What is \(0^0\), and who decides, and why does it matter? Definitions in mathematics.

By Art Duval, Contributing Editor, University of Texas at El Paso

How is \(0^0\) defined? On one hand, we say \(x^0 = 1\) for all positive \(x\); on the other hand, we say \(0^y = 0\) for all positive \(y\). The French language has the Académie française to decide its arcane details. There is no equivalent for mathematics, so there is no one deciding once and for all what \(0^0\) equals, or if it even equals anything at all. But that doesn’t matter. While some definitions are so well-established (e.g., “polynomial”, “circle”, “prime number”, etc.) that altering them only causes confusion, in many situations we can define terms as we please, as long as we are clear and consistent.

Don’t get me wrong; the notion of mathematics as proceeding in a never-ending sequence of “definition-theorem-proof” is essential to our understanding of it, and to its rigorous foundations. My mathematical experience has trained me to ask, “What are the definitions?” before answering questions in (and sometimes out of) mathematics. Yet, while we tell students that the definition needs to come before the proof of the theorem, what students apparently hear is that the definition needs to come before the idea, as opposed to the definition coming from the idea.

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Illuminating skills learned from teaching

By: Mary Beisiegel, Oregon State University

This past spring, I received an email from a graduate student who was concerned about applying for jobs in industry. The student wrote: “I’m having a difficult time trying to market my teaching experience. I’ve been teaching for three years now and I want to leverage that in my applications. I’m just not sure what to say beyond ‘improving communication skills’.”

Whether their interests are in academic positions or not, many graduate teaching assistants (GTAs) are concerned about the jobs they will find and whether they are prepared for those jobs. I have led the graduate teaching assistant training in the Department of Mathematics at Oregon State University since 2013. In that time, I have come to realize it is critical to help GTAs understand the professional skills they develop during their graduate careers, particularly as they learn to teach. My goal in this note is to unpack and describe some of the processes of teaching to help the GTAs appreciate the skills they learn through teaching, and see that these skills can be applied to a variety of jobs beyond academia.

I searched the internet for recent articles that describe the skills employers are looking for, now and in the future. In the list that follows, I highlight some of skills that were common across these articles and discuss how GTAs develop these skills through their teaching. This list is not meant to be exhaustive.

In providing this list, I want GTAs to see that teaching is much more than writing mathematics on a board, and that there is much to be learned through the processes of teaching. Illuminating the skills learned through the processes of teaching will help our GTAs reflect on their practices, help them to reflect on what they are doing as teachers, and inspire further exploration. This reflection in turn helps GTAs better describe their relevant experience in cover letters, on CVs, in their teaching statements, and in conversations about their work as teachers. I believe that explicit attention to these skills can contribute greatly to the professional development of GTAs.

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#thestruggleisREAL: Reflection in a Real Analysis Class

By Katharine Ott, Department of Mathematics, Bates College

Real Analysis is a rite of passage for undergraduate math majors. It is one of my favorite courses to teach, but I recognize that the course is challenging for students, and, for many, downright intimidating. In Fall 2017 I was scheduled to teach Real Analysis for the third time in my career. Prior to the semester starting, I knew that I wanted to alter the grading scheme of the course to de-emphasize exams in favor of effort. Ultimately, I wanted to promote a growth mindset and to help students identify their strengths and weaknesses independent of exam performance. During our annual summer visit, my good friend and graduate school classmate Matthew Pons described to me his new project with Allison Henrich, Emille Lawrence, and David Taylor called The Struggle is Real: Stories of Struggle and Resilience on the Path to Becoming a Mathematician. (For more information on their project, check out https://math.roanoke.edu/tsir/.) I loved their idea of gathering and sharing personalized stories around this topic and immediately thought of adapting the exercise for my students. Since I was teaching Real Analysis, I decided to include reflective homework problems and activities under the label #thestruggleisREAL. I was worried that the hashtag was too gimmicky, but decided that with the right sales pitch students would embrace the pun. In this post I describe how this well-trodden hashtag injected a great deal of reflection, and a bit of levity, into my students’ experience in Real Analysis. Continue reading

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Comparing Educational Philosophies

I have recently heard several mathematicians claim that the educational philosophies of Math Circles and the Inquiry Learning Community are essentially the same. I disagree. I will contrast the differences between these two approaches, along with two other common educational philosophies in the United States. All four approaches to math education differ significantly both in terms of the overall instructional goals and in terms of the primary methods used to achieve these goals.

In my experience, even the originators and staunchest advocates for specific philosophies incorporate the other approaches when putting their favored one into practice. Instructors should think carefully about the goals they have for a given set of students, and then choose a combination of approaches they believe most likely to meet those goals.

I would like to invite you to comment on any thoughts that you have about these lists.  A few questions that I have for readers include:

  • Which blend of these pedagogical approaches have you found congenial for specific audiences?
  • Are there other approaches that are essentially different that I did not include in this list?
  • What steps do you find helpful when transitioning students who are used to one pedagogical approach to another?
  • There is a tendency for students from high poverty schools to be exposed primarily to Traditional Math approaches. The Common Core State Standards represent an effort to slightly improve on this prevailing norm, by pushing teachers in the direction of Conceptual Math instruction. What do you think people who love math and teaching should do to improve access to high quality math education for these students?

I look forward to hearing your ideas!


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Ideas under construction: children saying what they know

Alli entered kindergarten quite skilled at mental addition and proud of her skill. Subtraction followed quickly. Near the end of her kindergarten year, Alli bounced into class and said that her father had taught her about negative numbers. To assure that I knew about them, she explained, “If you subtract 20 from 10, you get negative 10.” I asked, “And what if you subtract ten from seven?” She thought a second and chirped “Negative three.” Then she explained how to write a negative number—“Just put a minus in front”—and added “There are negative numbers and positive numbers.” And that was it. As with many conversations with 5-year-olds, this one ended as abruptly as it began. Continue reading

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Mathematical Practices

My work and that of my colleagues at Education Development Center has always put mathematical thinking—the habits of mind that are indigenous to our discipline—at the core of our work with teachers. What we’ve learned from expert teachers has led me to think more carefully about what it means to “work like a mathematician.” The attached essay details some of the things I’ve learned.

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Beyond Grades: Feedback to Stimulate Rethinking and Intellectual Growth

By Cody L. Patterson and Priya V. Prasad, Department of Mathematics, University of Texas at San Antonio

We frequently use writing assignments to encourage students to examine topics in greater depth than what we cover in class, and we emphasize to our students that writing assignments constitute one of the most important dimensions on which students’ thinking will be assessed. Yet in our early implementation of these assignments, we frequently received work that did not reflect students’ full potential for understanding the topic explored in the assignment. In these cases, because we were using a roughly linear scale to assign each submission a numerical grade, which would then become part of a student’s overall grade in the course, we faced a difficult decision.

  • We could assign low grades to work that did not meet our expectations; this would have the advantage of sending students a clear message about whether their work meets the standards of the course, but it might demotivate students or limit the potential of an otherwise competent student to earn a good grade in the course.
  • We could assign moderate-to-high grades to such work; this would lower the stakes of failure for students, but it would also require us to endorse work that does not meet a high standard.

Neither option, however, seemed to address our greatest concern: that some of our students had not explored and communicated about the topic of the assignment with the depth desired. Moreover, numerical grades allowed many students to decide that they had gained enough, grade-wise, out of the assignment, and did not need to take advantage of opportunities to revise their work. In this article, we’ll talk about our journey toward crafting and implementing a grading scheme for writing assignments that provides greater opportunity for student learning and growth. While we use writing assignments specifically in the context of content courses for preservice teachers, we believe much of our advice is adaptable to other mathematics courses. Continue reading

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Thinking Outside the Textbook

By Steven Klee, Contributing Editor, Seattle University

When two grandmasters face off in a chess tournament, they are faced with a complicated bit of game theory.  If you were in one of their positions, you would prepare for the match by studying your opponent’s games in great depth.  You would study board positions they had created, looking for weaknesses in their defenses and blunders their previous opponents (or they themselves) had made.  It would be safe for them to assume that you could have a strategy in mind to counter any of their strategies that had previously been successful.

Of course, your opponent would naturally study your body of work in the same way.  Therefore, by the time you sat down at the board, there would be a natural expectation that you know that your opponent knows that you know as much as you possibly could about them, and likewise they have the same expectation of you.

As a consequence, the natural strategy for determining who is the better player is to try to avoid these positions in the first place.  Don’t allow the board to get to a point where you have been defeated in the past.  Don’t allow the board to get to a point where you have been successful in the past because your opponent might know how to turn that position to their advantage.  Get away from what has been seen before and create a new position that truly tests the skill of each player.  There’s a term for this – chess players call it going “off book.”

To chess enthusiasts, this moment is exciting.  It’s the moment in the game when the board reflects a position that has never been recorded in a tournament.  It is an opportunity for observers to experience chess history and witness the creation of new knowledge or strategy.  Every move is new and the anticipation of what might come next is thrilling.

Why is this relevant to math education?

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On Being Imperfect

By Gizem Karaali, Pomona College

I have a secret: For the last year or so, my nine-year-old daughter and I have been trying to develop a meditation practice. This guy, Andy, who leads us daily through meditation sessions facilitated by a phone app, has become a familiar name between my daughter and me. (Even my five-year-old occasionally mentions Andy when going to bed at night; sleepy-time Andy tells us to lie on our backs and close our eyes and start by saying good night to our toes.) One day my daughter posed me a question. We had just completed our ten-minute session for the day. She was not willing to move on yet, it seemed, so I waited. She finally formulated her question and asked, very carefully, “Mom, is Andy perfect?”

This is a profound question even though it has an easy answer: “No”. If Andy is human, he is not perfect. Yet none of his flaws are really my business, because he is effectively teaching us to be better. He is consistently, with kindness, in good humor, and with no sign of condescension, telling us how we can do better.  In every session, or let me be honest, in most sessions, we learn from him.

Aren’t there always those we look up to who exemplify ideals we wish to uphold or those who represent the type of character that leaves us in awe? Reading Art Duval’s post on kindness in this very blog, listening to Francis Su’s talk on mathematics for human flourishing, digging into clear critiques of our community fearlessly dispensed by mathematicians such as Piper Harron and Izabella Laba, some of us might wistfully say: but I am not good enough. I am not as capable. I am not as kind. I am not as forgiving. I am not as insightful. I am not as brave.

Now let me rephrase that for you so as to be clear. All of the above are ways of saying the same thing: “I am not ready to be vulnerable.” All these amazing people are amazing partially because they are willing to put themselves out there, trying to live up to their own ideals. (And for some, an alternative may not even exist.) Do they ever falter? Maybe they do. It is not my story to tell. Again, like Andy, any of their possible faltering is none of my business. What is my business is what I learn from them.

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My “First” Mathematical Problem and What It Means

I am inspired, by several previous blog entries, to write about my own mathematical awakening, and what I’ve learned from reflecting on it.

I went to New York City Public Schools, in the Bronx.  I always enjoyed arithmetic and mastered it easily.  I remember not knowing what ‘fractions’ were, but don’t remember learning about them, any more than I remember learning to read. Continue reading

Posted in Active Learning in Mathematics Series 2015, K-12 Education, Mathematics Education Research, Student Experiences | Tagged , , , , | 4 Comments