The Role of Generalization in Advanced Mathematical Thinking

Generalization and abstraction both play an important role in the minds of mathematics students as they study higher-level concepts. In the second chapter of the Springer book Advanced Mathematical Thinking, Tommy Dreyfus defines generalization as the derivation or induction from something particular to something general by looking at the common things and expanding their domains of validity. As we teach our own math courses, we can look out for opportunities to introduce generalization and abstraction in order to help our students better understand the pattern behind what they are learning. Continue reading “The Role of Generalization in Advanced Mathematical Thinking” »

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False Positive vs. False Negative

A guest post by Long Nguyen:

Not all results of medical tests are absolutely correct. Scientists can make mistakes when they conclude that something is true when it is actually false or that something is false when it is actually true. False-Positive2When something is concluded true and it is actually false, we have a false positive or type I error. On the other hand, when something is false and it is actually true, we have a false negative or type II error.

Should we really be worried about a positive medical test for a rare disease?

Mammography is one way to detect breast cancer at its early stage, which helps patients to increase their survivalBN-HB837_HJourn_F_20150223124818 rates. In 2009, John Allen Paulos, a mathematics professor at Temple University, wrote an article, Manmogram Math, to discuss how frequent women should have their mammograms. If mammograms could help diagnose breast cancer, why wouldn’t we have them more frequently? According to Paulos, it is controversial whether women should have mammograms monthly since the tests could cause harmful effects resulting from radiation. Also, suppose you have a positive test, is this absolutely true that you have cancer? The answer is no. Continue reading “False Positive vs. False Negative” »

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Creating a Successful Learning Environment

Welcome to graduate school. Here, we will find ourselves immersed in a subject that inspires us. Here, we will become engaged in conversation with masters of the field. Here, we will be mentored into a community to which we will dedicate ourselves. Of course, there is a learning curve along the way. In most respects, we expect this. There will be an increase in work time and confusion. There will be difficult times in which we feel like there are not enough hours in the day. It will feel a lot like running a marathon with shoelaces tied together. We will feel inadequate, helpless, and isolated, but hey, it’s worth it. This is what graduate school is all about, right?

Having come to graduate school after teaching at a progressive secondary school, I questioned if it really had to feel this way. I moved out of the position of teacher—helping my students to feel positive about their progress and empowering them to actively participate in their education—and into the position of struggling student looking for assistance. It was during my reflection on the first semester that I began to give myself the same advice that I would give my students and take a more active stance in my education through self-advocacy. Continue reading “Creating a Successful Learning Environment” »

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Free Money: July 25th Deadline for AMS Travel Grants

As much as we might want summer to last forever, the Fall semester will be upon us before we know it! Luckily, with it comes a new batch of fantastic math networking opportunities at the four upcoming AMS Sectional Meetings:

The Sectionals are conveniently scheduled around weekends to minimize overlap with teaching and coursework. We enthusiastically encourage you to browse the program for each meeting (by clicking the four links above) and consider attending the AMS Sectional Meeting that is most appealing to you.

Best of all, the AMS is providing some $250 travel grants for graduate students (click the link for eligibility details). The application deadline is July 25th, 2016 at 11:59pm EDT (East Coast time).

Applications are only accepted through the link given below, and application results are given out nice and quickly by August 19th. The application is simple (no reference letters!) and well worth a little bit of your time. Remember: if you spend, say, half an hour filling out the application and you receive the award, then that’s equivalent to earning $500 an hour. Not bad!

Click here to apply for the AMS grad student travel grant for the Fall semester Sectional Meetings Continue reading “Free Money: July 25th Deadline for AMS Travel Grants” »

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Mathematics and Dance: Enhancing Understanding Through Movement

A guest post by Jenna Jensen:

Jenna Jensen_925513_assignsubmission_file_jensen math danceErik Stern and Karl Schaefer discuss the cross-curricular possibilities with math and the art of movement and dance in their video, Math Dance. I think it is appropriate to ask in what ways can art benefit the math classroom? Is dance the only method that we should consider? What benefits do students gain from movement in the classroom? Not only am I on track to become a math teacher at the secondary level, but I also am a dancer. How perfect is it to come across a Ted talk that addresses two of my greatest passions in life? Continue reading “Mathematics and Dance: Enhancing Understanding Through Movement” »

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