Tuesday, September 15, 2009

Pinoy Math wizards bag 102 medals

By JONATHAN M. HICAP
August 24, 2009, 4:50pm

SINGAPORE — Filipino math wizards won 102 medals, including 16 golds, in the 5th International Mathematics Contest (IMC) which was held here from August 20-24.

The IMC drew 557 contestants from Singapore, China, Hong Kong, Taiwan, Indonesia, Malaysia, India and the Philippines.

Dr. Simon Chua, head of delegation and president of the Mathematics Trainers Guild-Philippines, told the Manila Bulletin that the country won 16 gold medals, 27 silvers and 59 bronze medals for a total of 102, compared to 75 medals the country won last year in the IMC.

With the total medal haul this year, the Philippines placed second overall behind China.

“This is the best performance of our students in the competition. Almost all our contestants each won a medal,” said Chua.

Dr. Chua identified the gold medalists as Andrea Aiyanna Borbe of UNO High School, Christopher Kohchet-Chua of Saint Jude Catholic School, Mark Christopher Uy of Xavier School, Farrell Eldrian Wu of MGC New Life Christian Academy, Josh Thomas Clement of West Visayas State University, Miguel Lorenzo Ildesa of PAREF Westbridge School-Iloilo, Andrew Vince Lee of Xavier School, Raphael Villaluz of San Beda College Alabang, Andrew Brandon Ong of Chiang Kai Shek College, Czarina Angela Lao of Saint Jude Catholic School, Neil Phillip Poral of West Visayas State University, Ana Karenina Batungbakal of San Beda College Alabang, Allen Cedrick Domingo of San Beda College Alabang, Audrey Celine Lao of Saint Jude Catholic School, Camille Tyrene Dee of Immaculate Concepcion Academy, and Hazel Joy Shi of Philippine Cultural High School.

The silver medalists are: Kevin Brian Branzuela of Saint Jude Catholic School Sage Javis Co of Xavier School, Rainielle Maegan Cua of Saint Jude Catholic School, Luis Salvador Diy of Xavier School, Xavier Jefferson Ray Go of Zamboanga Chong Hua High School, Sedrick Scott Keh of Xavier School, Matthew Johann Uy of Xavier School, Alyssa Guevara of De La Salle Santiago Zobel, Clyde Wesley Ang of Chiang Kai Shek College, Jillian Therese Robredo of Unibersidad de Sta. Isabel, Felix Suarez Jr. of Oton Central Elementary School, Ramon Galvan III of Children Integrated School of Alta Tierra, Andrew Lawrence Sy of Saint Jude Catholic School, Raymond Joseph Fadri of San Beda College Alabang, Rafael Jose Santiago of PAREF Southridge School, Jasper John Segismundo of Pasig Catholic College, Jean Leonardo Abagat of Notre Dame of Greater Manila, Deanne Rochelle Abdao of Integrated Montessori Center, Jose Agerico Bacal II of Rosevale School, Ma. Christiana Guillermo of O.B. Montessori Center, Kaye Janelle Yao of Grace Christian College, Jerome Claude Palaganas of Angelicum College, Casey Oliver Turingan of San Beda College Alabang, Zixin Zhang of Grace Christian College, Lance Robin Chua of Bayanihan Institute, Alvin Ian Chan of St. Paul College of Ilocos Sur, and Gisel Ong of Grace Christian College.

Defining Geometric Figures

What is a two-dimensional figure?

A two-dimensional figure, also called a plane or planar figure, is a set of line segments or sides and curve segments or arcs, all lying in a single plane. The sides and arcs are called the edges of the figure. The edges are one-dimensional, but they lie in the plane, which is two-dimensional.

The endpoints of the edges are called the vertices or corners. These points are zero-dimensional, but they also lie in the plane, which is two-dimensional. The most common figures have only a few edges, the curves are very simple, and there are no "loose ends" - that is, every vertex is the endpoint of at least two edges.

If all the edges are segments, every vertex is the endpoint of two sides, and no two sides cross each other, the figure is called a polygon.1 Polygons are classified according to the number of sides they have, which equals the number of vertices. Here are some names of polygons.

Polygons often divide the plane into two pieces, an inside and an outside. The inside part is called the region enclosed by the figure. The name of the figure is also commonly used for this region, and the area of the region is commonly called the area of the figure.

When two sides meet at a vertex, they form an angle. Actually they form two angles, one inside the figure, and one outside. The one inside is called the interior angle at that vertex, or simply the angle at that vertex.

Mathematically speaking, a triangle consists of three vertices and three sides only. The interior is not included. When you want to refer specifically to the interior of a figure that does not have a name of its own, you can call it "the region of the plane enclosed by the figure" or "the figurate region": for example, the "triangular region." When you calculate the "area of the triangle" you are really finding the area of the region enclosed by the triangle.

1 polygon - from Greek polus, "many," and gonia, "angle." Although a polygon is defined as a figure with many sides, the word really means that it has many angles.


What is a three-dimensional figure?

A three-dimensional figure, sometimes called a solid figure, is a set of plane regions and surface regions, all lying in three-dimensional space. These surface regions are called the faces of the figure. Each of them is two-dimensional. The arcs of curves that are the edges of the faces of the figure are called the edges of the figure. They are one-dimensional. The endpoints of the edges are called its vertices. They are zero-dimensional.

The most common three-dimensional figures have only a few faces, the surfaces are very simple, and there are no "loose ends" - that is, every vertex is the end of at least two edges, and at least two faces meet at every edge.

If all the faces are plane regions, every edge is the edge of two faces, every vertex is the vertex of at least three faces, and no two faces cross each other, the figure is called a polyhedron.2 Polyhedra are classified according to the number of faces. Here are some names of polyhedra.

Polyhedra often divide space into two pieces, an inside and an outside. The inside part is called the region enclosed by the figure. The name of the figure is also commonly used for this region, and the volume of the region is commonly called the volume of the figure.

When two planar faces come together at an edge, they form an angle. Actually they form two angles, one inside the figure, and one outside. The one inside is called the dihedral angle (dihedral means "having two faces") at that edge, or simply the angle at that edge.

Mathematically speaking, a tetrahedron consists of four triangular faces, six edges, and four vertices only. The interior is not included. When you want to refer specifically to the interior of a figure that does not have a name of its own, you can call it "the region of space enclosed by the figure" or the figurate solid: for example, the "tetrahedral solid." When you calculate the "volume of the figure" you are really finding the volume of the region enclosed by the figure, or the "figurate solid."

2 polyhedron - from Greek polus, "many," and hedra, "base" or "seat." A polyhedron is thus a figure with many bases, or faces.


A note on dimensions

A point, which is 0-dimensional, can lie on a line, in a plane, or in space. A line, which is 1-dimensional, can lie in a plane or in space. A plane, which is 2-dimensional, can lie in space, and space is 3-dimensional.

Similarly, curves are 1-dimensional, but can lie in higher-dimensional objects, and likewise for surfaces, which are 2-dimensional.

Solids, which are 3-dimensional, can only lie in space.

Glossary of Mathematical Mistakes

By Paul Cox

This is a list of mathematical mistakes made over and over by advertisers, the media, reporters, politicians, activists, and in general many non-math people. These come from many sources, which will appear in parenthesis. I will try to find an actual example of each for learning purposes. Note: In this document, I attack errors made by popular social organizations. I am not attacking their important causes, only their mathematical errors. I try to find errors from all political and social views. Any suggestions for better examples and new topics can be e-mailed to me by clicking here.

Back to mathmistakes.com home


Aftermath Counting (A. K. Dewdney)- What the press has a tendency to do after a major disaster. It is the counting of known casualties from the police, paramedics, hospitals, and morgues, without considering duplication. Example: After the San Francisco Earthquake in 1989, reported deaths skyrocketed to 255 before finally settling on the number 64 dead. Interestingly, the Oklahoma City bombing of 1995 kept the body count at only the number of bodies actually recovered, though the press reported as many as 300 missing (the final tally was 163).

"All Disasters Come in Threes" Conjecture - Also called the "All Celebrities Die in Threes" Conjecture. Essentially it is the mistake of grouping what is essentially a random occurrence. See Cancer Cluster Syndrome, "Shooting the Barn" Statistics.

Astrology Amnesia - When your Astrological forecast comes true one day, you forget about the last three weeks when the forecast failed. Similar to Sample Trashing.

Cancer Cluster Syndrome - Making a lot of fuss over an above average number of cancer cases in a confined region. Note that all random functions have a tendency to cluster. For every reported cancer cluster there is also a cancer deficit region that does not get reported at all. This is not to say that all "cancer clusters" are just statistical abnormalities, there may be some toxic pollutant present in the area, but false reporting of cancer clusters is very common. See "Shooting the Barn" Statistics.

Circular Reasoning - see Recursive Arguments

Compound Blindness (Dewdney)- An "impressive" growth rate that does not take into account inflation, population growth, or other forms of natural compound growth. See also Law of Zero Return

Conspiratorial Coincidence (Paulos, 1995)- Given any two events, or any two people, it is highly likely there are strange commonalities. Unfortunately, sometimes these commonalities are put together to suggest or demonstrate a "conspiracy theory". Example: The commonalities between Abraham Lincoln, and John F. Kennedy have become rather famous. Lincoln was elected in 1860, Kennedy in 1960. Their names are seven letters long. Their assassins, John Wilkes Booth and Lee Harvey Oswald, both advocated unpopular political views, went by three names, and had 15 letters total in their names. Booth shot Lincoln in a theater and fled to a warehouse. Oswald shot Kennedy from a warehouse, and fled to a theater. But if we are to make anything from this, we should consider the strange commonalities between two other assassinated presidents: William McKinley and James Garfield. Both were Republicans, born and raised in Ohio. Both have eight letters in their last name. Like Lincoln and Kennedy, they both were elected on years ending with zero (1880 and 1900). Both had Vice Presidents from New York City who wore mustaches. Both were slain during the first September of their respective terms by assassins Charles Guiteau and Leon Czolgosz, who had foreign sounding names. So where are the conspiracy theories here? The Readers Submitted Examples page has more on this topic. This topic was featured as a Mistake of the Month.

Correlation Cause and Effect Problem - Statistical correlation is a comparison of how often event A happens when accompanied by event B, versus how often event A happens without event B. If the first happens more then the second, then event A and B are said to have positive correlation. For example, it has been shown that students with good self-esteem get better grades in school than students with poor self-esteem. So, self-esteem and grades correlate. The problem here is in deciding cause and effect. Either A helps to cause B, or B helps to cause A, or there is a third factor C that could help to cause both A and B. Unless there is definitive proof to decide, any one of the three can be the truth. Some educators believe that all they have to do is boost self esteem, and grade improvement would follow. But, it is just as likely that self-esteem is a result of good grades, not a cause. Furthermore, there could be a third factor, say a good family environment at home, that could be responsible for both. The Readers Submitted Examples page has more on this topic. This topic was featured as a Mistake of the Month.

Credit Card Games - An offer too good to be true from a credit card or loan company. Lets say that I use my credit card to buy a $2500 computer. But, this card charges me 16% interest. Another card charges only 10% but there is a 3% surcharge for using the card. Which card should I use? Obviously, the second card because 10% plus 3% is only 13%, but if you go with the obvious choice, you would be wrong! Lets say I decide to pay off the debt in 8 months by paying $300 per month. Under the first card, over that 8 month period, I will pay $2664.42 including interest for that computer. Under the second card, they will charge me 3% up front, so my loan to start is $2575. At 10% interest over those same months, I will pay $2680.68 including interest for that same computer. That is over sixteen dollars more! If you transfer your credit card balance to a lower interest credit card there usually is this kind of surcharge, so run the numbers before you do it. Better yet, do not use a credit card if you can help it. A $2500 computer only costs $2500, if you pay it all up front. Watch out for "0% interest for 6 months" deals, also. Unless you can pay it all off within those 6 months, they will charge you back interest accumulated during those 6 months, usually at a high interest rate. You are better off charging with a bank card that charges interest from the start, than to go with one of these deals. Compare Ratiocinitis.

Definition Errors- A category of errors in which a mathematical term is misused or confused, possibly to mislead. The three most common definition errors are:

  1. Profit (or earnings) - Could mean either gross profit or net profit, this is how "5 billion dollar" companies still have financial difficulty and have to reorganize, such as a former employer who shall remain nameless.
  2. Average - There are three different kinds of averages. Given a sample set: 2,2,2,3,5,6,8: The mode average (the most common answer) is 2; the median average (the one in the middle) is 3; and the mean average (the sum of all samples divided by the number of samples) is 4. Any one of which can legitimately be the average.
  3. Odds vs. Probability - Given a list of outcomes (rolling two dice) and a list of good outcomes (rolling a 7), you can calculate either the probability (good outcomes / all outcomes) to get 1/6. Or, you can calculate the odds (bad outcomes / good outcomes) to get 5:1. It is easy to get these two confused.

Dimensional Demensia (Dewdney)- 1. The confusion of the significance of dimensions. 2. The attempt to compare two objects of different dimensions. A foot has 12 inches, but a square foot has 144 square inches.

Example: Here is a comparison of the world population in three different ways:

* If every person in the world was lined up end to end, we would stretch four times longer than the orbit of the Moon around the Earth.

* If we all lived in a city with the population density of New York City, That city would cover the state of Texas.

* If every person was given a 20'x20'x20' apartment, the total volume would only fill the Grand Canyon half way.

Note that with each dimension added, the world population seems less significant. (Examples from Paulos) An additional question: Which dimension is the correct one to use? Since our number one need is food, and farm land is measured in acreage, it is safe to assume that the two dimensional measure is the best one. While we can live in apartment buildings stacked up on top of one another, farm land cannot be stacked. Another example: Back in high school, I was dancing close with a girl, when a chaperone came over and said, "You should be six inches apart." I told the chaperone, "We are. There is at least six cubic inches of space between us." Ok, so maybe you had to be there to think it was funny.

Dramadigits (Dewdney)- 1. The reporting of a number with more significant digits than what can be reasonably expected. Example: Advertisers love Dramadigits! Ivory soap likes to brag that it is 99.44% pure. It is impossible that each bar is exactly that impure. How do they get away with it? The "99.44% pure" statement is a trademark, not a statement of fact. 2. The reporting of a number with more significant digits than can be accurately calculated. Example: For decades it was thought that the normal body temperature was 98.6°F. This number was calculated from a study in Germany which reported normal at 37°C. What was not known was that this number was an average rounded to the nearest degree. In other words it was only accurate to two significant digits, not the three we have with 98.6. Scientists today know that normal is actually 98.2 plus or minus 0.6, that is to say anything in the range of 97.6° to 98.8° should be considered normal. Here is a dramadigit joke: A visitor at a natural history museum asks a guard how old the dinosaur skeleton is. The guard responds that it is 90,000,006 years old. He explains, "They told me it was 90,000,000 years old when I started working here, and that was six years ago." (Paulos, 1995) The Readers Submitted Examples page has more on this topic.

Election Paradox - In a very close election involving more than two candidates, it is possible to invent scenarios in which any candidate could have won. The only way to avoid the election paradox is to decide how votes are counted and how the winner is decided before the election is held. The 2000 Presidential Election demonstrated this paradox. It took the Supreme Court to decide the winner of Florida. When the votes were finally all recounted a year later Bush would have won anyways, but there were scenarios in which Gore would have won Florida and the Presidency. This topic was featured as a Mistake of the Month.

Factorectomy - 1. The failure to take into account factors that vastly affect the outcome. Inflation is a commonly left out factor, especially when comparing financial situations over time. Variety always publishes a list of the "Top Grossing Movies of All-Time", and according to the list Titanic is number one, but when you take into account inflation, Gone With the Wind, made in 1939, has made almost $300 million more than Titanic. In 1939, you could still go to the movies for 10 cents. When #2 Star Wars was released in 1977 it was 3 dollars a ticket. Still low compared to $7.50 ticket prices of 1998 when Titanic, came out, (which falls to number 5 when adjusted for inflation.)

2. The use of predictive models that do not fit past data or outcomes due to missing factors. Example: Global Warming calculations have come under serious attack due to their inability to predict past occurrences. It has been determined that these calculations failed to take into account the effect of sulfur particulates and other pollutants on global temperature. While the old factorectomized models predict global warming of up to 8°F by 2040, these new predictive models show the globe heating up no more than 1°F by the year 2040. Turn on your Air Conditioners! (Note: The February 1994 Scientific American has an excellent article by Robert Charlson and Tom Wigley on this topic. Also, some have pointed out that I am oversimplifying the global warming phenomenon. Of course, I am. If you want the complete weather picture, see the May 1997 Scientific American.) Compare Factoritis.

Factoritis - 1. Taking into account factors that are not really relevant in order to inflate the numbers. AIDS activist groups regularly report that there are over a million Americans with HIV. Current statistics show that it is around 740,000 - including the approximate number that have never been tested, but are positive any ways. The only way to justify over one million people is to include those that died from the AIDS since 1980 when it was first discovered. While some may call this valid, it only looks impressive as a Raw Number. To use this number as a comparison figure, you have to include all of the people that have died from other diseases during the same time period. Stated another way: To figure out the percentage of the population with HIV we divide 740,000 (the number of HIV positives still alive) by 260,000,000 (the number of Americans still alive), resulting in 0.29% or about 1 in every 340 people. If we include the 360,000 people who have died of AIDS in the numerator, we have to include the tens of millions of people who have died from all causes during the same time period in the denominator. The resulting percent is actually lower when calculated this way.

2. Taking into account factors that were already calculated earlier. In 1990, the Department of Education reported that school expenditures have more than tripled since 1960. Some education lobbyists produced the figure that education spending has actually gone down on a per pupil basis when you figure in inflation. The problem is, the Department of Education already figured in inflation in their figures (but they left out per pupil figures, see the double exposure graph example below for details). The declining education statistic is the result of inflation being taken into account twice. Compare Factorectomy.

False Positive Conjecture - Tests with very high, but not perfect, accuracy may actually produce more false positives than true positives. Let us suppose that 3% of the population uses illegal drug X, Let us also suppose that a test has been developed that is 95% accurate in determining who has been using drug X. Say 1000 people took this test, 30 of them being users, Since the test is 95% accurate, 29 of these users gets caught (the other one gets a false negative). At the same time, of the 970 remaining, 48 also show positives, even though they are false. In other words, 78 people tested positive for use of drug X, but only 30 were true positives. Try this out on AIDS tests that are 99.7% accurate, or for more fun try polygraph (lie detector) tests that are only 80% accurate!

Filter Counting (Dewdney) - An underestimate of reality caused by the deletion of important data. Similar to Factorectomy. See also Sample Trashing

Gambler's Fallacy - see Law of Averages Thinking.

Gambler's Ruin - If a gambler stays in a casino long enough he will eventually lose all of his money. This is why casinos can pay out millions in winnings and still stay in business. This topic was featured as a Mistake of the Month.

Graph Errors - Statistical graphs can be seen everywhere. It is based on the idea that a picture is worth a thousand words. Unfortunately, pictures can lie as well. Here are four common types of misleading graphs in the business: The Readers Submitted Examples page has more on this topic.

* The Magic Graph - A chart or graph that uses an optical illusion to make the information seem more significant.

Two magic graphs and a zoom graph.

* The Zoom Graph - A graph that does not start at (0,0) thus making small changes seem big. Useful in the study of trends, but often passed off as actual changes. Such graphs are very popular in financial circles. The graph below comes from an advertisement for Creative Labs Graphic Blaster3D.

Graphics Blaster Ad

* The Double Exposure Graph- Two graphs plotted on top of one another in order to make some kind of comparison. Again occasionally useful, but easily misleading. Note below how the combined SAT side is a zoom graph. This graph was developed by the U.S. Department of Education.

SAT vs Expenditures 1

Here is a more accurate portrayal of reality. Note that since SAT scores go from 400 to 1600 points it is best to demonstrate this range. Also Education Spending has been adjusted to a per pupil counting showing spending has doubled instead of tripled. The connection between school spending and falling SAT scores cannot really be made.

SAT vs Expenditures 2

Object d' Art Graph* The Object D'art Graph - A graph that lacks a standardized y-axis in order to hide what the graph is really saying. It is can be a pretty graph suitable for framing but in fact absolutely useless. To the right is an Object D'art Graph actually used in a stock report. Note that the 1970 figure is actually a negative number.

Innumeracy (From the similarly titled book by John Allen Paulos)- 1. The inability to deal with simple mathematical concepts. 2. The math version of illiteracy. This topic was featured as a Mistake of the Month.

The "Kevin Bacon" Game (a Hollywood trivia game) - During your life you have probably gotten to know at least a thousand people. Each of those thousand people, have gotten to know a thousand people as well. Taking into account overlap, at least 100,000 people knows someone who knows you, and possibly 10 million people know someone who knows someone who knows you. By this, it follows that everyone is connected to everyone else by no more than six degrees of separation. A movie with that title has inspired a game called the Kevin Bacon game. Take any famous person, and you should be able connect them to actor Kevin Bacon in less than six moves. OK, lets try Elvis Presley, who was married to Priscilla Presley, who was in Naked Gun with Leslie Nielsen, who was in Airplane! with Lloyd Bridges, who was in Blown Away with Tommy Lee Jones, who was in JFK with Kevin Bacon. So, the next time the complete stranger sitting next to you turns out to have gone to high school with your brother in law's boss, it really is not that big of a coincidence. Compare Post Occurrence Miracle. The Readers Submitted Examples page has more on this topic.

The Law of Averages thinking - A belief by gamblers that the more often you win or lose the more likely your luck will change on the next try. If you flip a coin and it lands heads 10 times in a row, what are the odds that it will land heads on the 11th try? Answer 1:1. What about after 100 times in a row? Again it is 1:1. The odds are the same on each toss! The Readers Submitted Examples page has more on this topic.

The Law of Zero Return (Dewdney)- Return on Investment = Inflation + Taxes. This topic was featured as a Mistake of the Month.

Loaded Questionnaire - Asking questions in such a way as to make the respondees feel foolish if they do not answer the questions the way the surveyor wants. Example: In 1995, the Corporation for Public Broadcasting surveyed the viewership of TV. The CPB had a certain interest in getting responses that favored their programming format so as to avoid federal budget cutters. One of many example questions:

"A recent study by a psychology professor at a leading university concluded that the amount of violence children see on television has an effect on their likelihood of being aggressive and committing crimes. From what you have seen or heard about this subject, do you agree strongly to that conclusion, agree somewhat, disagree somewhat, or disagree strongly?"

It is difficult to listen to such a question and not agree strongly. A more balanced question would include a contrary opinion, or better yet offer no opinions at all.

Logical Fallacies - If you want to deceive the majority of the people, use some of these in your arguments. (Note that these are used consistently by politicians, lawyers, and advertisers). The Readers Submitted Examples page has more on this topic.

* Fallacy of Ambiguity - Occurs when a word or phrase is used with one meaning in one premise, and with another meaning in another premise or conclusion. Example: People should do what is right + people have the right to disregard good advice = People should disregard good advice.

* Fallacy of Attacking the Messenger - Attempting to discredit a message by discrediting the messenger. Example: He is innocent, the cop who arrested him is a racist and therefore must have planted that glove.

* Fallacy of Composition - Confuses properties of a whole with properties of the parts. Example: This is a good class = Every student in this class is good.

* Fallacy of Emotion - An appeal to popular passions such as pity, fear, brute force, snob appeal, vanity, or some other emotion. Example: You would look sexy behind the wheel of this new $50,000 sports car.

* Fallacy of Experts - Quoting an expert in one field on a matter of another unrelated field. Example: Meryl Streep testifying before congress on the homeless because she played one in a movie once. Advertisers with celebrity endorsements sell more than with expert endorsements.

* Fallacy of the Complex Question - A loaded question designed to make the person answering look bad no matter how they answer. Examples: Have you stopped beating your dog? How does mind reading work? Are the people in your town still rude to visitors? Will that be cash or charge?

* Fallacy of Non Sequitur (Literally "it does not follow") - conclusions that do not follow from their premises. Example: John is sick + John needs some rest = I need some rest.

* Fallacy of False Cause - Example: I walked under a ladder + I was hit by a car = Walking under ladders brings bad luck. Most superstitions start out this way.

* Fallacy of Popularity - Example: The majority of Americans believe UFO's are real = Space aliens must visit America. Just because something is popular, does not mean it is correct. It is best to follow that old catch phrase, "If everyone jumped off a cliff, would you?"

* Fallacy of Innovation - If something is "new" or "different" then it is often assumed that it is better than the old and ordinary, which is rarely the case. Note that the word "New" is the word most often mentioned in advertisements, but only occasionally is it followed by the words "and Improved".

"Looks Like" Geometry - The tendency to find significance in insignificant geometric patterns. The best example is the "Face on Mars", that has recently been rephotographed. To the right are three images to compare, the first is the original "face", the second is the face as photographed April 5, 1998, the third is a negative of the second to change the angle of the light. Click on the image to see an enlargement. Martin Gardner has written an excellent article on this and other examples. This topic was featured as a Mistake of the Month. Face on mars comparison

Lottery - see Sucker Bet.

Multiple Comparisons Fallacy - (statistical epidemiology) Risk factor studies have a 5% chance of being too high and a 5% chance of being too low. Lets say a pre-election poll of 1,000 people shows candidate Smith with a 8% lead over Jones. If we did instead 20 polls of 50 people each, chances are at least one of those studies would show a slight lead in Jones' favor. In 1992, a Swedish study on the effects of power line radiation showed that children living close to power lines have a nearly four fold risk of childhood leukemia. But, upon closer examination, the Swedes did nearly 800 different studies in one. Other studies in the same report actually show a decrease of childhood leukemia from power line radiation. Studies with this many comparisons are not good for concrete results, they are best used to point out directions where future research should be done. (Frontline, "Currents of Fear")

Num (Dewdney)- A reported number with too few significant digits to be useful. These usually are round numbers like 1000 or 100,000. The term "six-figured salary" is an example, meaning any number between 100,000 and 999,999. Opposite of Dramadigit.

Number Inflation - 1. A gross overestimation or underestimation of reality. 2. A reporting of a statistic that is just not true. I used to call it "The Law of Five Times Reality" because of the tendency of political advocacy groups to over inflate their numbers by five times what is reality. Take the following examples:

Situation Reported Actual
AIDS Cases (1990) 1 million (ACT UP!) 200,000 (CDC)
Homeless (1990) 3 million (Mitch Snyder) 600,000 (Urban Institute)
Right to Life March (1989) 100,000 (Right to Life) 20,000 (Federal Park Police)
Spousal Abuse (1994) 6 million cases (NOW) 1.2 million cases (FBI)
Homosexuals (%) 10% (The Advocate) 2% (1993 sex survey)

It does not necessarily have to be five times reality. Besides reality is defined by the person doing the reporting. Some of these groups try to justify their numbers by taking in to account factors that should not be included (see Factoritis), or try to include numbers that we do not have enough information to make a good estimate (see Statistical Brick Wall). The Readers Submitted Examples page has more on this topic.

Number Numbness (Hofstadter)- The inability to fathom, compare, or appreciate really big numbers or really small numbers. Such as the difference between a million and a billion and a trillion. Politicians seem to make this mistake the most, noting the need to cut $164 million from the National Endowment of the Arts, while insisting that it is important to spend 'a mere $60 billion' on a missile defense system. The Readers Submitted Examples page has more on this topic. This topic was featured as a Mistake of the Month.

Opportunity Cost Error (expression from economics)- The failure to consider the real cost of doing something. Example: Spending a dollars worth of gas to drive across town to save fifty cents on soap. Another example is education. According to census figures, the average high school graduate earns $7000 more per year than the average high school dropout. Over a working life span of 40 years, that means a high school diploma is worth $280,000. That is the opportunity cost of dropping out of high school. The Readers Submitted Examples page has more on this topic.

Percentage Pumping Formula (Dewdney)- A non-standard of percent designed to increase the advertised percentage of discount or improvement. A normal discount formula is SAVINGS/NORMAL COST, a Percentage Pumping formula is SAVINGS/NON-SAVINGS. A 33% normal discount can become a 50% pumped discount. Advertisers use this to create as high as 200% savings which makes no sense at all. The Readers Submitted Examples page has more on this topic.

Post Occurrence Miracle - An unexpected coincidental event realized after the fact. The odds of a particular strange event, say the dream you had last night comes true, may be very small; but the odds that some strange event happening some time, like one night during your life you have a dream that comes true, is actually quite good.

Practically Zero Probability - A probability so small that it must be considered zero. Example: The probability of winning Powerball is 0.000000012. A statistician coming across such a number would not hesitate to call it 0 under most circumstances. The Readers Submitted Examples page has more on this topic.

A Rare Scare - A media report of a probable disaster (i.e.. death, earthquake, cancer risk from eating apples, etc.), where the probability is considerably lower than risks taken everyday (i.e.. getting in a car wreck on your way to school). The Readers Submitted Examples page has more on this topic. This topic was featured as a Mistake of the Month.

Rare Scaremongering - An attempt to avoid a rare scare that results in either a cost that exceeds the cost if the disaster happened, or increases the risk of a more likely and even worse disaster. Example: In Indonesia, eliminating pesticides with very small cancer risks has increased the population of malaria carrying mosquitoes. The Readers Submitted Examples page has more on this topic.

Ratiocinitis (Dewdney)- A tendency to forget the rules regarding the addition and subtraction of fractions, ratios, and percentages. Example: A store is offering 30% off everything. A fancy dress that normally sells for $100 has a tag saying 20% off. Does this mean the total savings is 50%? No, The actual savings is 44%. 20% off $100 is $80, and 30% off $80 is $56, which is 44% off of the original $100. Question: Will it make a difference if the store takes the 30% off first? Example 2: A Man is walking down the street, and stumbles on a five dollar bill, having $10 in his wallet he says to himself, "Hey, I just increased my money by 50%." Later he discovers that he had a hole in his pocket where his $5 bill was lost but thinks, "That's OK, first I gained 50% by finding the five dollars, and now I lost 33% by losing the five dollars, I am still 17% ahead." The Readers Submitted Examples page has more on this topic.

Raw Number - The reporting of an 'impressive' number that is meaningless without something to compare it to. Example: Everyday cars in America produce over a billion tons of pollution. Impressive? yes!, but let me state this statistic another way: Every day cars in America pollute 0.000000001% of our atmosphere. Impressive?, not really. Now this is a poor way to think about pollution, also. But, that is the problem with raw numbers: you have to compare them to something else to be meaningful. A better statistic: Cars are responsible for more than half of the carbon monoxide pollution.

Recursive Arguments - see Circular Reasoning

Regression Jinx - You probably have heard of the "sophomore jinx" popular in sports and music, a rookie athlete or musician performs poorly in their second year after an outstanding first year. Believe it or not, there is a mathematical explanation. When a star gains superstardom, it is partly because of talent, partly because of luck, or outside forces the star cannot control. When luck is extremely in the stars favor, it is only a matter of time that luck changes. Thus the "sophomore jinx" is not a jinx at all, but rather an expected outcome of statistical regression (the tendency of luck to move toward the norm). Alanis Morisette's first CD released in the US, sold 15 million copies. Despite critical acclaim, it is highly unlikely her second CD will sell even half that. An article in the March/April 1999 Skeptical Inquirer has more info and examples.

Sample Occulting (Dewdney)- A disregard for an enormous sample resulting in coincidences seeming "supernatural", requiring an "occult" explanation when there really isn't a need. Example: An Advertisement for a TIME-LIFE book on unexplained phenomenon mentions a daughter who touches a hot stove and a mother 3000 miles away feeling pain in her arm at the same instance. Is the mother tele-empathic? Probably not! Consider how many times people touch hot surfaces, consider also how many times older people get mysterious pains. The likelihood of these two things happening at the same time to two people who are related are very good. Now if these events happened consistently to the same two people, and this could be replicated in a scientific experiment, then one might look to the supernatural. See also Post Occurrence Miracle.

Sample Trashing - Throwing out perfectly good data as "unreliable" because it goes against what the statistics are trying to prove. Popular with ESP believers who point to a few studies with positive results, and ignore the majority of the studies with negative results. The Readers Submitted Examples page has more on this topic.

Selective Endpoints - The reporting or graphing of a change in naturally random functions (economic indicators, weather conditions, stock prices, natural disasters, etc.), by comparing an unusual low to an unusual high. Example: During the 1992 presidential race, the Democrats pointed to how the economy was not doing as well as it was five years ago (1987 was an unusually good year economically), while Republicans pointed out that the economy was better than it was 11 years ago (1981 was an unusually bad year economically).

Self Selecting Sample - The assumption that a group willing to take a survey represents a random sample. The Hite report (1976) on female sexual attitudes was based on surveys of 3019 women, unfortunately Shere Hite distributed over 100,000 surveys. All the report measured was the sexual attitudes of the 3% who were willing to fill out the survey. Another example can be found with 900 number polls on TV shows. These are only a representation of people who feel strongly enough to pay 75 cents a call and do not represent the real population.

Self-Fulfilling Prophecy - When an 'expert' predicts that a certain stock will go up in price, the increased demand for that stock from people following the expert advice pushes the price up. When your astrological forecast says you will have a good (or bad) day, and because this gives you a positive (or negative) outlook to begin with you end up actually having a good (or bad) day. This topic was featured as a Mistake of the Month.

"Shooting the Barn" Statistics - A story is told about a Texas sharpshooter who shot his gun into the side of a barn 30 times, then painted a circle around where most of the bullets landed, calling that his target. Collecting statistical data without first knowing what you are looking for results in bad statistics. Popular among business managers who want show their investors how they are doing, they set their productivity goals after already reaching those goals. Another variation is called "Spotlight Gag" Statistics, after a famous gag by Red Skelton where the spot light always shines where he was standing before (A Simpsons episode had Krusty the Clown do the same bit.) One place where I worked set its productivity goals based on the previous months performance. Each employee was expected to reach these goals for the next month. Natural seasonal fluctuation of demand made the following months goals either too easy or too difficult to achieve. The problem is that comparing past performance to present performance is impossible without a standard measure. Varying productivity goals so drastically made the percent of goal statistics invalid. You cannot measure height with a yardstick made of silly putty. See Cancer Cluster Syndrome, Multiple Comparisons Fallacy

Simpson's Paradox - A condition in statistics in which a "small" group seems to perform better in individual comparisons than a "big" group, but when overall performance is compared, the "big" group is better. For example, ACME Manufacturing is hiring two groups for their new division. Since ACME has had trouble with sex discrimination in the past, they do their bit to avoid it this time. They have two openings in group 1, they have 5 male applicants and 3 female applicants, and they hire 1 male and 1 female. This way 33% of the women applicants get hired compared to only 20% of the male applicants. In group 2 they have 15 openings, 20 men and 3 women apply, and they hire 2 women and 13 men. This way 67% of the women applicants get hired compared to 65% of the male applicants. Who could argue with that? A week later ACME gets hit with a sex discrimination suit by one of the women that did not get hired, because overall 56% of the male applicants (14 of 25) got hired compared to 50% of the female applicants (3 of 6). See Ratiocinitis

Statistical Brick Wall -A number that cannot be verified, or accurately estimated, because the statistical data does not exist. A good example is the statistic of endangered species. Some biologists have estimated that over 10,000 species go extinct every year. Actual verified extinctions are around one species a year, including insects. These highly reported statistic tries to take into account the number of undiscovered species that go extinct, a number that is impossible to calculate. Sure there may be species that go extinct without anyone noticing, but the statistical data does not exist due to the fact that it is impossible to obtain. It could be anywhere from two to 10,000. No one can prove otherwise. Other major victims of the Statistical Brick Wall are studies that involve "ruling out" all possible factors. In 1994, a study was done to show how dangerous particulate pollution is, the result is that people who live in cities with high particulate pollution shorten their average life span by about 2 years. The study compared life spans in non-polluted regions (rural small towns) with high polluted regions (big cities), then they had to "rule out" other factors that might contribute to shorter life spans. They eliminated dozens of factors where statistics exist (i.e. violent crime), but then they were unable to rule out dozens of other factors (i.e. lifestyle differences, eating habits, exercise) because these statistics do not exist. In such cases statisticians either assume all remaining factors add up to zero, or they make an educated guess base on trends. While this does not invalidate the study, it makes such studies less than reliable. The Readers Submitted Examples page has more on this topic. This topic was featured as a Mistake of the Month.

Statistical Rash - A judgment based on statistical data that does not take into account all of the factors that cause the data to result as it did. Example: Here are some actual statistics on accident rates based on the speed at which the cars were driving:

20 mph or less 2.0%
20 to 30 mph 29.7%
30 to 40 mph 30.4%
40 to 50 mph 16.5%
50 to 60 mph 19.2%
over 60 mph 2.2%

It would seem to the casual observer that it is safer to speed than to travel at the speed limit. In fact, the reason that only 2.2% of the accidents happen at over 60 mph is because at any given time only about 2.2% of the cars on the road are traveling at over 60 mph. These statistics do not say anything about speed and accident rates, only about how fast the average car is traveling. (Example from Marilyn Vos Savant) The Readers Submitted Examples page has more on this topic.

Sucker Bet (Dewdney) - A gambling wager in which your expected return is significantly lower than the wager. Lets say that LOTTO (the Arizona Lottery that chooses 6 out of 36 numbers) is paying out $2 million this week, a lottery ticket only costs $1, what is your expected return? To calculate expected return, use the following formula: EXPECTED RETURN = POTENTIAL WINNING * PROBABILITY OF WINNING - POTENTIAL COST * PROBABILITY OF LOSING. Since the probability of winning LOTTO is 0.00000019, the probability of losing is 1-probability of winning or 0.99999981. Therefore, EXPECTED RETURN = 2,000,000 * 0.00000019 - 1 * .99999981 = - 0.62 In other words, for every dollar you put into the lottery, you stand to lose 62 cents. The Readers Submitted Examples page has more on this topic. This topic was featured as a Mistake of the Month.

Technical Analysis (a stock market term) - The attempt to look for numerical trends in a random function. The stock market used to be filled with technical analysts deciding what to buy and sell, until it was decided that their success rate is no better than chance. Now technical stock analysis is virtually non-existent. The Readers Submitted Examples page has more on this topic.

Texas Sharpshooter, The tale of the - see "Shooting the Barn" Statistics


Sources and Further Reading: (clicking a title will take you to amazon.com)

Dewdney, A. K., 200% of Nothing: An Eye Opening Tour Through the Twists and Turns of Math Abuse and Innumeracy, John Wiley and Sons, New York: 1993.

Paulos, John Allen, Innumeracy: Mathematical Illiteracy and its Consequences, Hill and Wang, New York, 1988.

Paulos, John Allen, A Mathematician Reads the Newspaper, Basic Books, New York, 1995.

Hofstadter, Douglas, Metamagical Themas, Basic Books, New York, 1984.

Capaldi, Nicholas, The Art of Deception: An Introduction to Critical Thinking, Prometheus Books, Buffalo, New York, 1987.

Copyright ©1995-2001 by Paul Cox

How to Write a Solution

by Richard Rusczyk & Mathew Crawford

You've figured out the solution to the problem - fantastic! But you're not finished. Whether you are writing solutions for a competition, a journal, a message board, or just to show off for your friends, you must master the art of communicating your solution clearly. Brilliant ideas and innovative solutions to problems are pretty worthless if you can't communicate them. In this article, we explore many aspects of how to write a clear solution. Below is an index; each page of the article includes a sample 'How Not To' solution and 'How To' solution. One common theme you'll find throughout each point is that every time you make an experienced reader have to think to follow your solution, you lose.

As you read the 'How To' solutions, you may think some of them are overwritten. Indeed, some of them could be condensed. Some steps we chose to prove could probably be cited without proof. However, it is far better to prove too much too clearly than to prove too little. Rarely will a reader complain that a solution is too easy to understand or too easy on the eye.

One note of warning: many of the problems we use for examples are extremely challenging problems. Beginners, and even intermediate students, should not be upset if they have difficulty solving the problems on their own.

Table of Contents for How to Write a Solution

What is Problem Solving?

by Richard Rusczyk
Click here for a printable version of the article.

I was invited to the Math Olympiad Summer Program (MOP) in the 10th grade. I went to MOP certain that I must really be good at math… In my five weeks at MOP, I encountered over sixty problems on various tests. I didn’t solve a single one. That’s right – I was 0-for-60+. I came away no longer confident that I was good at math. I assumed that most of the other kids did better at MOP because they knew more tricks than I did. My formula sheets were pretty thorough, but perhaps they were missing something. By the end of MOP, I had learned a somewhat unsettling truth. The others knew fewer tricks than I did, not more. They didn’t even have formula sheets!

At another contest later that summer, a younger student, Alex, from another school asked me for my formula sheets. In my local and state circles, students’ formula sheets were the source of knowledge, the source of power that fueled the top students and the top schools. They were studied, memorized, revered. But most of all, they were not shared. But when Alex asked for my formula sheets I remembered my experience at MOP and I realized that formula sheets are not really math. Memorizing formulas is no more mathematics than memorizing dates is history or memorizing spelling words is literature. I gave him the formula sheets. (Alex must later have learned also that the formula sheets were fool’s gold – he became a Rhodes scholar.)

The difference between MOP and many of these state and local contests I participated in was the difference between problem solving and what many people call mathematics. For these people, math is a series of tricks to use on a series of specific problems. Trick A is for Problem A, Trick B for Problem B, and so on. In this vein, school can become a routine of ‘learn tricks for a week – use tricks on a test – forget most tricks quickly.’ The tricks get forgotten quickly primarily because there are so many of them, and also because the students don’t see how these ‘tricks’ are just extensions of a few basic principles.

I had painfully learned at MOP that true mathematics is not a process of memorizing formulas and applying them to problems tailor-made for those formulas. Instead, the successful mathematician possesses fewer tools, but knows how to apply them to a much broader range of problems. We use the term “problem solving” to distinguish this approach to mathematics from the ‘memorize-use-forget’ approach.

After MOP I relearned math throughout high school. I was unaware that I was learning much more. When I got to Princeton I enrolled in organic chemistry. There were over 200 students in the course, and we quickly separated into two groups. One group understood that all we would be taught could largely be derived from a very small number of basic principles. We loved the class – it was a year long exploration of where these fundamental concepts could take us. The other, much larger, group saw each new destination not as the result of a path from the building blocks, but as yet another place whose coordinates had to be memorized if ever they were to visit again. Almost to a student, the difference between those in the happy group and those in the struggling group was how they learned mathematics. The class seemingly involved no math at all, but those who took a memorization approach to math were doomed to do it again in chemistry. The skills the problem solvers developed in math transferred, and these students flourished.

We use math to teach problem solving because it is the most fundamental logical discipline. Not only is it the foundation upon which sciences are built, it is the clearest way to learn and understand how to develop a rigorous logical argument. There are no loopholes, there are no half-truths. The language of mathematics is precise, as is ‘right’ and ‘wrong’ (or ‘proven’ and ‘unproven’). Success and failure are immediate and indisputable; there isn’t room for subjectivity. This is not to say that those who cannot do math cannot solve problems. There are many paths to strong problem solving skills. Mathematics is the shortest.

Problem solving is crucial in mathematics education because it transcends mathematics. By developing problem solving skills, we learn not only how to tackle math problems, but also how to logically work our way through any problems we may face. The memorizer can only solve problems he has encountered already, but the problem solver can solve problems she’s never seen before. The problem solver is flexible; she can diversify. Above all, she can create.

Tips For New Math Teachers

by Gisele Glosser
Try not to frown on wrong answers. It discourages students from participating. Critical thinking and honest effort are more important than correct answers.
There is no teaching without control of your class. It is better to fall behind by a day or two early in the year to address discipline, than to have an uphill battle all year long over behavior.
Avoid talking over your students. If there is too much noise in the classroom, sometimes the best thing to do is to stop talking. (I am famous for the "Glosser Glare".)
Routine and structure are good, but too much of it can cause you and your class to fall into a rut. Try to vary activities from time to time.
Encourage active participation from your students. From time to time, call students to the board, or allow them to work in groups. Avoid giving teacher-directed lessons all of the time. (See our article on Cooperative Learning Techniques.)
Try to be flexible. Math can be a rigid topic, but you don't have to be. For example, I have a strict rule against chewing gum. But I close my eyes to it during a test.
Try to spell out what topics will be on the test. Telling your students to "Study Chapter 6" is not enough, especially if they have poor study skills.
In some schools, math is the only subject where students are grouped by ability (i.e. homogeneously). This makes it stand out more than other subjects. Parents may frequently ask: "Why didn't Johnny get an A in math? He got one in all his other subjects." Some parents may insist that their child be placed in the top math group, even when the child does not belong there.
It is important to get support from an administrator when it comes to difficult issues such as math groupings. Ask that they be present at conferences with difficult parents.
If a student was present for all the material taught, but is absent on the day of the test, then on the day the student returns, inform him/her of the make-up day and time. Don't let it go more than a day or two. However, if the student missed part or all of the material taught, you should give him a deadline by which to make up all missed work, and a new test date. It may be helpful to contact the parent in this case. A student should not be penalized for being absent. However, they can be penalized for failing to make up missed work.
I recommend a technique called "Front Loading". Students are most motivated to learn at the beginning of the school year. Rather than reviewing material from the previous school year, why not introduce a topic they haven't seen before?
Try to teach students good problem-solving skills. When your students enter the work place, their superiors will not give them a worksheet with 25 least common multiple (LCM) exercises. They will more likely have a scheduling problem that needs to be solved using LCM concepts.
To motivate students, give out awards for both good academics and for good effort.
Do your best to be fair to students. You will earn their respect this way.
The best motivator of all is connecting math to the real world. For example, when teaching the metric system, have students bring in empty cartons and bottles from their kitchen.

Test Anxiety by Virginia W. Strawderman, Ph.D.

Test Anxiety (the following information is loosely based on that of John Zbornik and Ellen Freedman)

Two major components seem to comprise test anxiety.

The cognitive aspect centers on worry which may include poor self-image, feelings of failure, or catastrophic thoughts.

The emotional aspects center on somatic disturbances such as stomach upset and headaches. Symptoms of nervousness such as shaking hands, sweating palms, dry mouth, shallow breathing, heart palpitations, and elevated blood pressure may also be present.

Behavioral responses may vary from focusing on one item, "checking out," hyper sensitivity to noise or other environmental stimuli, and "freezing up."

Assessing test anxiety should include, but not be limited to, getting a history from the student and parents, teachers, etc. and using rating scales.


Test Anxiety Assessment (based somewhat on John Zbornik’s work)

  1. What kinds of things happen in your body while you are taking a test?
  2. How is your breathing?
  3. How does your stomach feel?
  4. How does your head feel?
  5. Are you able to study the night before a test?
  6. How nervous do you feel when starting at test?
  7. Does the level of nervousness change and you progress through the test? How?
  8. Does your mind ever just go blank before or during a test?
  9. Even when you have studied a lot, do you still get nervous?
  10. Do you sometimes get stuck on a question or problem and can’t go on?
  11. Do you have trouble finishing tests?
  12. Does the subject matter of the test make a difference in your feelings while you are taking the test?
  13. What kind of things do you think about when you are taking a test?
  14. What thoughts go through your mind?

After a student has answered questions that are consistent with test anxiety, the next step is to help the student try to deal with the response.

[Note: The above questionnaire, with room for responses, can be printed here. .]


Suggestions to help students cope with math test anxiety

Teach students how to study for math tests by making note cards, working problems from classwork, homework, tests and quizzes.

Help students construct practice exams or practice tests that are available in books or through teachers.

Use other means to help the student "desensitize" by practicing test-like conditions.

Give positive reinforcement for good work and gentle correction for mistakes.

Teach students how to work backwards and/or eliminate answers on multiple-choice tests.

Help students practice doing the questions or problems in three waves: Easy, medium, and hard so they can maximize the time allowed.

Teach students about the physiology of test anxiety and to not be distracted by body responses.

Instruct students to eat meals with both carbohydrates and protein prior to the test.

Instruct students to try to exercise just enough to become a little bit tired prior to entering the testing situation (It lessens the affect of adrenaline caused by anxiety).

Help the students learn to have productive self-talk (rather than destructive self-talk).

"My job is to do the best I can on this test today."

Help students increase their ability to focus on the task of taking the test and every time attention wanders to refocus.



About The Author

Virginia W. Strawderman, Ph.D. did her dissertation on Math Anxiety. She runs Home Math Help, which has developed and produced the MathHELPS series of games and activities for young children.

Teaching Values Through A Problem Solving Approach to Mathematics

by Margaret Taplin
Institute of Sathya Sai Education, Hong Kong

For many reasons, the state of society has reached a stage where it is more critical than ever to educate people in the traditional values of their culture. In recent years there has been considerable discussion about whether it is the responsibility of schools to impart values education. There is growing pressure for all teachers to become teachers of values, through modelling, discussing and critiquing values-related issues.
There are many opportunities to teach the principles of values education through existing subjects and topics. The purpose of this article is to suggest one of the many ways in which values education can be incorporated into existing mathematics curricula and approaches to teaching mathematics. In particular, it will focus on ways in which values education can be enhanced by utilising a problem-solving approach to teaching mathematics. The articles include quotations, printed in italics, from the Sathya Sai Education in Human Values program, which originated in India and is now active in more than 40 countries around the world.
These quotations are concerned with the following values:
  • equipping students to meet the challenges of life
  • developing general knowledge and common sense
  • learning how to be discriminating in use of knowledge, that is to know what knowledge is appropriate to use for what purposes
  • integrating what is learned with the whole being
  • arousing attention and interest in the field of knowledge so it will be mastered in a worthy way

Why Can Values be Enhanced by Teaching Mathematics via Problem Solving?
Increasing numbers of individuals need to be able to think for themselves in a constantly changing environment, particularly as technology is making larger quantities of information easier to access and to manipulate. They also need to be able to adapt to unfamiliar or unpredictable situations more easily than people needed to in the past. Teaching mathematics encompasses skills and functions which are a part of everyday life.
Examples:
  • reading a map to find directions
  • understanding weather reports
  • understanding economic indicators
  • understanding loan repayments
  • calculating whether the cheapest item is the best buy

Presenting a problem and developing the skills needed to solve that problem is more motivational than teaching the skills without a context. It allows the students to see a reason for learning the mathematics, and hence to become more deeply involved in learning it. Teaching through problem solving can enhance logical reasoning, helping people to be able to decide what rule, if any, a situation requires, or if necessary to develop their own rules in a situation where an existing rule cannot be directly applied. Problem solving can also allow the whole person to develop by experiencing the full range of emotions associated with various stages of the solution process.
Examples:
  • The problem that we worked on today had us make a hypothesis. Through testing, our hypothesis was proven incorrect. The problem solving approach allowed our group to find this out for ourselves, which made the "bitter pill" of our mistake easier to follow.
  • I found this activity to be quite a challenge. I felt intimidated because I could not see an immediate solution,and wanted to give up. I was gripped by a feeling of panic. I had to read the question many times before I understood what I had to find. I really had to "dig down" into the depths of my memory to recall the knowledge I needed to solve the problem.
  • Seeing patterns evelop before my own eyes was a powerful experience: it had a stimulating effect. I felt that I had to explore further in a quest for an answer, and for more knowledge.

Extracts from a student teacher's journal after three separate problem solving sessions


The student who wrote the extracts above, has illustrated how interest rooted in the problem encouraged steady interest needed to master worthy knowledge. Experience with problem solving can develop curiosity, confidence and open-mindedness.


How To Teach Human Values By Incorporating Problem Solving Into The Mathematics Program.
This section will describe the types of problem solving which can be used to enhance the values described above, and will give some suggestions of how it can be used in the mathematics program.
There are three types of problems to which students should be exposed:
  1. word problems, where the concept is embedded in a real-world situation and the student is required to recognise and apply the appropriate algorithm/rule (preparing pupils for the challenges of life)
  2. non-routine problems which require a higher degree of interpretation and organisation of the information in the problem, rather than just the recognition and application of an algorithm (encouraging the development of general knowledge and common sense)
  3. "real" problems, concerned with investigating a problem which is real to the students, does not necessarily have a fixed solution, and uses mathematics as a tool to find a solution (engaging pupils in service to society).

Each of these problem types will be described in more detail below.


Problems which require the direct use of a mathematics rule or concept.
By solving these types of problems, students are learning to discriminate what knowledge is required for certain situations, and developing their common sense. The following examples have been adapted from the HBJ Mathematics Series, Book 6, to show how values such as sharing, helping and conserving energy can be included in the wording of the problems. They increase in difficulty as they require more steps:
Examples:
  • 7 children went mushrooming and agreed to share. They picked 245 mushrooms. How will they find out how many they will get each?
  • Nick helps his elderly neighbour for 1/4 of an hour every week night and for 1/2 an hour at the weekend. How much time does he spend helping her in 1 week?
  • Recently it was discovered that a clean engine uses less fuel. An aeroplane used 4700 litres of fuel. After it was cleaned it was found to use 4630 litres for the same trip. If fuel cost 59 cents a litre, how much more economical is the clean plane?
Sometimes it is important to give problems which contain too much information, so the pupils need to select what is appropriate and relevant:
Example:
Last week I travelled on a train for a distance of 1093 kilometres. I left at 8 a.m. and averaged 86 km/hour for the first four hours of the journey. The train stopped at a station for 1 1/2 hours and then travelled for another three hours at an average speed of 78 km/hour before stopping at another station. How far had I travelled?

To be able to solve these problems, the pupils cannot just use the bookish knowledge which they have been taught. They also need to apply general knowledge and common sense.
Another type of problem, which will encourage pupils to be resourceful, is that which does not give enough information. These problems are often called Fermi problems, named after the mathematician who made them popular. When people first see a Fermi problem they immediately think they need more information to solve it. Basically though, common sense and experience can allow for reasonable solutions. The solution of these problems relies totally on knowledge and experience which the students already have. They are problems which are non-threatening, and can be solved in a co-operative environment. These problems can be related to social issues, for example:
Examples:
  • How many liters of petrol are consumed in your town in a day?
  • How much money would the average person in your town save in a year by walking instead of driving or taking public transport?
  • How much food is wasted by an average family in a week?

Using a Fermi Problem to Promote Human Values
Ms. Lam wanted to teach her class of ten-year-olds about the value of money, and to appreciate what their parents were doing for them:
"I believe that students should be aware of this important issue and thus can be more considerate when a money issue raised in their own family, such as failure to persuade their parents to buy an expensive present. In solving the problems, I think that students can have a better understanding of the concept of money, not simply as a tool of buying and selling things.
"First I told the class a story about Peter's argument with his family. Peter failed to persuade his parents to buy expensive sportshoes as his birthday present and thought that his parents did not treat him well. The parents also felt upset as they regarded this son as an inconsiderate child. They thought that he should understand that the economy is not so good. They asked Peter if he knew about how much money was being spent on him throughout the whole year. Unfortunately, Peter could not produce the answer immediately. So I asked the class if they could help Peter. I asked them to find answers to the following problems:
  • How much money do your parents spend on you in a year?
  • How much money have your parents spent on you up till now?
  • How much money will your parents have spent on you by the time you finish secondary school?
  • How much money will be spent on raising children in the whole country this year?
"The students were formed into groups of 4 to find out the possible data that they need to know. Later, the groups were asked to present their data and the way of finding out the answer. Finally, I concluded that this is an open question as each person may have different expenditure along with some common human basic needs such as food, clothes and travelling fares. Anyway, the answer should be regarded as a large sum of money and thus give them a better understanding of their parents' burden."

Sometimes pupils can be asked to make up their own problems, which can help to enhance their understanding. This can encourage them to be flexible, and to realise that there can be more than one way of looking at a problem. Further, the teacher can set a theme for the problems that the pupils make up, such as giving help to others or concern for the environment, which can help them to focus on the underlying values as well as the mathematics.

Non-Routine Problems
Non-routine problems can be used to encourage logical thinking, reinforce or extend pupils' understanding of concepts, and to develop problem-solving strategies which can be applied to other situations. The following is an example of a non-routine problem:
What is my mystery number?
  • If I divide it by 3 the remainder is 1.
  • If I divide it by 4 the remainder is 2.
  • If I divide it by 5 the remainder is 3.
  • If I divide it by 6 the remainder is 4.

Real Problem Solving
Bohan, Irby and Vogel (1995) suggest a seven-step model for doing research in the classroom, to enable students to become "producers of knowledge rather than merely consumers" (p.256).

Step 1: What are some questions you would like answered.
The students brainstorm to think of things they would like to know, questions they would like to answer, or problems that they have observed in the school or community. Establish a rule that no one is to judge the thoughts of another. If someone repeats an idea already on the chalkboard, write it up again. Never say, "We already said that," as this type of response stifles creative thinking.
Step 2: Choose a problem or a research question.
The students were concerned with the amount of garbage produced in the school cafeteria and its impact on the environment. The research question was, "What part of the garbage in our school cafeteria is recyclable?"
Step 3: Predict what the outcome will be.
Step 4: Develop a plan to test your hypothesis
The following need to be considered:
  • Who will need to give permission to collect the data?
  • Courtesy - when can we conveniently discuss this project with the cafeteria manager?
  • Time - how long will it take to collect the data?
  • Cost - will it cost anything?
  • Safety - what measures must we take to ensure safety?
Step 5: Carry out the plan:
Collect the data and discuss ways in which the students might report the findings (e.g. graphs)
Step 6: Analyse the data: did the test support our hypothesis?
What mathematical tools will be needed to analyse the data: recognising the most suitable type of graph; mean; mode; median?
Step 7: Reflection
What did we learn? Will our findings contribute to our school, our community, or our world? How can we share our findings with others? If we repeated this experiment at another time, or in another school, could we expect the same results? Why or why not? Who might be interested in our results?
"The final thought to leave with students is that they can be researchers and producers of new information and that new knowledge can be produced and communicated through mathematics. Their findings may contribute to the knowledge base of the class, the school, the community, or society as a whole. Their findings may affect their school or their world in a very positive way" (Bohan et al., 1995, p.260).

Mathematical Investigations
Mathematical investigations can fit into any of the above three categories. These are problems, or questions, which often start in response to the pupils' questions, or questions posed by the teacher such as, "Could we have done the same thing with 3 other numbers?", or, "What would happen if...." (Bird, 1983). At the beginning of an investigation, the pupils do not know if there will be a suitable answer, or more than one answer. Furthermore, the teacher either does not know the outcome, or pretends not to know. Bird suggests that an investigation approach is suitable for many topics in the curriculum and encourages communication, confidence, motivation and understanding as well as mathematical thinking. The use of this approach makes it difficult for pupils to just carry out routine tasks without thinking about what they are doing.
Bird believes that investigational problem solving can be enhanced if students are encouraged to ask their own questions. She suggested that the teacher can introduce a "starter" to the whole class, ask the students to work at it for a short time, ask them to jot down any questions which occurred to them while doing it, and pool ideas. Initially it will be necessary for the teacher to provide some examples of "pooled" questions, for example:
  • Does it always work?
  • Is there a reason for this happening?
  • How many are there?
  • Is there any connection between this and.....?
The pupils can be invited to look at each other's work and, especially if they have different answers, to discuss "who is right".

Conclusion
This article has suggested some reasons why problem solving is an important vehicle for educating students for life by promoting interest, developing common sense and the power to discriminate. In particular, it is an approach which encourages flexibility, the ability to respond to unexpected situations or situations that do not have an immediate solution, and helps to develop perseverance in the face of failure. A problem-solving approach can provide a vehicle for students to construct their own ideas about mathematics and to take responsibility for their own learning. While these are all important mathematics skills, they are also important life skills and help to expose pupils to a values education that is essential to their holistic development.

References and Useful Reading
Bird, M. (1983). Generating Mathematical Activity in the Classroom. West Sussex, U.K.: West Sussex Institute of Higher Education. ISBN 0 9508587 0 6.
Bohan, H., Irby, B. & Vogel, D. (1995). 'Problem solving: dealing with data in the elementary school'. Teaching Children Mathematics 1(5), pp.256-260.

The ideas presented in this article suggest some ways in which teachers can explore the integration of values education into the existing mathematics program without needing to add anything extra. Further ideas have been presented in a book written by the author (Taplin, 1988). As well as giving teaching ideas, the book summarises recent research and suggests some questions for action research or discussion that teachers can use in their own classrooms. For further information about the article or the book, please contact the author at mtaplin@ouhk.edu.hk.

Some Questions For Discussion With Colleagues, or Action Research In Your Classroom