Overview - Introductions to Reasoning Modules

Module 1: Hypothetico-Predictive Reasoning

By all accounts, Hans was a very clever horse. According to German newspapers in the early 1900s, he could identify musical pieces, understand German, and was quite good at math. When his owner asked Hans to add numbers, he tapped out the answer with his hoof. For other questions, Hans pointed his head toward the appropriate pictures or objects. Understandably many people were skeptical. So a group of "experts" including two zoologists, a psychologist, a horse trainer, and a circus manager came to investigate. They watched closely as Hans answered question after question with near perfect accuracy. Hans could even reply correctly to questions posed by perfect strangers. If the questioner asked for the square root of 16, Hans confidently tapped his hoof four times. The experts could not discover any tricks, so they concluded that Hans was a very clever horse indeed.

Do you agree with the experts? If not, how could Hans correctly answer the questions? After reading the experts' report, a young psychologist named Oskar Pfunget came up with an alternative explanation. Suppose Hans could not think out the answers at all. Suppose instead he monitored subtle changes in the questioner's facial expressions, posture, or breathing that occurred when Hans arrived at the correct answer. Perhaps these cues told Hans when to stop tapping or moving his head. How could this body-language explanation be tested? Pfunget thought that in addition to an interrogator who knew the answers, he needed someone who did not know the answers. If Pfunget's body-language explanation is correct, and both types of interrogators ask questions, how should Hans reply? Consider the following argument:

If...Hans correctly answers questions by monitoring the interrogator's body language (body-language explanation),

and...Hans is interrogated by someone who knows the answers and then by someone who does not know the answers,

then...Hans should respond correctly to questions posed by the knowledgeable interrogator (presumably because he can monitor the interrogator's body language), but not to questions posed by the ignorant interrogator (presumably because there would be no body language to monitor).

Module 2: Probabilistic Reasoning

What are your chances of winning a $5,000 jackpot? What are your chances of being struck by an automobile while crossing the street? Do frogs with spots have a better chance of surviving than frogs with no spots? Does a farmer have a better chance of raising a good corn crop if he sprays with insecticide? The notion of chance is embedded in every part of our lives as well as that of every living organism. Therefore, understanding chance is fundamental to understanding life and the world in which we live. 

The Tennis Balls - Imagine a girl walking down a sidewalk bouncing two tennis balls -- a white one and a yellow one. Each time she bounces the two balls the yellow one bounces higher than the white one, even though she drops them both from the same height. As you watch, a second girl carrying a tennis racket runs up to the first girl and says, "Hey, Sis, give me back my yellow tennis ball. It's time for my tennis lesson." 

At hearing this, the first girl replies, "No. I want it. You can have the white tennis ball." The second girl exclaims, "I don't want the white ball. It doesn't bounce worth beans." To this the first girl says, "I'll tell you what. I'll hold both tennis balls behind my back -- one in each hand. If you guess which hand has the yellow ball, you can have it." The second girl replies, "OK, but hurry up. I'm late.... I'll pick the left hand." What are the second girl's chances of correctly guessing the hand with the yellow ball?

Module 3: Correlational Reasoning

Does smoking cause lung cancer? Does Laetrile cure cancer? Does drunk driving cause accidents? Do fatty foods cause high blood pressure? Does increased radiation increase mutations? Do small organisms have higher metabolic rates than large ones? 

You may think you know the answers to some of these questions. On what evidence and/or reasoning do you base your answers? The collection and analysis of evidence to determine whether or not two factors are "linked" are important components of scientific investigations. Finding such a link suggests the possibility of a cause-effect relationship. And cause-effect relationships are at the very heart of scientific understanding. 

Things That Go Together - Recall once again the girl walking down the sidewalk bouncing the white and yellow tennis balls. Well, her sister finally did get the yellow ball and left for her tennis lesson. As you continue to watch, the girl starts walking down the sidewalk until she comes to a bumpy dirt road. She keeps walking down the road and starts to bounce the white ball as she goes. You notice that each time she drops the ball it hits a different part of the road. Sometimes the ball hits a soft spot and a low bounce results. Sometimes the ball hits a hard spot and a higher bounce results. 

Because the height the ball bounces changes (varies) from bounce to bounce, the "height of bounce" is called a variable. If the height of the bounce stayed the same it would not be a variable. It would be a constant. Notice that the condition of the road also varies. Sometimes it is hard and sometimes it is soft. So "hardness of the road" would also be considered a variable. Of course, if the road were paved so that it was equally hard in all spots the "hardness of the road" would not be a variable. It would be a constant. 

In this example there are two variables: (1) the height the ball bounces after it hits the road and, (2) the road condition. The height that the ball bounces is either high or low and the road condition is either hard or soft. Because the two variables are "linked" (high bounces occur when the ball hits the hard spots and the low bounces occur when the ball hits the soft spots) we say that a correlation exists between the variables "height the ball bounces" and "road condition." A brief statement that summarizes this "linked" or correlational relationship would be - the harder the road the higher the bounce, or said another way -the softer the road the lower the bounce.

Module 4: Causal Reasoning

I once knew a woman who was married six times. Her first husband was killed when a truck struck his car. Her second husband was struck down in his prime by a sudden and unexpected heart attack. Her third husband drowned while swimming in the ocean. Her fourth husband met his untimely demise due to an overdose of sleeping pills. Her fifth husband contracted a rare and exotic disease and died while on a trip to South America. Her sixth husband slipped on a roller skate that had been left at the head of a long flight of stairs. The slip resulted in a fatal tumble to the bottom. Would you marry this woman? Sometimes a series of events just stretches the laws of chance too far. When this occurs we assume a hidden cause. But just what is too far? To answer this question lets take a little closer look at the idea of chance. 

Two in a Row; You Must Be Kidding! - When we last left our two girls with the tennis balls, the second girl had guessed that the yellow ball was in the first girl's left hand. Was she correct? Actually, the yellow ball was in the right hand so she was wrong. At seeing this, the second girl exclaimed, "Oh come on, I'm in a hurry so just give me the yellow ball." To this the first girl said, "No deal, but I will be nice and give you another try. But this time you must guess correctly twice in a row." 

What are the second girl's chances of guessing correctly twice in a row? Obviously, the chances are less than one out of two because it's tough to guess right once, but guessing right twice in a row is even tougher. Is it twice as hard?

Module 5: Identifying and Controlling Variables

One day at the lake two boys were overheard arguing about who could do more pushups. The bigger boy said, "There is no way that you can do more pushups than me. I'm bigger and stronger than you." To this the smaller boy replied, "OK, let's see you prove it. You go first. I'll bet you a dollar that I can do more than you." 

At hearing this, the bigger boy must have thought to himself, "What an easy way to win a dollar." He immediately got down on the beach and did 25 pushups before he tired and could do no more. The smaller boy watched patiently and appeared not the least bit shaken by the bigger boy's prowess. 

At the conclusion of the bigger boy's effort, the smaller boy smiled and jumped into the water up to his calves. He got down in the water and proceeded to do 26 pushups without breathing deeply. When he finished the bigger boy protested, "Hey, you can't do that. That's not fair. It's a lot easier to do pushups in the water than on the beach!" What do you think? Was it fair? Obviously it was not. One boy doing pushups on the beach and the other in the water is not a fair test to find out who can do more pushups. What would make it fair? 

The Tennis Balls Again - As the girl with the white tennis ball continued down the bumpy dirt road she noticed a boy walking toward her. The boy was bouncing an orange tennis ball. She noticed that the boy's orange ball seemed to bounce a lot higher than her white ball. 

After saying hello, the boy replied, "Hey your tennis ball is a real dud. My orange ball is a lot bouncier than yours." Not wanting to be outdone the girl responded, "Oh no it's not. Mine is bouncier than yours and I can prove it." 

To this the boy said, "Oh yeah let's see you prove it. Go ahead. Drop both balls and let's see which bounces higher." At this the girl held her white ball over her head and the boy's orange ball near her waist and dropped them both at the same time. The white ball bounced higher! To this the boy protested, "Hey that's not fair. You can't drop them from different heights." So the girl took the two balls again and dropped them from the same height. But this time she dropped them so that the white ball hit a hard spot in the road and the orange ball hit a soft spot. Again, the white ball bounced higher. Again the boy protested, "That's still not a fair test. My ball hit a soft spot and yours hit a hard spot. Do it again but this time don't drop them from different heights and don't let one hit a soft spot." 

Again the girl held up the two balls and released them. She did what the boy told her but this time she released them so the white ball hit the sidewalk and the orange ball hit the road. Again the white ball bounced higher than the orange ball. By this time the boy was getting rather upset. Again he protested, "You are messing up the test. Don't drop them both from different heights. Don't let one hit a hard spot and the other a soft spot and don't let one hit the sidewalk." "OK, OK," exclaimed the girl. "Let me try again." 

Again she held up the two balls and released them from the same height. They both hit hard spots in the road. But again the white ball bounced higher! This time she had cleverly spun the orange ball as she dropped it so when it hit the road it bounced at an odd angle and did not rise very high into the air. At seeing this, the boy was so upset at the girl's failure to conduct a fair test that he grabbed his orange ball, turned around, and went off down the road muttering. 

Suppose you were the boy. What could you have said to the girl to keep her from messing up the test?

Module 6: Proportional Reasoning - Flash Examples | English |   | Spanish |

If you have ever watched a good golfer you probably noticed that the harder he or she swings the club, the farther the ball goes. Things are often quite the opposite for most novices. The harder they swing, the shorter distance the ball goes. In fact, if the novice swings too hard, he or she may even miss the ball completely! Many other relationships such as these exist. The harder the road, the higher the ball bounces. The farther away an object is, the smaller it appears. The more food people eat, the more weight they gain. The faster a car travels, the more gas it uses each mile. The more a person exercises, the less his or her chances are of dying from heart disease.

As you know, such relationships are called correlations. The identification of correlations is crucial to understanding our world. Not only is it important to identify correlations, it is often important to quantify such relationships. For example, everyone knows that a correlation exists between how much of something you buy and the price you pay for it. The more you buy, the more you pay. But do you know if you get a better deal when you buy two gallons of ice cream for $1.95 or five gallons of the same kind of ice cream for $4.75?