We will seldom be able to change beliefs in a single year, but if we work on some general concepts, then over the course of a student's education we may be able to show students how the power of mathematical thinking can help them in their lives and bring a little more rationality into the world. I will indicate these general concepts throughout this talk, as I discuss specific examples of pseudomath and pseudoresearch.
Belief in the Paranormal: A Gallup Poll*
For each of the following items I am going to read you, please tell me whether it is something you believe in, something you are not sure about, or something you don't believe in.
|Believe||Not sure||Do not believe|
|In clairvoyance. or the powe r of the mind to know the past and predict the future||26%||24%||23%||51%|
|In astrology, or that the position of the stars and planets can affect people's lives||25%||29%||22%||54%|
|In ghosts. or that spirits of dead people can come back in certain places and situations||25%||11%||19%||56%|
|In telekinesis, or the ability of the mind to move or bend objects using just mental energy||17%||NA||24%||59%|
|That pyramids have a special healing power||7%||NA||26%||67%|
*selected items from 'Belief in Paranormal Phenomena Among Adult Americans: by George H. Gallup, Jr., and Frank Newport.
Today Is Saturday the 9th
Pseudomath refers to ideas and arguments that on the surface look to be mathematical but are not, just as pseudoscience refers to phenomena that are claimed to exist but under scientific scrutiny have never been verified-phenomena like ESP, clairvoyance, flying saucers, and so on. Likewise, pseudoresearch refers to data and arguments that look like good research but are not research at all.
A simple example of pseudomath was a subject of the Gallup poll I cited earlier. In that poll, 18 percent of the 1,236 respondents said they were somewhat or very superstitious. The three most common superstitions were: a black cat crossing one's path, walking under a ladder, and number superstitions such as Friday the 13th or things happening in threes. So 9 percent of 18 percent of 1,236 respondents, or 20 of the 1,236 respondents, mentioned mathematical ideas as superstitions. These people are going to be worried if you give them a test on Friday the 13th.
That's not a great number of people, and one that would not concern me, except that someone must believe the percentage of people bothered by the number 13 to be greater, for there is no floor numbered 13 in many hotels and some tall office buildings. Do you know anyone who believes that 13 is truly an unlucky number, anyone who will not do things on Friday the 13th? We laugh at such notions, but obviously the hotels do not laugh. Perhaps they have evidence that if you put people on the 13th floor, they are more likely to complain that things have gone wrong, that service is bad, that their room service food does not taste good, and so on.
Of course, if the hotel is tall enough, there is a 13th floor; it is just numbered 14. In Europe the same hotel would be numbered 12. If the number 13 were assigned to a different floor, these people would fear the new floor, because it is the name that presents the problem. Now you see why I titled this part of the talk "Today Is Saturday the 9th." If today had been Friday the 13th, you may have had a completely different reaction to it.
This is not to say that numbers do not have social significance. You can be dressed "to the nines" and be "behind the 8-ball." Recently, the phrase "to 411" has come into our language as a phrase that means to find information. But people who believe that 13 is bewitched have difficulty with Friday the 13th. They think the number has power or conveys power whenever it is identified with something. This is certainly pseudomath.
The number 666 is similar. A few years ago the owner of the building at 666 Lake Shore Drive, near Navy Pier in Chicago, felt that the address had to be changed. Why? Because the number 666 is the number of the devil, and the new condominiums weren't selling. Chicago has a coordinate numbering system, so there are certain addresses that will be 666 because of their location. But the numbering system in Chicago is arbitrary; more than hotel floors, the numbering sys- tem could start anywhere. So, carried to its extreme, if a building is hexed because it is numbered 666, then perhaps so are functions like f(x) = x2 - 10 which attain that value.
Some time ago, medical doctors began to take on a responsibility in addition to their historical responsibility for curing diseases: alerting the public about the dangers of certain practices such as smoking or drinking, about the benefits of a "healthy life style," and about other behaviors that fall under the heading of preventive medicine. A parallel question is appropriate for us to consider: are there perhaps interpretations about which mathematics teachers should be warning the public?
You might sway those who believe that 13 is magical by representing the number in other ways, as thirteen, or XIII, or 11012. Your students may be helped by knowing that numbers that are considered lucky in some cultures are unlucky in others. For instance, in China the number 4 is considered very unlucky, while 13 has no significance. In our culture, 7 is generally considered lucky; in China it is 8.
Some people are superstitious and try to behave in such away as to avoid bad luck or "jinxing" themselves, and others are not. How superstitious are you? Would you say you are:
|Not very superstitious||26%|
|Not at all superstitious||56%|
And what one or two superstitions affect you most?
|Black cat crossing path||14%|
|Walking under a ladder||12%|
|Numbers/Friday 13th/Bad things happen in threes||9%|
Skepticollnauirer 15: 137-146 (Winter 1991).
The Law of Averages
Here is another example of pseudomath, or should we call it "pseudostat"? You are listening to or watching the end of a baseball game. A batter comes up who has not had a hit in the game. The announcer says, "He is due for a hit."
I've never heard an announcer say, "He is due for an out," despite the fact that no batters in the major leagues bat over .500. Based on batting averages, all major league baseball players are due more for an out than a hit! The expression "He is due for a hit" is a corollary of the law of averages, and both the law and its applications are pseudomath.
You might try asking the following question in one of your classes.
Imagine that you take a quarter out of your pocket. You flip the quarter 9 times and each time it lands heads up. Which do you think describes the l0th toss?
(a) The coin is more likely to land heads up.
(b) The coin is more likely to land tails up.
(c) Heads and tails are equally likely.
Explain why you chose each answer.
Do you want to know the right answer? Well, there is no right answer, because two of the three answers are defensible by valid reasoning.
Choice (c) is defensible by deductive reasoning. If we assume the coin is fair, then heads and tails are equally likely regardless of what has happened in the past. We usually make this assumption because we believe the coin is not weighted to favor one side, because we believe the coin has no memory, and because we believe we cannot control the precise speed and angle at which we toss the coin enough to cause it to land on one side or the other.
Choice (a) is defensible by inductive reasoning. The coin has landed heads up each time it has been thrown. Thus there must be something that causes it to be landing heads up. In fact, the probability that a fair coin would land heads up 9 times in 9 tosses is less than .002. So there is evidence that the coin is not fair. So we decide to assume it is weighted to favor heads up.
Choice (b) is the choice obtained by using the law of aver- ages. It requires that one believe the coin remembers what it has done. It is not defensible by valid reasoning, statistical or otherwise.
In UCSMP texts, we discuss the fact that a fair coin does not exist in reality. Being "fair" is a theoretical construct because we can never determine with surety that a coin is fair. Tossingacoin1,000 times and finding out that there are 500 heads does not prove it, and tossing a coin 1,000 times and getting 501 heads does not disprove it. Some people believe that if we modify the question to say that the quarter is fair, then students will more often choose (b) rather than (c). I think that UCSMP students will not do this. Tell me if I am right or wrong.
The social importance of knowing that the law of averages is pseudomath goes well beyond baseball and other sports. Gambling in lotteries and casinos has become more and more common, as the recent approval of riverboat gambling in Michigan suggests. When gamblers are winning, they pick choice (a) because they believe that whatever they are doing will continue. This is not unreasonable. But when gamblers are losing, they think they are due to win, using the law of averages, essentially picking choice (b). They think "my luck is due to change." Choice (c), which is often the most realistic choice, never enters their mind: the game is a game of chance; it has relatively fixed probabilities of winning. The house always wins in the long run. So you are likely to lose in the long run.
When we do experiments in classes, we can seldom collect enough data to see the "long run." Computers and calculators can come in very handy for simulating the long run so that students can see how a small sample does not always indicate what will happen with a larger sample from the same population. You can do this by taking a sample of, say 500, and splitting it into 20 samples of 25. This is a critical concept: students need to see distributions of results.
Related to the Law of Averages are streaks. "That team is hot." "That player is cold." It is useful and interesting for students to see random sequences of Hs and Ts for heads and tails, or Ws and Ls for wins and losses, generated on a calculator. For instance, WWLLLWLWWWWL.... Use a long sequence, perhaps 200 or 300 or even 1,000 letters long, and count the number of single Ws, double Ws, triple Ws, and so on. You should find that each count is about half the previous. This indicates randomness at each juncture. What is not so obvious is that virtually all studies of wins and losses of teams have indicated that streaks occur quite randomly. Winning streaks tend to occur with about the same probability as those streaks would be expected to occur with a good team, and the corresponding happens for losing streaks. For instance, a team that wins 88 percent of its games, as the Chicago Bulls did during the regular season last year, would be expected to have some fairly long winning streaks during the 82-game winning season. All of this can be great motivation for students to study more statistics.
Lack of knowledge of distributions contributes to the next example of pseudomath, another example with broad implications.
The Lake Wobegon Effect and Its Corollaries
Most of us have heard of what is now called the Lake Wobegon effect. The name results from the statement of the famous public radio storyteller Garrison Keillor that in his fictional home town, Lake Wobegon, the men are all hand- some, the women are all strong, and the children are all above average. It was meant to be funny, of course, until some years ago, in a standardized test, every state in the union had a median score above the national median. Someone saw the connection, and the "Lake Wobegon effect" was born.
The impossibility of the Lake Wobegon effect seems obvious enough, but is not so obvious that people don't violate it all the time. Though it is obvious that we cannot bring everyone above the median, it is apparently not so obvious that we cannot bring everyone even above the 1Oth percentile. Percentiles are just like that: whatever we do, 10 percent of the people will be below the l0th percentile. Percentiles themselves are not pseudomath but percentiles as goals that can be reached by everyone are.
You may disagree with me. You may say: Suppose an old test, which was standardized in the past, is used. When a later population takes the test it can have a distribution quite different from the earlier population's, and so everyone can be above the 1Oth percentile on that test. I agree, but then what does the percentile mean? If a student is not part of the population for whom the test was meant, the percentile loses much of its meaning. I suppose we could be honest: "Joe, you scored in the 73rd percentile compared to students who took this test in 1988. But we don't know how you would score against today's students."
To avoid this problem, standardized test makers provide local percentiles. This is no better. As I mentioned earlier, schools differ by vast amounts, and what constitutes the 50th percentileatoneschoolmightbethe90thatanother. We at UCSMP are faced with this problem when we are called on the phone and asked what students should be selected for one of our courses, say Transition Mathematics, say in sixth grade. In our Teacher's Editions we say that the top 10 per- cent on some standardized test are ready for Transition Mathematics. The caller wants to know which test. I refuse to name any specific test; instead, I respond that from the time we first did this book, we had schools that picked students who were in the top 10 percent nationally and had them take the book, and it is always successful with that group, but that group had a traditional background, and we know that if a school district implements a stronger K-5 mathematics pro- gram like that of UCSMP, then many more students can be successful with Transition Mathematics in the sixth grade. It is my way of saying that percentiles are relative to the population that took them when the test was normed and should only be applied in that way. The general point is that percentiles should never be viewed as absolute scores and so probably should never be used as cutoffs for anything.
Related to the pseudomath of percentiles are grade levels. A month ago I examined a sample test to be used in teacher training in Illinois. Here is the last question on that test.
A sixth grade student attains a grade-equivalent score of 8.6 on a sixth grade standardized reading achievement test administered in February. A correct interpretation of this score is that the sixth grader:
A. achieved a score that eighth graders would be expected to achieve on a grade-appropriate standardized reading test administered in April.
B. can do eighth grade work as well as the aver- age eighth grade student in the sixth month.
C. performed as well as students at grade level 8.6 would perform if they took the same test that the sixth grader took.
D. reads as well as 80 percent of eighth graders.
Which of these four answers do you think is correct?
It may surprise you to learn that none of the choices is correct. When a test is normed with sixth graders, it is given to sixth graders. Seldom is the same test given to seventh graders, and it is virtually never given to eighth graders or fourth graders. So when a sixth grader scores at the fourth grade level or eighth grade level, it is a scaled score based on per- centiles. In fact, when a sixth grader scores at the sixth grade level, it is based on percentiles. We cannot bring everyone up to grade level because if we bring even one student up, some other student bas to go down.
The test from which this question was taken was written by a national organization that specializes in test construction. I f a corporation in the test-construction business does not know what grade levels mean, what can we expect of educators and others who are expected to interpret these numbers? The social consequence here is that we lead the public to believe that a score such as grade 7.8, or the 95th percentile, or an IQ of 100 bas some absolute meaning when it merely is a location in a distribution. Without telling people what that distribution is, we are leading the public astray. We are giving them false hopes or false despair.
Put another way, suppose you are doing a terrific job with your students, but there is some other teacher in your school who is not doing as well. On this year's local percentiles, your students will do well and push down the percentiles of the other teacher's students through no fault of those students.
A different application of the same idea occurs with at- tempts to improve teams in league sports. There is great pressure on coaches to do better this year than last. But the situation is what is known as a zero sum game-if I win 1, you lose 1, so the sum is 0. It is impossible for all the teams in a league to improve their records. In fact, if one team improves, then at least one other team has to do more poorly. It isn't pseudomathematics, but it is a case where the ignorance of the mathematics of the situation raises false hopes.
The 2n Scam
Pseudoresearch is subtle, so subtle that even the most honest of us can fall prey to it. Let me begin by discussing a practice that I call the 2n scam. I do not recall when I first heard of this scam, but it supposedly is practiced by unscrupulous brokerage houses and investment advisors.
A large number of newsletters, say 100,000, are sent to potential investors. Half of these newsletters predict that a particular stock, or perhaps the entire market, will go up. The other half predict that the stock or the market will go down. When there is a significant move in one of these directions, the 50,000 to which the correct prediction was made are sent a second newsletter with a second prediction. Of these second newsletters, half predict up and half predict down. When the move comes, a third newsletter is sent to the 25,000 to whom the correct prediction was made. Now the newsletter touts the record of correct predictions: we were right the first time and the second time; we cannot guarantee being right all the time, but invest with us. But half of the third newsletters predict up and half predict down. About 12,500 people will receive three newsletters that have three correct predictions, and of those who do not sign up, about 6,250 will get a fourth newsletter touting the track record of previously correct pre- dictions, and so on. Small brokerage houses do not need to have too many customers to make money, and this is apparently a highly effective scam.
A version of the 2n scam is prevalent in education. It proceeds as follows. A new idea is promulgated by a reform organization, and a number of teachers or schools or school districts are asked if they wish to join in on it. Some think about it, consider it deeply, and decide not to use the idea- maybe for good reason, maybe not, we do not know which. The first cut, like the first newsletter in the 2n scam, keeps in only those who decide to use the idea. Already we have a bias; we have lost those who found some reason right away not to use the idea. Next, the schools begin using the new idea, and in some the idea is effective and in others it is not. Those for whom the idea is effective are more likely to let the reform organization know how things are going than others for whom the idea is ineffective. This is the second cut; after this cut the only users of the new idea are those for whom it went well at least once. I f things went very well in succeeding years, these people may contact the reform organizations and become proselytizers, passing the third cut and maybe a fourth cut. They may present data indicating how well things went in their school district, not too dissimilar from the investment scam in which things went well a few times in a row due to fluctuations outside the control of the investor.
We have tried whenever possible to avoid the 2n scam in our interpretation of UCSMP results. We use all sorts of schools in our research on UCSMP, and we cannot avoid the fact that schools do not have to use our materials. Even so, we have always had teachers in our samples who have no idea of what they are going to be teaching. So, though we cannot avoid the first cut, we try to make it as small as possible. And from then on, over the history of the UCSMP secondary textbooks, we have tried to report all classes that have started in our studies, generally taking out classes only if the UCSMP and comparison classes did not match on performance at the beginning of the school year. In a couple of cases, teachers changed schools in the middle of the year, or students were mixed into other classes, and their classes not used, but we have tried never to remove a class from a study because of anything that occurred in the classes as a result of the materials used.
Selecting a Sample a Little Too Carefully
Suppose I were to say that I am going to convince you that if a number ends in 7,thenitisprime. Here is how I do it. I have a number, 67, and it is prime. And I have another number, 107, and it is prime. And here is 37, and it is prime, too. Do you believe that if a number ends in 7, then it is prime?
As I was writing this talk, I took a break and turned on my television set. The first thing I saw was a commercial for a weight-loss product. A woman was explaining that since she used the product she has lost 50 pounds. She was also using improper reasoning, and I don't mean merely that she constitutes a sample of size 1, or whatever size she was before or after the use of the product.
To identify the impropriety, let us analyze this commercial using logical language. We would like to know what hap- pens when we use this product.
Let p = use of product; let q = success with product.
The weight-loss product manufacturers want us to believe
use of the product => success with product,
or p => q.
But the commercial is only giving us
p and q.
In fact, it is likely that the woman was chosen because she was successful, so we really have begun with q and concluded with q, which is always true. In other words, the treatment was successful because we chose those cases where it was successful, just as the numbers I picked ended in 7 and were prime because I chose those cases in which that was so. It proves nothing except that the two things can happen at the same time.
It seems to be human nature for people to treat individual testimony as more credible than conglomerate data, but in having some idea of what might happen when a new treatment is initiated, conglomerate data is far more reliable. It is the difference between tossing a coin once or twice and tossing it fifty times; the latter gives you more information.
In our studies of classes using UCSMP materials, we have tried to have large enough samples so that our results are ro- bust; that is, other people replicating our studies should get the same results. There is great variability among classes; every teacher knows that two classes may be quite different even when they are supposed to contain students of the same background and ability. Between schools there are even larger differences, with classes in some schools knowing more at the beginning of the year than classes in other schools know at the end. So we always test classes in the same school. We try to minimize the variability due to differences in student population by equating classes by student results at the beginning of the year. By having reasonably large samples, we hope that these effects balance themselves out and that what is left are the effects of using UCSMP materials.
The general point here is that if you do not know the entire sample to which a treatment was given, it is impossible to interpret the results. Put another way: in a situation in which there are bound to be successes and failures, testimonials tell you only that there can be successes; they do not tell you the probability of success. Testimonials without corroborating evidence constitute pseudoresearch.
You will hear many testimonials over the next two days. W e cannot avoid that, and I think you would be disappointed if you came to a UCSMP conference and did not hear positive things. You also may hear some negatives, because we do not muffle our presenters or anyone in the question-and- answer or user sessions. But we can give you some general probabilities with respect to performance of students using UCSMP materials. On tests over content and ideas emphasized in the UCSMP materials, UCSMP classes almost always perform significantly better than comparable classes in the same school, which should be expected. On traditional tests, there have not been significant differences between the performance of UCSMP students and other students. Sometimes UCSMP classes perform better than other classes in the same school; sometimes it is the other way around; rarely are the differences statistically significant.
Considering Alternate Hypotheses
Suppose that you find that your school 's enrollments in advanced mathematics courses have increased since you began using UCSMP texts. You might attribute that to the UCSMP texts, but it also turns out that enrollments have gone up all over the country, so the trend is broader. In order to ascribe any sort of causality to a treatment, you also need to know the part of the population to which the treatment was not given. This is another reason why we have comparison classes in our studies.
In order to understand p => q, you need to know what happens with not-p. Here are three examples. You are teaching the triangle inequality:
If a, b, and c are Lengths of sides of a triangle, then
a + b > c, a + c > b, and b + c > a.
Here p is
a, b, and c are Lengths of sides of a triangle.
Try an example of not-p: Let a, b, and c be lengths of three sides of a quadrilateral. Must it be true that a + b > c? That kind of thinking is very useful for helping students under- stand mathematics.
The second example is about one of the most famous tri- angles: the Bermuda Triangle. Some people believe that if you are in the region known as the Bermuda Triangle, then an unexplained disaster is more likely to happen to you. What does this belief mean? Is there a region called the Bermuda triangle? If so, what are its vertices? What qualifies as a disaster? How many disasters have there been in that region? How many of these are unexplained? How many unexplained disasters have been elsewhere? Analysis of the data indicates that claims for such a region are not tenable.
The third example deals with coincidences. For instance, you are in an airport and you meet someone you haven' t seen in a long time. Is this really unexpected? Think of all the people you see in airports, think of all the friends you have, and think about all the times you have been in an airport and not seen anyone you know. Or you are thinking about a friend and all of a sudden the friend calls, and so you remark, "Your ears must have been burning." How many times have you thought about friends and they do not call?
The general point is that in a wide variety of situations, alternate hypotheses are worth considering. Without this consideration, it is too easy to believe in something with no reason for that belief.
The general idea underlying pseudomath or pseudoresearch is a lack of attention to evidence. Some of the evidence comes under the general category of logical cause and effect. We must emphasize that authority in mathematics comes from principles of reasoning, from within mathematics, not from books or teachers. We must emphasize that results follow in mathematics because they are assumed properties of objects or from the way the objects are defined; they do not just happen. It isn't magic that the sum of the measures of the angles of a quadrilateral is 360, and it isn't a mystery to good students why this is so. It isn't magic that there are infinitely many primes; it can be proved.
Other evidence, particularly in the case of statistics and probability, comes from observing large numbers of events. People have difficulty with the Lake Wobegon effect because they have not had much experience with distributions of numbers. They extrapolate from single numbers to sets of numbers. One student cannot raise his or her percentile score without some other student having his or her percentile lowered. A team winning a game means that another team loses. It seems so simple, but apparently it is not so obvious.
Examples of implications are present in every topic in the teaching of mathematics. From given information in a real situation or a figure, we deduce things. From an equation, we deduce its solution. It is natural to want justifications, not to do things mechanically. And in certain real-world con- texts, everyone is logical. A person says that a product helped him; does that mean that the product helped everyone that used it? It helps to collect data and carefully examine it.
Finally, it is important to realize that poorer students need to discuss these ideas at least as much as better students! In the past, mathematics courses for poorer students have stayed away from deduction and anything more than the simplest statistical reasoning. But thinking that mathematical results are unrelated, or magical, or mystical is far more likely to be held by those who are not as good at mathematics. In fact, keeping deduction and statistical reasoning from poorer students because we think they cannot handle it is a self-fulfilling prophecy; if we do not teach them, they are certain not to be able to handle it. We need to examine evidence with poorer students at least as much as we do with any other students. If we don't, who will?
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