Math Is Hard…

This originally was written in response to comments made to a Facebook posting by my granddaughter that “Math is hard”.  As one of my degrees is in mathematics, I obviously do not share that opinion.

It seems to me…

“The essence of mathematics is not to make simple things complicated, but to make complicated things simple.”  ~S. Gudder.

Everyone in the U.S. realizes that our math instruction system is “broken and must be fixed” if we are compete with established economic powers or emerging ones such as China.

Mathematics (from Greek μάθημα máthēma “knowledge, study, learning”) is the study of quantity, structure, space, and change.  For at least 4,000 years of recorded history, humans have engaged in the study of mathematics; the universe can only be explained and understood in terms of mathematics.

Math is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences.  In the 17^th century, the great scientist and mathematician Galileo Galilei noted that the book of nature “cannot be understood unless one first learns to comprehend the language and read the characters in which it is written”.  In the early 19^th century, the noted German mathematician Carl Friedrich Gauss called mathematics the “queen of the sciences” because it was so successful at uncovering the nature of physical reality.

Why do so many people claim math is a difficult subject when, in reality, it is the simplest subject offered?  Math makes the complex easy.  Math is the only basic subject that is totally subjective – you can be relatively certain that 2+2=4 regardless of the instructor’s preferences.  Additionally, the vocabulary required to understand math is remarkably small.

Students are told mathematics is “important in your daily life” but they do not have sufficient experience to appreciate why it might be important.  It also is generally accepted that if a student dislikes math, it probably is due to grade school trauma at the hands of a heartless teacher who themselves did not like math.  Although there is certainly some unfortunate history on how math has been taught in the past, it is simplistic to attribute all the challenges of math to a single cause.

To a certain extent, school mathematics traditionally is taught from the general to specific.  Typically, a concept is first introduced, then “practiced”, followed by some “word problems” at the end of the chapter where the solution process involved is applied.  While logical, this is the reverse of the way learning usually takes place.  We normally initially learn a solution to some specific problem and then progress to a general understanding of basic concepts.  Math should not be just memorizing equations, it is understanding equations.

We find much of nature to be interdependent upon other relationships of which without math we otherwise might not be aware.  Not only does a basic knowledge of math simplify what needs to be remembered, it integrates everything in a form that provides a much better understanding of how the world actually works.  Without this understanding, much of the natural beauty of our everyday lives remains invisible to us.

Rather than memorizing numerous formulas, it is much simpler to remember only basic relationships.  For example, the three laws of motion were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, first published in 1687.  The second law is that a force f is equal to the product of mass m and acceleration a or f=ma.  If we think of a derivative as being a measure of change:

• Velocity is the first derivative of distance with respect to time,
• Acceleration is the first derivative of velocity (the second derivative of distance) with respect to time…

 While only intended to illustrate the simple mechanical relationship between force, mass, velocity, and acceleration, other much more complex relationships either are based on or can be derived from this single formula using basic mathematics.  Which is easier, to remember one general formula and some basic concepts or dozens of special purpose formulas without any understanding of how they relate?

All math, no matter how difficult, is capable of being understood in the same way that 2+2=4 can be understood; and if your teacher cannot explain it to you, then he or she does not understand it.

To students who have not learned how to multiply quickly, who aren’t sure what a percentage is, and whose knowledge of fractions is meager, a problem can seem forbidding — homework assignments require too much reading.  If the student does not read well, as is frequently the case, they run into trouble even if their natural mathematical abilities are strong.  If no adult is around to walk them through the homework assignment, they probably will dash off a string of guesses and quickly head off to watch TV.

For a student to learn, the material has to make sense.  Students do not perceive abstract algebraic symbols as numbers.  Instructors need to teach an understanding of mathematical concepts and procedures – not only the “how” something works but the “why”.  It seems as if instructors do not keep in mind the intended goals of what they are teaching.  Students need to understand the information around us if they are to be able to navigate their lives in this complex modern world and they need to be adequately prepared for further studies in math and science.  Hopefully, students with an understanding of math’s basic concepts also would see some of its inherent beauty.

The majority of instructors still teach most subjects they way they were taught when they were in school.  Basic methods of instruction at almost every level of education has not changed substantially in hundreds of years.  This is unfortunate as there now are a remarkable array of tools available to instructors and new methods being developed to teach math and many other subjects.  Availability of personal computer notepads provides educational benefits never previously available to students and instructors.  Lessons are available on the Web.  While not endorsing any method of instruction, The Khan Academy (http://www.khanacademy.org/) provides access to much lesson material.

Unfortunately, to many teachers, school mathematics is not really about understanding.  It is about skills.  But being able to do something is not the same thing as understanding what one is doing.

The greatest challenge for learning math is trying to grasp the intuitive processes necessary for understanding how math works.  Failing to understand what a number is and trying to special case everything makes math seem much more complicated than it needs to be.  Students need to understand that integers, fractions, decimals, exponents… are all just numbers and treated the same regardless of any difference in appearance.  Too many students fail to see the forest for the trees.

Given adequate understanding of basic concepts, a much higher percentage of students would not only incorporate math into their lives, they would appreciate its beauty and develop a love for it.

That’s what I think, what about you?