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Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs, by Raymond Nickerson
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The development of mathematical competence -- both by humans as a species over millennia and by individuals over their lifetimes -- is a fascinating aspect of human cognition.
This book explores when and why the rudiments of mathematical capability first appeared among human beings, what its fundamental concepts are, and how and why it has grown into the richly branching complex of specialties that it is today. It discusses whether the ‘truths’ of mathematics are discoveries or inventions, and what prompts the emergence of concepts that appear to be descriptive of nothing in human experience. Also covered is the role of esthetics in mathematics: What exactly are mathematicians seeing when they describe a mathematical entity as ‘beautiful’? There is discussion of whether mathematical disability is distinguishable from a general cognitive deficit and whether the potential for mathematical reasoning is best developed through instruction.
This volume is unique in the vast range of psychological questions it covers, as revealed in the work habits and products of numerous mathematicians. It provides fascinating reading for researchers and students with an interest in cognition in general and mathematical cognition in particular. Instructors of mathematics will also find the book’s insights illuminating.
- Sales Rank: #2577176 in eBooks
- Published on: 2011-02-25
- Released on: 2011-02-25
- Format: Kindle eBook
Review
"Few authors who are not mathematicians have the understanding and sympathy for the subject and its practitioners that is shown in this book, whose author is an experimental psychologist and former VP of Bolt Beranek and Newman Inc. He surveys the nature, joys, usefulness, foundations, and teaching of mathematics, in its manifestations of pattern-finding, problem-solving, conjecture-making, and proof-devising. He concludes, "the supreme reason for acquiring some competence in mathematics is the door it opens to an immensely attractive and rewarding workspace―or playground―for the mind." This is a great book for nonmathematicians, one that a mathematician can praise." - Paul Campbell, Beloit College, USA
"Nickerson is a reliable and enthusiastic guide through a many-layered forest of ideas, leading us at each turn to a new appreciation of its topography – the box canyons and precipices and monumental vistas – of this very human adventure. His excellent references mark trailheads for those who would venture farther."�-�Peter Killeen, Ph.D., Arizona State University, USA
"A wide-ranging and insightful account of the nature of mathematics (its origins, utility, beauty) as well as of cognition and reasoning in general. This book is a welcome antidote to the notion that math is primarily a matter of computation." -�John A. Paulos, Ph.D., Temple University, USA
About the Author
Raymond S. Nickerson, retired senior vice president of Bolt Beranek and Newman Inc. and research professor at Tufts University, is a fellow of the American Association for the Advancement of Science, the American Psychological Association, the Association for Psychological Science, the Human Factors and Ergonomics Society, and the Society of Experimental Psychologists. He is the founding editor of The Journal of Experimental Psychology: Applied, the founding and series editor of Reviews of Human Factors and Ergonomics, an annual publication of the Human Factors and Ergonomics Society, and the author of several books, including Aspects of Rationality: Reflections on What it Means to be Rational and Whether We Are (Psychology Press, 2008).
Most helpful customer reviews
1 of 1 people found the following review helpful.
Do mathematicians think differently from you and me?
By Bruce E. Nevin
Charles Sanders Pierce urged mathematicians to present their results with demonstrations that showed how they actually got there, rather than the usual way, which is to tidy up a nicely polished pyramidal proof that leads enexorably, pontifically from axioms to Q.E.D., and then to throw away the scaffolding and sweep away all traces of the parallel paths, false starts, and tangents of the actual process. This suggests a difference between mathematical reasoning as practiced and mathematical reasoning as mathematicians feel it ought to be. We may also distinguish mathematical reasoning from mathematical knowledge, as for example the toolbox of rules and formulae that every high school student of algebra is asked to master.
Nickerson is a research psychologist inquiring into how reasoning is done in the diverse fields of mathematics as an outsider, as he presumes us, his readers, to be. This book is very readable, and also very thorough. One must be prepared to commit mental resources to taking in a lot, and one very much needs the readability lest the scope, variety, and depth of coverage appear to be too daunting. Nickerson's coverage is impressive, and his intent is very ambitious.
I think that mathematicians as well should benefit from reflecting on how they reason, perhaps agreeing at some times and then when they disagree it may be finding themselves on occasion brought to an unfamiliar but possibly fruitful self-reflection. Also, mathematicians naturally specialize, and a survey of the field can be like looking up from myopic concentration at a desk and gazing out the window to scan a distant and opening horizon.
I haven't yet read enough to comment on the success of Nickerson's project, but what I have read has led me to recommend it a few times, and I now recommend it to you.
2 of 2 people found the following review helpful.
The Human Side of Mathematics and Reasoning
By Douglas H. Harris
I enthusiastically recommend Mathematical Reasoning to anyone who is intellectually curious, enjoys being entertained while learning something important, and is interested in the human side of a subject that may have been approached with some apprehension in the past. In this book, Ray Nickerson provides an enriching experience, one for which previous knowledge or interest in mathematics is not necessarily required.
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