Mathsemantics: Making Numbers Talk Sense by Edward MacNeal (PDF)

Description

Here is a whole new way of looking at math that liberates math phobes from their anxiety, enables business people to do their jobs more effectively, challenges and informs math buffs, and provides educators with the tools to teach math easily and effectively. How can it do all that? By reuniting numbers and meaning, two subjects that should never have been separated in the first place. Entertaining, anecdotal, and immensely practical, this extraordinary book offers a revolutionary way of looking at math as a language, something that we’ve all heard before but which has never made sense until now. Mathsemantics is that rare book that will change the way you look at the world—and provide the most sensible and inspiring answer yet to the problem of American innumeracy.”Eye opening . . . a good antidote to innumeracy.”—Library Journal

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A book that not only talks about fun math-related ideas (in a casual/non-technical way) but also how the real world can throw a wrench in our plans no matter how elegant our abstractions seem on paper.The author is skilled enough that he could write about a trip to the gas station and make it engaging. I looked up his name intending to read his other works and was quite disappointed not to find any.I recommend it to friends who want a casual mathy book but aren’t math majors themselves.

⭐MacNeal makes a strong case that a proper understanding of mathematics requires thinking skills that our schools have not yet even identified. The author fills this book with examples of basic mistakes that “educated” people often make when using or even talking about numbers.MacNeal argues convincingly that using mathematics properly goes far beyond being able to manipulate numbers. Mathematics is a language that helps us understand the real world. Divorcing this language from spoken languages, such as English, fails to teach students how to use and think about issues that relate to mathematics. (And that certainly includes a multitude of topics!)”Mathsemantics” also provides a practical method of learning to make estimates. The author provides many examples of how to use information that we do have to make reasonable estimates regarding information that is unavaialable to us.If I were a high school or college instructor…of almost any subject…I would (easily) find a way to make “Mathsemantics” required, relevant reading. This book provides so much more value than most of the “best-selling” books that you will read and hear about.

⭐Priced right and fairly. it’s working out great for my son at school. Thanks

⭐I thoroughly enjoyed this book. MacNeal’s thesis was new to me and very interesting, and he explains it clearly and engagingly.Some reviewers have focused on how Mathsemantics might change the way we think about math, or teach it. That’s an understandable perspective for someone who teaches math. But MacNeal’s thesis is much broader than that. He shows (with actual data) how many people artificially separate mathematical concepts from their perception of the world, thereby misunderstanding decisions they try to make. To take one of MacNeal’s examples, if you said that three apples and four oranges together constitute seven pieces of fruit, few people would find that an incorrect proposition. But if you ask them, “What is 3 apples + 4 oranges” — phrasing it like a math problem — it turns out that a lot of people will reply, “You can’t add apples and oranges.” The phrasing of the question triggers math mode, and they stop thinking and instead attempt to blindly apply a rule.I was startled by that and some of the other all-too-common responses to seemingly simple questions that MacNeal documents. But MacNeal’s insight is that this is not just a problem in how we teach or think about math. It’s also a problem in how we teach and think about language. I would love to see “Mathsemantics” be an actual course on every student’s curriculum, but taught by the English department, not the Math department.

⭐Review: Edward MacNeal Mathsemantics: Making Numbers Talk Sense Penguin; 1994 ISBN 0 14 023486 1 (24 chapters, Tests, References and Index!; 310 pages)[From Prime Number (Mathematical Assocation of Victoria, Brunswick, Victoria, Australia) vol. 11, no. 2, 1996, p. 13)Mathsemantics, like John Allen Paulos’s stimulating Innumeracy (Penguin, 1988, about dealing comfortably with fundamental notions of number and chance), is easy to read, challenging, fascinating, and might change the way you teach and the way you think about teaching. Edward MacNeal discusses the connection between mathematics and the purpose and meaning of mathematics. According to MacNeal, many students experience difficulty because of confusion between formal mathematics and what the mathematics means, between the written symbols and the translated prose and the ideas they both represent. These examples of similar words have dramatically different mathematical meanings: “5 less than 10” is 5, while “5 less 10” is -5; “6 divided by 2” is 3, whereas “6 divided into 2” is one-third, yet “6 divided into 2 equal parts” is in fact 3 (pp 89 – 90). To resolve such confusion we need “mathsemantics” to make mathematics meaningful, and show how language can work mathematically.MacNeal’s discussion arises from the results of test questions uses to screen adult job-applicants on the basis of their ability to use mathematics and to think about what they are doing.MacNeal describes three vivid lessons. When he was about six years old, his accountant father challenged MacNeal and his brother to write whole numbers in sequence: 1, 2, 3, 4, …, offering a dollar for each number that had only a 1 following by zeros. The two brothers totalled about 60 hours of work, and discovered a great deal about number, positional notation, powers of 10, infinity, uniqueness, pattern, transformation, sequence, “perserverance, and the economics of diminishing returns to scale” (pp 103 -104). Another time their father expressed an interest in having the young brothers report to him the results of flipping a coin exactly 1000 times. As a father himself, MacNeal challenged his three- and five-year old children to find a pair of identical objects, rewarding them for each hopeful pair brought forward, but demonstrating (heartless father) the differences which always exist between any two things, however similar looking. The children were learning “that one and one make two different things, that aggregates always involve differences, that all we ever add are apples and oranges” (p 105).MacNeal and his brother also explored non-decimal-base arithmetic, discovering for themselves the gap between the mental idea, or its verbal or symbolic representation, and the thing, discovering that what you call something is arbitrary. Four strokes |||| can be represented as Roman IV, in base ten Hindu-Arabic as 4 or 4 or 4, in French as “quatre”, in German “vier”, in base 2 as 100, in base 3 as 11, using Dienes MAB blocks as four “minis”, on a hand as four extended fingers and a closed thumb, and so on. Yet beneath the symbols, the mental construct abstracted out of real distinct individual events remains the same. This explainswhy 2 times 2 equals 4, going beyond words to understanding that “times two” means “taking twice” or “putting together two lots of” whatever is being dealt with. “Proposition 15: Until you work it out for yourself, two times two makes four only because the teachers says so. You have to do multiplication before you can understand what it means.”Remember the rounding question at the beginning: round 0.098 to the nearest whole number? Many people have difficulty rounding to zero because they feel zero is not a number. Consider these possible replies to a typical mathematics question Q: “Does the equation have a number of solutions?” A: “No, only one”; or, A: “No, there is no solution”. These answers suggest 1 and 0 are not “numbers”. In fact the word “number” is used to refer to many different things, often related (wholes, cardinal, ordinal, fractions, evens, primes, etc.), “some of which are nearer the central, prototypical meaning … than others.” ( 129). Hence, zero is a number, but is not protypical of “number”. This confusion comes from our cultural inheritance of mathematics from the Ancient Greeks, who refused to work with “nothing”, and treated “one” as a unit” for measuring numbers, not a number in its own right, despite the later superficial overlay of the Hindu-Arabic “zero” on our fundamentally Western Graeco-Roman culture. Laws were passed in Medieval Europe banning the use of the Hindu-Arabic number system in favour of the hopelessly clumsy Roman system p 116! MacNeal says we should accept the naturalness and importance of zero, emphasising it in these ways: put a 0 at the left end of every ruler, start counting with zero, in print and orally, put “zero is a number” into school policy, display in every classroom a large dummy thermometer with a prominent zero and a moveable degrees pointer, speak of children younger than 1 as zero-year olds (pp 83 – 84). His research also suggests that people who use commas to separate millions from thousands and thousands from hundred handle mathematics tasks better than people who do not use commas (pp 178 – 179).Some book! Try it. As MacNeal stresses: we are “all going to be judged in the marketplace by the cost of our services to clients and by how well we [suvive] the witness stand” (p 178). Moreover, “Unfortunately, most of the serious threats to our species involve measurement, large numbers, or both” (p 146). Good mathsemantical thinking is more urgently needed than ever before! John Gough

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