Tomorrow is Halloween! Whether it’s watching a horror movie or jumping out from behind a door, many people find it fun to be scared or scary on this holiday. It’s also a good time to revisit what parents can do to keep their children from becoming afraid of math. Math anxiety is unnecessarily common in the United States. As a country we have focused intentily on promoting foundational literacy skills, but numeracy skills can also be fostered from an early age. At home, don’t just read to your toddler and pre-schooler; “math” to them also. Get down on the floor with your kids and some blocks to show them what “more” and “less” and different numbers actually mean. Make patterns out of just about anything age-safe such as different color socks or whole fruits and vegetables. When your kids are around 5-6 years old, playing with coins (counting and adding pennies, exchanging coins, etc.) on the floor is another fun activity.
Math outdoors is fun too! Go to the park and arrange rocks or wood chips into rows and columns. Draw shapes in the sandbox. Are there shapes that look the same no matter where you stand? Which shapes look like they’ve flipped or turned? Make shapes out of sticks. Compare the sticks. Which is longer? Which is shorter? Add sticks to go from a triangle to a rectangle to a pentagon and so on. If you keep adding sticks, what does your shape start to look like? (Don’t freak out, but you just did a little Calculus.)
Good old-fashioned floor time or outdoor time are excellent ways of cementing learning with positive memories. As your kids get older, make sure they are exposed regularly throughout their childhood and adolecense to adults who explain math well and enjoy it, especially if you don’t. And if you have math anxiety yourself, make that your best-kept secret. Math doesn’t have to be scary.
Now, what do you get when you take the circumference of a jack-o-lantern divided by its diameter? Answer: A pumkin π. Happy Halloween from IMACS!
Homeschooling is growing in the United States, having shifted from the “fringe” toward the mainstream. In January 2014, the National Home Education Research Institute (NHERI) estimated that there were 2.2 million1 K-12 home-educated students in the US, up from an estimated 1.5 million2 in 2007. According to the NHERI, one of the most common reasons that families choose homeschooling is to “customize or individualize the curriculum and learning environment for each child.” Not surprisingly, families with gifted children make up a significant part of the current homeschool movement.
If you thought that homeschooling meant having to teach your child all subjects on your own while sitting together at the kitchen table until you drive each other crazy, think again. Modern technology has opened up a world of educational opportunities. Online courses abound in every subject you could imagine. Social media allows homeschool parents to connect and form co-ops where they can share the work of teaching. Some public school districts offer classes taught by credentialed teachers several days a week at a homeschool campus. And numerous educational organizations offer local day classes for homeschoolers, such as the IMACS Homeschool Program. The job of teaching no longer has to be carried out solely by the homeschool parent.
Homeschooling also supports the need that many gifted students have to be self-directed learners. As eIMACS student and 2011 US Girls Junior (U21) Chess Champion Rachel Gologorsky said, “I recognized this as an excellent opportunity to have a say in my education.” In addition to allowing her to participate in the decision-making process, homeschooling provides Rachel with the flexibility she needs to develop her incredible chess talent, including travelling to national and international competitions.
Another oft-cited benefit of homeschooling is the freedom to custom-tailor an educational program that is matched, subject by subject, to a student’s abilities, learning style, and processing speed. Children can fully develop their strongest areas by going as deeply into a subject as they wish or by advancing as quickly as they are able while getting the appropriate level of support they need in other areas. This can be a good fit for many gifted children who, contrary to common misconceptions, are not always the fastest or highest achieving in every subject.
Consider what mathematician and hedge fund billionaire, James Simons, said in a recent New York Times article: "I wasn’t the fastest guy in the world," Dr. Simons said of his youthful math enthusiasms. "I wouldn’t have done well in an Olympiad or a math contest. But I like to ponder. And pondering things, just sort of thinking about it and thinking about it, turns out to be a pretty good approach." Today, a student like the young Dr. Simons can explore deep ideas in mathematics on his or her own schedule with self-paced, homeschool-friendly options such as eIMACS and Elements of Mathematics: Foundations.
A wealth of advice on homeschooling is available online, from helping families decide whether it is a feasible option to helping homeschooled high schoolers apply to college. Speaking of college, did you know that elite universities such as Stanford, MIT, Caltech, Princeton and Yale have sections on their websites dedicated to admissions information for homeschoolers? Homeschooled students are hot! As one Stanford Alumni Magazine article explained, “Stanford has found that the brightest homeschoolers bring a mix of unusual experiences, special motivation and intellectual independence that makes them a good bet to flourish on the Farm.”
While homeschooling has its advantages, it is not for every family. The benefits described above often come with meaningful sacrifices that should be considered carefully. Many families are already well-served by their local public or private schools. But for those who need a more customized education for any number of reasons, including giftedness, homeschooling today offers many options that make this path highly accessible.
If you are considering homeschooling your gifted child, start by visiting the following informative websites:
Families in the South Florida area may also find The Home Educator Magazine to be a helpful resource for local information.
1 From http://www.nheri.org/research/research-facts-on-homeschooling.html
2 From http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2009030
In The New York Times article, “But I Want to Do Your Homework,” author Judith Newman describes how she was driven, over her 12-year-old son’s objections, to help him ace his literature essay, only to earn a dismal grade. Her admission may sound familiar to many well-intentioned parents who believe they are doing what’s best for their child’s long-term success. Unfortunately, that’s not the case as Newman points out:
“Sociologists at the University of Texas at Austin and Duke University assessed the effect of more than 60 kinds of parental involvement on academic achievement. Read it and weep, helicopter parents: Across age, race, gender and socioeconomic status, most help had neither a positive or negative effect, and many kinds drove down a kidâ€™s test scores and grades. One of the biggest culprits? Homework help.”
It’s not a crime for parents with talented kids to envision them accumulating straight A’s and academic awards on their way to reaching their full potential. However, knowing that it’s okay to be wrong gives children the permission they need to take the kinds of intellectual risks required in order to achieve great things, as Newman learned:
“When I confessed my sins to Michael Goldspiel, my son’s beloved assistant principal … he summed up the problem better than I could. “Being wrong is part of the process of understanding,” he said. “Going out on a limb, being willing to take a chance, is a critical skill not just for homework, but for life.” He couldnâ€™t be more correct.”
It’s graduation time! The summer ritual of getting kids ready to send to college is around the corner. Mini-fridge, check. Shower caddy, check. Good study habits, hmmm. Surprisingly, developing good study habits before entering college is something that many talented children and their parents overlook. It’s easy to understand how this oversight happens when you realize that bright students often don’t need to study. That’s because their schoolwork isn’t challenging and requires minimal effort to receive high grades. Kids who are used to coasting like this hit reality in a big and stressful way when they encounter the rigor and higher expectations of college. But it doesn’t have to be like this. Helping your child build good study habits well before college is essential to their long-term success and happiness. You don’t develop your slice backhand by hitting tennis balls that are tossed straight at you. Likewise, you don’t develop strong study skills by “learning” easy material! It’s also important to let your child know that the effort he or she puts into intellectual pursuits, not just the outcome, matters to you. Failure should not only be an option but an understandable expectation when aiming high. As Thomas Edison said, “I have not failed. I’ve just found 10,000 ways that won’t work.” By encouraging a positive mindset and providing your child with challenging opportunities designed to stretch the talented mind, you’re well on your way to checking off the box for good study habits and many other skills that your child will need before, during and after college.
The following excerpt is from the new book by Mandee Heller Adler, From Public School to the Ivy League: How to get into a top school without top dollar resources, which is available at Amazon.com. Ms. Adler is the founder and principal of International College Counselors, a Florida-based firm that provides expert strategies for admission to undergraduate colleges, graduate programs, business schools, law schools, medical schools, dental schools and other postgraduate schools.
From Chapter 4: Writing Essays …
ANSWERING THE QUIRKY QUESTIONS
In recent years, a number of colleges have been adding quirky questions to their applications. These supplemental questions are considered a way to get students to stand out from the crowd.
These questions have included:
- Imagine you have to wear a costume for a year of your life. What would you pick and why?
- What is your favorite ride at the amusement park? How does this reflect your approach to life?
- What does Play-Doh have to do with Plato?
- What would you do with a free afternoon tomorrow?
- What was your favorite thing about last Tuesday?
- The Spanish poet Antonio Machado wrote, “Between living and dreaming there is a third thing. Guess it.” Give us your guess.
- According to Henry David Thoreau, “One is not born into the world to do everything, but to do something.” What is your something?
What colleges are looking for is your voice. Use this as an opportunity to demonstrate your “out-of-the-box” thinking. However, don’t go overboard. Admissions officers are looking to see if you’ll be an interesting person to have on campus. Interesting means imaginative, not crazy and not dangerous sounding.
GREAT FIRST SENTENCES
You need a great hook and a great first sentence. Opening sentences have the power to compel and fascinate. Some of our favorite student first sentences include:
- For eight years, I have celebrated polyester.
- I vividly recall coming home from school one day in Buenos Aires, Argentina, to find my house in disarray and my parents packing one suitcase after another.
- I’ll admit it: I have a thing for gavels—a thing for motions and seconds and the clarity that they bring to meetings.
- I eagerly reached into my Hello Kitty backpack.
- Max prances in place as we await our turn into the arena.
- Drip. Drip. Drip. Tick. Tick. Tick. As I lie in the hospital, waiting to be taken into surgery, I can only think that my IV drip sounds just like a metronome.
You want to read more, right?
FATAL ESSAY ERRORS
Application essays have been requested as part of the college application for the past umpteen years. The admissions teams have seen a lot of “creativity.” Here are their least favorite types of essays:
- Metaphor. Don’t compare yourself to a mango, a Ferris wheel, or any other objects.
- Death. Don’t write about a person or pet’s death unless it truly affected your life and you can use it to exemplify growth—for example, if someone died of cancer and you made it your mission to raise money/awareness, or if a death during high school affected your grades and caused you to stumble, but then you regrouped to overcome.
- Free verse essays, essays written as raps, limericks, etc. Don’t emphasize form over function.
- “Meta” essays where you talk about writing an essay, about the process of writing an essay, or about essays themselves.
Additionally, you should avoid writing about the topics below unless you have something extraordinary to say:
- A trip to Europe
- Generic admiration for your mom or dad
- The controversial rock star or movie star whom you idolize
- Overcoming an injury and making an athletic comeback
- Volunteering at a local community center
- Building homes in Costa Rica with Habit for Humanity
- Understanding the meaning of life from a fishing trip
Sorry, but thousands of students have beaten you to these topics and then beaten them to death. These are called “cliché essays” because the reader knows from the get-go just where you are going with it.
THE VIDEO COLLEGE ESSAY
A number of college admissions departments are formally accepting video college essays.
The first step for any student is to view recent videos and see what others have done. This will give you an idea of the range of possibilities.
When it comes to actually making your video, it’s important to be original but in a way that is comfortable for you. Do what works for you. Your main goal needs to be communicating your message.
- Start by identifying the question and any directions.
- Think about what you want to say.
- Write a script that is clear on the message and ideas you want to get across.
- Collect resources and props that you want to use in the video.
- Record the video until it’s as perfect as possible. Some students record the video themselves using a tripod while speaking directly into the camera; others enlist the services of a friend or family member.
- Review your video and collect feedback.
- Edit, edit, edit, and re-record if necessary.
- Get more feedback.
- Edit and re-record until it’s as perfect as it can be. Make sure it fits the requested length and meets all specs before sending it in.
To get a head start on preparing for college admissions, order a copy of From Public School to the Ivy League: How to get into a top school without top dollar resources from Amazon.com.
A recent study found that number sense in infants is a predictor of how well they later learn the symbolic math typically taught in school. Researchers observed how long 6-month-old infants looked at a screen with a changing number of dots compared with how long they looked at a screen with a constant number of dots in changing patterns. (See video above.) Three years later, these children were given an IQ test and several types of math tests. The infants who looked longer at the screen with a changing number of dots, which is indicative of stronger number sense, subsequently scored higher on the math tests, regardless of IQ. So what is “number sense,” and does this mean that parents should start packing dotted flash cards in the diaper bag?
What is Number Sense?
There are different definitions of number sense, but they generally agree on a few basic factors:
• Number sense is intuitive. In other words it is not arrived at through conscious reasoning.
• Number sense includes an understanding of magnitude of amounts, relationships between amounts, and how amounts change through natural actions.
Examples of number sense in an infant might include observing and understanding the following: There are a lot of stars on my blanket. Daddy has more food on his plate than Mommy. I have a bunch of blocks on my high chair table. If I knock some off, I have fewer blocks. From what current science tells us, pre-verbal infants cannot actually have these words running through their heads. Nonetheless, a primitive understanding is there as humans seem to be born with an innate ability to perceive quantity before they can speak about it and well before they can write about it.
Nurturing Number Sense
So should you start showing your baby flash cards with dots? No. While the study’s authors hope that their findings will lead to improved strategies for math education, they are careful to caution that “[O]ur infant task only explains a small percentage of the variance in young children’s math performance. But our findings suggest that there is cognitive overlap between primitive number sense and symbolic math. These are fundamental building blocks.”
Still, we know that there are benefits to talking to your baby as an essential part of the early learning process. It’s not unusual to hear a mother carrying on a one-sided conversation in which she emphasizes the phonetic sounds in the names of everyday objects. For example, “This is a banana. Buh buh banana.” The baby eventually catches on that the “buh” sound is associated with a banana, and that Mom’s lips do this strange “roll in-then-pop out” motion when she makes this sound. This happens before the baby ever learns to say “banana” or that there is this symbol we call the letter “b”Ãƒâ€šÃ‚Â that represents this sound. When the letter “b” is finally introduced, children can connect it back to the “buh” sound and objects such as a banana to which they were introduced in infancy.
But how frequently do you notice a parent holding up two different bananas and talk about which one is bigger, or which one has more brown dots? Or what happens to a bunch of bananas when you pull one off? Or when you cut that banana into pieces? The study’s lead author indicates that “We believe that when children learn the meaning of number words and symbols, they’re likely mapping those meanings onto pre-verbal representations of number that they already have in infancy.” If she is correct then adding number sense talk to those parent-baby conversations makes, well, sense!
Developing good study habits before entering college is an essential skill that many gifted and talented children and their parents overlook. Some parents are often surprised to learn that their bright child can ace a schedule of honors or Advanced Placement courses with little studying. They might assume that if their child is receiving top grades in the most advanced classes offered by their school, he will be well-prepared to handle the rigors of university courses. This is an unlikely outcome without good study habits, and waiting until college to learn how to study is much too late when one might already be dealing with living on one’s own for the first time. Here are three tips to help foster this important skill while you still can.
Find a Challenge That Requires Studying
If a talented child attends school in a structured setting, chances are she is already being asked throughout the school day to cover material she already knows or can learn quickly. To ask her to take time at home to study the same material is to double her frustration. You’ll have greater success instilling good habits if you ask your child to study material that actually challenges her.
The experience may take some getting used to by both child and parent because, if done correctly, it will involve:
• A healthy struggle to understand new ideas,
• Getting less than perfect scores,
• Not always being the smartest kid in the room, and
• (Drum roll, please) having to study to do well.
Sometimes subject or grade acceleration can help, but parents should keep in mind that even with acceleration, a gifted child is still being asked to learn material that was designed for the way a typical student’s brain works. A better alternative to getting through standard curricula faster is to find an alternative that will encourage your child to understand subjects more deeply by addressing the “Why?” questions bright children are so naturally inclined to ponder.
Encourage a Growth Mindset
Stanford psychologist, Carol Dweck, is noted for her ground-breaking research on praise and motivation. She found that children who believed that a person’s intelligence was fixed tended to believe that truly smart people don’t need effort in order to succeed. By contrast, those who believed intelligence could be developed were much more likely to credit hard work as a key factor in achievement. (Going back to our previous point, Dweck also found that children who were praised for their intelligence instead of their effort were more likely to avoid challenges for fear of failing and losing praise.)
Dweck’s later research showed that children can be taught this growth mindset when educated on how the brain gets stronger and smarter through the process of learning. In this later study, students who were taught about brain development in addition to study skills outperformed those who were taught only study skills. The latter group was not motivated to put those skills to use. So parents, make sure your child understands the positive impact he can have on his brain and save your praise for the effort he puts into learning and studying.
Turn the Tables and Have Your Child Quiz You
One way to make studying more fun and give your brain cells a workout at the same time is to have your child test you. Turn the tables by being a student again with your child as the teacher. Have her create, administer and grade an exam that you take. Bright children often enjoy discussing and sharing their knowledge and may be more than happy to “show you up.”
In the process of creating the questions, your child reinforces in her own mind the concepts on which she will be tested. When grading, your child must go through the analytical exercise of determining whether your answers are correct and why incorrect answers are wrong. Even if you know the material well, be sure to throw in some wrong answers and ask for explanations of the right answers.
There is no simple solution for helping a gifted child develop the study habits he will need in the complex world of university life. However, should he find himself facing true intellectual challenges for the first time without this basic learning tool, he may be at a distinct disadvantage relative to his classmates, regardless of the natural ability that used to take him so far. Like most good habits, studying is one best formed at a younger age when behavior and attitude are more malleable.
It’s the middle of September, so your school-aged kids are likely back in the rhythm of classes, extracurricular activities, homework, and maybe even part-time jobs. Busy schedules are taking shape, and the effort to fit everything in will likely mean some trade-offs. To the extent that your kids still look to you for help with their homework, take this moment to remember (or learn) this: Letting your children struggle at first will lead to better outcomes than if you simply provide the answer in the name of expediency.
In a study published earlier this year, education researchers compared the performance of math students in two groups. One group was given direct instruction when learning a new concept. In other words, they were told at the start how to solve a new type of problem. The second group was given the problem and asked to come up with as many approaches to solving it as they could before receiving any formal instruction. In several comparisons, researchers found that both approaches were effective at imparting basic knowledge of the new concept. However, students in the second group formed a much deeper conceptual understanding and were better able to transfer their knowledge to different situations.
This was the case even though the second group usually failed to come up with a completely correct solution. Researchers call this approach “productive failure.” Their study also showed that a higher number of incorrect solutions correlated to increased learning when those students were finally given instruction on the topic. The more we struggle to figure something out, the better we understand it when we finally do understand it. Why is that?
The theory behind the difference in performance has to do with the learning process. It seems that when we try to solve a new type of problem without new information, our brains have to lean on what we already know. When we are finally presented with correct approach, our brains make connections between the existing knowledge that was called up and the new knowledge. This seems to be a fundamental part of successful long-term learning.
Here’s the sad part for American students and where you, as parents, can try to help your kids. Math education in the US typically follows the direct instruction model. By contrast, consider how students in countries that outperform the US in math learn a new mathematical concept. The middle section of this short but eye-opening video discusses the TIMSS 1999 Video Study of eighth-grade mathematics and science teaching in seven countries. Consistent with the “productive failure” study, the TIMSSS videos showed that teachers in countries that outperform the US allow their students to struggle with new math concepts first.
In the best case, your child’s teacher already uses the “productive failure” approach. Keep up the good work at home when it comes to homework help. If the teacher uses direct instruction, then try encouraging your child to think of various ways of approaching the problem before attempting to explain the teacher’s way. Many of us at IMACS are parents too, so we know this is easier said than done with the busy lives that kids and parents lead today. In the long run though, it’s well worth the effort.
Unless you’ve been hiding under a rock for the past year, you’ll have noticed that the campaign to teach kids (and adults) how to code is everywhere you turn. As parents, politicians, and educators debate how to produce more graduates in technology fields, the push to introduce computing at an earlier age gets stronger. For example, MIT’s Lifelong Kindergarten group is collaborating on programming software aimed at kids in preschool to second grade. There are even board books for babies on HTML and CSS! We suspect that such novelties are more for tech parents’ enjoyment.
IMACS believes these efforts are well-intentioned and some, when implemented, will be well-designed. But before you click over to Amazon to buy your little drool monster a book on Web design, IMACS can offer you a few examples of how to introduce computational thinking to children through easy activities that are familiar to you, even if you think you’re computationally challenged.
Computational Thinking vs. Coding
This short and informative paper by Jeannette Wing, head of the Computer Science Department at Carnegie Mellon University, explains clearly what computational thinking is and is not. The following excerpted quote is a good summary of the focus of this blog post:
Learning to think abstractly is an essential skill if you want to succeed in computer science. It makes solving problems easier, which in turn makes working on those problems more fun. Notably, students can learn to think like a computer scientist without entering a single line of code into a computer. In fact, our experience in teaching CS is that writing computer programs is trivial for students who first develop computational thinking skills. Let’s see where in our daily lives we can show kids relatable examples to help them make the transition to abstract thinking.
Stacks and Queues
In computing, a stack is an object in which data expressions are stored and retrieved in such a way that the first data expression to be stored is always the last data expression to be retrieved. It is an example of a so-called Last-In-First-Out (LIFO) object. The same idea applies to various real-life constructs that young kids encounter, even babies who love board books.
Obviously, you can’t explain LIFO with words to a baby and expect the baby to understand, but you can certainly demonstrate the concept with your actions. Take the classic Fisher-Price Rock-a-Stack, for example. Start with the rings off the cone and then load them on in the intended manner with the blue ring going on first. Try to get just the blue ring off. Can you do it while the other rings are still on the cone? No, you have to take the rings off one by one with the blue ring coming off last.
Older kids can appreciate the same concept with examples they come across in their lives: unloading plates from the dishwasher into the cupboard, setting the table with said plates the next day, selecting a product such as cosmetics from a store shelf, putting said product back if you decide not to buy it. You get the picture.
A queue is similar to a stack in that it is an object used to store data expressions. In the case of a queue, however, the first data expression to be stored is always the first to be retrieved. Queues are examples of First-In-First-Out (FIFO) objects. Kids encounter them every time they go through a checkout line or a drive-thru. Switch that Rock-a-Stack cone for an empty paper towel roll, and you’ve got yourself a baby-friendly queue.
Sorting is one of the oldest problems in computer science. Although the end goal (an ordered list) is conceptually easy to understand, getting there can be complex. Add to that the need for sorting algorithms to be computationally efficient and you’ve got yourself an interesting abstract puzzle.
If your kids are old enough to know or learn how to put words in alphabetical order, then make a project out of sorting the books on their bookshelf. Decide on a sorting key such as title or author’s name. For this example, we’ll use title. For the first shelf, ask your child to try a simple bubble sort. Traverse the shelf from left to right, compare the titles of two books, and swap them if they are in the wrong order. Repeat this process until all books on this shelf are in the correct order.
For the next shelf, you can use a simple insertion sort. Take all the books off that shelf and put them in order one by one in a pile on the floor. Each time that you add a book to the ordered pile, be sure to put it in the right place relative to the books that were previously added.
Now that you have two properly sorted sets of books, you and your child can work together to sort all books. Sounds like a good time to use a merge sort. Move the sorted books off the first shelf into another pile on the floor while keeping them properly ordered and separate from the pile of books from the second shelf. Reshelve the books as follows: repeatedly compare the titles of the two books that are atop the two piles, selecting the one that goes first, and continuing until both piles are exhausted.
Object Oriented Programming
An “object” in computer programming is a complex structure containing data fields and instructions. These objects interact with each other to create even more complex computer programs. The beauty of object oriented programming is that you can reuse objects to do common computing tasks without having to reinvent them each time. Over time, programmers can build up a “library” of useful objects.
The following analogy certainly isn’t perfect, but it will help get the point across about these seemingly mysterious objects. If you’re planning an outing with kids, you’ll need a few things to help keep your sanity: nourishment, entertainment, and possibly a change of clothes. So grab three bags and make some objects! In the nourishment object, you’ll probably need fruits, carb snacks, a protein, and beverages. For the entertainment object, how about art supplies, books, sporting equipment, and portable gaming device? Kids are made to get dirty, and the weather may change, so pack a top, bottom, and outerwear in the clothing object. Throw those “objects” in your huge tote “library” and you’re ready to go!
Think Like a Computer Scientist
Planting the seeds of computational thinking, especially the ability to think abstractly, is really a matter of recognizing the examples in your life that can be used to foster discussion with your children. Like any new endeavor, remembering to look at events in a computational light takes practice. You might just find yourself thinking like a computer scientist when it comes to solving the data problems in your own adult life.
Milo Gardner may be retired, but he’s not letting his mind rest idle. This amateur code breaker chatted with IMACS about his mathematics background and how it influenced his life. After serving in the United States Army as a cryptanalyst, Milo went on to earn his B.A. in Mathematics with a minor in History of Economic Thought at California State University – Chico. What followed was a career in aerospace engineering and an M.B.A. from CSU – Fullerton. Years later in retirement, Milo’s curious mind, a letter to the editor, and a meeting at a coffee shop led him to his current passion – decoding ancient Egyptian math texts. IMACS blog readers can review his extensive list of online publications on the subject. Now let’s hear from Milo in his own words.
Please tell us about your mathematics background, including what interested you as a child.
Baseball and sports of all types filled my youth. In the 8th grade I was sports editor of our school newspaper. This meant computing box scores and other game statistics for the various teams and writing up short narratives. Several sports teams were enjoyed during high school. Basketball was my favorite. Math was a special high school interest, as well as Spanish, chess and the sciences. Algebra I, Algebra II, Geometry and Trigonometry were studied in the context of set theory. That is, math students determined in advance which number system(s) would solve particular problems.
Right out of high school, the US Army was joined with hopes of spending time in Europe. My high school language skills were tested first. Two entrance exams for the Defense Language Institute at Presidio of Monterey, California, were taken. Luckily, the second exam score was low. The Army’s code-breaking tests followed, which were passed. In a five-month code breaking school, language and number-based patterns were studied. Techniques identified one-to-one, two-for-one, and other substitution systems. Finishing third in a class of 20, Germany was selected as my two-year assignment. Lower ranked students were sent to Africa, Turkey, Korea and Japan. In Germany, ad hoc Russian language projects were assigned to cryptanalyst and linguist teams. A two-month side trip to Lebanon widened my foreign language exposure to include Arabic.
Upon discharge from the Army, a number of my co-workers went to work for the National Security Agency (NSA) in Washington, D.C. I went to college in Northern California and earned a Mathematics degree with a minor in History of Economic Thought, with an intention of applying to the NSA. Computer programming courses were taken, as I was preparing to be a high school math teacher as a back-up career. Upon graduation, wishing to be married and raise a family, a southern California aerospace engineer accepted. The pay was better than teaching. Equally important the location kept me and my wife-to-be close to both of our families.
What kind of work did you do in the aerospace industry?
The first aerospace position was at Vandenberg AFB as a systems analyst. Western missile test range issues input computer-generated data to hand drafted range safety charts. The charts factored in daily wind measurements and drag aspects of missile parts that allowed three-second delays for the missile safety officer to blow up errant missiles. Nearby populations were protected. Subsequently, college programming skills were applied working with a team that automated the hand drafted charts.
The second aerospace job was at Rockwell International in Fullerton, California. Failure patterns of Minuteman I, II and other avionics guidance systems were studied in terms of maintainability issues. While at Rockwell, evening business classes were taken towards an MBA degree. Upon graduation, a career change allowed our family move to northern California to be near our respective families.
You raised a daughter who went on to become a civil engineer. What advice can you offer to parents of mathematically talented girls on how to nurture their talent?
Missy, my CE daughter, was self motivated by the second grade. All three of my children played musical instruments. Parental guidance consisted of coaching youth softball and baseball teams up to age 12 and supporting each child’s active social life. Thereafter, tryouts for competitive teams were arranged. Missy made a traveling team at age 14 in an organization that her older sister played. My wife and I attended most of Missy and her sister’s weekend tournaments. Our children knew their parents were their biggest supporters.
Missy was strong willed and stood up to basketball, softball and volleyball coaches in high school. Her sports experience gave her confidence to confront one engineering professor in college who refused to call on women in the class. After receiving a failing grade at mid-term, she called a conference with the professor and asked why he had not taken even a single question from one woman in the class. Her strong position was understood and respected. Thereafter, in-class questions were answered sufficiently for her and other women to earn passing grades. Today, she is an engineer working for a private firm. Incidentally, she told us this story weeks after the situation had occurred. Missy had the confidence to handle it herself.
In retirement, you now pursue code breaking as a hobby with a specialization in ancient Egyptian math texts. How did you become involved in these pursuits?
In 1962, an upper division college history of math class told a fuzzy history of zero story. Zero, as we know it today, did not reach Germany until 1200 AD, in time for the birth of our base 10 decimal system in 1585 AD (that defined n^0 = 1), an unbelievable assertion. “Some day I’ll research that topic,” I told myself. In 1988, six months were spent studying the topic at two local university libraries. Medieval and older Near East numeration systems including Classical Greeks used zero, a round figure topped by two dots in clear ways. Babylonians and Egyptians used zero as a limit 1,500-2,000 years earlier. Zero was also the value of empty sets in an Egyptian double entry accounting system. The older uses of zero did not use placeholders. Care had to be taken to read the context of mathematical documents and inventories –- issues that I knew well as a military code breaker.
After completing the study of the longer history of zero, an unexpected event took place. Acting on the dehumanization of classroom math topics (e.g., omitting personal stories like how the pre-teen Gauss summed the addition of 1 to 100 by a formula), I wrote a letter to the editor of the Sacramento Bee on the weaknesses of the 1990 California Math Framework. The day after the letter was published a phone call from a retired electrical engineer was received. Noel Braymer asked that we meet at local coffee shop. I said yes, and my retirement world changed for the better.
Noel had worked on 1650 BCE Egyptian text called the Rhind Mathematical Papyrus (RMP) for 15 years. A 50 member RMP 2/n table was encoded. The table took up 1/3 of a papyrus that contained 87 other problems. Noel offered a modern number theory solution to the 2/n table. Number theory stresses prime numbers in ways that ancient and modern mathematicians parse divisors of composite numbers into primes. The ancient 2/7 was recorded as 1/4 + 1/28. What set of ancient rules were used in the entire 2/n table?
The 2/n table encoded 2/3, 2/5, …, 2/101 to concise unit fraction series in ways that scholars hotly debated during the 20th century. Robin-Shute published the Rhind Mathematical Papyrus text in 1987 and suggested one incomplete solution. Noel gifted the book to me, and asked if I would assist in publishing his work. I said yes, provided the ancient scribal methods were also decoded and published.
Ten years later, working with a linguist, a sister document to the RMP, the Egyptian Mathematical Leather Roll (EMLR), was decoded with modern number theory. Attempting to explain connections between the two documents, aspects of the RMP 2/n table construction methods slowly emerged. Another seven years passed. Finally, by considering Egyptian wages paid in commodities, a 2011 paper included a complete solution to the 2/n table problem as an appendix.
In your opinion, what mix of interests and skills makes a person well suited for a career in cryptography? What should kids who are interested in code breaking be doing now to prepare themselves for a career in this field?
Students should enjoy solving all types of puzzles. I have loved crossword and other puzzles since high school. Learn a foreign language and learn about new and old foreign mathematical issues. Enjoy competitive individual and team games. Chess and bowling took up much of my free time. Choose your games and puzzles wisely. To pursue a puzzle solving career, a student should learn to contribute to all sorts of teams. Team membership is an important skill in many aspects of the adult world.
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