Please, No Science! I Hate Math!
Math is hard!
—Teen Talk Barbie
Math is hard and SCARY too!
While standing outside the registrar’s office at their local college, the following conversation took place:
“Bubba, I would rather snorkel without a snorkel . . . have a root canal without Novocain . . . have to take Mary Jane Snodgrass to the hoedown . . . be shaved bald-headed with dull electric shears . . . or kiss a lip-sticked pig than have to take that college math course we are required to take . . . don’t you agree?”
There was a moment of silence.
“Hmmmmmm,” Bubba muttered, “well Buck, them there math problems can be real strange and hard to understand, for sure . . . and when they substitute them real numbers with that alphabet soup . . . WOW . . . what a mess!”
“I am not a math person,” Bubba said. “I like them shop classes that teach you real stuff . . . like how to overhaul a hotrod motor . . . or how to wrestle and skin an alligator . . . or fix a leaky roof than have to sit through the misery of that math class.”
“Amen to that, brother,” Buck replied.
There was a moment of silence.
Then Mary Jane Snodgrass, who had been standing behind the two young men strolled by and stopped in front of Bubba and Buck and said, “I heard you two from over there . . . and first of all, I wouldn’t go to a hoedown or anywhere else with someone who can’t do simple math. I’m not going to the hoedown . . . instead I’ll be home studying my math . . . dancing with decimals, limits, and intergrals, like you two should be doing.”
There was a moment of stunned silence.
With a huge toothy smile on her face, Mary Jane Snodgrass, looked at them both and went on to say, “If you two ever need any help with tuning a hotrod, respecting alligators, or replacing a roof or help with that mandatory math . . . just ask . . . I would be soooooooooooo happy to educate you two bozos.”
Another moment of silence.
And then Mary Jane Snodgrass walked off on her way to the library . . . still wearing that toothy smile and feeling real good about all things mathematical and otherwise, thank you very much!
Math by Listening
One of the major tools we have in learning is our ability to listen—to hear what we are being told or taught. Listening, of course, is a two way street—we take in information and we give out information. Seems simple enough, doesn’t it? Unfortunately, it is not that simple. We all basically know that in order to share information, coordinate projects, work in teams, or coach and empower others, we have to listen to what others say. However, listening is a vastly underdeveloped skill.
Why do we have so much difficulty listening? Maybe we don’t listen because we are bored. Maybe we are distracted—texting a friend while someone else is talking to us. Maybe a significant other is rambling on while we are reading the newspaper, watching television or just daydreaming. Maybe we are playing computer games while our teacher is lecturing. Maybe we do not listen to others because what we really want, is for them to listen to us—to acknowledge that we have something important to say; to acknowledge that we are funny or interesting; or, more importantly, to acknowledge that we exist. Maybe we don’t listen simply because we just don’t care about anything anyone else has to say. Sound familiar?
As teachers, we fully appreciate the need for listening. As teachers, we fully appreciate that all people are not listening. As teachers, we fully appreciate that there is a huge difference between listening and hearing. As teachers, we know that agonizing feeling of delivering a lecture that falls on deaf ears. As teachers, we know that when we lecture, the students might hear the words coming out of our mouths, but they may not really be listening to what we are saying. This problem is evident when we grade examinations and the simplest of the simplest questions are answered incorrectly.
This dilemma leads to some obvious questions: Why is there a universal listening problem? What is the problem with listening? We are not qualified to answer these questions in an all-encompassing and universal manner, but we can and do speak of the listening problem when it comes to learning.
Listening and hearing are critical to learning, but delivery of the message is just as important. For example, have you ever sat in a classroom and listened while the teacher presented his or her lesson to the class, and at some point been confused about the concept being taught? Haven’t we all experienced this situation sometime in our formal learning experience? Consider, for example, you are sitting in a math classroom where the instructor is using the chalkboard to explain an algebra problem. You follow and take notes as the instructor writes the problem and the solution on the chalkboard. At some point in the instructor’s explanation, you become confused—not sure of some particular point. So you raise your hand to ask for clarification. One of two things happens: The instructor ignores you (making you look like an idiot in front of your peers) or the instructor calls on you and abruptly tells you not to interrupt his or her presentation (the instructor does not want to lose his or her train of thought or simply does not want to answer); maybe the instructor explains that he or she will address your question after their presentation. After the presentation, what happens? You wait on the instructor’s explanation but it is never offered. So, after class is over, you have the audacity to ask again. The standard answer: “I do not have time to explain it right now . . . besides, all that is in the textbook . . . read your book.” So, the student, who does not want to be embarrassed again by this instructor (or any other instructor), clams up. He or she may pretend to get it, but of course, they go on not really understanding what they needed to understand; understanding is only for those who understand.
None of us can get through our formal educations without having experienced a situation like the one just described. However, there might be another reason the instructor does not want to answer the student’s question. Heaven forbid, maybe the instructor does not know the answer!
Note: If you have the misfortune of attending a class in which the instructor does not take the time and effort to explain math and science in its most basic terms, you might want to look elsewhere for a class and an instructor.
Based on years of personal experience, we have concluded that a person’s ability to learn such topics as mathematics and science depends on a number of factors—all of which have something to do with presentation, listening to the presentation, and actually hearing the presentation. However, as mentioned, presentation, listening, and hearing are just a few of the factors that make learning math and science difficult. In this chapter we discuss these factors and describe what can be done to alleviate their impact on the learning process.
“I Can’t Do Math”
Generally, those who experience difficulties with math are facing or have faced a problem or difficulty with some type of math operation. For example, when interviewing students for their selection of college majors, especially those that require the taking and successful completion of math and science courses, it is not uncommon to hear some students state their fear for anything involving math. One standard comment is “I have no problem with adding, subtracting, dividing, and multiplying—those are simple—but when it comes to anything higher than that, I just don’t get it—I just don’t have a clue.” Probing deeper for more detailed responses from the students who “just don’t get it,” we usually find that it is the math in the abstract (algebra and above) that befuddles and scares them away from math and science. Again, somewhere in their experience they have had a problem with some type of math operation, which, in turn, causes them to develop distaste for math or anything closely related to it.
This attitude about math (and therefore science) is widespread. To verify this, just ask a few friends or acquaintances if they like math, or, if they are good at it. Even those of us who like math have areas in math in which we are weaker. Why is this?
There are at least six reasons people have had or continue to have difficulties in mathematics. Each of these problems is addressed below and recommended resolutions are provided (Price 1991; Spellman 2006).
Fear of the Known and the Unkown
We feel that the ultimate limiting factor in what is decided to do or not to do comes down to fear. This is not to say that all fear is bad, because it is not. Our view, and that of the novelist Stephen King, is that fear is a good thing in that it is the one thing that works to make us safe. We do not walk down the center of a busy road or highway because of the fear of getting rundown or run-over by a motorcycle, car, or truck. We wear personal protective equipment such as eye protection when hammering or chiseling on something that can produce shrapnel-like flying objects that could damage our eyes, or worse. We wear life vests when boating in fear of accidental drowning. We keep an eye on the food cooking on the stove in fear of burning the house down. We check our food to make sure it is not spoiled to prevent food poisoning. We get an annual flu shot in an attempt to stay flu-free during the flu season. We lock our doors at night to make us feel safe from those things that go bang-bang or worse in the night. We are safe on the job in fear that if we are not, we might hurt ourselves, someone else, or lose our jobs for being unsafe. Some of us fear taking an examination because we haven’t studied for it, or, we expect it to be difficult even if we have studied, and because of this fear, we study and actually do pass the examination. Most of us feel this is a good thing—especially the exam taker.
Fear is something we live with everyday and if we have a bit of common sense, we will have just enough fear in our daily routines to protect ourselves and others from injury or accident. But, like anything else in life, fear can be overdone. Fear is a double-edged sword; on one side is the need to protect ourselves, and on the other side is irrational behavior. We say “irrational” in the sense that fear sometimes prevents us from doing things that will help us, make our lives better, such as learning math and science. We have found, for example, that fear has driven potential students away from those majors that could provide long-term, beneficial employment for them in the future. Long-term employment in careers that include math and science not only pay well, including benefits, but also help to ensure a prosperous future. To help make our point about the importance of an education refer you to recent projections from the Bureau of Labor Statistics (BLS). Table 8.1 shows that the employment change between 2016 and 2026 for those with a master’s degree will be the highest at 15.8 percent followed by those with a doctoral or professional degree at 13.4 percent. In contrast, for those with a high school diploma or equivalent, the change will only be 5.2 percent.
Table 8.1 Employment, Wages, and Projected Change in Employment by Typical Entry-Level Education (Employment in thousands)
Meanwhile, Table 8.2 from BLS shows that computer and mathematical occupations are expected to grow 13.5 percent from 2016 to 2026 while all occupations will only grow 7.4 percent. Additionally, computer and mathematical occupations have one of the highest annual wages at $82,830—only management occupations had a higher annual wage at $100,790.
Table 8.2 Employment by Major Occupational Group, 2016 and Projected 2026 (Numbers in Thousands)
Math and science subjects are the fine-meshed filters that screen out and prevent many students from passing through the gates of high school into the gates of college or technical schools where math and science are part of the curriculum. Words like “college algebra” and “calculus” fill many students with foreboding. Many students suffer through the dread of doing math homework that they view as boring. Even simple addition or subtraction drives many to their calculators. Many advanced training programs require proficiency testing in both math and science in order to qualify for entry. Thus, potential students lacking in math and science are unable to qualify for, or are restricted from entering into training programs or they shy away from even attempting to qualify—because of the fear factor.
Math has been vilified. From the earliest days in school students are told: boys are better in math than girls; math is not creative; it’s bad to count on your fingers; some people have a “math mind” and some don’t; math requires logic, not intuition; you must always know how you got the answer; math requires a good memory; and there is a magic key to doing math (Kogelman and Warren, 1979).
In America, in the past, it was possible to graduate from high school and enter the workforce in a factory as an assembly line or maintenance trainee, or fill some other entry level position. With hard work, experience, initiative, and continued training it was possible to advance to mid-level management positions within these types of industrial organizations. The problem is that at the present time it is difficult to find these types of jobs in America because many of them no longer exist—not in America, anyway.
Many young high school graduates choose the military option as a means of temporary employment and to gain training in some technical field that will enable them to find good paying jobs upon discharge from the service. Again, however, even in the military (and especially in the military in many cases), recruiters are looking for young men and women who possess the aptitude to be trained to operate sophisticated equipment and/or machinery, weapons systems, and to fill other technical billets. For example, one of the most technical specialties in the U.S. Navy is nuclear power reactor operations aboard naval ships and submarines. In order to perform the required duties for these positions, the operators must be extensively trained and skilled; they must have the aptitude and preliminary training in math and science. Because of the math, science and high aptitude requirements needed for the best training and jobs in the military, most enlisted people end up learning some other skill set that may or may not include the training and experience that civilian employers are seeking. One thing is certain, however; those young people who do not possess some form of advanced or technological training find themselves behind the curve in gaining beneficial employment; they become lost or non-digitized in this information age. When you consider that one of the most important assets young people possess is their seemingly endless supply of energy and their resiliency to bounce back, it is a shame to deflate their enthusiasm because there is no place for marginally trained or unskilled workers in an increasingly technical workplace.
It is in the formative years in middle school and high school where young people must be turned on and tuned in to math and science. There must be a switch to turn them on. There must be someone out there to turn on the switch that lights the path to young peoples’ future success. The switch has to be the parents, counselors, and teachers. In many cases, the parents, counselors, and teachers must not only be the switch to enlightenment, encouragement, and persistence but also the ones who turn on the switch—and leave it on.
The key to learning math and science is to overcome the fear of math and science. To do this someone must lead the way and provide counseling to the young person seeking guidance. Simply, the fear of doing math or science must be shifted to the fear of not doing math or science.
Out of Sequence
Have you ever baked cookies, bread, pies, or a cake—from scratch? Have you ever baked any of these goodies without using a recipe? If you did and you were successful, you must have memorized the step-by-step addition of the various ingredients. However, if not, you may have overlooked or left out a step, and the result is usually a disaster. We think all would agree that recipes are important in successful cooking and in the proper preparation of food.
If you have ever worked in a scientific, environmental, or school laboratory, and have been required to perform certain laboratory procedures using chemicals or biologicals, you have no doubt used a lab standard, a “recipe book” on how to perform the step-by-step procedure and analysis.
One professional whom we do not expect to be reading any kind of “recipe” while performing his or her work is the surgeon. There is little doubt that a surgeon must follow a step-by-step procedure in performing his or her surgery (we certainly hope this to be the case). But the surgeon follows a protocol of steps that he or she has learned and memorized through countless hours of practice.
The mathematician and scientist works like all three examples given above. But in the case of the mathematician, he or she is much more like the physician than the other two because the math steps are remembered or memorized, and followed as a normal function of performing math operations.
Math is a concept that builds upon itself; it is primarily sequential. And herein lays one of the biggest problems with learning mathematics. That is, if you miss a class, or a mathematical step, procedure, or rule, calculations that build on what you missed will suffer (similar to leaving out a step in baking bread from scratch). The dilemma snowballs; it gets out of control and you suddenly find yourself lost and not having a clue how to compute advanced operations. This occurs, of course, because you have weakened your foundational skills by missing certain operational steps. When this happens, it is rather easy to understand why the student concludes, “I can’t do math.”
Mathematical problem-solving skills must be learned in a structured, sequential and systematic manner. To be successful in math, you must follow step-by-step sequences. Knowledge in math is cumulative; leave no gaps in your understanding of the day to day examples, procedures, and exercises.
Not only is the learning of math based on learning step-by-step operations, but performing math operations is about understanding orders of operation—sequence of operations.
Recall the student who stated: “I have no problem with adding, subtracting, dividing and multiplying—those are simple,” etc. While it is true that we use these four basic math operations almost daily in our lives, it is not that often that we are called upon to use all of them to solve one problem. However, if we are required to work a problem involving all four operations to determine the proper solution to the problem, the solution is not as straightforward as you might think. In fact, the only way to correctly solve this type of problem is to follow the rules for performing sequence of operation type problems. For instance, consider the following math problem.
At first glance and first calculation we look at the above problem and say to ourselves: “Gee, what could be easier?” Our answer to your question? “A lot of math problems could be easier than this one.” Why? Simply because you can’t determine the correct answer unless you know the step-by-step rules for this type of operation. Let’s work this problem and we think you will see what we are talking about.
Again, at first glance, our intuition and basic math training tells us to simply plug the numbers and operators into our handheld calculators and we will get the correct answer?
Is this correct? Let’s find out.
Example 8.1
Problem:
Solve the following: 60 – 25 ÷ 5 + 15 – 100 ÷ 4 x 2 =
Solution:
Using the calculator and working left to right:
We enter 60 subtract 25 divide by 5 plus 15 – 100 divide by 4 multiply by 2 =
If your answer is 39, this is the wrong answer.
So, what is the correct answer?
To find the correct answer we must follow certain rules.
Rule: In a series of mixed mathematical operations, the rule of thumb is, whenever no grouping is given, multiplications and divisions are to be performed first and in the order written, then additions and subtractions in the order written. In a series of different operations, parentheses ( ) and brackets [ ] can be used to group operations in the desired order.
Thus, 60 – 25/5 + 15 – 100/4 x 2 = 60 – (25/5) + 15 – (100/4) 2
Working left to right and performing the divisions and multiplications first:
20 is the correct answer.
Again, math is primarily sequential—concept builds upon concept. Missing any concept makes performance of calculations based on those concepts difficult. Another point to consider is that you may have been taught math concepts in their proper sequential order, but you may be weak in their actual or proper use. To master concepts they must be practiced; this requires effort—ongoing effort.
The Missing Link
When building a brick foundation, the bricks are placed one upon the other (linked to each other) to form the entire foundation. Learning math is very similar to this brick-by-brick foundation-building procedure. When presented with a math concept, it is essential that you link it to something you already know. The idea is to construct a chain of math concepts whereby each link of the chain is linked to the other. It is important that each link be fully understood. If you do not understand a new concept, ask the instructor or consult other references. Without the linking of concepts, remembering the concept is difficult.
Understanding the Overall Concept
Learning how to perform math operations such as adding, subtracting, multiplying, dividing, fractions, ratio and proportion, percentages, and others is important. These operations are learned through practice and application. We have found that if students understand what the purpose of these and other math operations are—the big picture—they have a much easier learning process. Price (1991) points out that this “big picture” approach is not only integral to the learning process but also provides the skeleton on which all the details may be hung. This skeleton with its attached details helps to improve recall, because individual facts are part of a unified structure.
It is important to point out that gaining understanding of the “big picture” approach to learning and understanding math rests largely in the hands of the instructor and the textbook.
Use It or Lose It
We’ve all heard the line, “Use it or lose it.” Others say, “Well, most things we have not done for a while come back to us . . . you know, like riding a bike.” One thing we know for certain; operations in math and science are not like riding a bike. We know that things like square root, long division, and scientific formulas become more difficult once we’ve been out of school for a few years. Herold (1983) points out how easily or how long we remember something depends on how often we retrieve it. The more often we use it, the more accessible it becomes. This statement applies to math and science. The more we practice and use various math calculations and science concepts, the easier they become; in time they become second nature to us.
The Language of Math
Earlier we pointed out that one of the major limiting factors in learning science is fear. For those with a fear of math (and apparently there are a large number in this category), let’s take a look at how one seasoned math teacher puts it: “Those who have difficulty in math often do not lack the ability for mathematical calculation, they merely have not learned, or have been taught, the language of math” (Price, 1991). The natural language of science is mathematics. Stated differently, science is quantitative, and the natural way to express quantification is mathematics (Trefil, 2008). In our estimation and experience, the number one problem with people learning math skills is that they fail to learn the language of math. One can’t be expected to work beyond the basics of mathematics (simple addition, subtraction, multiplication, and division) unless they understand the terms used in mathematics.
The point just made about the language of math and its relationship to science is well taken, and can be expanded somewhat to “The language of math is the universal language of science.” This is logical when you consider that mathematical symbols have the same meaning to people involved in science speaking in many different languages throughout the world.
Mathematics is numbers and symbols. Math uses combinations of numbers and symbols to solve practical problems. Every day, you use numbers to count. Numbers may be considered as representing things counted. The money in your pocket or the power consumed by an electric motor is expressed in numbers.
As mentioned, the greatest single cause of failure to understand and appreciate mathematics is not knowing or understanding the key definitions of the terms used. In mathematics, more than in any other subject, each word used has a definite and fixed meaning.
The following basic math terms and definitions will aid you in understanding that the language of math is important in comprehending math operations.
• An integer, or an integral number, is a whole number. Thus 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 are the first 12 positive integers.
• A factor, or divisor, of a whole number is any other whole number that exactly divides it. Thus, 2 and 5 are factors of 10.
• A prime number in math is a number that has no factors except itself and 1. Examples of prime numbers are 1, 3, 5, 7, and 11.
• A composite number is a number that has factors other than itself and 1. Examples of composite numbers are 4, 6, 8, 9 and 12.
• A common factor, or common divisor, of two or more numbers is a factor that will exactly divide each of them. If this factor is the largest factor possible, it is called the greatest common divisor. Thus, 3 is a common divisor of 9 and 27, but 9 is the greatest common divisor of 9 and 27.
• A multiple of a given number is a number that is exactly divisible by the given number. If a number is exactly divisible by two or more other numbers, it is a common multiple of them. The least (smallest) such number is called the lowest common multiple. Thus, 36 and 72 are common multiples of 12, 9, and 4; however, 36 is the lowest common multiple.
• An even number is a number exactly divisible by 2. Thus, 2, 4, 6, 8, 10, and 12 are even integers.
• An odd number is an integer that is not exactly divisible by 2. Thus, 1, 3, 5, 7, 9, and 11 are odd integers.
• A product is the result of multiplying two or more numbers together. Thus, 25 is the product of 5 x 5. Also, 4 and 5 are factors of 20.
• A quotient is the result of dividing one number by another. For example, 5 is the quotient of 20 divided by 4.
• A dividend is a number to be divided; a divisor is a number that divides. For example, in 100 ÷ 20 = 5, 100 is the dividend, 20 is the divisor, and 5 is the quotient.
• Area is the area of an object, measured in square units.
• Base is a term used to identify the bottom leg of a triangle, measured in linear units.
• Circumference is the distance around an object, measured in linear units. When determined for other than circles, it may be called the perimeter of the figure, object, or landscape.
• Cubic units are measurements used to express volume, cubic feet, cubic meters, etc.
• Depth is the vertical distance from the bottom of the tank to the top. This is normally measured in terms of liquid depth and given in terms of sidewall depth (SWD), measured in linear units.
• Diameter is the distance from one edge of a circle to the opposite edge passing through the center, measured in linear units.
• Height is the vertical distance from the base or bottom of a unit to the top or surface.
• Linear units are measurements used to express distances: feet, inches, meters, yards, etc.
• Pi (π) is a number in the calculations involving circles, spheres, or cones: π = 3.14.
• Radius is the distance from the center of a circle to the edge, measured in linear units.
• Sphere is a container shaped like a ball.
• Square units are measurements used to express area, square feet, square meters, acres, etc.
• Volume the capacity of the unit, how much it will hold, measured in cubic units (cubic feet, cubic meters) or in liquid volume units (gallons, liters, million gallons).
• Width is the distance from one side of the tank to the other, measured in linear units.
When the basic math terms are understood, the actual calculation process is easier. In addition to knowing and understanding the terms listed above, the user needs a standard method or “recipe” to follow in making calculations—a set of calculation steps. A commonly used standard methodology used in making mathematical calculations includes:
1. If appropriate, make a drawing of the information in the problem.
2. Place the given data on the drawing.
3. Ask, “What is the question?” This is the first thing you should ask along with, “What are they really looking for?”
4. If the calculation calls for an equation, write it down.
5. Fill in the data in the equation—look to see what is missing.
6. Rearrange or transpose the equation, if necessary.
7. If available, use a calculator.
8. Always write down the answer.
9. Check any solution obtained. Does the answer make sense?
Important Point: Solving word math problems can be difficult. Solving word problems is made easier, however, by understanding a few key words.
Math Problem Key Words:
• of: means to multiply
• and: means to add
• per: means to divide
• less than: means to subtract
Scientifically Literate without Math
From the material just presented you may think our intention is to teach you mathematics. We can assure you, however, that is not our intention. We are simply throwing some basic concepts, key terms, and definitions at you to demonstrate that if you know the fundamentals or the “grammar” of mathematics, you (and anyone else) can perform the math operations.
This does not mean, however, that you need to master the concepts of mathematics or actually perform math operations to be scientifically literate. So, what are we saying here? Simply, you do not need math to understand science; you need math to perform science. Most concepts of science are rather simple and straightforward; they can be understood by the non-mathematician. If you are confused, blame the scientists or the scientific community. You see, scientists speak and think in “scientese.” For example, one of the scientists we know cannot compose a standard email message without putting his message into spreadsheet format—each word, sentence, and paragraph must be presented in an absolutely logical manner—to him, at least. We can assure you that this scientist is a brilliant person but he is not capable of communicating at most people’s level. Recall that earlier we stated that one of the main problems with scientific illiteracy is the failure to communicate. The truth is most scientists, with Carl Sagan being one of a few outstanding exceptions, are not good communicators.
Let’s test our ability to communicate a few laws of science to you, hopefully in an understandable way. In so doing, we have selected, arguably, three of the most important laws in science (especially in physics and classical mechanics). First, we present Sir Isaac Newton’s Laws of Motion in basic scientese. They are:
1. In the absence of force, a body either is at rest or moves in a straight line with constant speed.
2. A body experiencing a force F experiences an acceleration a related to F by F = ma , where m is the mass of the body. Alternatively, force is proportional to the time derivative of momentum.
3. Whenever a first body exerts a force F on a second body, the second body exerts a force –F on the first body. F and –F are equal in magnitude and opposite in direction.
In scientese, these laws may be difficult to understand or fully comprehend, but they are actually quite easy to understand when written in plain English.
1. Unless you do something to an object, it will not move.
2. F = ma (simply, the harder you push on something, the faster it goes)
3. If you push an object, it will push back.
From the example above, it should be clear that to be scientifically literate and understand even the most important laws in physics, you do not need the math.
There is no single solution to the problem of illiteracy in math and science. Carl Sagan (1996) sums up this issue by pointing out that the responsibility for solving the problem of illiteracy in science is broadly shared—“at every level teachers complain that the problem lies in earlier grades. And first-grade teachers can with justice despair of teaching children with learning deficits because of malnutrition, or no books in the home, or a culture of violence in which the leisure to think is unavailable.”
References and Recommended Reading
Bureau of Labor Statistics, 2018. Employment Projections: 2016–2026. https://www.bls.gov/emp/
Herold, M., 1983. You Can Have a Near Perfect Memory. New York: NTC/Contemporary Publishing Company.
Kogelman, S., and Warren, J., 1979. Mind Over Math. New York: McGraw-Hill.
Price, J. K., 1991. Basic Math Concepts: For Water and Wastewater Plant Operators. Lancaster, PA: Technomic Publishing Company.
Sagan, C., 1996. The Demon-Haunted World. New York: Ballantine Books.
Spellman, F. R., 2006. Mathematics for Environmental Engineers. Boca Raton, FL: CRC Press.