Art and Craft of Mathematical Problem Solving

Course No. 1483
Professor Paul Zeitz, Ph.D.
University of San Francisco
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Course No. 1483
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Course Overview

One of life's most exhilarating experiences is the "aha!" moment that comes from pondering a mathematical problem and then seeing the way to an elegant solution. And many problems can be solved relatively quickly with the right strategy. For example, how fast can you find the sum of the numbers 1 + 2 + 3 up to 100? This was famously answered in the late 1700s by the 10-year-old Carl Friedrich Gauss, later to become one of history's greatest mathematicians. Young Gauss noticed that by starting at opposite ends of the string of numbers from 1 to 100, each successive pair adds up to 101:

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101

and so on through the 50th pair,

50 + 51 = 101

Gauss was already thinking like a good problem solver: The sum of the numbers from 1 to 100 is 50 × 101, or 5,050—obtained in seconds and without a calculator!

In 24 mind-enriching lectures, The Art and Craft of Mathematical Problem Solving conducts you through scores of problems—at all levels of difficulty—under the inspiring guidance of award-winning Professor Paul Zeitz of the University of San Francisco, a former champion "mathlete" in national and international math competitions and a firm believer that mathematical problem solving is an important skill that can be nurtured in practically everyone.

These are not mathematical exercises, which Professor Zeitz defines as questions that you know how to answer by applying a specific procedure. Instead, problems are questions that you initially have no idea how to answer. A problem by its very nature requires exploration, resourcefulness, and adventure—and a rigorous proof is less important than no-holds-barred investigation.

Think More Lucidly, Logically, Creatively

Not only is solving such problems fun, but the techniques you learn come in handy whenever you are presented with an unfamiliar problem in mathematics, giving you the confidence to try different approaches until you make a breakthrough. Also, by learning a range of different problem-solving approaches in algebra, geometry, combinatorics, number theory, and other fields, you see how all of mathematics is tied together, and how techniques in one area can be used to solve problems in another.

Furthermore, entertaining math problems sharpen the mind, stimulating you to think more lucidly, logically, and creatively and allowing you to tackle intellectual challenges you might never have imagined.

And for those in high school or college, this course serves as an enriching mathematical experience, equal to anything available in the top schools. Professor Zeitz is a masterful coach of math teams at every level of competition, from beginners through international champions, and he knows how to inspire, encourage, and instruct.

Strategies, Tactics, and Tools of Math Masters

The Art and Craft of Mathematical Problem Solving is more than a bag of math tricks. Instead, Professor Zeitz has designed a series of lessons that take you through increasingly more challenging problems, illustrating a variety of strategies, tactics, and tools that you can use to overcome difficult math obstacles. His goal is to give you the persistence and creativity to turn over a problem in your mind for however long it takes to reach a solution.

The first step is to come up with a strategy—an overall plan of attack. Among the many strategies that Professor Zeitz discusses are these:

  • Get your hands dirty: Dive in! Plug in numbers and see what happens. This is a superb starting strategy because it almost always shows a way to keep on investigating. You'll be surprised at how often a pattern emerges that takes you to the next step.
  • Think outside the box: Break the bounds of conventional thinking. Professor Zeitz shows you the original think-outside-the-box problem, in which the key idea is to disregard the boundaries of an implied box. He also explains why he prefers to call this strategy "chainsaw the giraffe."
  • Wishful thinking: Turn a hard problem into an easy one by removing the hard part. For example, substitute small numbers for big ones. This is a confidence-builder that often gives you a partial solution that shows you how to solve the original problem.
  • Change your point of view: Every problem has a natural point of view, such as a time or place where something is happening. Step back and try a different point of view. This could mean recasting an algebra problem as one in geometry, or vice versa.

The next step after choosing a strategy is to find a suitable tactic. For example, suppose you live in a cabin that is two miles north of a river that runs east and west, and your grandma's cabin is 12 miles west and 1 mile north of your cabin. Every day you go to visit grandma, but first you stop by the river to get fresh water for her. What is the length of the route that has the minimum distance?

You start with the "draw a picture" strategy. Once you have something to look at, you realize that the "symmetry" tactic will give you the shortest distance. Here's how it works: Imagine an alternate you on the same errand but on the south side of the river, in a mirror image of the situation on the north side. By drawing a line connecting your real cabin with the alternate grandma's cabin, and another line connecting the real grandma's cabin and the one belonging to the alternate you, you find an intersecting point at the river that is the perfect place to stop.

On some problems you also need a special-purpose technique—a tool. For example, the 10-year-old Gauss's trick of pairing numbers in the earlier example is a tool whose underlying idea—symmetry—can be applied to a wide range of problems. You learn the strengths, as well as possible pitfalls, of such tools.

Prepare for an Exhilarating Experience

Professor Zeitz compares this systematic approach to problem solving—in which you deploy strategies, tactics, and tools—to the mountaineer's quest to reach the top of a high peak. The mountain may seem insurmountable, but there is always a way to conquer it by proceeding one step at a time.

Looking at an impressive mountain, you can't but feel a sense of awe at the prospect of climbing it. Math problems, too, can produce this same reaction. But don't be daunted: You are more ready than you think. So sharpen your pencil, get some paper, and prepare for the exhilarating experience of The Art and Craft of Mathematical Problem Solving.

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24 lectures
 |  Average 30 minutes each
  • 1
    Problems versus Exercises
    Solving a math problem is like taking a hike, or even climbing a mountain. It's exciting, challenging, and unpredictable. Get started with three entertaining problems that plunge you into thinking like a problem solver and illustrate two useful strategies: "wishful thinking" and "get your hands dirty." x
  • 2
    Strategies and Tactics
    Learn the difference between strategies, tactics, and tools when applied to problem solving. Try to decipher a puzzling reply to a census question, and determine whether three jumping frogs will ever land on a given point. x
  • 3
    The Problem Solver's Mind-Set
    Delve deeper into the psychological aspects of problem solving—especially concentration, creativity, and confidence—and ways to enhance them. Learn to avoid overreliance on very narrowly focused mathematical tricks, and investigate a number of "think outside the box" problems, including the original that gave the name to this strategy. x
  • 4
    Searching for Patterns
    Brainstorm an array of problems with the goal of building your receptiveness to discovery. See how far you can go by just letting yourself look for interesting patterns, experiencing both conjectures that work as well as cautionary examples of those that don't. The core of the lecture is an investigation into trapezoidal numbers and a search for patterns in Pascal's triangle. x
  • 5
    Closing the Deal—Proofs and Tools
    Learn how to "close the deal" on some of the outstanding conjectures from the previous lecture by using airtight arguments, or proofs. These include deductive proof, proof by contradiction, and algorithmic proof—along with the narrow (and often overestimated) power of specific tools or "tricks," such as the "massage" tool, to make a mathematical expression simpler. x
  • 6
    Pictures, Recasting, and Points of View
    Explore three strategies for achieving a problem-solving breakthrough: draw a picture, change your point of view, and recast the problem. Try these strategies on a selection of intriguing word problems that almost magically yield an answer, once you find a creative way of analyzing the situation. x
  • 7
    The Great Simplifier—Parity
    Applying the problem-solving tactic of parity, test your wits against an evil wizard, an open-and-shut row of lockers, and other colorful conundrums. Then see how parity leads naturally into graph theory, a playground for investigation that has nothing to do with conventional graphs. x
  • 8
    The Great Unifier—Symmetry
    Having used symmetrical principles to tackle problems in earlier lectures, take a closer look at this powerful tactic. Discover that when symmetry isn't evident, impose it! This approach lets you compute the shortest distance to grandma's when you first have to detour to a river to fetch water. x
  • 9
    Symmetry Wins Games!
    Devise winning strategies for several fun but baffling combinatorial games. One is the "puppies and kittens" exercise, a series of moves and countermoves that can be taught to children but that is amazingly hard to play well; that is, until you uncover its secrets with symmetry and a few other ideas. x
  • 10
    Contemplate Extreme Values
    Take your problem-solving skills to extremes on a variety of mathematical puzzles by learning how to contemplate the minimal or maximal values in a problem. This "extreme" principle is a simple idea, but it has the nearly magical ability to solve hard problems almost instantly. x
  • 11
    The Culture of Problem Solving
    Detour into the hidden world of problem solvers—young people and their mentors who live and breathe nontraditional, nontextbook mathematics such as what you have been studying in this course. The movement is especially strong in Russia and eastern Europe but is catching on in the United States. x
  • 12
    Recasting Integers Geometrically
    Delve deeply into the famous "chicken nuggets" problem. In brief, what's the largest number of nuggets that you can't order by combining boxes of 7 and 10 nuggets? There are many roads to a solution, but you focus on a visual approach by counting points in a geometric plane. x
  • 13
    Recasting Integers with Counting and Series
    Apply the powerful strategies of recasting and rule-breaking to two classical theorems in number theory: Fermat's "little" theorem and Euler's proof of the infinitude of primes. x
  • 14
    Things in Categories—The Pigeonhole Tactic
    According to the pigeonhole principle, if you try to put n + 1 pigeons into n pigeonholes, at least one hole will contain at least two pigeons. See how this simple idea can solve an amazing variety of problems. Also, delve into Ramsey theory, a systematic way of finding patterns in seemingly random structures. x
  • 15
    The Greatest Unifier of All—Invariants
    To Professor Zeitz, the single most important word in all of mathematics is "invariants." Discover how this granddaddy of all problem-solving tactics—which involves quantities and qualities that stay unchanged—can be used almost anywhere and encompasses such ideas as symmetry and parity. x
  • 16
    Squarer Is Better—Optimizing 3s and 2s
    What is the largest number that is the product of positive integers whose sum is 1,976? Tackle this question from the 1976 International Mathematical Olympiad with the method of algorithmic proof, in which you devise a sequence of steps—an algorithm—that is guaranteed to solve the problem. x
  • 17
    Using Physical Intuition—and Imagination
    Draw on your skills developed so far to solve a tricky problem about marbles colliding on a circular track. Martin Gardner's airplane problem and a question about how many times a laser beam reflects between two intersecting mirrors help you warm up to a solution. x
  • 18
    Geometry and the Transformation Tactic
    Focusing on geometry, consider some baffling problems that become almost trivial once you know how to apply rotations, reflections, and other geometric transformations of your normal point of view. This clever tactic was pioneered by the 19th-century mathematician Felix Klein. x
  • 19
    Building from Simple to Complex with Induction
    Sometimes a problem demands a different type of proof from the ones you learned in Lecture 5. Study cases in which proof by mathematical induction is the only feasible approach. These typically occur in recursive situations, where a complicated structure emerges from a simpler one. x
  • 20
    Induction on a Grand Scale
    Continuing your use of inductive proof, calculate the probability that a randomly chosen number in Pascal's triangle is even. This problem is surprisingly easy to investigate, but it requires sophistication to resolve. But by now you have a good grasp of the methods you need. x
  • 21
    Recasting Numbers as Polynomials—Weird Dice
    Is it possible to find weird dice that "play fairly"? These are two dice that are numbered differently from standard dice but that have the same probability of rolling 2, 3, 4, and so on through 12. Learn that, amazingly, the answer is yes. x
  • 22
    A Relentless Tactic Solves a Very Hard Problem
    In a lecture that Professor Zeitz compares to walking along a mathematical cliff edge, use the pigeonhole principle to find patterns within apparently random and mind-bogglingly large structures. You'll discover there is no limit to what the intrepid problem solver can do. x
  • 23
    Genius and Conway's Infinite Checkers Problem
    No course on problem solving is complete without a look at the checkers problem, formulated by contemporary mathematician and puzzle-master John Conway. Also learn about two other icons in the field: Paul Erdos, who died in 1996, and Évariste Galois, who lived in the early 1800s. x
  • 24
    How versus Why—The Final Frontier
    Professor Zeitz reviews problem-solving tactics and introduces one final topic, complex numbers, before recommending a mission to last a lifetime: the quest for why a solution to any given problem is true, not just how it was obtained. He closes by sharing some of his favorite examples of this elusive intellectual quest. x

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Your professor

Paul Zeitz

About Your Professor

Paul Zeitz, Ph.D.
University of San Francisco
Dr. Paul Zeitz is Professor of Mathematics at the University of San Francisco. He majored in history at Harvard and received a Ph.D. in Mathematics from the University of California, Berkeley, in 1992, specializing in ergodic theory. One of his greatest interests is mathematical problem solving. He won the USA Mathematical Olympiad (USAMO) and was a member of the first American team to participate in the International...
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Art and Craft of Mathematical Problem Solving is rated 4.3 out of 5 by 34.
Rated 5 out of 5 by from Fantistic! I'm a math major and this course covers problem solving techniques I never encountered in my coursework! A unique meta-mathemtical gem.
Date published: 2017-08-05
Rated 5 out of 5 by from Enjoyable This course was fun and eye-opening. Not being a mathematician, I finally see why some find math fun to do. it inspired me to use more math problems with elementary age children and to allow time for thinking and experimenting with numbers. I loved learning about math circles and how some cultures talk about math being part of every day life with puzzles and games. The presentation by the teacher was well paced and easy to follow, even for someone not into math. I'm very glad I purchased this course.
Date published: 2016-12-18
Rated 4 out of 5 by from Good to prepare for math competitions If you are a "Mathlete" or participate in math clubs, math circles, or math competitions, then Dr. Zeitz is the right person to take you to the next level on mathematical problem solving. I liked the beginning lectures on strategies vs tactics and the problem solver's mindset and there were many great nuggets along the way. The flow was that a lecture would talk about a specific strategy/tactic (e.g., the pigeon hole principle) and then there would be a few problems solved using that technique. There are also some additional problems in the course guide where you can try to solve additional problems using that technique (i.e., get your hands dirty). Probably because I'm currently more of a math practitioner, I wanted more of a problem solving framework to apply to any new/hard problem where I don't know how to start to solve the problem (e.g., the four-step problem solving method). I felt like there were 24 lectures, with good nuggets, but I wanted a unifying framework to pull it all together so when faced with a new problem, I have an approach. To summarize, If you are into math clubs/competitions, this will likely be a valuable course for you.
Date published: 2016-06-02
Rated 2 out of 5 by from Lots of info but In Dr. Zeitz's intro, boy, did I ever feel like an old dumb person. Shame on me for being curious about the world and thinking the The Teaching Company might get a professor who doesn't make your lousy early education experience seem dismal compared to his genius. Way to go Dr. Zeitz and The Teaching Company. Sincerely, Just an Average Bloke
Date published: 2016-05-01
Rated 4 out of 5 by from He'll make a puzzle geek out of you Professor Zeitz is the super-geek. You can easily see how he would take a group of young math whizzes and hone their skills until they became ace competitors in problem-solving tournaments. It's not easy to communicate something as weird as a useful strategy for attacking a difficult math brain-teaser, but Zeitz often manages to do it. On the one hand, I got the impression that his mind simply goes places mine won't go, but on the other hand, he did manage to get across concepts that even I can use. I didn't give the course full 5-star ratings only because the pace was a bit uneven. In some sections, I was glued to my chair and trying (unsuccessfully) to think of someone I could call to gush about it over. (I'm listening to this great math lecture . . . . No, really, you'd love it . . . .) In other sections, I bogged down a bit. Maybe that's just an indication of what problems grab me and what problems don't. Or maybe he was sometimes talking over my head. Not that he assumes you know a lot of higher math, but he does talk at rather a challenging level, at least from the perspective of this amateur. While remaining perfectly dispassionate and avoiding any cheap dramatics, Zeitz nevertheless communicates a deep love of and enthusiasm for puzzles and innovative problem-solving.
Date published: 2013-03-29
Rated 1 out of 5 by from The Art and Craft of Mathematical Problem Solving Dr. Zeitz has too much time in his hands... If the student wants to spend lots of time, buy the program otherwise spend time in other math courses.
Date published: 2012-12-17
Rated 5 out of 5 by from Motivational for children Great experience. Wealth of techniques to learn. Wide range of difficulty levels of the problems allow listeners with more/less math background and ability to find a solution and feel good about themselves once in a while. Very motivational for children. My kids, 10 and 12, referred on several occasions to material covered in the lectures.
Date published: 2012-12-17
Rated 5 out of 5 by from Very Useful - Great Professor This one is definitely worth the purchase. This professor is outstanding. His love of the subject is contageous. His ability to think clearly and focus on what is important is impressive. All the while he speaks in a very captivating, interesting and pleasantly mild humorous way and makes us feel like fellow human beings. There were times, as always with math, that I wasn't fully following along, but the beginning and ending alone are well worth the money and I have taken a few critical concepts away that I am grateful for. I am ready for the next one by Paul Zeitz and hope it is Complex Analysis. Not only have I been dying for a good course on that topic even before this, but then he leaves us at the end with several reasons why Complex Analysis is so important. I am sure he would do an excellent job presenting that topic. Thanks Paul! Looking for Complex Analysis now! Oh, I can't forget to mention that the setting for this one was also wonderful and so superior to the old brick wall / fake ivy background that some of the courses have. This setting is sleek and professional and not distracting. I do have one suggestion for improvement. Can the instructor have access to some kind of physical board for writing / typing / displaying? Maybe I'm just old, but I think it would be better than the somewhat cold separate graphics. Also, please include some of the graphics in the books - even just still photos give some reminder of what happened in the course. Thanks.
Date published: 2012-12-14
Rated 5 out of 5 by from Alternately loved and hated this course I liked the way the hard parts of the course were proceeded and/or followed by motivating lectures. I was enthusiastic at the start of the course but gradually became depressingly aware of how stupid I am and how little I can think. About the time I felt like giving up, there would be another motivating lecture. Overall the course was very helpful, I have doubled the amount of time I will think about a problem before giving up, and I have tried his suggestion of having some problems on mental back-burners, and after a couple weeks I realized the solution to one of the problems which was a really thrilling experience. This course taught me to feel more relaxed about taking time (hours, days, weeks) to think, whereas before I felt a lot of pressure to understand things fast (which probably just slowed me down).
Date published: 2012-08-25
Rated 5 out of 5 by from Thinking outside the box This was a very interesting course. It is unlikley to be of value to the average TC student but high school math students or anyone interested in math will find it fascinating. None the less, I could not stop watching it! The professor is very well informed, clear, and has an excellent presentation style. It was incredible to see math problems be transformed to pictorial, diagramatic, or graphic problems and be solved within a few seconds. If one is interested in what outside-the-box thinking looks like, this is it.
Date published: 2012-06-13
Rated 5 out of 5 by from Great teacher and motivator Dr. Zeitz reinvigorated my desire to solve mathematical problems. I watched the course and solved many of the additional problems in the supplement. You have to be willing to "Get your hands dirty" with solving problems if you want to benefit from this course. It was a profound pleasure for me to watch him and his enthusiasm for mathematics. I also think that his remarks about the different cultures in which students learn mathematics are very thought provoking.
Date published: 2012-02-27
Rated 2 out of 5 by from Too speciaized to be a general interest course Started out promisingly but quickly lost my interest with too many unrealistic examples after the first few lessons. It would be helpful to relate these examples to real-world situations or begin with such examples. Too many asides on his own nerdiness and contributions to mathematical contests. We got the point early on. More graphics while presenting a problem would make grasping ideas better than a several minute monologues leading up to the problem at hand. I managed the entire 24 lessons by use of the fast forward button when the monologues became too much.
Date published: 2011-12-25
Rated 5 out of 5 by from This is not your fathers math Professor Zeitz addresses concepts in math that are typically not taught in school. You will be amazed by some of the concepts and how powerful they are in solving riddles. After a slow start in lecture 1 - it quickly picks up speed where the math is challenging - yet fun.
Date published: 2011-01-30
Rated 5 out of 5 by from A jewel. One of TTC's best courses. I checked this course out at my local library before coughing up the jack so as to spend my limited dollars wisely. This is one of the BEST math courses that TTC has offered. The teacher, Paul Zeitz, does an excellent job in introducing one to problem solving skills. Note that I say skills *not* a how-to. Yes, you guessed correctly - one CAN learn problem solving skills if one is willing to put in the sweat and tears. I recall in college one of the common myths making the rounds with regards to maths was "you either have it or you don't." And most of us fell into the "don't" category, myself included, who managed to get through Calc II and Matrix Algebra via late nights and sheer work. Here Dr. Zeitz, shows that one can in fact learn the skills of effective problem solving. Granted an "average" person may not get to be super but she will be able to improve her analytical skills beyond all expectations. I had to view the lessons two and three times to really get the ideas and am still lost in some areas. One caveat. Zeitz's book by the same name was a bit of a disappointment to me because it did not have answers to exercises. I checked out the book at a local University book store. One more thing. Zeitz describes the problem solving categories in the first five lectures and then goes to specific problems beginning with the sixth lecture. On the second viewing I first got to ONE problem solving category and then viewed the specific lecture that dealt with the specific category in detail. That helped me imbibe the stuff faster. When this goes on sale and if the price is reasonable I will grab this course hook, line, and sinker. An EXCELLENT investment.
Date published: 2010-09-23
Rated 5 out of 5 by from The Course for All Problem Solvers This course will introduce you to problem solving with real problems discussed and explained by one of the premier problem solvers and expositors of our time. This is certainly the best course that I have bought from the Teaching Company and I have purchased almost every mathematics course in the catalog. Whether you are already a talented problem solver or just beginning, there is a wealth of information to be gained by joining Paul Zeitz in his exploration of this field of interest. No matter how much actual talent you have, you will become much more confident of your ability in the craft of problem solving. You will come away from the course with a much better understanding of why some individuals love mathematics and spend their entire lives exploring its problems. His book with same title "The Art and Craft of Problem Solving" is a wonderful companion piece, but being able to actually see and hear him discuss the problems and techniques is priceless. Such charisma is what makes him such an outstanding teacher as well as problem solver.
Date published: 2010-06-05
Rated 5 out of 5 by from Fantastic topics and presentation Around two lectures into this course, I couldn't help but saying to myself "WOW". This course is fantastic for several reasons: 1. It uses plenty of pictures, animations and graphs. This really helped the understandings of the problems and their solutions. This is my fifth or sixth TTC course, and I really love the presentation of this course (through the use of the animations, pictures etc.) compared to the DVD versions of the other courses I have seen (which relied too much on the lecturers' verbal commentaries). 2. Every lecture always has at least one problem to be solved - i.e., practical. 3. Prof Zeits also almost always explains WHY the problems are solved in such ways, rather than just the HOWs. 4. Some of the mathematical problems are very interesting and fun. Overall though, most parts of this course are quite (mathematically) hard-core and require mastery of at least most of high school geometry, algebra, trigonometry and calculus to fully appreciate the courses' nuances. I have loved maths from a very young age, and there were around 3 - 5% of the problems/examples i did not fully get. One recommendation I would like to make to Prof Zeits and TTC is that I noticed small / a few typos in the notes as well as in the written texts on the DVD, which occurred (I noticed) around where the problem's climax is at! (which could get annoying). But these are very minor in frequency. For instance, on page 27 of the course guidebook, point I.B.3 should end with "8Tn + 1" instead of "8Tn". Personally, I loved the following problems (& the strategies and tactics to solve them) - not in any order: a. The pill problem (in lecture 1). b. "Determine, with proof, the largest number that is the product of positive integers whose sum is 1976" - using 2s and 3s - amazing! (and the importance of e) (in lecture 16). c. The pigeonhole (and intermediate pigeonhole) tactics (in lecture 14). I think it has great applications - for instance, problem 2 on page 60 of guidebook. d. Handshake problem (in lecture 10) - contemplate extreme values. e. Wythoff's Nim or "puppies and kittens" problem (in lecture 9) - solved by plotting positions and contemplating symmetries. f. And lastly, the problem in lecture 20: "What is the probability that a randomly chosen number in Pascal's triangle is even?" I love the process of answering this and the final patterns and answer! Overall, this is a superb superb course and I highly recommend it, but only for the hard-core math enthusiasts.
Date published: 2010-04-10
Rated 5 out of 5 by from Best Course Evah, and I have bought quite a few... I have worked for many years in various areas that require both mathematical and non-mathematical problem solving, although I (newly regret that I) never tried Math Olympiads as a kid. More than just technical insight, this course has given me a lot of psychological insights and tools that i am putting to use in my professional and personal life. For example, I have spent a lot of time studying advanced math from books on my own, struggling mightily with abstract problems - it was very helpful for me to realize, through this DVD, that math is very much an oral tradition and that in doing math problems I should expect to spend a lot of time feeling "lost" because that's a normal condition for mathematicians. Now I realize that on some topics I'm going to wind up needing to ask questions and discussing them with others, and that the fact that I spend a long time staring at a problem without "getting it" is just par for the course. Aside from all that, the problems and overview of solution strategies, tactics, and tricks is fun and well presented. Also, I've got a gifted 4th-grader who is starting to do some MO problems through his school, and I am showing some of these lectures to him as well.
Date published: 2010-03-21
Rated 5 out of 5 by from Teach? Or awaken? To teach the art of solving mathematical problems may seem impossible. Perhaps it is. Perhaps all you can do is awaken a capacity that has lain unused and unsuspected. But if even that much is possible, this course has a good chance of doing it. Paul Zeitz has spent years training "mathletes"--students who compete in elite tests of problem-solving. I suspect they could clean my clock, and yet what he teaches echoes and extends what I have learned over the years from Martin Gardiner's columns and books, a four-year engineering degree, and continued if occasional interest in matters mathematical. The point, you see, is not so much to teach mathematics as to teach you to think just a little bit like a mathematician. When you come face to face with a problem, what do you look for? What guides your approach to it? What approaches do you have? To teach even this much is a huge task, but by a good set of examples and steady review of "what we tried and why it worked" Zeitz goes quite far down the road, dividing the subject into tools, tactics, strategy, and culture. Culture matters because culture is the climate in which problem-solving acumen withers or flourishes. Zeitz provides glimpses of the culture of mathematics and the people, great and lesser, who aspire to do well at it. Those glimpses also hold the hope of awakening something in the student. There is very little algebra; most of the algebra presented earns a place either by its importance as a tool or the penetrating insight it illustrates. The emphasis is on a variety of tools and tactics, not surpassing strength in one or two. And only at the end does Professor Zeitz offer the domain of complex numbers as a topic for study, one too broad to take up in the course, though deeply connected with several of the tactical approaches we examine. The final lectures include vignettes of several great mathematicians and a discussion of "how versus why." These are worth the price of the whole course, but only if you have reached a certain level of understanding. The rest of the course, its challenges and insights, are the price to pay to reach that point. Professor Zeitz is somewhat soft-spoken. On the one hand, this means that he does not overshadow or distract from his material. On the other, it meant that I sometimes had to marshal my full attention at the opening of each lecture. Most of the visuals are very good to excellent. The exception are the "subtitle" notes that appear on a band across the bottom of the screen. On a small screen they are harder than necessary to read. I would encourage The Teaching Company to make such titles about two fifths again as large as these. Fortunately, only a few are of central importance to the course. The most important visuals, those that present and illustrate problems and how they can be solved, are well designed and very clear and legible. It appears that the Teaching Company has gone to a new backdrop or "stage" for their presentations. It is much deeper and somewhat more opulant than the older ones, and I found it a little distracting. It may be that the speaker simply does not stand out as well against its colors. They also are using a new lectern, one that surrounds the speaker on three sides. This shields the speaker from view, making it harder for the eye to recognize a human form and, in my judgement, harder for the speaker to hold the viewer's attention. I suggest to The Teaching Company that they reconsider these choices.
Date published: 2010-03-07
Rated 5 out of 5 by from a little different Like previous reviewers I was not sure what to expect with these lectures. The intial lectures were a little disjoint and Prof Zeitz did labor on the difference between tactics and stategy too much. In the end, it did not seem to me the distinction mattered that much. Then each lecture addressed a specific type of problem and showed techniques to solve these problems. With each technique the appropriate mathematical back ground was explained well. However, there are some jumps in logic i did not follow and you will need to hit pause a lot for some of the explanation to sink in. Overall, this is the first set of lectures I've seen that attempts to address problem solving. For his creative approach Prof Zeitz should be congratulated. If you like complex maths problems then you will enjoy this series.
Date published: 2010-02-22
Rated 5 out of 5 by from Truly Exceptional! Get your graph paper, some colored pens, some plain paper and bring along your imagination for an intellectual adventure of the first order! If you just want a lecture series where you listen and watch and/or generally prefer a very rudimentary treatment of the subject matter, this is probably not the course for you. On the other hand, if you enjoy "getting your hands dirty" with some experimentation, and are ready to actively engage in some creative thinking while learning some of the art and craft of mathematical problem solving, then this course is a "must have". Professor Zeitz presents overall strategies (many of which are very helpful for non-mathematical problem solving as well) and specific tactics in solving a wide variety of very interesting problems touching on several areas of mathematics. In addition to being a master at mathematical problem solving, he's an excellent teacher as well--the content, delivery and development of the subject is superb. The problems selected are very engaging and nicely illustrate, and illuminate, the strategies and tactics presented in each lesson. In addition, there's interesting material on the culture of mathematical problem solving, some motivational information, and a great list of additional resources to continue your mathematical journey. I intend to do just that. Thank you, professor Zeitz and The Teaching Company for job very well done!
Date published: 2010-02-17
Rated 4 out of 5 by from Art and Craft of Mathematical Problem Solving I have purchased most of the science and math courses offered by the Teaching Company. While the others courses served to reinforce what I learned in high school and college, this course presented mathematical material from a completely new perspective. Professor Zeitz has some incredible mathematical talents, however I found myself frustrated when unable to solve his problems using the techniques he presented. The skills he presents are to be acquired with time and practice, and more of an art than the black-and-white material usually associated with a math course. If I had one complaint regarding this course, it would be Professor Zeitz’ lethargic tone. Perhaps I have been spoiled by the boisterous presentations from Arthur Benjamin and Edward Berger (see Teaching Company’s Joy of Math and Intro to Number Theory). Professor Zeitz’ voice seems more suited to a hypnotherapy session, such that if he counted down from three and snapped his fingers I’d instantly fall into a somnambulistic state. I would recommend this course to anyone who considers themselves a math nerd. Anyone seeking new and advanced tactics to tackle seemingly complex problems would find this course useful.
Date published: 2010-02-15
Rated 5 out of 5 by from I'm Loving It Maybe it is premature to review this when I am only five lectures into it, but I plan to stick with the course, even though it is confusing at times. Since it is full of discrete examples of what Professor Zeitz calls "tricks" or "tools", I guess I have to realize that I never mastered those to which I was long ago introduced, but they aren't hard to learn, and ultimately they are not the essence of what it takes to appreciate the process. My education has been sketchy, and I would say that I dipped into the math and science parts of it with a sense of ever-greater confusion, after having been considered a good student at one time. There is something rather forgiving in the nature of this course. Not only can you push the pause button, you can push the stop button and go back to some of the earlier material. Although things move fast, I liked how the professor set up the course, with an invitation to be perplexed and get comfortable with that state of mind as you work towards understanding. There's no Darwinian strategy for eliminating the unfit, which largely prevailed in classes of my youth, whether explicitly or implicitly. Women were first on the chopping block then, which is silly when you see the halls of MIT today teeming with females (many from China and India where they did not seem to get the memo that girls can't hack it). My best experience was in middle school, with an old Navy man, who loved numbers for their own sake and introduced us to all sorts of number systems and really tried to get us to change our perspectives. It was also the year I was studying art, and learning to draw in perspective. My brain was waking up from those boring and perfect years in elementary school arithmetic drills. The next year, unfortunately, I drew an algebra teacher who insisted on lots of quizzes and perfect answers (which is indeed part of math). The loose quest I remained on after that disaster (I had to learn to be content with "C"'s in math afterwards) was not all to the bad, because it took me through places I would return to, only not on such a clunky tour bus. Eloquently, Zeitz has said that an explorer is someone who is lost. He also reminds us that excessive rigor in the real world is, while definitely for some of us, much less fun than getting out in that place where you do not know the language, and trying to communicate with the natives. He repeatedly uses the phrase "playground" to describe the places we can visit in what I would call "Mathemagic Land" (it's a good Donald Duck movie). So many cultures have developed and used math to grasp the complexity of the world. We can feel a sympathy with distant Mayans or Babylonians, and can even begin to read their codes. Perhaps my old Navy man teacher who had sailed around the world, who cried when we learned in that class that JFK had been shot - perhaps he passed on a kind of spark which said the light would stay on for me. Perhaps it was in an otherwise scary Catholic school classroom when I took refuge, waiting to be called on to stand recite my times table problem, that I could find patterns in the table and make it my salvation. In those days, I thought a lot about the Trinity - and nine, magical nine. We acquire mathematics in different ways, but pattern is our friend. That is for sure. This course is a friendly course. I even solved a few problems in the down time when I pushed pause. Quelle surprise! There's an athlete in everyone. There's also a mathematician in everyone. A professor I once knew socially, a Greek by birth, wrote that at some time in the Renaissance or even before, "techne" and the humanistic side had parted company. He wanted them to become reconciled, so that everyone knew one another again, not shunning each other's world view. I am not a nerd, but I know and love several nerds. Now I am learning a bit about the language they speak, and it's not such a big scary deal.
Date published: 2010-02-09
Rated 3 out of 5 by from I'm not sure what to make of this course This is a very unusual course. As the tile says, It is based on Mathematical Problem Solving. In fact that is all it is about. Zeitz, who proudly describes himself as a "math nerd" has been involved in numerous math competitiions and math "olympics" around the world, and coaches young teams in math contests.The TTC course is basically a collection of the types or problems/puzzles in these competitions and an overview of how he teaches his students to solve them. He gives the viewer numerous opportunities to solve the puzzle ("Use the pause button", he says). Then he gives an explanation of one approach, usually involving graphics (algebra, number theory, graphing, symmetry, etc.). He also give a lecture of the "culture" of math problem-solvers, especially in Bulgaria. I may be a nerd, but I'm not a math nerd, and was quickly lost. I was able to approach some problesm intuitively but my answers were usually incorrect. I wanted to try to keep up but just felt defeated. Zeitz suggested problems that he says can take hours and days to solve. Clearly, this course became quickly obscure and impractical. There was no "carry-over" to"real-life" or other math courses, or even much "content" to the course. Now, I suspect that for those viewers who like a challenge and like math, this is a really "fun" course. Or rather, a collection of puzzles,because it isn't really a "course." I think a lot of viewers would enjoy it. But I would think that many others would be befuddled..
Date published: 2010-02-01
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