Head Mentoring at Girls’ Angle

Thanks to a grant from the Mathenaeum Foundation, Girls’ Angle is now in the process of hiring a full-time Head Mentor for a minimum of two-years. We are looking for a mathematician who loves to work with K12 students, especially girls.

To give prospective applicants a better idea about the position, the following describes in more depth the educational philosophy of Girls’ Angle and explains the critical role of the Head Mentor.

Girls’ Angle’s approach to math education

We believe that everyone benefits by studying more math, not just those with a special interest in math. Mathematics is a fabulous vehicle for improving one’s ability to think and solve problems, for no other subject shows up errors in reasoning so well. And the best way to obtain these benefits of studying math is to do math.

Consequently, Girls’ Angle welcomes all girls, currently in grades 5-12, to our club. Our members have diverse relationships to math. Some joined because they love math and can’t get enough of it. Others joined because they feel weak in math but would like to improve. And, yes, there are members who hate math and attend only because their parents want them to. For all these members, we aim to provide a safe, friendly, comfortable environment where they can feel at ease and not self-conscious, so they can focus on math without distraction.

To ensure the best experience for such diversity, our mentors need to be flexible, because what works for one girl may fail miserably for another. Some girls are perfectly happy to be given a challenging math problem and then given lots of space to think on it. Other girls need more guidance. Some members are motivated by competition, others by collaboration. Some members gravitate toward the concrete, while others revel in abstractions. Members have diverse personalities and hold a variety of different interests. Consequently, the stumbling blocks each member naturally encounters in the process of studying math are unique and fascinating. It’s a rich tapestry that also changes with time, even for the same member.

The role of the Head Mentor

Our Head Mentor is responsible for sorting out all this diversity and figuring out what would be best for each member, as an individual, to grow in thinking ability, problem-solving ability, mathematical knowledge, and mathematical understanding as effectively as possible. It is a big job, but it is extremely rewarding, and the Head Mentor has a lot of tools at her disposal to accomplish this task.

The first and most important tool is our group of super mentors, the heart and soul of Girls’ Angle. From the beginning, Girls’ Angle has been blessed with fabulous mentors who are excellent role models. They range from undergraduates majoring in math and related fields, to graduate students in math and related fields, to postdocs in math – each of them possesses strong fundamentals for their respective level in academia. The Head Mentor recruits, coordinates, and works with the mentors to deliver the highest quality math education we can muster.

Second, there is the enormous breadth and depth of mathematics itself. In our view, what is more important than the specific math being studied is to study math in the first place. Rather than insist that a member learn a particular piece of math, we prefer to help a member find some aspects of or problems in mathematics that resonate with her. The beauty of this approach is that mathematics is highly interconnected so that if a person gets hooked on some nice piece of mathematics (and it could be something that is never taught in the standard curriculum), it won’t be long before they branch out and pick up all the standard material. So we have all these members each navigating a unique path through the world of mathematics under the guidance of our mentors, who are, in turn, all coordinated by our Head Mentor.

Should a member become serious about mathematics and begin to contemplate making mathematics a profession, then it does become important for her to develop the discipline to learn important material that may not immediately appeal. When and how to develop this discipline is another matter that the Head Mentor must sort out. Ideally, the student’s own desires provide sufficient motivation to put the nose to the grindstone, but there certainly can be a region of transition.

Why do we need a mathematician for the Head Mentor position?

This is an important question, and one that is not easy to answer completely in a blog post because there are multiple reasons.

Members are a diverse group and represent many different stages in mathematical development. While it is not necessarily immediately appropriate for all members, one of the ideals we aim for is to help members develop into independent and capable thinkers who can solve the yet unsolved. We hope that members who spend some years with us are equipped with the tools and attitude to go into this world and contribute to the solutions to hard problems that have so far stymied us. Solving the unsolved requires creativity, persistence, and an ability to handle psychologically trying conditions. Having tackled unsolved problems and succeeded in creating new mathematics, mathematicians possess these qualities, and because we aim to impart these qualities to our members, we need a Head Mentor who possesses them and knows how to convey them, as well as help and/or facilitate our mentors to do the same.

Often, members don’t yet possess the vocabulary or language skills needed to express their thoughts well, but they do have precious thoughts. Our mentors have to have a radar for member thoughts, however ill-formed they may be, and be able to encourage them to pursue those thoughts. It may begin with helping a member to sculpt her thoughts into something mathematically actionable, or helping them learn to break a question down into tractable pieces, etc. A mathematician is practiced in this art. What we do not want at Girls’ Angle is for a member to have the inkling of an idea, try to express it, but then somehow lose that thought in the wind.

In fact, this also explains why we need excellent mentors for all our members, whether they excel at math or are floundering. It often takes a great deal of mathematical insight to figure out what a struggling member is having trouble with and how to help the member find a more effective and productive path; and it is so important for struggling members to get fundamentally sound guidance. When a member is bewildered, it does not help for her to have to deal with added layers of confusion created by poor instruction. A mathematician has thought about math to an unusual depth, and with that depth comes greater perspective for the relative importance of various concepts, an understanding of which descriptions of ideas have more generality than others, a knowledge of ideas that may seem expedient but lead to long-term confusion, etc. Our Head Mentor marshals her understanding of math to help our mentors help struggling members in ways that give them the best chance of future success.

At the same time, we need a Head Mentor who knows how to help our members turn their thoughts into theorems. As a concrete example, last year, Allie (we use pseudonyms, which are set in bold font, to refer to our members in public), one of our 6th graders, was playing a dice game we call “Cliffhanger”. This involves the rolling of dice and using the numbers that come up to create an arithmetic expression that evaluates to, or comes close to, a target number. In one round, the dice came up 1, 1, 1, 1, 2, while the target was 34. Allie exclaimed, “There’s no way we’ll be able to make a number that big with all those ones!” Instead of giving hints, the mentor said, “Well, try your best!” While contemplating the challenge, Allie muttered, “What’s the biggest number I can make?” Now there is a mathematical opportunity that must not be ignored! Allie had come up with a mathematical question that is loaded with mathematical potential. So we quickly responded, “Hey, I love that question! I can’t wait to hear your answer.” With encouragement, Allie persisted. From this, Allie managed to formulate the following question: What is the largest number you can express using N ones, addition, multiplication, and parentheses? That became her project for the better part of last year. By mid-October, she had a clear conjecture, but was having great difficulty proving it. Since induction was strongly implicated and induction is an important mathematical proof technique, we took a detour with her and taught her the concept and technique of induction using standard examples and exercises unrelated to her conjecture (such as using induction to prove formulas for the sum of the first n perfect squares, or proving the arithmetic-geometric mean inequality). She eventually had a great handle on the technique and returned to her conjecture. About six meet hours later, she was able to produce a clean, well-organized, detailed proof, which you can read in our magazine, the Girls’ Angle Bulletin, Volume 11, Number 4, starting on page 22.

There are many other such examples. Math is alive at Girls’ Angle. Mathematics is not a fossilized subject. It is a creative, conceptual art, and our Head Mentor must not only be an ambassador for this art, she must know how to practice it herself and be able to pass this art on to the next generation in an effective way.

 

 

 

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Girls’ Angle Bulletin, Volume 11, Number 6

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

It’s always nice to hear from old friends. Katy Cook née Bold was the first author for Math In Your World, a column about applied math. She contributed 8 installments before turning over the reins to Katherine Sanden. The cover features a rug that Katy designed together with Sara Eizen. What’s its mathematical significance? If you’ve made a math-themed carpet, we’d love to see pics. If we get enough, we’ll devote an installment of Math Buffet to them.

Rachel Pries gives us a wonderful interview for this issue. Rachel is a Professor of Mathematics at Colorado State University and received her doctoral degree in mathematics from the University of Pennsylvania under the supervision of David Harbater. Also, she is an alumna of Cambridge Rindge and Latin, which is just down the road from Girls’ Angle.

Next up, the fourth installment in Deanna Needell‘s series The Needell in the Haystack, about big data algorithms. Big data generally means high dimensions. For example, a very small 100 pixel by 100 pixel image can be thought of as a vector in a 10,000-dimensional space. In this installment, Deanna explains an unintuitive feature of high-dimensional spheres.

Addie Summer gives us an example of how she entertains herself in idle moments with mathematics in Systematic Counting, Part 1.

Then, we follow Emily and Jasmine on their continuing exploration of stacked circles. This time, their on a quest for stacked circles whose radii form a harmonic progression, but they run into a massive snag.

We conclude with solutions to the four Summer Fun problem sets of the previous issue. The best way to learn math is to do math, and tackling math problems is a good way to start doing math. But also, tackling math problems is a good way to develop your problem-solving skills. There is so much more to math problem-solving than being familiar with mathematics. I have seen students who know all the math involved with a particular problem, yet fail to solve the problem because of organizational issues. There are also psychological factors to learn to deal with. I don’t know how many times I’ve seen, and been victim to, an unwillingness to explicitly work out the simplest case of something, thinking that it is too trivial to pay attention to, and have that turn out to be a key obstacle. Once the reluctance is overcome and the seemingly trivial example is carefully worked out, the solution to the rest suddenly appears.

Special thanks to our four 2018 Summer Fun contributors, Matthew de Courcy-Ireland, Laura Pierson, Whitney Souery, and Vicky Xu!

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some 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!

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Girls’ Angle Bulletin, Volume 11, Number 5

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

This issues’ interview is with Duke University Assistant Research Professor Heekyoung Hahn. Prof. Hahn had a remarkable journey from a rural farm in South Korea to mathematician. For most of her student years she would walk an hour each way to school. She had no toys. Instead, she would spend hours playing with math. Today, not only does Prof. Hahn play with math as a professional researcher, she also created a summer program for high school girls called SWiM.

Deanna Needell’s third installment to her column The Needell in the Haystack is about classification algorithms. How does an ATM machine read the checks you deposit into it to determine the correct amount of the deposit? It must look at the often handwritten digits and decide which digit each is. That is an example of a classification algorithm. In this case, written symbols are classified as various numerical digits. She concludes by describing some of the latest work she’s done with her colleagues on the topic. These days, the importance of such algorithms cannot be overstated.

Anna solves the math problem she began investigating two issues ago: Reduce Pascal’s triangle modulo 3 and interpret each row as a ternary number. In this way, one obtains a sequence that begins 1, 4, 16, 28, 112, 448, etc. How many of the first 2018 of these numbers are odd? This was one of the problems at this year’s SUMIT 2018.

Emily and Jasmine figure out a nice shape into which circles with radii in arithmetic progression can neatly be stacked. For a hint as to what this shape is, take a look at this issue’s cover.

Our June/July issue also contains this year’s batch of Summer Fun problem sets. As Prof. Hahn said in her interview, “Math is not something to be afraid of. Rather, it is something that you should enjoy playing with.” The Summer Fun problem sets provide an opportunity for you to play with math. This year, we have contributions from Texas A&M undergraduate Whitney Souery who is currently enrolled in Harvard’s Neonatology Summer Student Research Program at Boston Children’s Hospital and Beth Israel Deaconess Medical Center, two Harvard undergraduates, Laura Pierson and Vicky Xu, and one from newly minted Princeton PhD Matthew de Courcy-Ireland. They provide a wealth of problems to start playing with math on topics ranging from generating functions, step functions, probability, and Markoff triples. As with all Summer Fun problem sets, all members and subscribers to the Bulletin are welcome to send in their solutions and comments!

Also in this issue, a Meditate to the Math on Napoleon’s theorem and some Notes from the Club featuring the second end-of-session Math Collaboration created by mentors at Girls’ Angle, this one made by Northeastern graduate student Jacqueline Garrahan and MIT undergraduate Elise McCormack.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some 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!

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Girls’ Angle Bulletin, Volume 11, Number 4

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

Betsy Stovall, assistant professor of mathematics at the University of Wisconsin-Madison, kicks off Volume 11, Number 4 with a wonderful interview. Her story of how she became interested in math is inspiring. She also describes one of the nifty problems that sparked her interest in math when she was a student. If you haven’t seen it before, it’s well worth thinking about.

Deanna Needell adds a second installment to her column The Needell in the Haystack, this time about filling in missing entries of a matrix. She describes an application to the study of Lyme disease. To fully understand Prof. Needell’s articles, one does have to be familiar with matrices. For those of you who don’t know about matrices, we included a very brief introduction to matrices intended to help reader’s unacquainted with the concept at least feel good enough about them to read Prof. Needell’s Bulletin contributions. Her articles are very much worth reading even if you don’t know about matrices.

Emily and Jasmine embark on a new adventure resulting from a graphic design project themed on the circle. The assignment inspired Emily to create a design of radiating circles. To create the design, she discovered some neat math. This new journey was inspired (in a somewhat indirect way) by work of students at the Buckingham, Browne, and Nichols Middle School.

Anna makes progress on her current problem by characterizing those m for which {2m} \choose m is congruent to 1 modulo 3.

Lightning Factorial authors an installment of Math In Your World, analyzing how to intercept an incoming tennis shot as quickly as possible.

Next, Girls’ Angle member Allie presents her wonderful inductive proof of a conjecture she came up with at the Girls’ Angle club regarding the largest number expressible using N ones, addition, multiplication, and parentheses.

We round out the issue with some Notes from the Club.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some 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!

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Girls’ Angle Bulletin, Volume 11, Number 3

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

Can you figure out what the cover represents?

Volume 11, Number 3 kicks off with an interview with Rhonda Hughes, Professor Emeritus of Mathematics at Bryn Mawr College. In it, Prof. Hughes describes one of the most interesting paths into mathematics, and she gives lots of good advice to students.

Next up is the concluding half of Partitions from Mars, by Pamela E. Harris, Alexander Pankhurst, Cielo Perez, and Aesha Siddiqui. They conclude with an open problem about partition functions. Can you solve it?

Last issue, we interviewed UCLA Professor of Mathematics Deanna Needell. In this issue, Prof. Needell returns with a nifty article on topic modeling, explaining the technique of non-negative integer matrix factoring. Every day, people produce something on the order of 10^{19} bytes of data every day, and this rate is accelerating. But all this data is useless if we don’t have the tool to extract information from it. In The Needell in the Haystack, Prof. Needell shares with us her expertise and teaches us the basics of large-scale data analysis.

Anna’s Math Journal returns after a brief break to tackle a question about numbers formed by interpreting rows of Pascal’s triangle, modulo 3, as ternary numbers. Specifically, in the resulting sequence, which numbers are odd? Computing how many of the first 2018 of these numbers are odd was a problem at SUMIT 2018, a big math event we ran at the Broad Institute earlier this month.

In Errorbusters! we discuss a powerful way to minimize numerical computation errors: use variables. The use of variables delivers several benefits besides just reducing the number of dicey numerical computations. We illustrate by tackling problem 15 from the 2017 AIME I contest.

I’ve now been working with students on math for decades, and sometimes, I meet students who treat all facts equally, as if each were another isolated item to learn about this world we live in. But what is amazing about math is that some facts follow logically from others. Building connections between facts through implication enables us to understand a lot with little knowledge and helps us to retain facts because we can then understand them from many angles. To progress in mathematics, it is crucial to understand this. So I tried to address it directly with the article Implications.

We conclude with a Learn by Doing problem set on Fibonacci numbers, a note on the notation used for functions, and some notes from the club.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some 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!

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Happy New Year 2018!

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Girls’ Angle Bulletin, Volume 11, Number 2

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

You could call this issue an ode to the triangle because half the content pertains to that most basic of geometric shapes, including the cover.

However, we open with an interview with UCLA Professor of Mathematics Deanna Needell. One of Prof. Needell’s mathematical interests is about how to get accurate views of things with less information, such as getting useful MRI scans in less scan time.

Next up is a nifty way to introduce vectors via partition functions by a group from Williams College, led by Pamela E. Harris. They motivate the whole concept by imagining a fictitious martian currency which behaves just like our dollar, except that there’s more than one unit of currency and they cannot be interchanged. The setup inevitably leads them to begin computing what is known as Kostant’s partition function for A_3. In fact, the authors dedicate the article to the memory of Bertram Kostant who passed away earlier this year.

Inscribed equilateral hexagons in an isosceles right triangle.

Milena Harned and Miriam Rittenberg are back explaining some results they found and proved this past summer about inscribing equilateral hexagons in triangles in such a way that every side of the triangle is flush with at least one side of the hexagon. In fact, for every triangle, they found a continuous family of such equilateral hexagons and they “rotate” about inside the triangle. You can explore these inscribed equilateral hexagons using a JavaScript app on the Girls’ Angle website.

Girls’ Angle mentor Ashley Wang pays tribute to triangles through her Math Buffet.

After an installment of Meditate to the Math (which asks you to interpret Sierpinski’s triangle as a sequence of binary numbers and contemplate its patterns), we present a self-referential test created by four Girls’ Angle members: Ghost Inthehouse, HolAnnHerKat, Katnis Everdeen, and Shark Inthepool. Earlier this fall, they solved Jim Propp’s self-referential multiple choice test and enjoyed it so much, they wanted to create their own. Their work was primarily overseen by Girls’ Angle mentor Rachel Burns.

Emily and Jasmine’s epic search for nice triangles comes to a heady conclusion. They succeed in putting all the pieces of the puzzle together to get a fairly complete classification of triangles with various numbers of integer side lengths and angles that measure rational multiples of \pi. Their journey took them through some lattice combinatorics, rational parameterizations of curves, and some basic Galois theory and a study of the values of cyclotomic polynomials at the square root of -1. And it all began with a fake triangle.

We conclude with some notes from the club.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some 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!

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