In this chapter we prepare ourselves to study the nature of our world by laying the foundations needed in concepts of measurement, namely a review of or introduction to what is commonly called the "metric system." Additionally, we must at times use functional relations in perhaps several different forms to describe phenomena. While we certainly make every effort to avoid bogging down our studies with mathematical details, we do take the time to discuss problem solving itself and techniques involved therein. We are not planning to solve mathematical problems associated with the ideas explored in this text. But since the processes of problem have some global applications reaching far outside our investigations, we do take this opportunity to explore them.

        To be able to describe, and then to explain the world that we live in, we need a consistent reference frame and set of descriptors to use in it. We choose for our purposes of measurement to use the SI, or the French "Le Systeme International d'Unites" (International System of Units)) which is most recently identified as the "metric system." Coupled with this, and naturally so, we also will be employing the Power of Ten notation as much as possible to be able to make reasonable comparisons of objects, concepts and ideas in a straightforward manner.

        The concept of a metre (yes, that is the way the USA has agreed to spell it even though private enterprise has chosen the "meter") was not introduced until 1540, when Mouton, a Jesuit suggested that a standard base of measurement be based on one-millionth of the circumference of the earth. Even so, it was not developed and used until about the time of the French Revolution. Until then, many bizarre systems were in effect where a foot was described as the average of the first twelve men exiting a church, or a yard as the distance from the tip of one's nose to the tip of a finger. Depending on whose arms one used and whether you were buying or selling, the systems were not very consistant.
 
 
 

        The English System, (today we should appropriately call it the American system since even the British have converted to metric) is based on the number two rather than the decimal. In this manner, everything is doubled. Did you know that:
 
 

        Actually this system does have a formal basis (powers of 2) , but it fails to allow straight forward conversion between volume and weight. For example, notice the Gallon. If we keep doubling the Gallon eight times, or 2 to the 8th power, we get 256. Well, a gallon of water weighs about 8.3 pounds, so 256 gallons weighs about 2124.8 pounds. Whew! Thus the system is closely (??) related to a common base of water, but not quite equal to a ton, or 2000 pounds. Looking at the above relations, it is no wonder that we get confused ourselves, and we grew up with it.

        The metric system's simplicity and utility depends on its decimal application. Consequently, we never saw any Roman numerals such as MDCXXI.XXII They didn't use a decimal system at all.

        During the French Revolution, along with heads rolling, so did the many and varied systems of weights and measures that were in effect. In fact, at one point even a decimal year, day, hour and second were instituted. Concurrently, in the United States, Thomas Jefferson, then Secretary of State, was directed by President Washington to develop or select a reasonable system of Weights and Measures. This was required by the new Constitution. Now Thomas Jefferson was a genius and was successful at most every venture he attempted. He suggested to the Congress that the new US adopt a system that was
 

        He even suggested a metric yard, foot, and inch. Well, even though we had just come out of a war with Great Britain, we still traded with her extensively and to adopt a system not used there was unthinkable.

        It wasn't until 1865 that the question was raised formally again, this time by President Lincoln. He commissioned a congressional study, and (even though he wasn't around to follow through on it) in 1866 the USA formally adopted the metric system (as known internationally then) as the official system of the United States. It has never been repealed and today it is still the only system legal for trade.

        In 1875 we were formally a part of the Treaty of the Metre. We agreed with the rest of the world to use the standard established then so world trade could be simplified. However, Great Britain didn't adopt it and we still traded with them. We (the USA) received a platinum bar (Number # 17) that was the standard of the metre and a kilogram weight. An International Bureau of Weights and Measures was established, just outside of Paris, where it still resides. We agreed to meet periodically with the other nations of the world to reaffirm our commitment and confer on the metric system. Wars, as well as domestic difficulties, frequently interferred with these meetings.

        The other nations, including Great Britain, Japan, Canada, etc., have all converted. On Dec 23, 1975, President Gerald Ford signed the "Christmas" Metric Bill assuring that we would convert in ten years. Today, it still seems we are just "inching" our way to the SI.

        It is important to recognize that a consistent system is necessary to describe our physical world. To say that the speed of light is about 1 foot per nanosecond or that the speed of sound is about one foot per millisecond may in fact be true, and even allow one to estimate the time for sound or light to travel a certain distance, say the three feet across a table, it would in reality be confusing for many. Consistency allows for comparisons more readily.

        Since the SI is based on the decimal, it is a relatively simple matter to use the powers of ten to expand or contract from a base unit. For example, consider the following prefixes denoting larger multiples of the base system:

        Note that the prefixes describing multiples of the base are all harsh, Greek prefixes. And those describing the fractional parts are all soft, Latin prefixes.

        Once a base unit is established for the various physical quantities, then sizes can be described in terms of combinations of the above prefixes and base units. In some instances it is also desireable to use some derived units to describe quantities that involve several base units in combinations.

        Our place in the universe can be viewed in terms of these quantities. We must recognize that we are in a particular place in the universe at a particular time in the history of it. Our experiences reflect interactions with the limited quantities we observe. It is a goal of mankind, especially through scientists, to be able to described the world with a cohesive picture of the universe. This is not so easy. For example, If the standard model of the universe is reasonably correct, the universe has cooled off dramatically since the Big Bang. What we observe is how it is now, and since these changes are occurring over millions and billions of years, even the history of mankind itself is not long enough to even observe how it changes. So to describe the universe with the limited tools available to us, and especially our limited experiences, is a staggering challenge.

        We are meeting that challenge. In fact, trying to describe what the world is, how it became that way, and the ultimate destiny of the universe is but one of our objectives in this course. That is not exactly a humble task. To start at it, though, one needs to recognize our place in the universe. Consider, if you will, the following sizes and where we fit in.

"When you can measure what you are speaking about, and can express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science."

        Physics is considered an experimental science. Ever since the days of Galileo we have recognized the importance of performing experiments to determine the validity of our models of the physical world. The very nature of observation, which helps us in developing models, also requires that experiments be performed. Remember, we observe the world from a limited perspective, one instant in time, in the middle of the very large and the very small. To be able to adequately describe the characteristics of the quantity we are observing, we must often perform experiments to see how that quantity behaves as we control the environment in which we are observing it.

        I recall an individual I saw who was crawling on his hands and knees on the lawn in front of his house. Being curious, I asked him what he was doing. He answered that he was looking for a lost ring. I offered to help and asked if he knew exactly where it was dropped. He replied that he lost it in the living room of his house. After regaining my composure I asked why he was looking out here for it. He said that the light was betterl Sometimes we perform experiments not because they are appropriate, but rather because they are easy to do. Some things we shouldn't bother to measure and sometimes we avoid the necessary because they are difficult to measure.

        When we observe the world around us, we can do so in a qualitative manner. This may result in an evaluative judgment, but it would be difficult to describe well. Creative writers are good at this. They can describe the flight of a butterfly, but it does little to help us in attempting to model our world. We need to quantitatively measure certain quantities and describe them in terms that are well defined so that the rest of the world knows exactly what you are describing.

        As we make measurements, we are collecting data. We do not sporadically collect information, but very carefully measure certain quantities. We find that in fact the data collected relates several things together. We call these things variables. For example, temperature, color, size, mass are all variables. These are not necessarily related yet, but when we compare things we develop a relationship between them, usually two at a time.

        Mathematicians call these FUNCTIONAL RELATIONS. For example, we could relate the pay you receive to the number of hours worked or the grade on a test to the number of hours a student studied for it. Note that in effect the data has more meaning, because the relationship established between the variables is clearly defined.

        Mathematicians study functional relationships for their own sake, but scientists usually study them to better understand the principles describing our physical world. There are two major ways to describe functional relations, sometimes with a picture of the relationship- we call these graphs, and sometimes with a mathematical statement called an equation. We examine both in the following paragraphs.

Note that the data itself is not easily interpreted. We need to describe the relationship either through a picture of the data; ie a graph, or through an equation which succinctly describes the phenomena.

Consider the graph of the above data for a ball free falling as a function of time.
 
 

        Functional Relations are tools of the scientist, just as any other piece of equipment. They allow us to describe and explain our world better. For the above set of data, we could also write the equation:

which tells us how the data is related. We can use either the graph or the equation to predict what the distance that the ball falls in 3.5 seconds. The table of data hardly is any good for that. Note too, that the graph may not be as useful as the equation for predicting values beyond the graph, say at ten seconds. Each is useful in its own right.

        It is important to point out that we can readily perform the process of interpolation between data points.
This is easy to do with the pictorial representation, or the graph. Performing extrapolation is a shakey procedure at bests It is used when no other alternatives are available and its results must be questioned as to their validity.  Sometimes the model may describe the phenomena beyond therange of known data. At other times this is not true.

         In relating the variables, we have one variable that is controlled and its effect on the other measured. We call the control variable the INDEPENDENT VARIABLE. The variable that depends on the other is called the DEPENDENT VARIABLE. We traditionally plot them on the x and y axes respectively. How we collect some of this data, form the functional relations and use it in modeling our world to help us better understand it is a goal of this course. With these relations we can be successful in our endeavor. Keep in mind that we will use these relationships primarily in the laboratory sessions. Yet, to make comparisons and keep our perspectives
proper, we must be able to use them otherwise.

The Nature of Problem Solving

        Bloom's Taxonomy of Educational Skills categorizes learning in three major areas - the Cognitive, Affective, and Psycho-Motor Domains respectively. Look at Table 2-7 for a list of skills in the specific areas of interest. Much of schooling, especially in the classroom, concerns itself with the Cognitive Domain, even though the other areas are equally as important to the development of the whole individual. Skills in the Psychomotor Domain may be achieved  during laboratory activities such as fine motor manipulation of lab equipment, etc. One concern among many educators is still in the area of the Cognitive Domain, that of Problem Solving. According to Bloom, this is a mid-level skill, ranking well above simple knowledge or fact acquisition
 (which, unfortuneately, is where much teaching takes place) and well below Evaluation, Judgement and Synthesis levels.
 
 

 

1. Knowledge	- remembering previously learned material. Defines, identifies, states, selects
2. Comprehension	- the ability to grasp the meaning of material. Estimates, explains, summarizes, gives examples
3. Application	- the ability to use learned material in new, but still basic situations. manipulates, solves math problems, predicts, uses
4. Analysis	- the ability to break down material into component parts so that the organizational structure is better understood.diagrams, differentiates, distinguishes, outlines
5. Synthesis	- the ability to put parts together to form a new whole. categorizes, combines, plans, rewrites, creates
6. Evaluation	- the ability to judge the material of others for given purposes, based on specific criteria. appraises, concludes, supports, interprets, contrasts

Scientific Theory

After learning how to measure time by a clock, you decided to use your new knowledge to do some scientific research. After recording the time of sunrise for 7 consecutive days, and taking into account that there are uncertainties in these measurements caused by weather, reading clock, etc. here is one hypothesis you can generate, using the information collected:

The sun rises every morning in St. Paul at 6:30am ± 3 minutes !

Question: Does your work qualify as scientific research? and does your statement qualify as a scientific theory? Why ? What's the difference?

        The ability to solve problems is a necessary requisite for good judgement and evaluation. It is the basis for making good decisions. If we' are to take the knowledge from a science course and apply it to activities of daily living we should then have a basis for problem solving.

Do you believe in Bigfoot? Have you had the local pet psychic try to contact your deceased cat Fluffy? Do you believe in UFOs? Do you consult your horoscope before making a move?

If so, you’re not alone. Ghosts, aliens, angels, devils, Loch Ness monsters, plane- and ship-sucking triangles in the ocean—many people believe in things that cannot be proven scientifically.

Centuries ago, people believed in such things as a flat Earth. They believed that the Sun and all the planets revolved around the Earth and that flies were created from raw meat.. People believed that evil spirits caused sickness, and that carrying cloves of garlic, burning candles, or smoking pipes could keep you healthy. It seems ridiculous today, in the light of science and technology advances, but back then, most people believed these things. However, throughout history there have been individuals who wanted more proof, scientists who conducted tests and experiments to prove or disprove popular ideas. They didn’t accept many common beliefs unless they could establish a factual basis for them. But skeptics weren’t the norm. Even today, polls show the number of people who believe in ghosts and extraterrestrial visitors has risen in the last 30 years

        Richard E. Mayer (1:350) points out that " ~..there is no overwhelming evidence that global skills can be learned independently of specific fields ...Successful courses in problem solving are courses that emphasize specific knowledge and strategies applicable within a specific domain..." Thus, applying problem solving skills that are succesful in one area such as social studies, may not be as effective in solving problems in the sciences. With this in mind, let us consider a "generic" scientific problem solving algorithm.

        Computer scientists are familiar with the term algorithm. It is essentially a recipe to follow to accomplish a specific task. For a baker this might involve the processes of combining ingredients along with subsequent mixing and baking steps resulting in an edible cake. For scientists the scientific method itself is a methodological process that can be described as an algorithmic process. For problem solving, let us consider the following four basic steps:
 
 
 
 

        Solving problems in the real world is typically very different than word problems. The problem solver must decide what the problem is, what is the desired quantity to be known, what variables must be measured, etc. There is a lot of extraneous information always at hand. word problem seldom include these "confusion factors." Sometimes we are not even sure of what we are looking for. No wonder confusion and anxiety, perhaps even mystery, surround the process of solving problems.

        Actually, the process of solving problems is not so mysterious. While there is no easy way to solve every problem, the following procedure will generally work in most cases. Students from Junior High School, through college, and even graduate school have found success using this system.

        Before we can solve a problem we must thoruoghly understand what the problem is and what connection exists between the known and the unknown. Sometimes this is given, sometimes it is assumed that the problem solver should know it. Articulating this information is called Precisely Formulating the Problem. It may take several forms, but one very successful, almost mundane, method is to write down what is known and what is to be found; i.e., a Given and a To Find. Frequently it is also useful to draw a picture of the phenomena. Now, we know that few of us are really decent artists, but drawing a sketch with essential forces and known quantities helps us to visualize the problem and more precisely formulate it.

        Let us follow through the following example. We do this to show the process rather than suggest that this is a real problem the student will be expected to solve in this course. Suppose that one wants to find the speed of a 1 kilogram mass dropped from a 19.6 metre tower as it impacts the ground. This is not an unrealistic question, but like the real world, some of the information is unnecessary and some is assumed, not given. A knowledge (low level cognitive) of free fall reminds us that the motion of an object in free fall is independent of its mass. (Remember Galileo and his tower experiment.) Thus knowing that the object has a mass of 1 kg is extraneous. In a pure word problem this information could be not only confusing, but misleading as well. In the real world, if we needed to know the mass, we would have to weight the object.

        We must assume that the initial velocity is zero since no other information was given. We were told only that the object was dropped (not thrown downwards or upwards with any initial velocity.) Thus we can write:
 

Given:    Initial Velocity, V = 0 m/s
 Acceleration of Gravity, g = 9.8 m/s2
 Distance of fall, h = 19.6 m

To Find: Speed at impact, Vf
 

If one includes a sketch, it might look like:

        Mathematical models are useful in allowing us to make predictions about the phenomena they describe. They are necessary when  a numerical solution to a problem is required. In our example problem, the model  of motion of a free falling object is more complex than one initially suspects. All too often, textbooks show solutions by merely inserting the intial values into an equation without displaying the logic involved in getting there. Our model involves relationships between time, distance and speed. For one-dimensional motion the model looks like:

        With these two relationships serving as our model, we proceed to perform the Algebra to solve for the speed at impact. Note that we cannot use either the initial or the final equation for neither really is sufficient. Solving equation (2-2) for time and substituting this value into equation (2-3) we get:

We complete our Algebra by substituing in known values for the initial speed, VO = 0. We get:

Now, this is where it is appropriate to use our calculators and perform the Arithmetic step of the process.

        We have tip-toed through a solution of a straight forward problem. It is important to emphasize again at this point that the reader will not be expected to solve this kind of problem. Our commitment to avoiding mathematical details is still a priority, but the reader must be able to follow through certain processes so that he or she can accept and understand ideas, where they came from and what they mean. The reader will not be expected to derive results, but should be able to follow the flow of information so that so that he or she understands that modern concepts depend on scientific theories, not stories or "magic."

1. What are three basic criteria for developing a system of weights and measures? If these are good around the planet Earth, would they be good throughout the solar system, or in other galaxies? Why?
 

2. What are variables and why are they necessary for us to use them?
 

3. Distinguish between independent and dependent variables.
 

4. Describe reasons, pro or con about why the U.S. should adopt the metric system.
 

5. Contrast the Problem Solving Algorithm with what is traditionally known as the Scientific Method. How do they compare?
 

6. Describe at least one problem that is non-scientific that might be solved with the Problem Solving Algorithm as a global type process.

        There is a story of two physicists marooned on a desert island. One day one of them saw what turned out to be a mirage, but looked like a distant oasis. After crawling a great distance to it, he/she found out that it was indeed a false image. Upon returning to their camp the physicist discussed the phenomena with the first one who replied that he/she had discovered that yesterday. "Why didn't you tell me that?" the first said to the second, who replied " Who wants to publish negative results?"
 
 

1. Mayer, Richard E., Thinking, Problem Solving, Cognition, 1983.

2. Flower, T.F., An Evaluation of a Metric Teacher Inservice Training Program, University o Wyoming, 977.

3. Gronlund, Norman, Measurement and Evaluation, The Macmillan Company, New York, 97 .