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E18. Alphabetical ordering of whole numbers

Suppose the integers from 1 to 10,000,000,000 are ordered alphabetically as spelled out in American English, according to the rules below. Find the first odd number in this ordering.

(1) Numbers are spelled out formally, using neither "and" nor common shortcuts. For example, 2400 is two thousand four hundred, not twenty-four hundred.
(2) To alphabetize, ignore any spaces or hyphens. For example, sixteen comes before six thousand.

This problem is one of the easier ones in Peter Winkler's excellent collection, Mathematical Puzzles: A Connoisseur's Collection. It is an excellent problem to give to students working together in small groups. Usually students will have to make many attempts along the way to finding the solution, giving everyone a chance to participate. Note the specification of American English. Many people don't realize that there is a difference between American English and British English in some number names. For example, "billion" has different meanings in the two forms of English.

It might make an interesting exercise to write an algorithm for converting an integer (say in the range specified in this problem) to the spelled out form.

Curriculum for Overcoming Math Anxiety Course

(c) Peter. F. Ash, Ph.D. 2011
The following is the Curriculum for my Overcoming Math Anxiety course offered at Cambridge Center for Adult Education February 23 – March 9, 2011 over three two-hour classes:

1.Let's get personal
What brings you here? Why do you need to overcome math anxiety? When did your dislike or fear of mathematics first develop? Start keeping a "math journal".

2.Everyone can learn math
Is there something in your brain that means you can't learn math? What is dyscalculia? Overcoming handicaps.

3.Math phobia?

A serious fear of math may be a phobia, and may require treatment. A treatment you can do yourself, called TAT (Tapas Acupressure Technique) can help you. Our special guest lecturer shows you how.

4.One size doesn't fit all (even if your teacher thought it did)
Different people have different learning styles. If you know your preferred learning style you can learn math better. Are you a quantitative or a qualitative learner? Learning through different modalities: visual, aural,  or tactile/kinesthetic.

5.Math myths

If you were taught with traditional methods, you probably learned that being good at math required prodigious memory and the ability to regurgitate what the teacher told you. You may believe that there is one way to solve a math problem, and that math must be done while sitting still and keeping quiet. Wrong!

6.The new way to learn math
Modern reform mathematics suggests that math instruction be focused on solving interesting complex problems which can be solved in different ways, that students work in groups and communicate their ideas to one another, and that students learn to do mathematics with deep understanding, not by rote.

7.A sound mind in a sound body
Research shows that regular aerobic exercise helps you to beat stress, improve memory, and sharpen your thinking. Schedule your exercise before doing your math and watch what happens.

8.Learning is not all in your head
Learning cannot be separated from movement. The fact that proper movement leads to optimal learning underlies Brain Gym®,  We'll practice basic Brain Gym exercises to help reduce stress and make learning easier.

9.The mind-body connection
Learn to reduce stress and improve focus with meditation-based techniques. Use Zen meditation, yoga, TM, the relaxation response, or simple diaphragmatic breathing to reduce stress and empty your mind of chatter so you can learn better.

10.Music hath charms…
Playing certain classical music in the background can help you energize and focus. I'll play the CD and you'll hear if it helps.

11.Manipulate and understand
Learn what a mathematical manipulative is and how it helps visual and tactile/kinesthetic learners understand math concepts. Experience the power of multiple representations in math.

12.So, can I really do math?
Sure you can! You'll investigate a few math problems working in a group. Try out your new-found math anxiety reduction skills and enjoy some interesting open-ended problems.

13.Help! I need to take a math class
How to tell if you have a good teacher. What to do if you don't. Important study skills
14."Teach your children well"
What you can do so your children learn to like math, not to fear it.

15.Where do you go from here?
I'm here to help. Send me an email if you'd like a bibliography on math anxiety and math learning. Contact me if you are interested in math tutoring or math classes.