A quick Google search on mathematics metacognition returns more than 300,000 results. What is metacognition and why should we care about it? The Merriam-Webster dictionary defines it as

Awareness or analysis of one’s own learning or thinking processes.

I find that often, students are not processing the information being given to them in useful ways. For instance, when learning a new mathematical concept, many students just memorize the concept. However, reflecting upon the foundation of that concept will serve them better by providing them with a framework for the material.

Perhaps this is because students are not taught to reflect upon their learning and thinking enough. If this is the case, how can we encourage students to do this self-reflection? I am using Dr. Cynthia Young’s *Precalculus with Limits* textbook this semester. One of the features I really like about the book is that it lists objectives at the beginning of each chapter and each section. This makes it quite easy for students to go through the list and reflect upon their understanding of the core ideas covered in each section.

I have started trying a new strategy myself (I believe it is called the Cornell style). I take all of my notes on my iPad. I asked one of my friends to create the template to the left for me. I take my notes in the space between the two blue lines and label in the margin if something is a theorem, definition, example, etc. Later, I go back and summarize what is on the page in the five lines below the bolded blue line.

What do you do to reflect on your learning and thinking? Do you have strategies for getting your students to practice metacognition in and outside of the classroom?

]]>I always enjoy seeing mathematics in pop culture. When a friend of mine sent me a txt message about an article talking about celebrities who enjoyed mathematics, I was all over it. (Tangent – The title of the article, *10 Secret Celebrity Math Geeks, *is a bit indicative of the societal attitude towards mathematicians. It is sad that folks who enjoy math need to be “secret.”) The celebrities included on this list were:

- Danica McKellar (The Wonder Years)
- Mayim Bialik (Blossom)
- Lisa Kudrow (Friends)
- Huey Lewis (Musician)
- Montel Williams (Talk show host)
- Terrence Howard (Hustle & Flow, Iron Man)
- Dr. Dan Grimaldi (The Sopranos)
- Art Garfunkel (Musician)
- Cindy Crawford (Actress and supermodel)
- Tom Hands (Actor)

There are other lists around the internet of celebrities who have excelled in STEM fields as well (a quick Google search will give you many results). I wish this was more well-known to fans, especially younger students who are interested in STEM areas but afraid of being socially outcast due to their “geeky” interests. There is a shortage of females in the STEM fields and many theories on why this may be. (The diversity of practitioners in STEM fields will be saved for another post.)

My wish, though, is that students of all ages, genders, races, etc., who are interested in STEM fields, will set aside the stigma and explore the topics that truly interest them. I also wish that celebrities who are interested in these fields will publicly “make a big deal” out of what is happening in these fields. Maybe this will help with people understanding, it is okay to be a little geeky – study what you enjoy.

]]>Logarithms – this mathematical staple is celebrating its 400^{th} birthday this year. But how much do you know about the development of logarithms or the man behind them? Edinburgh born John Napier, the inventor of logarithms, is in danger of fading into the shadows of the scientific landscape. In the new book *John Napier: Life, Logarithms, and Legacy*, Julian Havil does a marvelous job of bringing Napier back into the spotlight.

Havil’s book is a self-proclaimed “scientific biography” – biography because it describes the life of Napier and scientific because it describes his work. Though the book focuses more on Napier’s scientific innovations, the first two chapters are dedicated to his life and his analysis of the Revelation. Not much is known about Napier’s life due to the fact that little about it was recorded and many of those records were lost. Despite this, Havil is able provide a basic sketch of Napier’s life, using historical facts to add perspective to the narrative. Though Napier is best known today for his development of logarithms, in his own time he was equally renowned for his scrutiny of the Revelation (the final Book of the New Testament). Napier’s work on the Revelation was immediately and widely embraced. Now a relic of history, Havil notes of Napier’s work on the Revelation:

Gone, but not entirely forgotten, it remains the epitome of its type providing the historian and theological scholar with important material on which to reflect.

The majority of the book focuses on Napier’s mathematical contributions, especially his development of logarithms. It is interesting to note that the logarithms Napier originally conceived are not the logarithms we know today. Havil presents Napier’s logarithms with enough detail to satisfy a mathematician without getting too bogged down in the details. Napier’s logarithms are not presented in a vacuum – Havil describes the motivating problems that drove Napier to develop logarithms, as well as how they were developed. In addition to logarithms, Napier’s Rods (or Bones) and his Promptuary are also presented.

The book ends with a survey of how Napier’s findings have influenced his successors. The most notable and famous of these is, of course, the development of the slide rule. However, Napier’s Rods and his Promptuary served as the inspiration for some analog computing devises. Napier also had a fruitful (if brief) collaboration with Henry Briggs, leading to the Briggsian logarithms and eventually the natural logarithms as they are known today.

If you’ve ever wanted to learn more about logarithms or the man who developed them, Havil’s book *J**ohn Napier: Life, Logarithms, and Legacy* is a great read. As Havil so aptly puts it, “John Napier deserves better than obscurity.”

The Directorate for Mathematical and Physical Sciences (MPS) at the National Science Foundation (NSF) is now accepting supplemental requests to support one (additional) Ph.D. student per award, as long as the graduate student is a United States Veteran. The proposed MPS-GRSVs will afford Veterans an opportunity to conduct research towards a doctoral degree with an NSF MPS Directorate active grantee.

There’s no dollar amount mentioned, but this one year award can be extended up to three years. Here’s to hoping your advisor has an NSF grant!

]]>We generally ask that editors post at least one time each month. However, if you don’t want to commit to writing something each month, you can still be a contributor to the blog.

You can email your posts or any questions you have to thomas.clark973@topper.wku.edu.

I look forward to hearing from you!

]]>I really like that it allows me to compile documents on the go whether I am connected to the internet or not. It also connects with Dropbox, so I can include all of my packages in a Dropbox folder and it will allow me to edit TeX documents I have saved in Dropbox. One issue I am having is I have not figure out how to select text in it yet (I think this might just be a glitch with iOS 8, though).

What TeX editors do you use on your iPad or other tablet? What do you like about them? What do you dislike about them?

]]>The method of markers is a fair division method which is used when

- There are more items to be divided than there are players in the game.
- The items are reasonably close in value.

The method (for N players and M discrete items) can be described by the following process:

**Preliminaries** – The items are lined up in a random order. For convenience, the items are labeled 1 through M, going from left to right.

**Step 1 (Bidding)** – Each player independently divides the array of items into N segments by placing N-1 markers along the array. These segments are assumed to represent the fair shares of the array in the opinion of that player. In order to keep the method fair, the players must all place their markers at the same time. One way of achieving this is if all players submit sealed envelopes with the positions of their markers. Everyone is guaranteed to get a fair share because the segments for a particular player never overlap.

**Step 2 (Allocations)** – Scan the line of items from left to right until the first *first marker* is located. The player owning that marker (let’s call him or her P1) goes first and receives the first segment in his or her bid. (If there is a tie, it is broken randomly.) Since P1 has received his/her fair share, the rest of his/her markers are removed. We continue scanning the line of items from left to right, looking for the first *second marker*. The player owning that marker (let’s call him or her P2) goes second and gets the second segment in his or her bid (which corresponds to the segment between his/her first and second markers). Continue this process, assigning one segment of his/her bid to each player. The last player gets the last segment in his/her bid, which is to the right of his/her last marker.

**Step 3 (Dividing Leftovers)** – If there are items left over after every player has received a fair share, divide the leftovers among the players by some form of lottery. For example, the players could take turns choosing leftover items or, if there are many more leftover items than players, the method of markers could be used again.

Consider the following example. Three children (Abby, Bryan, and Chloe) are dividing the array of nine candy pieces shown in the following figure using the method of markers. The players’ bids are indicated in the figure (with A for Abby, B for Bryan, and C for Chloe).

In this example, Abby would consider any one of these shares to be fair:

{1} (Abby’s 1^{st} segment)

{2, 3, 4, 5} (2^{nd} segment)

{6, 7, 8, 9} (3^{rd} segment)

We start at the left-hand side of the line of candy and look for the first *first marker* (in this case, it is A1). Abby will go first and will receive the first segment in her bid:

Notice that the rest of Abby’s markers have been removed since she has now received her fair share. We now continue scanning the line of items from left to right, looking for the first *second marker*. The next marker we come across is B1. However, since this is not a *second marker* we ignore it. The first *second marker* that we come across is B2. Bryan will go second and will receive the second segment in his bid:

Now we are down to the last player, Chloe. Chloe will get the last segment in her bid, which is to the right of her last marker (i.e. to the right of her C2 marker):

In this example, the final allocation of candy is: Abby gets piece 1, Bryan gets piece 3, Chloe gets pieces 8 and 9, and pieces 2, 4, 5, 6, and 7 are all leftovers (which can be divided among the children using some form of lottery).

Despite its elegance, the method of markers can be used only under some fairly restrictive conditions. In particular, the method works best if the items are roughly equivalent in value to each other and if the players’ preferences are fairly close. This is almost impossible to accomplish when there is a combination of expensive and inexpensive items, but is perfect for dividing items that are of small and equal value (like candy). So go out and enjoy, knowing you are now equipped with a method to fairly split the spoils from trick-or-treating. Happy Halloween!

Sources:

Tannenbaum, Peter. *Excursions in Modern Mathematics (Sixth Edition)*. Upper Saddle River, N.J: Pearson Prentice Hall, 2007.

In the wake of mathematical enlightenment a profound understanding of basic notions bridges the gap between the conceptual and concrete. In many cases, problems that have an exterior of simplicity exploit the boundaries of comprehension and provide insight into extensive associations. From the mind-stretching inclinations of geometry and algebra emerges the intricate framework from which these connections form. Piece by piece, generalizations are built from the material of empirical understanding fabricated by the process of asking intrinsic questions.

Questions of this nature are entwined in reticent patterns found across the full spectrum of mathematics. Many of these inquiries encompass and ascertain the properties of special functions. Mappings such as Euler’s Totient function provide a strong basis for further investigation into characteristics of positive integers. Specifically, this function denoted by counts the number of positive integers less than or equal to a positive integer such that the positive integers counted and have only 1 as their common divisor (in other words they are relatively prime to , denoted such that ). In example, = because there are two positive integers less than or equal to 3 that are relatively prime to 3 (namely 1 and 2, given ). Euler’s Totient function is distinguished by several other properties as well. For instance, if is a prime number, then . It is also multiplicative, in the sense that if then = . By virtue of these attributes, several open problems in the field of number theory involve .

The mathematician Robert Carmichael proposed one such conundrum in 1907 that still remains unsolved. Basically, Carmichael conjectured that for every positive integer there exists a positive integer such that and . As a consequence, with the given properties, the conjecture is certainly true for odd numbers. This can be seen by letting be a positive odd integer and in the fact that = , which it follows

= = .

However, as easily proved as the conjecture is for the positive odd integers, the statement has not been shown true for the positive even integers. Maybe a clever argument will come from a thorough investigation of basic notions. Perhaps, rather, it will be stumbled upon in search of greater abstractions. Whatever the case of discovery may be, a resolution will certainly be achieved, if at all, by asking insightful questions.

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1. Go watch Prof. Mahajan’s teaching course. Take notes. Seriously. I cannot stress enough the excellence of the advice he dispenses. I have returned to these videos many times in the past four years.

2. Be flexible. It’s easiest to manage a course or recitation when you maintain a mental model of the student’s learning. Prof. Mahajan discusses many techniques for building such a model which I have summarized in a previous post.

I almost derailed my vector calculus class in its first week. I assumed that because the students completed linear algebra, they would be comfortable reasoning about lines and planes in three dimensions. After receiving feedback from the students, I quickly realized that one of the goals of a vector calculus class is to develop spatial intuition. I corrected the mistake by devoting an extra lecture to the basic material, which brings me to my next point.

3. Less is more. If you are lecturing, you will cover one page of handwritten notes per ten minutes of class. Often less. Find your ratio and treat it as law.

4. Be clear and direct; doubly so when discussing course policies. Each character you write or figure you draw will be copied into 129 notebooks. Every quiz or exam will be read by 129 people. Your syllabus will be interpreted 129 different times. The exercises will be discussed in dozens of small study groups.A not insignificant fraction of students will attempt to bend a rule or ask for a more favorable interpretation of a grading policy. If you’re unsure how to respond, then don’t. Ask them to submit their case via email and sleep on it.

5. Work on your shtick. It sounds silly, but a gimmick can help break down the communication barriers which develop after twelve years of schooling. Some of the social norms students pick up are downright toxic to learning!

I learned about the importance of your shtick by observing a UCSB professor known for his elaborately choreographed, 850-student calculus lectures. But it didn’t really hit home until I went through my teaching evaluations with a fine tooth comb. My first time as an instructor, 88 students completed an evaluation. Eleven of them mentioned my beard or included a little picture of me, accentuating my beard. Ten of them talked about my effort to learn names. There were only a handful of comments with a greater “hit rate”.

I thought I was bad at learning names until I forced myself to try. Every time a student asks a question I either ask them their name or, if I think I know it, address them by name. The remainder of my system is based on two principles. First, I have a pretty good spatial memory. Second, students tend to sit in the same location every day. Give it a try!

6. When you begin lecturing, you will have to choose a side: slides versus chalk. I’ve written before about some of the issues with slides. For the record, out of those 88 evaluations, seven liked my use of the chalkboard and four suggested I use slides.

7. Spend a little bit of time reading research in mathematical education. At a minimum, this will give you a language to describe your existing habits in the classroom. This is the first step in improving or changing those behaviors.

8. Keep a journal. I’ve written before about my love of journaling; I view it as an integral part of the scientific method. Here’s a more literary take on keeping a diary. At a minimum, this record of your thoughts and experiments will provide great material for the teaching statement you will eventually have to write.

9. Be knowledgeable about your department. Know answers to common administrative questions and know where to direct students when you don’t know an answer. Inform students of the resources available to them. New students, especially, need to be reminded about office hours, drop-in tutoring opportunities and review sessions.

10. Use teaching to practice your public speaking. As a former software developer, I can tell you that this is an integral part of a professional career both in and outside of academia.

11. Go watch Prof. Mahajan’s videos. Seriously.

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