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What is Mathematics?

To most people, mathematics is about calculating numbers. Some may even expand the definition to include the study of quantity (arithmetic), space (plane and solid geometry), and change (calculus). But even this definition does not encompass the many areas where mathematics and mathematicians are found. A broader definition of mathematics comes from W. W. Sawyer. In the 1950s, he
described mathematics as the “classification and study of all possible patterns.” He explained that pattern was meant “to cover almost any kind of regularity that can be recognized by the mind” (Sawyer, 1982).

Other mathematicians who share Sawyer’s view have shortened the definition even further: Mathematics is the science of patterns. Devlin (2000) not only agrees with this definition but has used it as the title of one of his books. He explains that patterns include order, structure, and logical relationships and go beyond the visual patterns found in tiles and wallpaper to patterns that occur everywhere in nature. For example, patterns can be found in the orbits of the planets, the symmetry of flowers, how people vote, the spots on a leopard’s skin, the outcomes of games of chance, the relationship between the words that make up a sentence, and the sequence of sounds we recognize as music. Some patterns are numerical and can be described with numbers, such as voting patterns of a nation or the odds of winning the lottery. But other patterns, such as the leopard’s spots, are visual designs that are not connected to numbers at all.

Devlin (2000) further points out that mathematics can help make the invisible visible. Two thousand years ago, the Greek mathematician Eratosthenes was able to calculate the diameter of Earth with considerable accuracy and without ever stepping off the planet. The equations developed by the eighteenth-century mathematician Daniel Bernoulli explain how a jet plane flying overhead stays aloft. Thanks to Isaac Newton, we can calculate the effects of the unseen force of gravity. More recently, linguist Noam Chomsky has used mathematics to explain the invisible and abstract patterns of words that we recognize as a grammatical sentence.

If mathematics is the science of patterns and if visible and invisible patterns exist all around us, then mathematics is not just about numbers but about the world we live in. If that is the case, then why are so many students turned off by mathematics before they leave high school? What happens in those classrooms that gives students the impression that mathematics is a sterile subject filled with meaningless abstract symbols? Clearly, educators have to work harder at planning a mathematics curriculum that is exciting and relevant and at designing lessons that carry this excitement into every day’s instruction.

I will leave the discussion of what content to include in a Pre K to 12 mathematics curriculum to experts in that area. My purpose here is to suggest how the research in cognitive neuroscience that we have discussed in the previous chapters can be used to plan lessons in mathematics that are more likely to result in learning and retention.


Devlin, K. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. New York: Basic Books.

Sawyer, W. W. (1982). Prelude to mathematics. New York: Dover Publications.

  1. December 8, 2012 at 12:14 am

    Reblogged this on True Story.

  2. Annette Tiff
    May 18, 2013 at 3:01 am

    Hi Dr. Sousa I have a 8 year old autistic son and was wondering if you had any plans of writing a book on teaching autistic students. I read your book “How the Special Needs Brain Learns’ and found it very informative and eye-opening. My son has been mainstreamed but the problem is most teachers do not know how to teach to autistic students there their brains learns. I have looked for books on this subject but have not found any that take your approach about the Neuroscience of education. It would be so beneficial for so many parents and teachers if there was a book based on how the brain of an autistic student learns with strategies use. I hope you will consider it. Thank you.

  3. Hassanat
    October 6, 2013 at 11:39 am

    Some student feel mathematics is a difficult subject because of following: lack of interest, unqualify teacher, lack of instructional materials and proper uses of it

  4. Marcia Livingston
    April 20, 2017 at 12:45 am

    Hi Dr. Sousa, your work and writings on the brain especially How The ELL Brain Learns has been very useful and informative. I have adopted your book as a primary text for two of my courses and have raved about your work to the extent that I have been asked to contact you to see if you would be available to speak at an Educators conference that will be hosted at our university in Tulsa Oklahoma September 29-30, 2017. Please advise of the possibility. I am sorry I did not learn of you when I lived in South Florida. Thanks

  1. December 6, 2012 at 9:39 pm

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