Announcement: Statement by AMS Board of Trustees

By Benjamin Braun, Editor-in-Chief

Due to connections with mathematics education, some of our readers might be interested in the following statement issued by the members of the Board of Trustees of the American Mathematical Society regarding the Executive Order on Immigration issued by the President of the United States.  To read the statement, see this link: http://www.ams.org/news?news_id=3305

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Announcement: Active Learning Article in AMS Notices

By Benjamin Braun, Editor-in-Chief

Some of our readers might be interested to know that the February 2017 Notices of the American Mathematical Society contains an article on active learning that is based on the six-part series on active learning published on this blog in Fall 2015.  See the Notices article here:

What Does Active Learning Mean for Mathematicians?” Benjamin Braun, Priscilla Bremser, Art M. Duval, Elise Lockwood, Diana White. Notices of the American Mathematical Society, Vol 64, Number 2, February 2017.

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What is an Experimental Math Course and Why Should We Care?

By: Lara Pudwell, Valparaiso University

What is the first meaningful mathematics problem you remember solving? For me, it was the nine dots, four lines puzzle. When my fourth grade teacher assigned it as an extra credit problem, I spent several days of recess scribbling out attempted solutions in the sandpit, erasing, and trying again until, at last, I found a solution!

I believe this geometric puzzle still sticks out in my memory nearly three decades later because it was one of the first experiences I had with trying to answer a question that didn’t simply involve mimicking previous work. For practitioners, informed trial-and-error is a key step in doing mathematics, so the idea of “thinking out of the box’’ (or in my case, literally thinking in the sandbox…) to build intuition seems natural. However, this is a far stretch from the view of many students who see mathematics as an opportunity to memorize formulas and execute repetitive tasks.

Where do students learn the process of refining mathematical conjectures? Certainly, teaching (via) inquiry in the mathematics classroom has generated much discussion, but often the conversation about inquiry is attached to particular material in the curriculum, with an inquiry-based approach to calculus or statistics, for example. Despite being fundamental to doing mathematics, the majority of the time the inquiry process is a means to an end, rather than a focus of an entire class, and it’s rarely addressed directly. In this environment, some students internalize the inquiry process by indirect exposure. Others finish their education without a true sense of how mathematics is actually developed.

Experimental mathematics courses are one answer to the need to celebrate and study inquiry for the sake of inquiry.  In particular, an experimental mathematics course is not a course about a particular set of material; it is a course about a particular approach to doing mathematics.

Courses in experimental mathematics have been offered by at least 7 different colleges and universities [1].  Outside of those who have taught or taken these courses, there is not widespread understanding of what “experimental mathematics” means in the undergraduate curriculum. My goal in this post is to give a better idea of what such a course looks like.

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Integrating Computer Science in Math: The Potential Is Great, But So Are The Risks

By Emmanuel Schanzer, Bootstrap

Recent calls to bring Computer Science to K-12 schools have reached a fever pitch. Groups like Code.org and Girls Who Code have become household names, having raised tens of millions in funding from Silicon Valley luminaries and small donors alike. In February of 2016, President Obama announced the “CSforAll” initiative, and asked for $4 billion of funding from Congress to pay for it. Even in today’s divided climate, this initiative found bipartisan support, and mayors and governors from coast to coast have made sweeping commitments to bring CS Education to all students.

This effort has serious consequences for math education. Adding a new subject is easier said than done: recruiting, training, hiring and retaining tens of thousands of new CS Teachers will take decades and cost billions, and the finite number of hours in the school day and rooms in the school building make it difficult to find space for these courses. To meet these commitments, many schools and districts have employed three strategies: (1) take time out of existing math classes for CS, (2) take math classes out of a teacher’s schedule, and instead have them teach a CS class, and/or (3) have CS classes count as a math credit [1]. All of this is done because there’s a widespread misconception that “computer science is just like math”, and that skills from one will transfer to the other. Unfortunately, most of the programming languages being taught in these classes have little to do with mathematics, and embrace concepts that are explicitly math-hostile. In this article, I will discuss some of the challenges and opportunities faced by K-12 mathematics educators in our efforts to develop an authentic incorporation of CS into the K-12 curriculum. Continue reading

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If You Don’t Talk To Your Students About Math, Who Will?

By Steven Klee, Contributing Editor, Seattle University

During my freshman year of high school, my geometry teacher came into class one day and challenged us to trisect an angle with a compass and a straight edge. Anyone who was successful would receive an A in the class for the rest of the year. We wouldn’t have to do any more homework or take any more tests. Nothing. Of course, this should have seemed too good to be true. But I was in ninth grade and didn’t know any better, so I set off to solve this seemingly innocent problem.

I came up with a dozen or so false proofs, all of which included reasoning like “well, now you just move the compass a bit over here and then you draw this line, and it works!”   Of course it didn’t work, but this is the kind of non-proof you would attempt to make if you had only just learned what a proof is.

But rather than simply tell me I was wrong and insist that I was doomed to failure, my teacher let me share the ideas behind every failed proof so that I could see the shortcomings in my arguments.   He sat with me and we talked more broadly about what does and does not constitute a proof. He knew I was going to be wrong. He knew this was an impossible assignment. But he still listened.

My teacher’s openness to hearing my ideas inspired me to keep working and to keep trying new approaches. As I learned more math, I kept coming back to this problem. I tried using trigonometry. I tried using calculus. I tried making up a unit distance that I would call “1.” After watching Good Will Hunting, I decided that it would probably help if I drew all of my diagrams on mirrors. None of these things helped. Along the way, I learned about quantifiers. I learned about proofs. I learned to identify the errors in my attempted proofs on my own. Ultimately, I think I shed a tear of joy when I finally saw the proof of impossibility in my graduate algebra class.

This story can lead to a lot of different discussions. Ben Braun wrote a beautiful article for this blog about the value of having students work on difficult and unsolved problems, which I highly recommend. Instead, I’d like to explore the value of talking about mathematical ideas informally, especially when they are ill-formed and possibly incorrect; the value of encouraging our students to share such ideas with one another; and the value of participating in these discussions with our students. Continue reading

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Inverse Functions: We’re Teaching It All Wrong!

By Frank Wilson, Chandler-Gilbert Community College; Scott Adamson, Chandler-Gilbert Community College; Trey Cox, Chandler-Gilbert Community College; and Alan O’Bryan, Arizona State University

What would you do if you discovered a popular approach to teaching inverse functions negatively affected student understanding of the underlying ideas? Would you continue to teach the problematic procedure or would you search for a better way to help students make sense of the mathematics?

A popular approach to finding the inverse of a function is to switch the \(x\) and \( y\) variables and solve for the \(y\) variable. The strategy of swapping variables is not grounded in mathematical operations and, we will argue, is nonsensical. Nevertheless, the procedure is so ingrained in textbooks and other curricula that many teachers accept it as mathematical truth without questioning is conceptual validity. As a result, students try to memorize the strategy but struggle to “accurately carry out mathematical procedures, understand why those procedures work, and know how they might be used and their results interpreted” (NCTM, 2009; Carlson & Oehrtman, 2005). As we will illustrate, this common process for finding the inverse of a function makes it harder for students to understand fundamental inverse function concepts.

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The National Science Foundation Has Resources to Help You Improve the Teaching and Learning of Undergraduate Mathematics

By Ron Buckmire, TJ Murphy, John Haddock, Sandra Richardson, and Brent Driscoll

This article is intended to serve as a rough “proof” of the statement, “There exist many resources and opportunities supported by the National Science Foundation (NSF) to improve the teaching and learning of undergraduate mathematics.” We present a curated, annotated list of projects funded by the NSF’s Divis­ion of Undergraduate Education (DUE) that readers of this blog might be interested in. Additionally, we demonstrate the remarkable diversity of projects and institutions that are funded by DUE to improve the teaching and learning of mathematics, and share professional opportunities for people who share these goals.

The NSF is an independent federal agency tasked by the United States Congress to “promote the progress of science.” With a budget of 7.5 billion dollars in fiscal year 2016, NSF received approximately 50,000 proposals and made almost 12,000 awards. NSF is organized into seven Directorates that support research in various disciplines in science, technology, engineering and mathematics (STEM) as well as in education. Each of the Directorates is further organized into Divisions. For example, the Division of Mathematical Sciences (DMS) is situated in the Directorate for Mathematical and Physical Sciences (MPS). The Directorate for Education and Human Resources (EHR) houses DUE, which manages the awards that are the primary focus of this article.

DUE’s current signature program is Improving Undergraduate STEM Education (IUSE). IUSE is the latest incarnation of DUE’s programmatic efforts to actualize its mission “to promote excellence in undergraduate STEM education for all students.” Former DUE programs include “Transforming Undergraduate Education in STEM” (TUES) and “Course, Curriculum and Laboratory Improvement” (CCLI). The current IUSE solicitation is 15-585, and the next deadline for full proposals is January 11, 2017.

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On What Authority? – Considering Implicit Messages in Our Teaching

By Brian Katz, Augustana College

I think that mathematics draws in some people and repels others in large part because of the distinctive role of authority in our discipline and teaching, especially when we act as content experts and discussion leaders in the classroom. For instance, consider the following phrases from students, distilled from my interactions with college students over the past 15 years.

I’m not a math person. I learn best when you show me a bunch of examples and then I practice them. It’s true, so why do I have to prove it? That’s just how my last teacher told me to do it. I always liked math because there was one right answer. I just want to teach high school; why do I have to learn this? Wait, what – you want me to ask my own question!? Do I have to simplify my fractions? Well, that’s what the computer said was the answer. The test was unfair because it had problems we didn’t discuss in class. ~silence~

I expect that these comments are also familiar and painful to the reader. I think that each of these comments is in part a symptom of ways students have internalized a relationship with authority from our teaching. In this post, I will illuminate the role of authority in mathematics teaching, argue that taking a more overt stance toward it can better support both the students we repel and the ones we attract, and offer a handful of strategies for taking such a stance.

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Creating Momentum Through Communicating Mathematics

By Matthias Beck, San Francisco State University, and Brandy Wiegers, Central Washington University

Given five minutes, can you turn to the person next to you and describe your research? How about over 15 minutes in front of a class of 10th graders? Thinking of one of your research graduate students, how would you prepare her/him to make such an activity equally beneficial for her/him and the 10th graders? For many of us, these are skills only nurtured through conference talks and time within the profession. The SF-State (CM)²: San Francisco State University Creating Momentum through Communicating Mathematics program worked to change this, creating a program that developed mathematics graduate students who could have this conversation and were better engaged in why they were studying mathematics and what role they wanted to have in the future of our profession. As an NSF GK-12 program, (CM)2 ran from 2009-2014, working with master’s level students in mathematics to engage them in mathematical discourse while also supporting their research and professional development. Over the course of the five-year program, a total of 43 SF State mathematics graduate students were involved in the project, spending considerable time and energy on K-12 activities in 13 schools in the greater San Francisco area. A key goal was to strengthen the graduate students’ communication, teaching, outreach, and teamwork skills by immersing them in mathematics classrooms and the San Francisco Math Circle. A second key goal was to make mathematics, especially algebra, and its career connections more relevant and explicit for 6-12th grade teachers and students. This post will share successes, lessons learned, and resources for you as a faculty member to build aspects of outreach, teaching, and professional development programs for your own students. Continue reading

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Conventional Courses are Not Enough for Future High School Teachers

By Yvonne Lai, University of Nebraska – Lincoln and Heather Howell, Educational Testing Service

Consider how you would respond to two different versions of a question. In the first, you are asked to solve a high school mathematics problem. In the second, some high school students’ solutions to that problem are shown to you. You are asked to assume the role of the students’ teacher and to evaluate the mathematical validity of the students’ different approaches. What knowledge, if any, do you need in the second situation that you don’t need in the first situation?

Some would argue that the second situation is just about knowing math. If you, yourself, can solve the high school mathematics problem correctly, and you are very capable in high school mathematics, then this should be enough to evaluate a high school students’ solution. Yet others might say that this question is about teaching. If you can’t interpret students’ work, you can’t judge it accurately. Still others might say that this question targets something in between straight math and teaching. We would say that this scenario assesses a blend of all of these things that previous scholars have named mathematical knowledge for teaching (MKT). We ask the reader to join us in considering, as some have argued, why MKT is a form of applied mathematics – and why mathematicians have a stake in thinking about MKT in this way.

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