Ok, it's trivia time. How many of these questions can you answer?
- What do mathematicians call a regular polygon with eight sides?
- What mathematical symbol did math whiz Ferdinand von Lindemann determine to be a transcendental number in 1882?
- What number is an improper fraction always greater than?
- How many equal sides does an icosahedron have?
- What two letters are both symbols for 1,000?
- What Greek math whiz noticed that the morning star and evening star were one and the same, in 530 B.C.?
- What handy mathematical instrument's days were numbered when the pocket calculator made the scene in the 1970s?
See this Math trivia page for more questions and answers to the above questions.
One followed by 3 zeros is called a thousand.
One followed by 6 zeros is a million.
One followed by 9 zeros is a billion.
Can you keep going?
Answers at the Tricks and Trivia page.
The Blogosphere contains a number of Math-related blogs. One that caught my attention was The Math Less Traveled (an obvious pun on "The Road Less Traveled"). Author Brent Yorgey is a software developer and former Math and Computer Science instructor. Yorgey's blog has a number of delightful posts that arouse curiosity about Math. The blog is aimed at High School students although adults will enjoy it as well. The blog reads like a good book of Mathematical excursions - many posts serve as self-contained sparks of inspiration towards exploration. Yorgey explains mathematical concepts clearly and relates very well to his young audience. Clearly Yorgey delights in communicating ideas to students.
Excursions topics include:
- Fibonacci numbers and the related golden mean
- Patterns in Pascal's triangle
- Triangular numbers
- Sums of sequences
- Tetrahedral numbers
I was delighted to discover Math Less Traveled as Yorgey and I share the sense of joy and beauty in mathematical exploration. And, we both take pleasure in sharing the joy, in inspiring others to get psyched about Math.
"As long as algebra is taught in school, there will be prayer in school. " -- Cokie Roberts
Since writing my last post I got a copy of Mark Wahl's Math for Humans: Teaching Math Through 7 Intelligences. Wahl is brilliant. He provides inspiration, explanation, and techniques (with plenty of examples) for using Gardner's Multiple Intelligences to teach Math to students who may excel in one kind of intelligence over another. The 1997 edition of the book I have was written when Gardner's theory included 7 intelligences. The theory now includes 8 intelligences and Wahl's book has been updated to reflect that.
Wahl leads in his introduction with a great story of a second grader he met who was a budding artist, possessing very high spatial intelligence. She couldn't, however, learn (i.e. memorize) her addition tables. Judgment aside about the value of her teacher and parents forcing that she learn the tables in the timetable and not hers, Wahl undertook the challenge to teach her the arithmetic table. He had her make pictures of the symbols in "8 + 7 = 15" on a large index card and then had her make cards for other Math facts. This student was able to learn Math facts by making her own artistic flash cards. She got to learn by using her high spatial intelligence.
The book provides examples of how to "season", as Wahl likes to say, Math lessons with the multiple intelligences (MI). He gives examples for:
- Linguistic Intelligence (writing about what you learned or experienced)
- Spatial Intelligence (diagrams with symbols, flow charts for procedures, visual mnemonics, charts, mind maps, graphs, Venn diagrams, branching trees)
- Musical-Rhythmic Intelligence (clapping, singing, humming, rhythmic movement, rhythmic words, jingles)
- Bodily-Kinesthetic Intelligence (cubes, blocks, Cuisenaire Rods, calculators, and other manipulatives)
- Intrapersonal Intelligence (sharing of thoughts, feelings, and ideas with others)
- Interpersonal Intelligence (discussing of cross-cultural and historical aspects of Math topics)
- Logical-Mathematical Intelligence (deepening the analysis of concepts, posing questions, making connections, furthering exploration)
Wahl discusses brain hemisphericity, how to use left and right brain skills together to do Math. He discusses learning styles, how to use an understanding of a student's temperament to create an environment for succeeding in Math. He explains the value of students working cooperatively toward solving problems. And, Wahl presents a chapter on dealing with Math anxiety.
While the first half of the book presents and illustrates quite a bit of theory, the second half grounds us with concrete examples, quite a number of activities, that apply what we've learned earlier to real-world classroom and tutoring experiences.
All in all, Math for Humans is a fabulous book, chock full of very inspiring information and ideas with tremendous power to help teachers, tutors, coaches, and parents to teach Math in ways that students can really get and enjoy.
Some of you reading this blog might wonder how anyone could like Math. Or, maybe you really enjoy Math but a friend, child, or student of yours hates Math. What can you do to increase your appreciation and enjoyment of Math or to help someone else increase their appreciation?
Here are some suggestions:
1. When you are in a helping role - as teacher or tutor or friend - the relationship is EVERYTHING.
People who don't like Math are often afraid of Math and likely have shame about not being good at Math. Do whatever it takes to not judge the person's struggles with Math. Be supportive. Applaud their successes no matter how insignificant those little triumphs may appear to you.
2. Be aware of your preferred learning style.
Different people learn differently. Sounds obvious, doesn't it? Well, schools don't seem to get it. The traditional lecture benefits mainly the auditory learner which comprises only 20-30% of the school-aged population. I could never learn Math via lecture. I'm very kinesthetic and visual. I need to do Math in a very hands on way and I need to draw lots of pictures to cement my understanding. If you're in a class that's taught in a mostly auditory way and you're not a mostly auditory kind of person then takes lots of notes, draw lots of pictures, use different color pens to stimulate your brain and do lots of sample problems. Here's a good read on learning styles.
3. Beyond learning styles, understand multiple intelligences.
In 1983 Harvard Professor of education Howard Gardner developed the theory that the traditional emphasis on IQ was a very limiting way of measuring intelligence. He proposed that there are 8 different kinds of intelligences that promote a broader sense of human potential. You can read more about Gardner's multiple intelligences in this great article. Understanding Gardner's work gives you 8 different ways to teach or learn Math. The work of Mark Wahl is all about Gardner's multiple intelligences.
Harvey Mudd College is renowned for its Math department and for its overall education. Harvey Mudd students perform remarkably well on the William Lowell Putnam Mathematical Competition, a very grueling 6-hour 6-question exam taken by roughly 3,600 undergrads in the U.S. and Canada every year.
Given Mudd's commitment to excellence in mathematics education I was quite delighted to stumble upon the Mudd Math Fun Facts page, created and authored by Harvey Mudd Math Professor Francis Su. I am mentoring a couple of gifted high school students in Math. I am guiding them in explorations to increase their comfort with challenging Math contest problems and, more importantly, I'm helping them to grow in their appreciation of all things Math. I'll definitely be using a number of the Mudd Fun Facts in our sessions as they all wow me with their elegant and often surprising statements and many draw me in to try to understand and explain why they're true.
Mudd Math Fun Facts are perfect for inquisitive high school students. They come in easy, medium, and advanced levels of difficulty so there's something for everyone. Each Fact contains a description, suggestions for guiding student exploration, the Math behind the Fact plus references for further study. With 190 Mudd Math Fun Facts as of this posting curious students can spend many many fun-filled hours in joyful exploration.
My very favorite Fact (so far) is Chords of a Unit Circle. Su states that you get an interesting results if you do the following:
- Take a unite circle (i.e. a circle of radius 1)
- Mark off n equidistant points along the circumference of the circle. (n=6 in the illustration.)
- Select one of the points
- Draw chords from the chosen point to the n-1 other points along the circumference
- Multiply together the lengths of the n-1 chords
See if you can find a pattern for the product of the n-1 chords for circles with n points by starting with n=2 (just one chord) and increasing n.
Phi, also known as the golden ratio or the divine proportion, is one of the great mathematical constants. It is equal to a little more than 1.6 and is a most interesting irrational (but not transcendental) number. Phi has a fascinating connection with the Fibonacci series, it can be derived by solving a simple quadratic equation, and it reveals itself in simple but deep geometric constructions.
http://goldennumber.net provides the familiar background material on Phi and then goes much deeper, showing startling examples of how the golden ratio appears in art, architecture, music, poetry, proportions of the human body, and other surprising places.
A fun example of Phi appearing in unexpected places is in the dimensions of a credit card. The ratio of the two sides is very close to Phi.
Another surprising example, at the microscopic level, is the DNA molecule. Each double helix spiral is in the proportion of the golden ratio.
Check out http://goldennumber.net for more than you could every want to know about Phi, all beautifully illustrated.
How many of you remember doing geometry proofs in High School? How many of you enjoyed writing them? I don’t know about you but I’ve always preferred pictures to words when it comes to understanding how something works.
“Proofs Without Words: Exercises in Visual Thinking” by Roger B. Nelsen is a wonderful book that provides visual insights into how one might go about proving mathematical theorems. The Pythagorean Theorem has always been a mystery to me. How are the squares of the sides of a right triangle related to its hypotenuse? “Proof Without Words” has five clever illustrations that guide readers in writing their own proofs.
If you ever doubted that algebra and geometry were related, the diagrams demonstrating how to compute sums of series will produce aha! experiences.
Writing proofs when one is guided by visual cues is a much more fulfilling endeavor than stringing together dry facts from memory. This book delivers much fulfillment in exploring theorems in geometry, algebra, trigonometry, sequences, and other aspects of Math.
Well, this post is not so “mathy.” It’s about anagrams, which are permutations of the letters of one or more words, and about a fun site to play with generating anagrams.
If you take the letters of my name, Sol Lederman, and rearrange them one of the many possible phrases is this post’s title, Male Nerd Sol. A less complimentary anagram of my name is Lame Nerd Sol.
Many fun anagrams have been found, many undoubtedly with help from a computer. Some fun ones are:
Desperation = A Rope Ends It
The Morse Code = Here Come Dots
Slot Machines = Cash Lost in’em
Clothespins = So Let’s Pinch
A Domesticated Animal = Docile, as a Man Tamed it
Snooze Alarms = Alas! No More Z’s
The anagrams above, and others, are listed at the Anagram Hall of Fame. Wordsmith.org is also the site where I found anagrams of my name.
Enter your name into the Internet Anagram Server and see what you come up with:
Male Nerd Sol