The winner was selected randomly from correct entries and has requested to remain anonymous.

Here’s a solution.

Many people used the hook-length formula to compute the answer. It might be interesting to note that all the people who did not use the hook-length formula got the correct answer, but only half of those who did use the formula were correct.

Since you can read about the hook-length formula elsewhere, here we’ll deduce the answer using basic counting techniques.

Instead of using word titles, we’ll use the numbers 1 through 10 and put them into the diagram so that the numbers increase left to right across any row and down any column.

The number 1 must go in the upper left square.

Now, let’s fill in squares until the square to the right of the 1 is filled, then stop.  There are many ways to proceed, so we split into cases. The diagram below shows all the cases. The first number that is put into the second column must go at the top of that column, which is the square to the right of the 1, and that’s when we stop. So we end up with 4 cases.

In each of the 4 cases, notice that we are free to choose any 3 numbers from the numbers that haven’t been placed yet for the remaining 3 squares in row 1. These numbers must be placed in row 1 in ascending order, and the numbers still remaining must be placed in the unfilled squares in the first 2 columns. The number of ways the unfilled squares in the first 2 columns can be filled is quite limited and can be counted directly.

Case 1: .

Case 2: .

Case 3: .

Case 4: .

Adding these up, we get 280 + 175 + 60 + 10 =  525, which is the answer.

The cover was made by Prof. Laura DeMarco, the subject if this issue’s interview.  To make it, she used the Dynamics Explorer Tool, a program developed by Suzanne Lynch Boyd and Brian Boyd.  One topic that Prof. DeMarco researches is stability.  So that readers can actually roll up their sleeves and do some experimentation with this concept, we explain how to construct the famous Mandelbrot set in the article right after the interview.

Illustration by Julia Zimmerman from The Stable Marriage Problem, Part 4

Next comes the concluding part of the 4-part series by Dr. Emily Riehl on the Stable Marriage Algorithm. We hope you have enjoyed this series of articles and her accompanying WIM Video. It was a huge treat for Girls’ Angle. Also, special thanks to Julia Zimmerman for illustrating the series and to Grace Lyo for editing it.

Math Doctor Bob continues his series on Fermat’s little theorem with a discussion on how to find the multiplicative inverse of a number, modulo n.

This issue’s Math In Your World was inspired by a game of Scrabble played in front of Au Bon Pain in Harvard Square. If two people play a single game, what is the probability that the winner of this single game is, indeed, the better player? Following up on this single question leads to a lot of mathematics and statistics. The importance of this mathematics perhaps becomes more evident if put in the context, say, of choosing a medicine or treatment option: if Medicine A does a better job of helping a patient than Medicine B in one single case, does that mean that Medicine A is better?

Coach Barb continues her conquest of fractions, this time by revisiting the way ancient Egyptians expressed them.

Finally, in the Notes from the Club section, we describe Support Network Visits from Iris Ortiz, of Cambridge Systematics, Crystal Fantry, of Wolfram Research, and Ashlee Ford-Versypt of MIT’s Department of Chemical Engineering.  Every semester, professional women who use math in their work visit the Girls’ Angle club and work directly with our members. These “Support Network” visitors serve as role models and provide our members with yet another reason to study math. We also describe an activity using knots that is easy to implement in a classroom and is designed to underscore the difference between active and passive learning.

We continue to encourage people to subscribe to our print version, so we have removed much content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

Last month, Girls’ Angle affiliates Rediet Abebe, Jordan Downey, and Lauren McGough hosted a mathophile raffle contest at the Girls’ Angle booth in the 2013 FIRST Robotics Career Expo in New York City. Nobody succeeded in solving the first question on that raffle, so we’re reposting it here. To enter, please send your name, solution, and contact info to girlsanglepuzzler “at” gmail.com by April 30, 2013. We will draw randomly from the correct entries received and the winner will receive some chocolate from L. A. Burdick. For this one, if you have a doctoral degree in mathematics, you’re not eligible for the prize.

The Problem.

The menu of links above is taken from former Girls’ Angle mentor, now graduate student at the University of California, Berkeley, Maria Monks’ website.

How many ways can the link titles be placed into the squares (one per square) so that they are in alphabetical order down any column and across any row (going from left to right)? (Note: Her current layout does not obey this property. Also, “Home” is not required to go in the upper left square.)

Good luck!

(We will not use your contact information for any purpose other than to deliver your prize, should you win. After the winner has been selected, all emails received will be promptly deleted. At the winner’s discretion, we will let you know who won. Anyone who makes more than one submission will be disqualified! Sorry! Also, this offer is only valid in those states in the United States where such things are legal. There is no fee to enter this puzzle contest.)

That was a terrific question! Unfortunately, I didn’t have time to discuss it as well or as fully as I would have liked. A lot of math hovers around it, so I decided to amplify my in-club remarks in a blog post.

First, there’s a common misconception that I’d like to expose. The misconception is that the Pythagorean theorem is a statement about the relationship between the lengths of the sides of right triangles found in the real world. It is not. It is a statement about the relationship between the lengths of the sides of a mathematical concept known as a right triangle.

If this comment strikes you as off, it could mean that you have not understood the distinction between mathematics and science. Mathematics is a universe of abstractions, not physical objects. People often confuse mathematical abstractions for actual things because natural observations inspired so much mathematics. Perhaps when one thinks of a plane, one imagines a table top or a tennis court or a sheet of glass. But we know that the physical objects are imperfect models of the pristine, abstract concept of a plane, infinitely thin, absolutely flat, and extending without end. Perhaps the surface of a frozen lake inspired the mathematical concept of the plane, but the concept is an abstraction endowed with idealized properties. Mathematical theorems are about these abstracted objects. Science and applied math model actual things with these abstracted objects. And the Pythagorean theorem is a mathematical theorem, not a scientific hypothesis.

When a land surveyor applies the Pythagorean theorem to deduce some length, that surveyor is assuming that distances in the real world obey the relationship given by the theorem. This assumption is extra-mathematical and is part of what distinguishes math from science. In order to use math in science, scientists have to make some assumptions that connect the mathematical world to the natural world. But while science concerns itself with the “real world,” mathematics, strictly speaking, does not.

And yet, time and time again, people improperly call real world illustrations of the Pythagorean theorem, “proofs” of the Pythagorean theorem. No doubt you’ve seen the so-called “water-based ‘proof’ of the Pythagorean theorem.” (Such as this one or this one. Both are nice demos, but the first title is, strictly speaking, inappropriate.) If you understand this misconception, then you realize that such demonstrations are not proofs. (By the way, of these demonstrations, one can also note that many illustrate only a single instance of the Pythagorean theorem because they involve a very specific right triangle, whereas the Pythagorean theorem is a statement about all right triangles. That is a valid observation, but that is not the reason for the misconception I am now addressing. This problem of being too specific to capture the full generality of the theorem can be a problem with mathematical proofs too, such as if one were to prove the Pythagorean theorem in the specific case of an isosceles right triangle, then erroneously conclude that, therefore, the Pythagorean theorem is valid for all right triangles.)

The mathematical proof of the Pythagorean theorem does not depend on the geometry of the space of the world that we inhabit. Instead, it is about whether a geometric statement follows logically from some set of axioms.

The question of whether distances in the “real world” obey the Pythagorean theorem is a scientific hypothesis and can be checked experimentally. You could regard the “water-based proof” as just such an experiment. Like all scientific hypotheses, this one could be disproved some day, but it will never be proven.

If the water-based “proof” of the Pythagorean theorem isn’t really a proof, what is?

When a mathematical statement is proven, it is only proven relative to some set of axioms, which are statements made without any proof at all.  Axioms are assumptions upon which a theory is woven.

Euclid’s Elements contains a valiant attempt to provide such a deductive proof by clearly stating axioms that are accepted as true and then showing a way to logically deduce, step by step, the Pythagorean theorem.  Although subsequent analysis of the Elements has shown that not all assumptions used were explicitly stated, the procedure used is the right mathematical one: Axioms are stated, logical implications of these axioms are explored. And any statement that is logically deduced has been proven, relative to those axioms.

In the case of the Pythagorean theorem, many proofs have been supplied. Alexander Bogomolny has collected 99 of them on his extensive website Cut The Knot. Now, most of these proofs do not supply detailed logical deductions all the way down to foundational axioms of some geometry. Doing so would make reading the page as unpleasant and frustrating as swimming through peanut butter. Instead, they bridge the theorem logically to other mathematical results. The presumption is that the reader either accepts that these other results have been proven (and proven without using the Pythagorean theorem), or can prove or find a proof of them.

The important point is that no theorem can be proven true or false in the absence of axioms.
The next time you’re asked to prove something in math, perhaps the first thing you should say is, “what am I allowed to assume?”

And this leads to the following modification of the student’s question:

Are there any interesting collections of axioms from which the Pythagorean theorem can be proven to be false?

The short answer is “Yes!” The long answer is the story of non-Euclidean geometry.

There’s so much more to say, and others say it so much better than I. So here I’ll refer interested parties to two books:

Worlds Out of Nothing by Jeremy Gray

Non-Eucliden Geometry by H. S. M. Coxeter

There’s much more out there, but hopefully, these two will serve as a good place to start.

Teacher: To solve 7x + 3 = 12, first subtract 3 from both sides to get 7x = 9.  Next, divide by 7 on both sides to get x = 9/7.

Student: Can you start by dividing by 7 on both sides first?

Teacher: No, you have to subtract 3 first.

Oops!

The Truth: If you apply the same function to equal quantities, you will get equal quantities.

So, yes, you can definitely start by dividing both sides by 7. You’d get x + 3/7 = 12/7, and now you can subtract by 3/7 on both sides to solve for x.

There’s a tremendous variety of functions, so there’s a great many ways of solving equations.  Keep this in mind if you’re just beginning your study of equation solving. Because there are generally many ways to solve an equation, don’t waste efforts trying to memorize specific derivations. Instead, focus on what a manipulation accomplishes and whether it moves toward the desired goal. Just for fun, you might even challenge yourself to come up with multiple derivations.

This particular teacher, in this particular case, is wrong. Nobody is always right, so don’t just trust everything you hear. Think about everything you learn at school and decide for yourself whether it rings true or not. If you’re still unsure about whether a mathematical concept is correct or not, feel free to ask us at girlsangle@gmail.com.

The cover pertains to a rather hot topic at the club of late: divisibility.  Can you see how the cover relates to divisibility?

Much of this issue’s content pertains to divisibility. There’s an explanation of how all the standard divisibility rules learned in school (typically for 2, 3, 4, 5, 8, 9, 10, and 11) are all instances of a uniform method, and this method is used to get divisibility rules for less common divisors like 7 and 27. Test your understanding of the material with a problem set on divisibility rules. Can you quickly tell what remainder results when you divide

334,554,321,123,322,334,554,321,123,211

by 7? (And extra credit if you can guess what inspired this sequence of numbers!)

Math Doctor Bob continues his series on Fermat’s little theorem, which is also about divisibility. He gives Euler’s generalization of Fermat’s little theorem and presents a beautiful proof of the result due to James Ivory.

The second half of our interview with Radmila Sazanović touches on some of her views on the relationship between art and math.

Emily Riehl provides proofs of theorems pertaining to the stable marriage algorithm…again adorned by drawings of Julia Zimmerman. Dr. Riehl also describes the results she proves in this part in her WIM Video.

Coach Barb continues her conquest of fractions, this time by looking at reciprocals in more depth. 3/7 makes a claim without proof, but not to fear, Anna B. provides a proof in Anna’s Math Journal.

Also inside, we describe two different solutions to part of the fifth Invention from the previous issue, one solution due to Girls’ Angle mentor Luyi Zhang.

We continue to try to encourage people to subscribe to our print version, so we have removed much content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

Did you feel an urge to reach for your calculator to compute that? If so, please continue reading this post.

I was actually working with a student who had exactly that computation to perform. The problem was to find the measures of the base angles of an isosceles triangle with an apex angle measuring 40 degrees. This student always reached for her calculator to perform any computation, and she did the same here. However, she made a calculator typo and punched in “180 – 50 =”. She proceeded to get the wrong answer of 65 degrees for the measure of each base angle.

After having observed this student reach for her calculator for every single computation, I asked her if she really needed her calculator. She said, “yes.” I asked her to try to do the problem without her calculator. She protested, “I can’t.” I insisted that she give it a try. We spent about 3 minutes going back and forth like this. Finally, she gave in and tried without the calculator.

I was expecting to see her substitute scratch paper for her calculator and set up the subtraction problem 180 – 40. But to my surprise, she figured out the answer in her head and said, “180 minus 40 is . . . 140?” and then proceeded to get the correct answer of 70 degrees for each base angle.

“You’re better without your calculator!” I announced.

“What do you mean?” she countered.

“Well, you got the wrong answer when you used the calculator.”

Okay, that last bit might have been a tad facetious, but, in this case, I think it is true that this student would fare better if she left her calculator at home. And, there are important reasons for this student to try to avoid her calculator beyond the simple fact that she was faster and more accurate without it.

What is education all about, after all? I like to make education about improving one’s ability to think and learn. Mathematics is a fabulous subject by which to achieve those goals. Errors in one’s thinking reveal themselves in contradictions and the subject offers material to think and wonder about that provides a super-astronomical spectrum of difficulty levels.

To get better at thinking, a good strategy is to find things to think about that are challenging, but not impossibly so. Also, higher-level thinking ability requires competence at lower levels of thinking ability. So to reach a higher-level of thinking ability where one is not yet capable, it’s a very good idea to work with and master concepts and problems that require a lower level of thinking ability.  Furthermore, abstractions in mathematics will not be meaningful without knowledge and understanding of the concrete examples which inspire the abstraction.  And so many abstract concepts trace their roots to properties of numbers.

Therefore, playing with numbers sets the stage for many more advanced concepts and avoiding this number play will make it harder, perhaps even impossible, to achieve a sophisticated understanding of mathematics.

The human mind has this remarkable ability to notice things and make itself more efficient at doing things if given the chance. Reaching for that calculator on every computation is depriving the mind from having the chance to notice and learn useful ideas that may seem very tiny, but form the building blocks of ever grander concepts and abstractions.

Every arithmetic computation can be viewed as a tedious chore…or it can be viewed as an opportunity to notice something, reaffirm something, test something, refine something, or practice something.

For example, let’s take a look at that computation the student confronted: 180 – 40. Noticing that one can ignore the 0′s and compute 18 – 4 to get 14, then tack on a 0 to get 140 is something that is worth noticing. It may seem awfully small, but these awfully small observations are like little grains of sand that fill in the holes of our understanding. If one does a lot of computations, perhaps this observation will first be made empirically.  Soon, a point in time will come where one wonders why it works, or if it really always works that way. This wondering will make the distributive law a friend. But the key point is that this kind of observation cannot be made if one always reaches for the calculator.

I’d like to give 2 real-life examples to show how calculator addiction can be holding you back.

Example 1. The problem: Concentric circles of radii 1, 2, 3, 4, and 5 comprise a target for a dart game. What fraction of the dartboard is contained within the 3rd ring?

I watched as a student reached for a calculator and punched in the following:

ENTER STORE A ENTER.

ENTER STORE B ENTER.

A / B ENTER ToFRAC ENTER.

If this student uses this calculator approach to solve such problems for the rest of his schooling years, he’ll miss out on seeing live examples of the power of simplification, for a student more adept at math might write

and see that the ‘s cancel. Entering 3.14 into the calculator is thus proven to be a total waste of time. It’s just more opportunity to make a mistake. However, to be able to notice the cancellation, one must think about the computation before jumping for that calculator. There is also the matter of using an approximate value for , which calculator addicts seem to do as a matter of habit, so much so, that I have met some who seem to believe that . Even more, computations like this one are instances of the general geometric concept that the ratio of the areas of similar figures is equal to the square of the similarity factor. By actually doing the computation without a calculator, the student is creating a bed of experience onto which the more abstract geometric concept can rest soundly.

Example 2. This example comes from an exercise that a student was given as part of a lesson on arithmetic progressions: Compute 1.2 + 1.6 + 2.0 + 2.4 + 2.8.

Again, I watched a student punch into a calculator:

1.2 + 1.6 + 2.0 + 2.4 + 2.8 ENTER,

and move on never noticing, or needing to notice, that the numbers were in arithmetic progression, thereby defeating the point of the exercise in the given context.

If this student were forced to do this computation without a calculator, probably the student would evaluate the expression left to right, adding the numbers one by one in that order. That would no doubt feel tedious. But if done properly, a series of such “tedious” questions could be posed that would induce the student to prefer to seek a pattern, and when they see it, the point of the lesson will really sink in, along with possibly a deeper understanding of the notion that the sum of a list of numbers does not depend on the order in which they are added up. A really strong experiential understanding of that principle could translate into a profound psychological difference in reaction upon hearing the commutative law stated for the first time. To someone who always reaches for that calculator, the reaction might be a sense of unfamiliarity and discomfort with the abstraction (“there’s nothing to put into a calculator…”). But to someone who has knowingly or not used the law to aid in computation, the reaction might be one of familiarity and even the feeling that the statement is “obvious.” And these kinds of psychological differences can mean the difference between going much further in one’s study of mathematics and quitting it as soon as possible.

The calculator can be a very useful tool, to be sure. But if you are using your calculator to avoid thinking about things, hide your calculator and remove its batteries because while your calculator may be saving you time for the moment, it’s costing you huge amounts of time in your future and it is preventing you from playing with things that are just what you need to play with to prepare yourself for more advanced mathematics.

They [student chefs] go, “I won’t do anything like that…” And that is, I think, this is one of this…this awful American syndrome of “Fear of Failure.” And if you’re gonna have a sense of fear of failure, you’re just never gonna learn how to cook because cooking is, well, lots of it is, one failure after another. And that’s how you finally learn! For instance, you’ve got to have- develop what the French call “je-m’ens-foutisme“- I don’t care what happens! The sky can fall and omelettes can go over…all over the stove- I’m gonna learn! I shall overcome! If you’re not gonna be ready to fail, you’re not gonna be ready to learn to cook.

- Julia Child

That is great advice for learning math too! When you’re trying to solve a problem or understand some mathematical concept and you think you have an inkling of what to do or how it all works, try it out and see! Let’s not care if it doesn’t work! Let’s not care if your scratch paper ends up covered with number rubble, diagram debris, or symbol nonsense! The goal isn’t to avoid error. The goal is to gain understanding. If you’re not gonna be ready to fail, you’re not gonna be ready to learn to do math!

Interview Introduction by Manuel de León
Interview, Part 1
Interview, Part 2

I would like to thank Ms. Timón and the ICMAT for giving me this opportunity and for taking such care with the interview and the translation (by Jeff Palmer of the Technical University of Catalonia).  The topic of gender and mathematics is a challenging one full of subtlety.

If you’re curious about the interview, but, like me, don’t know Spanish, here is the original English version, posted with permission from Ms. Timón:

ICMAT: Which are your main goals?

Thank you for this opportunity to talk about Girls’ Angle.

We aim to be a comprehensive, quality approach to math education particularly for girls in grades K-12. We view math education rather broadly, so one could say that we aim to help girls improve their ability to think using math as an ideal vehicle for that purpose. What we hope is that girls who attend Girls’ Angle all the way through high school, will emerge with the skills and confidence to tackle the yet unsolved. That’s why we have a postdoctoral mentor at every meet. People who have earned doctoral degrees in math have proven original theorems and understand that math is a creative endeavor. They know how to create new math and have successfully endured the psychological challenges of exploring the unknown.

We also aim to produce useful, quality math educational content. For example, we produce a bimonthly magazine, the Girls’ Angle Bulletin, that contains articles on math and applied math, expository articles on math, collections of math problems, math inspired art, and interviews with women in math. For example, we have had interviews with both Ingrid Daubechies and Sophie Morel, the first women ever tenured in the mathematics departments at Princeton and Harvard, respectively. We also recruit mathematicians to write content for the Bulletin. In fact, last summer we received a modest grant from MathWorks expressly for developing Bulletin Content, and we are using these funds mainly to entice mathematicians to contribute. We make videos which can be viewed on our YouTube channel and website, such as Emily Riehl’s presentation on the stable marriage algorithm. And we create special mathematical activities, such as our Math Treasure Hunts, a concept born and developed at Girls’ Angle, and which we now host at schools. I’m currently writing a book on how to make and run Math Treasure Hunts, which I hope will be available by next summer.

Yet, though we’ve been around for just over 5 years, we’ve only realized a fraction of the full vision. Our next major goals are to be able to rent a permanent place of our own where we can hold our meets and offer classes and hire a full-time Head Mentor. Currently, we have to set up and take down Girls’ Angle each meet and we rely on charity to have space to offer courses like Math Contest Prep (which is sponsored by the Microsoft New England Research and Development Center). Head Mentors would be women with doctoral degrees in mathematics who are skillful at and care very much about teaching the next generation. I see no contradiction in the concept of a math professor whose educational duties apply to the K12 level instead of the university level, and these Head Mentors would ideally be mathematical researchers who oversee the teaching of our members. Of course, the only obstacle between where we are now and this vision is a lack of money. But I think there’s hope because the amount of money I think it will take to realize this dream is just a mere fraction of the amounts of money that are routinely moved around in many financial transactions that take place today. We have been trying to develop revenue sources that have the potential to realize sufficient income to achieve these next goals, but donations are much needed and always welcome.

ICMAT: Where does this club come from?

A major source of motivation is probably easiest to explain if I briefly describe my own history with math. I obtained a doctoral degree under the supervision of George Lusztig at MIT and then was a Benjamin Peirce assistant professor at Harvard. So I was walking a path in academic mathematics. But I left academia for oil painting. As my savings dwindled, I started freelancing as an editor in math educational publishing to pay the bills. There I saw first hand just how bad the state of math education has become in the US. Conceptual errors have become rampant in math educational materials produced today, and, even if they are noticed, they are often impossible to weed out. I was shocked into action.

While I was freelancing, I volunteered to lead a modular origami project for an organization called Science Club for Girls. We made geodesic spheres, one of which was on display at Boston Children’s Museum for half a year. It was the first time I worked with an all-girl group and the experience opened my eyes to gender issues in science and math education. I began to see subtle ways in which math education is biased in favor of boys. When I was in academia, it was impossible not to notice the huge gender disparity in mathematics, but I had never really thought about its causes before.

I think there are a number of excellent co-educational math programs. That is the status quo. And the status quo leads to a mathematics profession that is dominated by men. But there are and were very few, if any, comprehensive math programs for girls. So I decided to create one.

I discussed the idea with several people, partly to refine the concept and partly to gauge its feasibility. Two of these people, Lauren Williams and Elisenda Grigsby, were both undergraduate math majors at Harvard when I was on the faculty there and had continued on in mathematics and are now holding tenure-track positions, one at UC Berkeley and the other at Boston College. Those two along with a friend of mine from college and graduate school, Ray Sidney, joined me to form our first Board of Directors. Since then, Bianca Viray, a mathematician at Brown University, has joined the Board.

It’s a peculiar strength and weakness of our Board that every member holds a doctoral degree in mathematics. But I came to believe that one of the reasons for the deterioration of US math education is that there is disconnect between the math educational world and the mathematical world. I felt it important to have leadership that fully understood math as creative art. I saw the many slippery slopes that have dragged down the quality of math education and I wanted leadership that can decisively avoid such slippage.

Girls’ Angle also has a strong Board of Advisors whose members have generously made themselves available as consultants to ensure that Girls’ Angle remains true to its mission. Our Board of Advisors includes two professors at MIT: Gigliola Staffilani and Bjorn Poonen.

ICMAT: Why girls needs clubs?

This question leads to the heart of any controversy surrounding the notion of a math club for girls. It is an important question, though I often wish it were phrased like this: What is the value of a math club for girls? The word “need” complicates matters. In a strict interpretation of the word “need,” the answer would have to be that there isn’t a need. Mathematics is not like oxygen, water, or food.

Our point in establishing a math club for girls is not that “girls need a math club.” Instead, our point is that there exist girls who greatly benefit from an all-girl math educational environment and that there are enough of these girls to justify the existence of several girls’ math clubs.

The human race is diverse. It almost seems that there is an exception to every generalization. For instance, one might assert that boys are XY and girls are XX. But then you take a scientific look and find people who are XXY, XYY, XO, XXX, and even XXYY. One of the challenges in discussing this topic is that there are always exceptions. I guess you could say that not only is the human race rather exceptional, it is also full of exceptions.

Keeping exceptions in mind, I would assert that the modes of instruction best suited to teaching mathematics, especially in the K12 arena, are gender differentiated. I’ve often had conversations with people who don’t seem to understand the complication of exceptions, because they retort, “well, my daughter is in a co-ed math class and she’s the best student in her class!” In fact, I’ve met many women who are successful in mathematics and are highly skeptical of a math club for girls. They think, “I didn’t need a math club for girls!” That was true for them, but these women are exceptions and not the rule. They comprise a distinct minority. A survey by the American Mathematical Society in 2004 found that less than 6% of tenured faculty at 10 top math departments in the United States were women.

That gender disparity is so pronounced that it begs for explanation. Indeed, many scholars research this disparity and there are few, if any, definitive answers. At Girls’ Angle, we believe that boys and girls are equally capable of doing mathematics at the highest levels, and that the way math is currently taught in the US at the K12 level tends to favor boys.

One way in which this preference for boys manifests itself is in cultural messages. Even just last year, a T-shirt company marketed “Allergic to Algebra” shirts just for girls. A few weeks before that, another girls’ shirt read, “I’m too pretty to do my homework, so my brother has to do it for me.”

Another way is that it is hard for girls to find women role models in mathematics. A girl can graduate high school never once having met a woman who uses math in her professional work. In fact, in the US, there is evidence that girls’ math anxiety can sometimes be traced to having a female teacher with math anxiety. At Girls’ Angle, girls work directly with women mentors who love math and understand it deeply at their respective level of the math career path, and we have mentors at all levels, from undergraduate, to graduate, to postdoctoral. Girls also meet professional women who use math in their work in a vital way through our Support Network. These are women like Dr. Elissa Ozanne, who uses statistics to help women afflicted with breast cancer, Dr. Pardis Sabeti, who uses statistics to probe the human genome, and Dr. Karen Willcox, who uses math to design better aircraft.

There are also differences in modes of instruction. Again, acknowledging the ever present exception, roughly speaking, especially in grades 5-10, competitive events resonate more with boys than with girls. Yet, the US math educational landscape is flooded with math competitions. There are dozens upon dozens of math competitions, and many of them test for characteristics that are unimportant in mathematics, yet whose results, nevertheless, strongly influence participant’s self-perception of mathematical ability. At Girls’ Angle, we develop new modes of math instruction that are particularly effective for girls’ math education. Of course, there is great variety among girls, so one cannot point to a single such mode of instruction that works for all girls, but a good example of the kind of thing I’m referring to is our Math Treasure Hunt or Math Collaboration. The basic scenario for such events is that participants are thrown into a predicament from which they must extricate themselves by solving a collection of math problems. Because the participants all win together or lose together, there is no incentive to withhold observations or ideas from each other. We’ve run about 20 such events so far, including a large one, called “SUMiT,” in conjunction with MIT’s Undergraduate Society of Women in Mathematics. Of these, one was a co-ed event and time will tell if it was merely a coincidence that the co-ed event was the only one where there was a scuffle between two participants. Two boys fought over one of the math problems.

ICMAT: What have been your best experiences?

Running Girls’ Angle has given me many experiences that I’ll forever cherish, but my favorite moments tend to be those where girls are learning a lot of math and laughing at the same time.

A good example of that happened soon after the club opened for the first time. We had the girls write, in as much detail as they could, an algorithm for how to eat a banana. The purpose of this exercise is to gain awareness of assumptions as well as provide an exercise in precision communication. When the girls were finished with their algorithms, a mentor would read the algorithm to me and I would perform the algorithm verbatim. Normally, I end up with a lot of banana skin in my mouth, so if anyone reads this and decides to try it, I highly recommend using organic, thoroughly washed bananas. Some girls laughed so hard they started rolling on the floor when they saw the effect of gaps in their algorithm. (I was recently filmed doing this at the recent Math Prize for Girls Games Night.)

Every Math Treasure Hunt that we’ve run affords many such moments. At the second one we did at the club, the girls had to find the combinations to two locks by solving math problems. After much good hard work and with just a few minutes to spare, they thought they had gotten the combinations and rushed over to the locks. When both locks clicked open, a huge, well-deserved, cheer of celebration erupted.

There have also been many amusing moments that don’t directly involve our members. For example, at the 2010 FIRST Robotics Conference in New York City, we hosted a small raffle contest using math problems that were either created or solved by our members. Toward the end of the event, a man who said he was a math major in college came up to our booth asking us how to solve one of the problems and I was able to hand him an article written by one of our 12 year old members showing how she solved it. (If any of your readers are interested in trying this problem, it is to find all solutions to the equation:

.

For our member’s solution, please see page 29 of Volume 3, Number 3 of the Girls’ Angle Bulletin.)

I also love it when a girl arrives at a meet eager to show us something mathematical that she did in between meets. At the club, we don’t have mandatory homework and part of the mentor’s charge is to try to inspire members so much that they will go home and do math on their own initiative. When that happens, it feels like the club has been a success.

The cover shows Red Sea Pearls by Radmila Sazdanović.

The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

The cover features Red Sea Pearls, by Radmila Sazanović, a postdoctoral fellow in mathematics at the University of Pennsylvania.  You can also check out the WIM video she did presenting an interesting geometric problem.  We continue to try to encourage people to subscribe to our print version, so we have removed most of this interview, as well as Part 2 of the Stable Marriage Algorithm and Coach Barb’s Corner from the electronic version of the Bulletin.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.

In this interview, Radmila makes the strong case that mathematics is an art.  It takes a lot of creativity to invent new math.  Unfortunately, math seems to have a reputation for being cut and dry.  To help counter this, toward the end of this issue, there is a set of 8 problems collected under the title Invent!  These problems are math problems where two people are very likely to come up with completely different, yet valid solutions.  Each problem admits multiple solutions.

Detail from illustration by Julia Zimmerman
for the Stable Marriage Algorithm Part 2

Julia Zimmerman magnificently illustrates the Stable Marriage Algorithm using a comic script written by Larry Guth, Grace Lyo, Amy Pasternak, Elizabeth Wood, and Julia Zimmerman.  They took the example Emily Riehl used in Part 1 of this series and turned it into a story full of romance, intrigue, and back-stabbing! Keep an eye out for Part 3 in the next issue.  This work and Robert Donley’s series on Fermat’s little theorem are supported in part by a generous grant from MathWorks.

A cone inscribed in a square pyramid.

Taotao and I write about basic properties of cones in Math in Your World and a separate article on the volume of a cone.  This story touches on curvature, Cavalieri’s principle, calculus, and Hilbert’s 3rd problem and can serve as a launch point for investigating each of these topics in greater depth.

Robert Donley, aka Math Doctor Bob, delves more deeply into Fermat’s little theorem, providing a non-inductive proof of the result.  (In the previous issue, he gave an inductive proof utilizing the binomial theorem and properties of Pascal’s triangle.)

Coach Barb’s inimitable character 3/7s returns to lay down the wisdom on average speed.

Also, if you like cracking secret codes, there’s one for you at the back of Notes from the Club.  Good luck and Happy New Year!