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 17th 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 19th 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=maIf 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?

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About lewbornmann

Lewis J. Bornmann has his doctorate in Computer Science. He became a volunteer for the American Red Cross following his retirement from teaching Computer Science, Mathematics, and Information Systems, at Mesa State College in Grand Junction, CO. He previously was on the staff at the University of Wisconsin-Madison campus, Stanford University, and several other universities. Dr. Bornmann has provided emergency assistance in areas devastated by hurricanes, floods, and wildfires. He has responded to emergencies on local Disaster Action Teams (DAT), assisted with Services to Armed Forces (SAF), and taught Disaster Services classes and Health & Safety classes. He and his wife, Barb, are certified operators of the American Red Cross Emergency Communications Response Vehicle (ECRV), a self-contained unit capable of providing satellite-based communications and technology-related assistance at disaster sites. He served on the governing board of a large international professional organization (ACM), was chair of a committee overseeing several hundred worldwide volunteer chapters, helped organize large international conferences, served on numerous technical committees, and presented technical papers at numerous symposiums and conferences. He has numerous Who’s Who citations for his technical and professional contributions and many years of management experience with major corporations including General Electric, Boeing, and as an independent contractor. He was a principal contributor on numerous large technology-related development projects, including having written the Systems Concepts for NASA’s largest supercomputing system at the Ames Research Center in Silicon Valley. With over 40 years of experience in scientific and commercial computer systems management and development, he worked on a wide variety of computer-related systems from small single embedded microprocessor based applications to some of the largest distributed heterogeneous supercomputing systems ever planned.
This entry was posted in Algebra, Calculus, Carl Friedrich Gauss, Education, Friedrich Gauss, Galileo Galilei, Khan Academy, Mathematics, Philosophiæ Naturalis Principia Mathematica, Science, Sir Isaac Newton, Teaching and tagged , , , , , , , , , , , , . Bookmark the permalink.

13 Responses to Math Is Hard…

  1. Emily says:

    In my own defense, I never said “Math is hard”. Lewis did, and my mother did. Your wording at the top of the page makes it seem as though I was the one saying such things. I can assure you, I was not, it is not, and it will probably never be. (Also, “Math is hard” is a running gag in my household…. It was an attempt at humor on my parents part.)

    • lewbornmann says:

      I did recognize the remarks by your mom and Lewis where intended as humor but decided to take advantage of the opportunity to make a general statement of my own. Re-reading the introduction (“…was written in response to comments made to a Facebook posting by my granddaughter“), I now admit it might not have been sufficiently clear that it was in response to their comments to your posting – not yours – but if you reread my introduction, that intent hopefully will be apparent. Please accept my apology if there was any confusion as to my intent. You always have been an excellent student and I never have questioned either your capabilities or dedication.

  2. berlioz1935 says:

    I love your exchange with your granddaughter Emily. Inter-generational communication is to be encouraged. I keep a Facebook page for just that reason. Alas, not all my grandchildren are willing to communicate.

    Your blog is informative and to the point. I will take advantage of the website you are mentioning. I always loved math but did never get further than Year 10. People are not interested in math because the majority is simply innumerate.

    Two examples of non understanding: the tax system and the time factor in evolution.

    • lewbornmann says:

      Like most grandparents, I think my grandkids are fantastic and consider myself extremely fortunate to have a good relationship with them. Emily, now fifteen, always has had the highest grades in her class and does extremely well in math. She is gifted and has the potential to succeed in whatever field she decides to pursue. Yep, I’m very proud of my grandkids as I’m sure you also are of yours…

  3. Lovely website! I am loving it!! Will be back later to read some more. I am taking your feeds also.

  4. Hey there! I could have sworn I’ve been to this site before but after reading through some of the post I realized it’s new to me. Anyhow, I’m definitely delighted I found it and I’ll be bookmarking and checking back often!

    • lewbornmann says:

      Thank you. The blog you responded to was actually created in response to something my granddaughter said. I think she was somewhat surprised to read replies originated in other countries around the world.

  5. test1234 says:

    Greetings I am so excited I found your blog, I really found you by error, while I was browsing on Digg for something else, Nonetheless I am here now and would just like to say thanks a lot for a marvelous post and a all round exciting blog (I also love the theme/design), I don’t have time to go through it all at the minute but I have saved it and also added in your RSS feeds, so when I have time I will be back to read more, Please do keep up the fantastic work.

  6. berlioz1935 says:

    You might interested to know there is a book out describing Carl Friedrich Gauss AND Alexander von Humboldt lives. It is “Measuring the World” by Daniel Kehlmann and it is available as a Kindle ebook. ISBN-13 978 1 84724 114 6

    Kehlmann, Daniel (2010-10-04). Measuring the World (p. 2). Quercus. Kindle Edition.

    It turns out that Gauss was not an easy person to get along with. He did not have a high opinion of us lesser people or other sciences.

  7. lewbornmann says:

    I’m admittedly interested in Gauss (one of my degrees is in Mathematics) but am not familiar with that book. Since Friedrich Gauss and Alexander von Humboldt lived in different periods and were not in the same field, it would be interesting to see why they were grouped in the same text.

    • berlioz1935 says:

      Well, the title of the book is the give-away. The lived at the same time and knew of each other. According to the book they met too. They were both interested in the earth magnetism. Perhaps Gauss provided the formulas. Wherever A. von Humboldt went he measured everything. Gauss worked at times at a land surveyor. I think rather than being grouped together their lives ran in parallels.

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