I had read most of Marilyn's books but not this one until yesterday. I think I have to consider it the best I've read so far on why we need to change math education. I kept reading it thinking, I'm so GLAD someone with as much of a voice as she has is saying this. So much of what she expresses is what I've felt myself as I've changed the way we "do" math in our home.
Rather than summarize, let me quote from parts that really resonated with me:
"The main premise of this book is that what was good enough for us in learning mathematics is not good enough for our children. Despite the reality that learning math was a bust for so many of us, we have pressed on with ineffective teaching approaches that clearly don't work. If they did, math phobia wouldn't be rampant today.
Even in the face of widespread failure in learning mathematics, we seem to want to cling to educational methods with a nostalgia for them that has long outlasted their usefulness and has perpetuated failure. The way we've traditionally been taught mathematics has created a recurring cycle of math phobia, generation to generation that has been difficult to break. We start young children with counting and move them along through arithmetic, then on to algebra, geometry, trigonometry, and so on. The 'and so on' depends on whether or not the student sticks with math, which means not falling off the ladder of math progress in school. But an alarming percentage of the people in our country have fallen off the ladder and feel like mathematical failures. And once people fall off the ladder, there seems to be no way for them to get back on.
. . . Children must be helped to learn mathematics in a better way than we were, so that mathematical limits do not shut them out of certain life choices and career options."
Chapter 1 does a better job of describing the true relevance of math in our daily lives than most books I've read. One anecdote that I loved: "A friend of mine is an interior decorator. ' I was never good at math,' she once told me 'I'm so grateful that I have work to do that doesn't rely on doing math.'
I looked at her in amazement. To do her work, she has to measure the dimensions of rooms for floor covering and wallpaper, figure yardage for drapes and upholstery, calculate the cost of the goods, and prepare invoices for clients, figuring in the percentage for her services. I've watched Barbara size up rooms, her eyes darting as she takes in the floor area, the ceiling height, the placement of windows and doors,; suggest the right amount, size and scale of furniture; prepare an estimate that is amazingly close to actual cost. I experienced this firsthand when we remodeled our house. No math? What was she talking about?
I asked her. 'Oh that,' she said, 'That's easy. It's those pages of math problems in the book I never could do.'
That's the problem. People think that doing math in books is what doing math really is. It's not. What we need to do for our children is get math off the textbook and worksheet pages so they can see reasons for doing math and gain experience using math to solve problems."
There is a very interesting portion of this chapter that sizes up the impact of years of pencil and paper arithmetic instruction when we hardly ever use this skill in real life - we usually either estimate in our heads, or use calculators. Kids don't get the point of "estimate first, then get the 'right' answer" because all that seems to count is getting the right answer. But in real life, the estimate is what matters, because so many "calculations" have too many variables to be accurate.
There is much, much more of course. Chapter 6, Calculators - Crutch or Tool? is very interesting to me because I've had a hard time explaining why I've felt there is way too much emphasis on pencil and paper arithmetic, on skills we *don't* hardly, if ever, use in real life.
This is also the book I recommend to rebut those that recognize the problem but suggest more rigor in arithmetic instruction as the solution. I agree with her, it's just throwing more bad after good. She promotes rigor in learning to problem solve and to *think*.
I highly, highly recommend this book even if you are not math phobic yourself but are looking for reason and validation to change from a traditional math approach. Her writing is clear and easy to follow.
> I have a question. I certainly see Marilyn's argument for using calculators more and the paper and pencil less. But in order to understand how to program a computer, the programmer must understand the need to borrow tens, hundreds, thousands, etc. >
Borrowing is a term for an algorithm, a process (when your son said he hated borrowing, that's what he meant, right?) All programmers today use programs that have basic arithmetic operations built into them and they do not in fact need to be able to tell the computer to "borrow."
It is actually quite possible to never "borrow" or "carry." Some of us are "adders". For example, in the book Marilyn points out, if you double 38, how do you do it? Many will double 40 and subtract 4; but other will double 35 and add 6, or double 30 and add 16. Others will perform the borrowing algorithm mentally in their heads. Same thing if I were to subtract 18 from 40 - I think of subtracting 20, then adding back 2. I don't borrow at all.
If a person can manipulate numbers like this in their head, then numbers that become too cumbersome will either be dealt with through a calculator or a computer, with pencil and paper being a last resort if neither is handy. That's just a fact. In 13 years of working in national public accounting firms, I almost never used pencil to paper math, and I was highly successful.
In her chapter on this, she relates a discussion with an engineer who also points out he rarely if ever uses this skill, but he could not shake his belief it was necessary, and she highlights this illogical connection quite well I think.
> What do you think? Isn't there a balance between living math and traditional math? Are we in danger of throwing the baby out with the bath water? >
I think common sense, if allowed to reign, will prevail. It's when we allow ourselves to pendulum swing in fear one way or another that we do this. Taking steps based on ideas that make sense to us, taking in constant
feedback as to the results we get, continuing to be open to making changes wins the day.
Also, some things I feel need to be said strongly to combat such deeply held beliefs such as "doing math" = working pencil and paper problems. There is little danger I really feel that by making changes we will be tempted to completely swing the other way. For one thing, when I'm working *with* my kids, pulling out pencil and paper is the only way I can *show* them how I think. It obviously has a purpose, as writing things down vs. narrating has a purpose.
Julie Brennan
July 30, 2004
KENDALL HAVEN is a nationally recognized master storyteller and the author of numerous books, including Marvels of Math, Write Right!, and Close Encounters with Deadly Dangers. A former research scientist, he is based in Fulton, California.
Book Description: When did the concept of zero originate? Who discovered negative numbers? How were algebraic equations first created? These answers and more are all here. Each story has a list of terms to learn, related discussion questions, and experiments and activities that amplify the story's math theme.
Excerpt from Introduction: Mathematics has evolved to meet specific human problem-solving needs. People created every bit of our number and math systems. They invented it. They created the math that they needed to solve everyday, practical problems and to describe everyday, real-world phenomena--from the first invention of whole numbers to describe the members of groups to the recent development of chaos theory and surreal numbers. Even more exciting, new math concepts and approaches are still being discovered and invented. For example, surreal numbers, a nifty way to count beyond infinity, were invented in 192. Much of the math learned over your lifetime is less than one hundred years old. Likely as not, some math that you will learn and use has not yet been created.
Math is a language, a language used by science. Math is a short-hand version of English. A formula is much shorter and easier to write than English. More than any other field math development is a group effort. We teach the math and ignore the fascinating people and stories associated with the development of the field. Mention math and story in the same sentence and most people shudder in terror, thinking of the dreaded math story problem. (If John and Carol meet in Chicago, and John leaves Denver at 8 am on a train at 60 mph .) Those are not stories.
But there is another type of math story. These stories are gripping, fascinating, and illuminate the development, history and purpose of our modern math tools and concepts. These historical stories concern the real-life drama of the development of math and of the mathematicians who invented it. Any topic, math included, becomes more accessible and understandable when we tell the human story behind the development topic. Stories make subjects real and purposeful. They create context and relevance. They provide a foundation from which students can understand and appreciate math, rather than merely memorize a series of rote exercises.
This book focuses on significant developments in mathematics history. However, all stories are about characters, not concepts, and these stories are no exception. The stories are here because they tell about characters at the moment of significant developments in math concepts and principals upon which we depend on every day. The stories are divided into four groups: the development of our number system, the development of geometry, the development of mathematical concepts and applications and the development of devices to aid in mathematical computations.
Each story is historically accurate as available research permitted. If I found quotes recorded in diaries or letters, I used them for dialog. Otherwise, I inferred dialog from known personality traits, written accounts, the mathematician's writings and essays, and from known interactions and events. To the extent that I could reasonably establish the emotional states of the personality, I used the knowledge to bring the character to life.
The book has 16 chapters. Each chapter has 4 sections:
At a Glance - Brief overview of the history of the concept.
Terms to Know - List of math terms with definitions.
Story - The story is told.
Follow-On Questions and Activities
Excerpt from Chapter 5, Elementary Elements - The Invention of Euclidean Geometry by Euclid in 295 BC
"Twenty-year-old Theoclese (THEE-oh-kleez) raced into his family's small house at the edge of their grain fields. This year's crops were planted. Green shoots sprouted everywhere repainting the dark brown dirt of their upper Nile Valley farm soft green. "Father! Father! I know what I must do."
"Do? You mean tomorrow morning when we start weeding?"
"No father, it's the most wonder book imaginable." Theoclese had dropped to his knees, his hands resting on his father's thigh as he pleaded. "It's The Elements. Euclid explains everything about mathematics and geometry. I want to go to Alexandria.I have to go to Alexandria to study with Euclid."
Dentus tensed, his body rigid with shock, his voice now barely a whisper. "Alexandria is over thirty leagues away. You'd never be able to journey there, talk with this Euclid fellow, and be back to help run the farm."
Thoeclese's voice now sounded stronger, more resolute. "No, father. I must leave the farm and become a full-time student at the Alexandria Museum. I must."
As if frozen in a stone etching the two remained motionless while this revelation slowly washed over and through the father. Finally Dentus' hand crept forward to rest on his son's shoulder. Dentus slowly shook his head, still bewildered by Theoclese's words. "My son, a mathematician. Who could have guessed?"
Blessings, Tuesday
July 23, 2004
(OOP, but used copies are available as of this review - I have the 1967 version)
It has clear writing like the Isaac Asimov books, but a little more flair. The book is packed - it could actually function as a high school text for one year for a print-oriented learner that doesn't need a lot of practice problems. Most of it is anecdotes and examples. Similar to the Harold Jacobs MAHE text, the scope is very comprehensive, but it is designed to be read, whereas the Jacobs book IS a textbook with more focus on working practice problems. All illustrations are b&w, but are very clear, there is enough white space in the book to keep it from seeming crowded.
Here are the chapter headings:
Numerals and Numeration (History and use of numerals)
Systems of Numerations (one of the best descriptions of other bases I have ever seen)
Some Remarkable Properties of Numbers
Number Giants (large nos, history including Archimedes contribution)
Number Pygmies
There's Secrecy in Numbers (cryptography, codes, etc.)
The Arithmetic of Measurement
Simple Calculating Devices (abacus, soroban, etc.)
Rapid Calculations ("human calculators", math tricks)
Problems and Puzzles (actual cases)
How the Number Magician Does It
Algebra & Its Numbers (the algebra chapters all contain many problems in real applications)
The Algebra of Number Giants and Pygmies
The Grammar of Algebra
Algebra, Boss of Arithmetic
Algebra Looks at Installment Buying
Chain Letter Algebra
Streamlining Everyday Computation (logs)
The Bankers Number - Jack of All Trades (net present value, future value, compounding in other ares such as growth and aging)
How to Have Fun with Lady Luck (probability)
The Thinking Machines
Postoffice Mathematics (relativity, probability, geometry)
New Worlds for Old (zero dimension, Flatland, geometry meets
arithmetic and algebra)
Passport for Geometric Figures (geometry and movement)
Man's Servant - The Triangle
The Triangle - Man's Master
Circles, Angles & an Age Old Problem (pi)
The Mathematics of Seeing (optical illusion)
The Lost Horizon (curvature of the earth, natural illusions)
The Shape of Things (geometry beyond circles and triangles)
The Size of Things (three dimensions, area of geometric figures)
Escape from Flatland (solid geometry)
How Algebra Saves Geometry (applications of geometry & algebra)
Cork-Screw Geometry (distances between points in solid geometry)
Mathematics, Interpreter of the Universe (geometry in astronomy / geology, physics)
The Firing Squad & Mathematics (math and movement)
Of Math and Magic (more math & motion, time and distance)
The chapters encompass about 700 pages, after which there is an extensive appendix of tables, definitions, and answers to problems posed in the book.
This is a really good resource, and relatively inexpensive for now while there are copies available. I haven't seen anything comparable IN print for a while, except possibly the Birth of Numbers book I have been using for the high school math history course - but this one is less scary :o). What I really like about it is that the text is virtually all applied, rather than just talking about math in the abstract. It is also conversational - i.e., just opening up to page 389, "Let the reader imagine that he is a Flatlander, living in a two-dimensional world. As we found out when we examined this type of world, he will be able to move around only in a plane, as in the space between two plates of glass. Moreover, he cannot see what is above him or below him and could only imagine a world of three dimensions. A Flatlander may know everything about flat figures such as lines, triangles, squares, and circles, but he will have to use his imagination to picture a cube or a sphere. Suppose a sphere has suddenly descended upon Flatland. . "
It would also be an excellent refresher book for an adult that may have been good at math in high school but has not reviewed math for a long time. This may not be good for a math-phobic parent trying to overcome math anxiety - it really does assume a fairly strong high school math base, even as it explains things well, it does not treat topics as if they are being introduced to real beginners.
Hope that helps someone else, I was really surprised at finding this resource and more surprised when I found there are copies available.
Julie in San Diego - Reviewed 7/24/05
The Math Instinct is a must read for anyone who struggles with math and for parents whose children struggle with school math. According to Devlin, not only do we as humans have inborn math instincts but so do many other species. The beginning half of the book is devoted to interesting stories illustrating the performance of various math principles including a dog and a frisbee chase as well as an ant's daily
travels.
The second part of the book is a discussion of how people use math and his defining of different types of math. According to Devlin, the subject taught to children in schools is ABSTRACT math. Innate math such as real world math is NATURAL math. Both are just mathematics. "The distinction lies in how the math comes to get done. Abstract math is symbolic and rule based. To do abstract math you have to learn what the symbols stand for & you have to learn how to follow the rules. Natural mathematics arises, well, naturally."
(This part of the book is fuzzy in my memory so the details may be somewhat different when you read the book.) He shares a scientific experiment where researchers interviewed people in the grocery store and asked them to carry out various math calculations when comparison shopping. The researchers found that the participants were much better doing the math when they used their own calculation methods to arrive at an estimation and when the situation felt relaxed and informal. When the situation was more structured in terms of providing exact answers and the participants felt their math was being tested or evaluated (as experienced while in school) the higher their anxiety and the worse they performed. It was discovered that “when asked for approximate answers to math questions the greatest brain activity occurs in the brain regions related to number sense & support spatial reasoning skills. For questions requiring an exact answer the brain activity took place in the frontal lobe where speech is controlled. This is also the area of the brain that specifically seeks meanings.” As it was pointed out, most “school math” is focused on exact answers whereas in “real life math” the focus is usually more on estimation. "The problem is that humans operate on meanings. In fact, the human brain evolved as a meaning-seeking device. A computer can be programmed to obediently follow rules for manipulating symbols with no understanding of what those symbols mean. People do not function this way. Mastery of school arithmetic involves the acquisition of some kind of meaning for the object involved & the procedures performed in them."
Devlin also shares a math research project involving young Brazilian street vendors and their "numerical dexterity". According to Devlin, when the vendors were totaling up orders they showed “considerable familiarity with numbers. To simplify computations, they used properties of the specific numbers they were faced with. Their standard approach was to find a way to transform the problem into one involving numbers & arithmetic operations they could recognize & handle. Sometimes involved rounding, sometimes they would break the initial problem into 2 or more subproblems.” What was interesting was that after completing a transaction, the researcher would ask the vendors to solve the exact same problem in abstract form, meaning just asking the vendors to add certain numbers (which were the exact same numbers and computation the vendors had just performed in relation to the produce). The vendors were unable to do math this way. “An obvious key factor for the street traders is that within
the computations, both the numbers & the operations they performed on them had meaning & the operations made sense. Indeed the children were surrounded by physical meanings of the arithmetical procedures they performed...In short, street mathematics is all about carrying out meaningful operations or meaningful objects, whereas school mathematics is about carrying out purely formal
manipulations of symbols whose meaning, if any, is not represented in those symbols. For most people, $27.99 means something, but 27.99 doesn’t – it is just a number.”
The underlying conclusion shared among all the math research projects described in the book is that, “It’s not that someone who gives a ridiculous answer to a math problem can’t see how silly it is when it is pointed out. If asked to repeat the same calculation in a context or fashion with immediate practical
significance, they generally get the right answer, or at least an answer that is plausible. And they do even better if presented with a “real life” problem that they can handle using “street math” – arithmetic methods they have developed themselves “on the job”. In some cases there is almost no procedural difference between a street method and a schoolroom counterpart. And yet the difference
caused by the absence of meaning in school math is significant. Complexity is not an issue since most can cope with considerable complexity in their mental arithmetic.”
“In contrast to street math, the essence of school math is that it is entirely symbolic. In performing a standard school procedure for addition, etc. you carry out the very same actions, in exactly the same order, regardless of what the actual numbers are and what they measure. This is the whole point. The methods taught in school are supposed to be universal. Learn them once and you can apply them in any circumstance, whatever specific numbers are involved.”
For students to succeed in school math, the student has to understand what the symbols represent. If they understand what they are doing (what is going on) they don’t forget the procedure. For example 3,844 means 1.62 squared. Because of this meaning to someone (the relationship between the numbers), the better s/he is at math. People good with numbers have relationships with them – the numbers have meaning. "Just how successful a person is at mastering school mathematics is largely a matter of how much meaning they can construct for the symbols manipulated & the operations performed on them. The problem many people have with school arithmetic is that they never got to the meaning stage, it
remains forever an abstract game of formal symbols."
Devlin has great respect for school math and believes, “In the hands of a person who can master the abstract, symbolic procedures taught in school, those procedures are extremely powerful. Indeed they underlie all our science technology & modern medicine & practically every other aspect of modern life.
But that doesn’t make them easy to learn or to apply….Since the arithmetic procedures taught in school are designed to be universal – to apply in all cases, whatever the actual numbers – the first think a student has to do in order to master those procedures is to learn to ignore temporarily any possible
meanings in terms of actual numbers or real objects in the world. The second thing the student must do to acquire mastery is to construct a different, more abstract kind of meaning.”
Devlin delves into the differences of math test scores among the various countries and believes that within countries such as America, England and much of Western Europe there is a math language barrier which further complicates the math process. According to Devlin, the various scores remain pretty equal throughout the younger grades but when higher numbers are introduced the disparity in test scores begins. “In Chinese & Japanese their number words are much shorter & simpler making them easier to recite, either aloud or internally & that in turn makes them easier to learn. The Chinese rule for making number words for numbers past ten is simple: 11 is ten one, 12 is ten two, 13 is ten three, two ten for 20, two ten two for 22. American kids encounter the various special rules for forming number words. Chinese children don’t, they simply keep applying the same ones that worked for 1 to 12. Also, Chinese language rules closely follow the base-10 structure of the Arabic system. A Chinese pupil will see from the linguistic structure that the number two ten five (i.e. 25) consists of two 10’s & one 5. American kids have to remember that “twenty” represents two 10’s.”
The Math Instinct concludes with a 4 step method of how to improve your math skills. These are:
“1. Be aware that math activity is a natural thing that occurs all the time in nature. Knowing that math is natural should help overcome any math fear. Note: this is what Devlin attempts to do in the beginning of the book which makes it invaluable for math phobics.
2. Approach abstract (i.e. school) math as merely a formalized version of your innate mathematical abilities – that is, as formalized common sense. In math as in most things in life, your approach can make all the difference in how well you do.
3. Recognize why the school methods were developed, what their advantages are & what it is about them that makes them hard to learn. Knowing why something is done a certain way generally helps us to handle it.
4. Practice. Given sufficient repetition, our minds can become skilled at performing practically any new task i.e. swimming, speaking a foreign language, etc. For the human brain familiarity breeds a sense of concreteness. And there we have the key to learning how to handle abstract entities: become sufficiently familiar with them for them to become (i.e. to seem) more concrete. No particular skill is required to do this. All it takes is sufficient repetition. It is only by repetition that the brain can be taught a new skill or to regard the abstract as concrete. Practice basic arithmetic, at least up to decimals & fractions, until you become proficient. Although boring, keep in mind it is a remarkable feature of the human brain that it can acquire such a wide range of new skills.”
Reviewed by Sally Leathers 11/4/05
Another resource to recommend for families wanting to understand exactly *what* their child can learn from math literature.
Most of the books are for PK through say 4th/5th. It is a bit dated (copyright 1998) so many books I have are not listed. But I know that some people ask themselves, what are they learning as they read this math book? So the categorizing, rating and concept identification can help people desiring this.
I find it hard to categorize books because I see so many concepts that can be learned even if the book targets one.
The main chapter headings are:
Early Number Concepts
Number - Extensions and Connections
Measurement
Geometry and Spacial Concepts
Series and Other Resources (in this chapter they discuss specific
books in the I Love Math series, Mathstart, and some others).
All resources are rated highly recommended (3 stars), recommended (2) or acceptable (1).
I don't agree with all of their ratings. I can see that books that are written with more obvious math connections that can fit into standards did get higher ratings, books with softer connections got less. There is an emphasis on books with obvious concepts, and there is little soft math. But I did find most books that I would expect to be there that were in print at the time this was updated are listed.