The Math Formula
If any discipline has embraced the 7 step lesson plan, it is most assuredly mathematics. I know this because I am a consultant teacher and spend my days observing instruction in a school district of over 20,000 students. I have witnessed lessons from the 4th grade through the 12th grade and in content areas as diverse as science, language arts, social science and mathematics. I can say with some authority that the 7 step lesson plan is alive and well in California schools and no where is it more firmly entrenched than mathematics.
The 7 step lesson plans has the following or similarly titled steps:
Elements of Lesson Design
Anticipatory Set
A mental set that causes students to focus on what will be learned.
Objective and Purpose
Not only do students learn more effectively when they know what they’re supposed to be learning and why that learning is important to them, but teachers teach more effectively when they have that same information.
Input
Students must acquire new information about the knowledge, process, or skill they are to achieve.
Modeling
“Seeing” what is meant is an important adjunct to learning.
Checking for understanding
Before students are expected to do something, the teacher should determine that they understand what they are supposed to do and that they have the minimum skills required.
Guided Practice
Students practice their new knowledge or skill under direct teacher supervision.
Independent Practice
Independent practice is assigned only after the teacher is reasonably sure that students will not make serious errors.
A principal once told me that in all his years observing math classes he had only seen one lesson, but he’d seen it hundreds of times. I prefer to call it the Math Formula. The version of the 7 step lesson plan that is most often observed in math classes would have these re-named steps:
Warm-up
The warm-up activity, the most ubiquitous of all math strategies, is normally used for one of two purposes. The problems are either a review of recent learning and are often critical elements of the day’s lesson, or, in very rare cases, they forecast new learning in some insightful way.
Homework Review
Most teachers ask the class if there were any difficulties with the previous night’s homework. Teachers, or selected students, solve the troublesome problems step by step.
New Learning
This is the lesson for the day. Teachers typically engage in direct instruction and begin to explain step by step a sample problem.
Modeling
The teacher will solve two or more additional problems, on the board, overhead projector, or smart board, that illustrate the new concept.
Checking for Understanding
The teacher will question students to establish the degree to which they understand the new concept. Sometimes volunteers are sent to the board to work a problem or two. More often the teacher continues at the board, working the problem as he questions the students.
Guided Practice [Problem set]
The teacher will assign a manageable problem set which students will work on alone, sometimes in pairs or trios, with the teacher directly supervising their work.
Independent Practice [Homework]
The teacher will assign a longer problem set for independent practice. This usually cannot be completed by the end of the class period and then becomes that night’s homework.
I must signal a caveat here. My background is in language arts and I did spend 30 years teaching high school English. However, I have also spent 5 years in full-time release from the classroom to work as a mentor to new teachers. I have worked with over a dozen middle school and high school math teachers. In the course of this work, I have had occasion to observe many veteran math teachers as well. Regardless of my content area background, I have been tasked by the county office of education to look for the teaching standards in the classrooms I observe. The comments I will make, though they may sound critical of math instruction, are grounded in my understanding of the teaching standards.
When you compare the 7 step lesson with its incarnation in math classrooms, what you find most often missing is step 2: “Objective and Purpose.” Most math teachers appear to feel that the objective of the learning is embedded in the design of the textbook and the corresponding lessons. I have rarely seen a math teacher explicitly address this. On the contrary, on numerous occasions, I have heard the instructor say some variation of the following: “Yesterday we completed 6.2 and so today we are moving on to 6.3.” There is no mention even of the topics these numerals represent. But students have the text and instructors seem to feel that they can check these topics for themselves. This is not to say that no math teacher ever explicitly addresses the objective and the purpose. Certainly they do. But often it is only at the beginning of a unit of study or an individual concept. Often, for days, there are discrete lessons with problems sets to follow and little reference to the larger context of the learning. In all of my observations, though, I have never seen students complain about this. It is either not a problem for most of them, or they have just accepted it as the way math is taught.
Most math teachers I have discussed this with emphasize the need for students to continually be working problems, that it is the repetition that will cement the concept. Any activity that distracts students from problem-solving runs the risk of leaving them unprepared for higher math classes or for exams to test their competencies. Whenever I suggest the history of mathematics as a fascinating topic, I am usually reminded that any reading about math is not solving math problems and steals from the time students need to perform enough repetitions to become predictably competent in math.
My own instincts move me in exactly the opposite direction. New learning embeds longer if it hangs from already existing cognitive hooks. It is crucial for students to place today’s learning in a context significantly high enough on the abstraction ladder to distinguish it from other learning conceptually. I would argue that the history of mathematics, which is both fascinating and illuminating, is an excellent avenue for providing the kind of cognitive hooks that will embed the lesson and will have the ancillary benefit of instilling a lifelong interest in, and respect for, mathematics.
This activity would also be an excellent way to support reading across the curriculum. At the present time, the only reading students are doing in schools is self-chosen material as part of sustained silent reading programs. Very little content area reading is occurring in secondary schools. The goal of every school should be to dramatically increase the amount of content area reading. The best way to accomplish this is with content area monographs which are both interesting and well-written.
I am, therefore, advocating two changes in the math formula:
1. The objective and purpose for each lesson concept should not only be explicitly referenced, but should also be moved high enough up the abstraction ladder to place it properly within the hierarchy of mathematical understanding. Where does this concept fit in the over-all schema of mathematics? What is its function?
2. Mathematics department should purchase high-interest, well-written monographs on the history of mathematics, as well as biographies of famous mathematicians, or monographs on engineering, architecture, astronomy, NASA, or any number of related subjects, for students to read as an anchor activity in their math classes. Students work at various rates and most students complete guided practice problems ahead of the slower learners. If students were trained to take out their monograph on mathematics whenever they finish early and read, the potential benefits would be profound.
a. Over the course of their secondary careers, students would increase by hundreds of hours their exposure to non-fiction content area text.
b. They would gain a deeper understanding of the history and significance of mathematics and would have a larger context within which to place and value their learning.
As a way of testing the viability of the second recommendation I visited the public library and scanned the shelves in the math section. Inside of 20 minutes I located several high-interest, well-written books on the history and organization of mathematics as an academic discipline. I also did readability analyses of the small excerpts I typed from the monographs. I have arranged them from 8th to 12th grade. I also included the Flesh Reading ease indicator. This is an inverse scale. The higher the number, from 1-100, the easier the read. As the numbers fall from 50 toward 0 the difficulty of the reading increases. A book written at the 9th grade level will have a reading ease of around 60. A college level text will have a reading ease in the 20-30 range.
1.
Zero, The Biography of a Dangerous Idea, Charles Seife, Viking Press, New York, N.Y. 2000
Chapter 0
Null and Void
“Zero hit the USS Yorktown like a torpedo.
“On September 21, 1997, while crusing off the coast of Virginia, the billion-dollar missile cruiser shuddered to a halt. Yorktown was dead in the water.
“Warships are designed to withstand the strike of a torpedo or the blast of a mine. Though it was armored against weapons, nobody had thought to defend the Yorktown from zero. It was a grave mistake.
“The Yorktown’s computers had just received new software that was controlling the engines. Unfortunately, nobody had spotted the time bomb lurking in the code, a zero that engineers were supposed to remove while installing the software. But for one reason or another, the zero was overlooked, and it stayed hidden in the code. Hidden, that is, until the software called in into memory – and choked.
“When the Yorktown’s computer system tried to divide by zero, 80,000 horsepower instantly became worthless. It took nearly three hours to attach emergency controls to the engines, and the Yorktown then limped into port. Engineers spent two days getting rid of the zero, repairing the engines, and putting the Yorktown back into fighting trim.
“No other number can do such damage. Computer failures like the one that struck the Yorktown are just a faint shadow of the power of zero. Cultures girded themselves against zero, and philosophies crumbled under its influence, for zero is different from the other numbers. It provides a glimpse of the ineffable and the infinite. This is why it has been feared and hated – and outlawed.” [pp. 1-2]
Flesh Reading Ease = 59.2
Flesh-Kincaid Grade Level = 8.4
2.
The Problems of Mathematics, by Ian Stewart, Oxford University Press, Oxford and New York, 1987
“One of the biggest problems of mathematics is to explain to everyone else what it is all about. The technical trappings of the subject, its symbolism and formality, its baffling terminology, its apparent delight in lengthy calculations: these tend to obscure its real nature. A musician would be horrified if his art were to be summed up as ‘a lot of tadpoles drawn on a row of lines’; but that’s all that the untrained eye can see in a page of sheet music. The grandeur, the agony, the flights of lyricism and the discords of despair: to discern them among the tadpoles is no mean task. They are present, but only in coded form. In the same way, the symbolism of mathematics is merely its coded form, not its substance. It too has its grandeur, agony, and flights of lyricism. However, there is a difference. Even a casual listener can enjoy a piece of music. It is only the performers who are required to understand the antics of the tadpoles. Music has an immediate appeal to almost everybody. But the nearest thing I can think of to a mathematical performance is the Renaissance tournament, where leading mathematicians did public battle on each other’s problems. The idea might profitably be revived; but its appeal is more that of wrestling than of music.
“Music can be appreciated from several points of view: the listener, the performer, the composer. In mathematics there is nothing analogous to the listener; and even if there were, it would be the composer, rather than the performer, that would interest him. It is the creation of new mathematics, rather than its mundane practice, that is interesting. Mathematics is not about symbols and calculations. These are just tools of the trade – quavers and crochets and five-finger exercises. Mathematics is about ideas. In particular, it is about the way that different ideas relate to each other. If certain information is known, what else must necessarily follow? The aim of mathematics is to understand such questions by stripping away the inessentials and penetrating to the core of the problem. It is not just a question of getting the right answer; more a matter of understanding why an answer is possible at all, and why it takes the form that it does. Good mathematics has an air of economy and an element of surprise. But, above all, it has significance.” [pp. 5-6]
Flesh Reading Ease = 57.7
Flesh-Kincaid Grade Level = 8.6
3.
A History of the Circle, Mathematical Reasoning and the Physical Universe, Ernest Zebrowski, Jr. Rutgers University Press, New Brunswick, New Jersey, 1999
“This book is about something that doesn’t physically exist. Try to find a circle in nature, or even in the artifacts of human civilization, and you’ve embarked on an impossible hunt. A car tire? No, every tire is a bit flat on the bottom. A quarter? No, quarters have little ridges around their rims. A doughnut? Too bumpy, if we look closely. Even the full moon has mountains and craters around its perimeter, and we need only a decent pair of binoculars to see them . . .
“But isn’t this just an abstract philosophical issue? Surely, a physical circle is often close enough to a true circle that for all practical purposes we can consider it to be one. Indeed, if this weren’t the case, a whole host of devices we take for granted would not be possible, from bottle caps to automobile transmissions. Why concern ourselves about the nonexistence of true circles, when real but approximate circles seem to serve our human needs so well?
“The answer which forms a thread running through this book, is that approximate circles don’t always serve our human needs, nor are shapes that appear to be approximately circular always best described as mathematical circles. The question How round is round enough? it turns out, does not have a simple answer, and deciding when the geometry of “perfect” circles has something to say about real physical objects has historically often presented considerable intellectual challenges. . .
“This book is an odyssey through two worlds: one the world of round physical entities, the other the world of true circles and their mathematical cousins. At points where these two worlds intersect, we’ll use the opportunity to step from one to the other. Along the way, we’ll explore some of the marvels of ancient and modern engineering, we’ll examine some principles of architecture and art, we’ll dwell on a few questions of geography and astronomy, and we’ll consider questions of the grand design of the universe. We’ll look at some ancient abstract ideas that have stood the test of time, and some more modern ideas that have failed. Throughout, the question ‘How round is round enough?’ will remain one of our guides.” [pp. 1-2]
Flesh Reading Ease = 58.5
Flesh-Kincaid Grade Level = 9.0
4.
Fermat’s Enigma by Simon Singh, Walker and Company, New York, 1997
“The story of Fermat’s Last Theorem is inextricably linked with the history of mathematics, touching on all the major themes of number theory. It provides a unique insight into what drives mathematics and, perhaps more important, what inspires mathematicians. The Last Theorem is at the heart of an intriguing saga of courage, skullduggery, cunning, and tragedy, involving all the greatest heroes of mathematics.
“Fermat’s Last Theorem has its origins in the mathematics of ancient Greece, two thousand years before Pierre de Fermat constructed the problem in the form we know it today. Hence, it links the foundations of mathematics created by Pythagoras to the most sophisticated ideas in modern mathematics.” [Preface, p. xv]
Absolute Proof
“The story of Fermat’s Last Theorem revolves around the search for a missing proof. Mathematical proof is far more powerful and rigorous than the concept of proof we casually use in our everyday language, or even the concept of proof as understood by physicists or chemists. The difference between scientific and mathematical proof is both subtle and profound, and is crucial to understanding the work of every mathematician since Pythagoras.
“The idea of a classic mathematical proof is to begin with a series of axioms, statements that can be assumed to be true or that are self-evidently true. Then by arguing logically, step by step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem.
“Mathematical theorems rely on this logical process and once proven are true until the end of time. Mathematical proofs are absolute. To appreciate the value of such proofs they should be compared with their poor relation, the scientific proof. In science a hypothesis is put forward to explain a physical phenomenon. If observations of the phenomenon compare well with the hypothesis, this becomes evidence in favor of it. Furthermore, the hypothesis should not merely describe a known phenomenon, but predict the results of other phenomena. Experiments may be performed to test the predictive power of the hypothesis, and if it continues to be successful then this is even more evidence to back the hypothesis. Eventually the amount of evidence may be overwhelming and the hypothesis becomes accepted as a scientific theory.” [pp. 20-21]
Fermat’s Last Theorem, as it is known, stated that
x+ y= z has no whole number solutions for n greater than 2.
. . . Fermat had claimed that even if all the mathematicians in the world spent eternity looking for a solution to the equation they would fail to find one. . .
For over 300 years many of the greatest mathematicians had tried to rediscover Fermat’s lost proof and failed. As each generation failed, the next became even more frustrated and determined. In 1742, almost a century after Fermat’s death, the Swiss mathematician Leonhard Euler asked his friend Clerot to search Fermat’s house in case some vital scrap of paper still remained. No clues were ever found as to what Fermat’s proof might have been.
Fermat’s last Theorem, a problem that had captivated mathematicians for centuries, captured the imagination of the young Andrew Wiles. In Milton Road Library, ten-year-old Wiles stared at the most infamous problem in mathematics, undaunted by the knowledge that the most brilliant minds on the planet had failed to rediscover the proof. Young Wiles immediately set to work using all his textbook techniques to try to recreate the proof. Perhaps he could find something that everyone else, except Fermat, had overlooked. He dreamed he could shock the world.
Thirty years later Andrew Wiles stood in the auditorium of the Isaac Newton Institute. He scribbled on the board and then, struggling to contain his glee, stared at this audience. The lecture was reading its climax and the audience knew it. One or two of them had smuggled cameras into the lecture room and flashes peppered his concluding remarks.
With chalk in his hand he turned to the board for the last time. The final few lines of logic completed the proof. For the first time in over three centuries Fermat’s challenge had been met. A few more cameras flashed to capture the historic moment. Wiles wrote up the statement of Fermat’s Last Theorem, turned toward the audience, and said modestly, ‘I think I’ll stop here.’
Two hundred mathematicians clapped and cheered in celebration. Even those who had anticipated the result grinned in disbelief. After three decades Andrew Wiles believed he had achieved his dream, and after seven years of isolation he had revealed his secret calculation. While a general mood of euphoria filled the Newton Institute, everybody realized that the proof still had to be rigorously checked by a team of independent referees. However, as Wiles enjoyed the moment, nobody could have predicted the controversy that would evolve in the months ahead.” [pp. 32-33]
Flesh Reading Ease = 48.6
Flesh-Kincaid Grade Level = 10.8
5.
A Beautiful Mind by Sylvia Nasar, A Touchstone Book, New York, N.Y., 1998
“The young genius from Bluefield, West Virginia – handsome, arrogant, and highly eccentric – burst onto the mathematical scene in 1948. Over the next decade, a decade as notable for its supreme faith in human rationality as for its dark anxieties about mankind’s survival, Nash proved himself, in the words of the eminent geometer Mikhail Gromov, ‘the most remarkable mathematician of the second half of the century.’ Games of strategy, economic rivalry, computer architecture, the shape of the universe, the geometry of imaginary spaces, the mystery of prime numbers – all engaged his wide-ranging imagination. His ideas were of the deep and wholly unanticipated kind that pushes scientific thinking in new directions.
“Geniuses, the mathematician Paul Halmos wrote, “are of two kinds: the ones who are just like all of us, but very much more so, and the ones who, apparently, have an extra human spark. We can all run, and some of us can run the mile in less than 4 minutes; but there is nothing that most of us can do that compares with the creation of the Great G-minor Fugue.’ Nash’s genius was of that mysterious variety more often associated with music and art than with the oldest of all sciences. It wasn’t merely that his mind worked faster, that his memory was more retentive, or that his power of concentration was greater. The flashes of intuition were non-rational. Like other great mathematical intuitionists, . . . Nash saw the vision first, constructing the laborious proofs long afterward. But even after he’d try to explain some astonishing result, the actual route he had taken remained a mystery to others who tried to follow his reasoning. Donald Newmann, a mathematician who knew Nash at MIT in the 1950s, used to say about him that ‘everyone else would climb a peak by looking for a path somewhere on the mountain. Nash would climb another mountain altogether and from that distant peak would shine a searchlight back onto the first peak.’” [pp. 11-12]
Flesh Reading Ease = 44.2
Flesh-Kincaid Grade Level = 12.8
Am I the only one who finds these books utterly fascinating? Most students of mathematics will not become professional mathematicians. We desire certain competencies in our culture, but isn’t instilling a wonder about mathematics and physics and biology, for that matter, history and philosophy, as well, just as important? I am not asserting that a respect for mathematics as a discipline replace mastery of mathematical concepts and reasoning. But the reverse is also true. Computational accuracy should not be exclusive of the wonder that produced such formulas. Can’t students have both and won’t having both serve their competencies?
A recent issue of Newsweek Magazine, from December 5, 2005, contained this article under the byline of Mary Carmichael entitled “Analyze These!” on p. 49. It references several popular recent titles on physics and math primarily to highlight a new work by Mario Livio entitled The Equation That Couldn’t Be Solved. Here is an extended excerpt from the review:
“Considering how few people use higher math in their lives, or even remember much of it from high school, the popularity of books on chaos theory and number theory and higher-dimensional geometry is, well, a paradox. Brian Green’s bestselling The Elegant Universe, on string theory, kicked off the most recent outpouring in 1999, followed by John Derbyshire’s Prime Obsession, Steven Strogatz’s Sync and Janna Levin’s How the Universe Got its Spots, a gorgeously written collection of unsent letters to her mom on cosmology and topology.
"The latest entry is Mario Livio’s The Equation That Couldn’t Be Solved, a wide-ranging exploration of the phenomenon of symmetry, focused on, well, a seemingly unsolvable equation. Specifically, it’s the “quintic,” one step up from the dread quadratic equation that gives so many kids fits in algebra.
"But at the center of The Equation That Couldn’t Be Solved is the story of two mathematicians who independently invented much of modern math by coming up with ways to crack the unsolvable equation. They suffered similar – dare we say symmetrical? – tragic fates, dying before their discoveries brought them acclaim. Niles Henrik Abel succumbed to tuberculosis in 1928, at the age of 26, a few years after hitting on the basic tenets of group theory. His counterpart, Evariste Galois, is less well known, but Livio eloquently defends Galois’s contributions to group theory while bringing us the story of how the 20 year old half-mad mathematician died in a duel over a woman. There’s math, yes, but there are also tales of love, violence, history – and the whole, in this case, turns out to be greater than the sum of those parts.”
Recent monographs on complex theories of space and time and the advancement of mathematics have achieved success with adult readers. I would assert that they are missing their primary audience, those exposed to daily doses of mathematical concepts and nightly homework involving the solving of complex problem sets. Adult readers are, for the most part, reaching far into their past to understand advancements in mathematics. But at least most adult readers have broad contexts in which to place these ideas. Secondary students need these contexts every bit as much and providing them will serve all involved in the process: society, curriculum specialists, teachers and students.
I recently observed a math lesson in which an instructor cleverly employed what he termed journaling. He introduced a dialectical journal and modeled it for his students. On the left hand side he displayed each step in the solution of a problem. On the right side he wrote a description of what he had done and the reasoning behind it. This sounds like a proof in geometry, but it was much more. The ideas were written out in sentences and the logic was on display, in language, not just in symbol. The intent, of course, was to help these students embed the learning and become more proficient at solving subsequent problems, but it achieved so much more. This teacher intuitively understood that mathematics is language-based. For those math teachers who think all math consists of solving equations on the board, this was an implicit acknowledgement that what teachers do as they write their lines of code is convert them to language concepts. They explain each step using language which is, itself, grounded in the logic of grammar. The teacher seemed to realize, however, how much instructional time such journaling took and wanted to move things along. But I would argue that time spent journaling in math, or science, or social science, is time well spent. If the learning goes deeper and lasts longer, if you take the long view, no time is really lost.
Journaling also supports the school-wide, or district-wide, goal of writing across the curriculum, just as reading content-based monographs supports the parallel goal of reading across the curriculum. Often content area teachers complain that reading and writing should be taught in language arts. But the reading and writing described here supports curricular goals first and cross-curricular goals second.
I would speculate that students, from as early as grade 4, who are engaged daily in content based reading and writing and who are routinely introduced to new concepts via text will have competencies far surpassing the minimum requirements our state-wide testing programs are trying to achieve.
[Editor's Note: While I was a full-timed released mentor to new teachers in Ventura County I worked with a 1st and 2nd year Math teacher at Newbury Park High School named Michael Weingarden. I recently sent him this article for his review. Here is a link to his considered response: Response to "The Math Formula"
The 7 step lesson plans has the following or similarly titled steps:
Elements of Lesson Design
Anticipatory Set
A mental set that causes students to focus on what will be learned.
Objective and Purpose
Not only do students learn more effectively when they know what they’re supposed to be learning and why that learning is important to them, but teachers teach more effectively when they have that same information.
Input
Students must acquire new information about the knowledge, process, or skill they are to achieve.
Modeling
“Seeing” what is meant is an important adjunct to learning.
Checking for understanding
Before students are expected to do something, the teacher should determine that they understand what they are supposed to do and that they have the minimum skills required.
Guided Practice
Students practice their new knowledge or skill under direct teacher supervision.
Independent Practice
Independent practice is assigned only after the teacher is reasonably sure that students will not make serious errors.
A principal once told me that in all his years observing math classes he had only seen one lesson, but he’d seen it hundreds of times. I prefer to call it the Math Formula. The version of the 7 step lesson plan that is most often observed in math classes would have these re-named steps:
Warm-up
The warm-up activity, the most ubiquitous of all math strategies, is normally used for one of two purposes. The problems are either a review of recent learning and are often critical elements of the day’s lesson, or, in very rare cases, they forecast new learning in some insightful way.
Homework Review
Most teachers ask the class if there were any difficulties with the previous night’s homework. Teachers, or selected students, solve the troublesome problems step by step.
New Learning
This is the lesson for the day. Teachers typically engage in direct instruction and begin to explain step by step a sample problem.
Modeling
The teacher will solve two or more additional problems, on the board, overhead projector, or smart board, that illustrate the new concept.
Checking for Understanding
The teacher will question students to establish the degree to which they understand the new concept. Sometimes volunteers are sent to the board to work a problem or two. More often the teacher continues at the board, working the problem as he questions the students.
Guided Practice [Problem set]
The teacher will assign a manageable problem set which students will work on alone, sometimes in pairs or trios, with the teacher directly supervising their work.
Independent Practice [Homework]
The teacher will assign a longer problem set for independent practice. This usually cannot be completed by the end of the class period and then becomes that night’s homework.
I must signal a caveat here. My background is in language arts and I did spend 30 years teaching high school English. However, I have also spent 5 years in full-time release from the classroom to work as a mentor to new teachers. I have worked with over a dozen middle school and high school math teachers. In the course of this work, I have had occasion to observe many veteran math teachers as well. Regardless of my content area background, I have been tasked by the county office of education to look for the teaching standards in the classrooms I observe. The comments I will make, though they may sound critical of math instruction, are grounded in my understanding of the teaching standards.
When you compare the 7 step lesson with its incarnation in math classrooms, what you find most often missing is step 2: “Objective and Purpose.” Most math teachers appear to feel that the objective of the learning is embedded in the design of the textbook and the corresponding lessons. I have rarely seen a math teacher explicitly address this. On the contrary, on numerous occasions, I have heard the instructor say some variation of the following: “Yesterday we completed 6.2 and so today we are moving on to 6.3.” There is no mention even of the topics these numerals represent. But students have the text and instructors seem to feel that they can check these topics for themselves. This is not to say that no math teacher ever explicitly addresses the objective and the purpose. Certainly they do. But often it is only at the beginning of a unit of study or an individual concept. Often, for days, there are discrete lessons with problems sets to follow and little reference to the larger context of the learning. In all of my observations, though, I have never seen students complain about this. It is either not a problem for most of them, or they have just accepted it as the way math is taught.
Most math teachers I have discussed this with emphasize the need for students to continually be working problems, that it is the repetition that will cement the concept. Any activity that distracts students from problem-solving runs the risk of leaving them unprepared for higher math classes or for exams to test their competencies. Whenever I suggest the history of mathematics as a fascinating topic, I am usually reminded that any reading about math is not solving math problems and steals from the time students need to perform enough repetitions to become predictably competent in math.
My own instincts move me in exactly the opposite direction. New learning embeds longer if it hangs from already existing cognitive hooks. It is crucial for students to place today’s learning in a context significantly high enough on the abstraction ladder to distinguish it from other learning conceptually. I would argue that the history of mathematics, which is both fascinating and illuminating, is an excellent avenue for providing the kind of cognitive hooks that will embed the lesson and will have the ancillary benefit of instilling a lifelong interest in, and respect for, mathematics.
This activity would also be an excellent way to support reading across the curriculum. At the present time, the only reading students are doing in schools is self-chosen material as part of sustained silent reading programs. Very little content area reading is occurring in secondary schools. The goal of every school should be to dramatically increase the amount of content area reading. The best way to accomplish this is with content area monographs which are both interesting and well-written.
I am, therefore, advocating two changes in the math formula:
1. The objective and purpose for each lesson concept should not only be explicitly referenced, but should also be moved high enough up the abstraction ladder to place it properly within the hierarchy of mathematical understanding. Where does this concept fit in the over-all schema of mathematics? What is its function?
2. Mathematics department should purchase high-interest, well-written monographs on the history of mathematics, as well as biographies of famous mathematicians, or monographs on engineering, architecture, astronomy, NASA, or any number of related subjects, for students to read as an anchor activity in their math classes. Students work at various rates and most students complete guided practice problems ahead of the slower learners. If students were trained to take out their monograph on mathematics whenever they finish early and read, the potential benefits would be profound.
a. Over the course of their secondary careers, students would increase by hundreds of hours their exposure to non-fiction content area text.
b. They would gain a deeper understanding of the history and significance of mathematics and would have a larger context within which to place and value their learning.
As a way of testing the viability of the second recommendation I visited the public library and scanned the shelves in the math section. Inside of 20 minutes I located several high-interest, well-written books on the history and organization of mathematics as an academic discipline. I also did readability analyses of the small excerpts I typed from the monographs. I have arranged them from 8th to 12th grade. I also included the Flesh Reading ease indicator. This is an inverse scale. The higher the number, from 1-100, the easier the read. As the numbers fall from 50 toward 0 the difficulty of the reading increases. A book written at the 9th grade level will have a reading ease of around 60. A college level text will have a reading ease in the 20-30 range.
1.
Zero, The Biography of a Dangerous Idea, Charles Seife, Viking Press, New York, N.Y. 2000
Chapter 0
Null and Void
“Zero hit the USS Yorktown like a torpedo.
“On September 21, 1997, while crusing off the coast of Virginia, the billion-dollar missile cruiser shuddered to a halt. Yorktown was dead in the water.
“Warships are designed to withstand the strike of a torpedo or the blast of a mine. Though it was armored against weapons, nobody had thought to defend the Yorktown from zero. It was a grave mistake.
“The Yorktown’s computers had just received new software that was controlling the engines. Unfortunately, nobody had spotted the time bomb lurking in the code, a zero that engineers were supposed to remove while installing the software. But for one reason or another, the zero was overlooked, and it stayed hidden in the code. Hidden, that is, until the software called in into memory – and choked.
“When the Yorktown’s computer system tried to divide by zero, 80,000 horsepower instantly became worthless. It took nearly three hours to attach emergency controls to the engines, and the Yorktown then limped into port. Engineers spent two days getting rid of the zero, repairing the engines, and putting the Yorktown back into fighting trim.
“No other number can do such damage. Computer failures like the one that struck the Yorktown are just a faint shadow of the power of zero. Cultures girded themselves against zero, and philosophies crumbled under its influence, for zero is different from the other numbers. It provides a glimpse of the ineffable and the infinite. This is why it has been feared and hated – and outlawed.” [pp. 1-2]
Flesh Reading Ease = 59.2
Flesh-Kincaid Grade Level = 8.4
2.
The Problems of Mathematics, by Ian Stewart, Oxford University Press, Oxford and New York, 1987
“One of the biggest problems of mathematics is to explain to everyone else what it is all about. The technical trappings of the subject, its symbolism and formality, its baffling terminology, its apparent delight in lengthy calculations: these tend to obscure its real nature. A musician would be horrified if his art were to be summed up as ‘a lot of tadpoles drawn on a row of lines’; but that’s all that the untrained eye can see in a page of sheet music. The grandeur, the agony, the flights of lyricism and the discords of despair: to discern them among the tadpoles is no mean task. They are present, but only in coded form. In the same way, the symbolism of mathematics is merely its coded form, not its substance. It too has its grandeur, agony, and flights of lyricism. However, there is a difference. Even a casual listener can enjoy a piece of music. It is only the performers who are required to understand the antics of the tadpoles. Music has an immediate appeal to almost everybody. But the nearest thing I can think of to a mathematical performance is the Renaissance tournament, where leading mathematicians did public battle on each other’s problems. The idea might profitably be revived; but its appeal is more that of wrestling than of music.
“Music can be appreciated from several points of view: the listener, the performer, the composer. In mathematics there is nothing analogous to the listener; and even if there were, it would be the composer, rather than the performer, that would interest him. It is the creation of new mathematics, rather than its mundane practice, that is interesting. Mathematics is not about symbols and calculations. These are just tools of the trade – quavers and crochets and five-finger exercises. Mathematics is about ideas. In particular, it is about the way that different ideas relate to each other. If certain information is known, what else must necessarily follow? The aim of mathematics is to understand such questions by stripping away the inessentials and penetrating to the core of the problem. It is not just a question of getting the right answer; more a matter of understanding why an answer is possible at all, and why it takes the form that it does. Good mathematics has an air of economy and an element of surprise. But, above all, it has significance.” [pp. 5-6]
Flesh Reading Ease = 57.7
Flesh-Kincaid Grade Level = 8.6
3.
A History of the Circle, Mathematical Reasoning and the Physical Universe, Ernest Zebrowski, Jr. Rutgers University Press, New Brunswick, New Jersey, 1999
“This book is about something that doesn’t physically exist. Try to find a circle in nature, or even in the artifacts of human civilization, and you’ve embarked on an impossible hunt. A car tire? No, every tire is a bit flat on the bottom. A quarter? No, quarters have little ridges around their rims. A doughnut? Too bumpy, if we look closely. Even the full moon has mountains and craters around its perimeter, and we need only a decent pair of binoculars to see them . . .
“But isn’t this just an abstract philosophical issue? Surely, a physical circle is often close enough to a true circle that for all practical purposes we can consider it to be one. Indeed, if this weren’t the case, a whole host of devices we take for granted would not be possible, from bottle caps to automobile transmissions. Why concern ourselves about the nonexistence of true circles, when real but approximate circles seem to serve our human needs so well?
“The answer which forms a thread running through this book, is that approximate circles don’t always serve our human needs, nor are shapes that appear to be approximately circular always best described as mathematical circles. The question How round is round enough? it turns out, does not have a simple answer, and deciding when the geometry of “perfect” circles has something to say about real physical objects has historically often presented considerable intellectual challenges. . .
“This book is an odyssey through two worlds: one the world of round physical entities, the other the world of true circles and their mathematical cousins. At points where these two worlds intersect, we’ll use the opportunity to step from one to the other. Along the way, we’ll explore some of the marvels of ancient and modern engineering, we’ll examine some principles of architecture and art, we’ll dwell on a few questions of geography and astronomy, and we’ll consider questions of the grand design of the universe. We’ll look at some ancient abstract ideas that have stood the test of time, and some more modern ideas that have failed. Throughout, the question ‘How round is round enough?’ will remain one of our guides.” [pp. 1-2]
Flesh Reading Ease = 58.5
Flesh-Kincaid Grade Level = 9.0
4.
Fermat’s Enigma by Simon Singh, Walker and Company, New York, 1997
“The story of Fermat’s Last Theorem is inextricably linked with the history of mathematics, touching on all the major themes of number theory. It provides a unique insight into what drives mathematics and, perhaps more important, what inspires mathematicians. The Last Theorem is at the heart of an intriguing saga of courage, skullduggery, cunning, and tragedy, involving all the greatest heroes of mathematics.
“Fermat’s Last Theorem has its origins in the mathematics of ancient Greece, two thousand years before Pierre de Fermat constructed the problem in the form we know it today. Hence, it links the foundations of mathematics created by Pythagoras to the most sophisticated ideas in modern mathematics.” [Preface, p. xv]
Absolute Proof
“The story of Fermat’s Last Theorem revolves around the search for a missing proof. Mathematical proof is far more powerful and rigorous than the concept of proof we casually use in our everyday language, or even the concept of proof as understood by physicists or chemists. The difference between scientific and mathematical proof is both subtle and profound, and is crucial to understanding the work of every mathematician since Pythagoras.
“The idea of a classic mathematical proof is to begin with a series of axioms, statements that can be assumed to be true or that are self-evidently true. Then by arguing logically, step by step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem.
“Mathematical theorems rely on this logical process and once proven are true until the end of time. Mathematical proofs are absolute. To appreciate the value of such proofs they should be compared with their poor relation, the scientific proof. In science a hypothesis is put forward to explain a physical phenomenon. If observations of the phenomenon compare well with the hypothesis, this becomes evidence in favor of it. Furthermore, the hypothesis should not merely describe a known phenomenon, but predict the results of other phenomena. Experiments may be performed to test the predictive power of the hypothesis, and if it continues to be successful then this is even more evidence to back the hypothesis. Eventually the amount of evidence may be overwhelming and the hypothesis becomes accepted as a scientific theory.” [pp. 20-21]
Fermat’s Last Theorem, as it is known, stated that
x+ y= z has no whole number solutions for n greater than 2.
. . . Fermat had claimed that even if all the mathematicians in the world spent eternity looking for a solution to the equation they would fail to find one. . .
For over 300 years many of the greatest mathematicians had tried to rediscover Fermat’s lost proof and failed. As each generation failed, the next became even more frustrated and determined. In 1742, almost a century after Fermat’s death, the Swiss mathematician Leonhard Euler asked his friend Clerot to search Fermat’s house in case some vital scrap of paper still remained. No clues were ever found as to what Fermat’s proof might have been.
Fermat’s last Theorem, a problem that had captivated mathematicians for centuries, captured the imagination of the young Andrew Wiles. In Milton Road Library, ten-year-old Wiles stared at the most infamous problem in mathematics, undaunted by the knowledge that the most brilliant minds on the planet had failed to rediscover the proof. Young Wiles immediately set to work using all his textbook techniques to try to recreate the proof. Perhaps he could find something that everyone else, except Fermat, had overlooked. He dreamed he could shock the world.
Thirty years later Andrew Wiles stood in the auditorium of the Isaac Newton Institute. He scribbled on the board and then, struggling to contain his glee, stared at this audience. The lecture was reading its climax and the audience knew it. One or two of them had smuggled cameras into the lecture room and flashes peppered his concluding remarks.
With chalk in his hand he turned to the board for the last time. The final few lines of logic completed the proof. For the first time in over three centuries Fermat’s challenge had been met. A few more cameras flashed to capture the historic moment. Wiles wrote up the statement of Fermat’s Last Theorem, turned toward the audience, and said modestly, ‘I think I’ll stop here.’
Two hundred mathematicians clapped and cheered in celebration. Even those who had anticipated the result grinned in disbelief. After three decades Andrew Wiles believed he had achieved his dream, and after seven years of isolation he had revealed his secret calculation. While a general mood of euphoria filled the Newton Institute, everybody realized that the proof still had to be rigorously checked by a team of independent referees. However, as Wiles enjoyed the moment, nobody could have predicted the controversy that would evolve in the months ahead.” [pp. 32-33]
Flesh Reading Ease = 48.6
Flesh-Kincaid Grade Level = 10.8
5.
A Beautiful Mind by Sylvia Nasar, A Touchstone Book, New York, N.Y., 1998
“The young genius from Bluefield, West Virginia – handsome, arrogant, and highly eccentric – burst onto the mathematical scene in 1948. Over the next decade, a decade as notable for its supreme faith in human rationality as for its dark anxieties about mankind’s survival, Nash proved himself, in the words of the eminent geometer Mikhail Gromov, ‘the most remarkable mathematician of the second half of the century.’ Games of strategy, economic rivalry, computer architecture, the shape of the universe, the geometry of imaginary spaces, the mystery of prime numbers – all engaged his wide-ranging imagination. His ideas were of the deep and wholly unanticipated kind that pushes scientific thinking in new directions.
“Geniuses, the mathematician Paul Halmos wrote, “are of two kinds: the ones who are just like all of us, but very much more so, and the ones who, apparently, have an extra human spark. We can all run, and some of us can run the mile in less than 4 minutes; but there is nothing that most of us can do that compares with the creation of the Great G-minor Fugue.’ Nash’s genius was of that mysterious variety more often associated with music and art than with the oldest of all sciences. It wasn’t merely that his mind worked faster, that his memory was more retentive, or that his power of concentration was greater. The flashes of intuition were non-rational. Like other great mathematical intuitionists, . . . Nash saw the vision first, constructing the laborious proofs long afterward. But even after he’d try to explain some astonishing result, the actual route he had taken remained a mystery to others who tried to follow his reasoning. Donald Newmann, a mathematician who knew Nash at MIT in the 1950s, used to say about him that ‘everyone else would climb a peak by looking for a path somewhere on the mountain. Nash would climb another mountain altogether and from that distant peak would shine a searchlight back onto the first peak.’” [pp. 11-12]
Flesh Reading Ease = 44.2
Flesh-Kincaid Grade Level = 12.8
Am I the only one who finds these books utterly fascinating? Most students of mathematics will not become professional mathematicians. We desire certain competencies in our culture, but isn’t instilling a wonder about mathematics and physics and biology, for that matter, history and philosophy, as well, just as important? I am not asserting that a respect for mathematics as a discipline replace mastery of mathematical concepts and reasoning. But the reverse is also true. Computational accuracy should not be exclusive of the wonder that produced such formulas. Can’t students have both and won’t having both serve their competencies?
A recent issue of Newsweek Magazine, from December 5, 2005, contained this article under the byline of Mary Carmichael entitled “Analyze These!” on p. 49. It references several popular recent titles on physics and math primarily to highlight a new work by Mario Livio entitled The Equation That Couldn’t Be Solved. Here is an extended excerpt from the review:
“Considering how few people use higher math in their lives, or even remember much of it from high school, the popularity of books on chaos theory and number theory and higher-dimensional geometry is, well, a paradox. Brian Green’s bestselling The Elegant Universe, on string theory, kicked off the most recent outpouring in 1999, followed by John Derbyshire’s Prime Obsession, Steven Strogatz’s Sync and Janna Levin’s How the Universe Got its Spots, a gorgeously written collection of unsent letters to her mom on cosmology and topology.
"The latest entry is Mario Livio’s The Equation That Couldn’t Be Solved, a wide-ranging exploration of the phenomenon of symmetry, focused on, well, a seemingly unsolvable equation. Specifically, it’s the “quintic,” one step up from the dread quadratic equation that gives so many kids fits in algebra.
"But at the center of The Equation That Couldn’t Be Solved is the story of two mathematicians who independently invented much of modern math by coming up with ways to crack the unsolvable equation. They suffered similar – dare we say symmetrical? – tragic fates, dying before their discoveries brought them acclaim. Niles Henrik Abel succumbed to tuberculosis in 1928, at the age of 26, a few years after hitting on the basic tenets of group theory. His counterpart, Evariste Galois, is less well known, but Livio eloquently defends Galois’s contributions to group theory while bringing us the story of how the 20 year old half-mad mathematician died in a duel over a woman. There’s math, yes, but there are also tales of love, violence, history – and the whole, in this case, turns out to be greater than the sum of those parts.”
Recent monographs on complex theories of space and time and the advancement of mathematics have achieved success with adult readers. I would assert that they are missing their primary audience, those exposed to daily doses of mathematical concepts and nightly homework involving the solving of complex problem sets. Adult readers are, for the most part, reaching far into their past to understand advancements in mathematics. But at least most adult readers have broad contexts in which to place these ideas. Secondary students need these contexts every bit as much and providing them will serve all involved in the process: society, curriculum specialists, teachers and students.
I recently observed a math lesson in which an instructor cleverly employed what he termed journaling. He introduced a dialectical journal and modeled it for his students. On the left hand side he displayed each step in the solution of a problem. On the right side he wrote a description of what he had done and the reasoning behind it. This sounds like a proof in geometry, but it was much more. The ideas were written out in sentences and the logic was on display, in language, not just in symbol. The intent, of course, was to help these students embed the learning and become more proficient at solving subsequent problems, but it achieved so much more. This teacher intuitively understood that mathematics is language-based. For those math teachers who think all math consists of solving equations on the board, this was an implicit acknowledgement that what teachers do as they write their lines of code is convert them to language concepts. They explain each step using language which is, itself, grounded in the logic of grammar. The teacher seemed to realize, however, how much instructional time such journaling took and wanted to move things along. But I would argue that time spent journaling in math, or science, or social science, is time well spent. If the learning goes deeper and lasts longer, if you take the long view, no time is really lost.
Journaling also supports the school-wide, or district-wide, goal of writing across the curriculum, just as reading content-based monographs supports the parallel goal of reading across the curriculum. Often content area teachers complain that reading and writing should be taught in language arts. But the reading and writing described here supports curricular goals first and cross-curricular goals second.
I would speculate that students, from as early as grade 4, who are engaged daily in content based reading and writing and who are routinely introduced to new concepts via text will have competencies far surpassing the minimum requirements our state-wide testing programs are trying to achieve.
[Editor's Note: While I was a full-timed released mentor to new teachers in Ventura County I worked with a 1st and 2nd year Math teacher at Newbury Park High School named Michael Weingarden. I recently sent him this article for his review. Here is a link to his considered response: Response to "The Math Formula"