Starting grad school has been a bit of a roller coaster and everyone seems to say that the first year is the hardest. So far, I still enjoy showing up everyday so here is a list of advice I have received and have tried to incorporate into my lifestyle that might help if you find yourself overwhelmed.
If the first two answers are “no” and the last question doesn’t have a clear or good answer, she stops doing it. The key here is to be realistic with the things you spend your energy on to be sure you’re spending it where it needs to go.
I am still working on taking my own advice but these are the things I look at when I’m overwhelmed. Feel free to comment any other suggestions or strategies that you find useful!
]]>Saturday was the Pi Day of a lifetime – 3/14/15 9:26:53! Hopefully all of you were able to share some delicious pie with your friends and family. When slicing up your pie how can you make sure everyone received a fair share? It’s the LoneDivider Method to the rescue!
The LoneDivider Method for N players dividing a pie can be described by the following process:
Preliminaries – As the name suggests, one of the players is chosen to be the divider, D, while the other N1 players act as choosers. It’s always better to be one of the choosers than to be the divider – the divider is guaranteed a piece worth exactly of the total value of the pie whereas the choosers have a chance to get a piece worth more than of the total value. Since we want this method to be fair to all players, everyone should have an equal chance of being the divider (which can be done randomly).
Step 1 (Division) – The divider, D, divides the pie into N pieces (). The divider knows that he/she will receive one of these pieces, but at this point does not know which one. This is important because it forces D to divide the cake into N slices that to him/her have equal value.
Step 2 (Bidding) – Each of the N1 choosers submits a bid that lists every piece he or she considers to be a fair share (i.e. worth or more of the total value of the pie). These bids are made secretly and independently of the other choosers.
Step 3 (Distribution) – The bid lists are opened. Depending on these lists, there are two different ways we may have to proceed.
Case 1 – If there is a way to assign a different slice of pie to each of the N1 choosers, then that should be done. The divider (who equally prefers all of the pie slices)gets the last unassigned slice. At the end, the players may choose to swap pieces if they want.
Case 2 – There is a standoff! In this case, there are two choosers bidding for the same piece of pie (or K choosers bidding for fewer than K pieces). This case is complicated and we give a brief overview of how to proceed when a standoff occurs. First, we set aside any of the pieces of pie that were involved in the standoff. In addition, we temporarily separate any players involved in the standoff from the others. It is possible for multiple standoffs to occur and each standoff is handled individually. For ease of explanation, we assume only one standoff has occurred. Once the standoff players and pie slices are isolated, each of the remaining players (including the divisor) is assigned a fair share from the remaining pie pieces. All the remaining pie pieces (those involved in the standoff) are then recombined into a new “pie” S to be divided among the players involved in the standoff and the process starts all over again.
We will consider two examples to illustrate the two cases that can occur in the distribution step. Four friends, Abby, Jenny, Ellie, and Maddie, want to share a pie for pi day. Abby is randomly chosen as the divider and cuts the pie into four pieces ( and ). The following table shows how each of the friends values the four pieces.
s_{1} 
s_{2} 
s_{3} 
s_{4} 

Abby 
25% 
25% 
25% 
25% 
Jenny 
40% 
20% 
20% 
20% 
Ellie 
25% 
20% 
40% 
15% 
Maddie 
20% 
20% 
30% 
30% 
In this example, Jenny’s bid list is {} only, Ellie’s bid list is {}, and Maddie’s bid list is {}. It is clear that Jenny must get – there is no other option. This forces the rest of the distribution. Piece must then go to Ellie and piece goes to Maddie. Finally, the remaining piece of pie, goes to the divider, Abby.
In our next example, we see what happens when a standoff occurs. Again our four friends, Abby, Jenny, Ellie, and Maddie, want to share a pie for pi day, with Abby being named the divider. Abby cuts the pie into four pieces ( and ). The following table shows how each of the friends values the four pieces.
s_{1} 
s_{2} 
s_{3} 
s_{4} 

Abby 
25% 
25% 
25% 
25% 
Jenny 
40% 
20% 
20% 
20% 
Ellie 
35% 
20% 
22% 
23% 
Maddie 
20% 
20% 
35% 
25% 
In this example, Jenny’s bid list is {} only, Ellie’s bid list is {} only, and Maddie’s bid list is {}. It is clear that we have a standoff – both Jenny and Ellie want piece ! The first step of the distribution process is to set piece aside and assign Abby and Maddie a fair share from and . Notice that Maddie could be given either piece or . She would prefer to have , but that’s not for her to decide at this point – a coin toss can be used to determine which piece Maddie receives. Let’s say Maddie ends up with . Then Abby could be given either piece or . Another coin toss determines that Abby receives piece . Now we must determine what Jenny and Ellie receive. At this point, we recombine the remaining pie pieces ( and ) into a single piece to be divided between Jenny and Ellie using the dividerchooser method. Since is worth 60% to Jenny and 58% to Ellie, regardless of how this final division plays out they are both guaranteed a final share worth more than 25% of the pie. When all is said and done, every player ends up with a fair share.
This method works equally well for cake and pizza (or any other continuously divisible items). So go out and enjoy, knowing you are now equipped with a method to fairly divide your pie, cake, and pizza! Happy Pi Day!
Sources:
Tannenbaum, Peter. Excursions in Modern Mathematics (Sixth Edition). Upper Saddle River, N.J: Pearson Prentice Hall, 2007.
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http://www.inquirybasedlearning.org/?page=conference
Call for Papers
The program committee invites abstracts for the 2015 Legacy of R.L. Moore/Inquiry Based Learning Conference, to be held June 25 – 27 in Austin, TX. We are especially interested in presentations which fit with this year’s theme: “Empowering with IBL,” and highlight how inquirybased learning gives students and instructors the space to realize their own talents.
Following the success of last year’s meeting, we encourage the submission of abstracts for ACTIVE contributed sessions. We hope that the longer (25 min) blocks will bring more handson, mindson sessions which are focused on the practices of IBL instruction. We recognize that not everything of value easily lends itself to an activity or workshop; all presentations will be considered, and presentations that lend themselves less well to handson activities for large groups should consider the academic poster session.
We welcome proposals concerning any facet of IBL teaching and learning, but we encourage submission of proposals to fit these general sessions:
Call for Posters
The conference planning committee has decided to emphasize poster sessions this year. Poster sessions are becoming increasingly popular as a means for disseminating information in a format that allows for greater interaction between the presenter and the audience. Conference attendees who would like to present a topic not on listed are highly encouraged to present a poster.
As usual, time will be reserved for 5 minute talks. The organizers encourage 5 minute talks about “How I got here” which give short personal histories with IBL.
In addition to presentations, the organizers are planning to set aside networking times for the following areas: for experienced users, for new users of IBL to share their experiences, for Math Circle practitioners, and for potential IBL SIGMAA members.
Abstracts received by Wednesday, March 18, 2015 will be given full consideration. Submissions after that date will be considered on the basis of space available.
]]>In September, 2012, Laura Zirbel wrote a post about her experience with using i>Clickers in the classroom. I used to be very against i>Clickers. Oftentimes I found they were used solely as a means of determining attendance and not as a tool to help students learn course material. I am teaching a combined precalculus and trigonometry course this semester and have incorporated i>Clickers into my lectures and am loving the results.
The class I teach is part of the EXCEL and COMPASS programs (loosely, programs designed to retain STEM majors and encourage students to declare a STEM major, respectively) at UCF. It meets MWF for 1 hour and 50 minutes each time. My 142 Students (and I) sometimes have difficulty remaining focused throughout this long period of time unless they have something to keep them actively engaged.
I use Beamer to create slideshows with a question and choices (I usually stick to AD, but sometimes have AE or even fewer options). Depending on the question, I may insert a \pause command into Beamer so that students cannot see the answer choices right away. Students are allowed and are encouraged to work together on these questions. This part gets the students talking to one another about math and helps them determine what they are and are not understanding. After polling has closed I show how many students chose which answer. I try to choose the wrong choices I give the students so that I can see what mistakes students are making (this can be tough at times). I then show the correct answer and the next slide has an explanation of why that is the correct answer. If a majority of students answer the question incorrectly, I make sure to go through the explanation in detail. If the majority answer it correctly, I briefly talk through the explanation and then leave it for the students to go through it after I have posted it on Canvas for them to view. (An example from Chapter 2 of my course can be viewed here.)
My students who are already declared STEM majors take a STEM seminar class. Part of this class is they have reviews for their math class the week of or the week before their exam. We also have an in class review before each exam. I conduct these reviews solely with i>Clicker questions. This allows the students extra time to practice problems. It also gives me an idea of the types of material with which they are struggling. While students are working through the problems, I walk around the classroom and listen to their discussions about the problems. This gives me time to ask individual students or groups of students why they are answering questions the way they do, giving me a better idea of their degree of understanding.
The i>Clicker grades are 5% of the final course grade. Each question is worth 2 points. Students receive 1 point for answering the question and another point for answering the question correctly. At the end of each class, I just have to go into the iGrader software and click which answer is correct for each question. It also assigns 1 participation point to students who answered a certain percentage of questions during that class period (you can choose what percentage you want). UCF has i>Clicker integrated into Canvas, so then I just have to click a few buttons and the grades are automatically imported into Canvas for students to see. One thing I found early in the semester is that some (maybe 2 or 3) would just click whatever button without thinking at all (e.g. they clicked E when E was not even an answer choice). The iGrader software will still give a point for all choices AE unless you tell it otherwise. After I told students that selecting an answer that was not a choice would give them no credit for that question, that situation stopped happening.
About 75% of my students already had an i>Clicker for another class. This prompted me to allow the i>Clicker GO application (seen in the photo at the top of this post) in my classroom. It allows students to answer the questions using their phone, tablet, laptop, etc., at a lower cost than having to purchase an actual remote. One of my worries was, “will students abuse this freedom to use their devices in my class?”. From my experience, it has had little to no change on the number of students who are texting, Facebooking, etc. in class.
There is one thing I have found that I wish was better about the i>Clicker system. In iGrader, I have not been able to easily see how each student responded to each question. It shows me the grades students have for each class period. I can easily see which questions were asked and what were the correct answers and the class percentages. However, sometimes I want to look at individual student responses. This may be a feature already, but I have not been able to find it.
Are you using i>Clickers in your classroom? What have you found about them to be effective and to be not effective? What is your grading policy? What are your thoughts regarding allowing i>Clicker GO to be used in the classroom?
]]>A quick Google search on mathematics metacognition returns more than 300,000 results. What is metacognition and why should we care about it? The MerriamWebster 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 selfreflection? 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:
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 wellknown 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 selfproclaimed “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 John 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 MPSGRSVs 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!
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