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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More Linear Algebra, Please

By Drew Armstrong, Associate Professor of Mathematics, University of Miami

Anyone who teaches mathematics in the US knows that the quality of education could be better, but we also know that the problems are complicated and defy easy solutions. I grew up in Ontario, Canada, where I attended high school and completed an undergraduate degree in mathematics. Afterwards I completed a Ph.D. in the United States and I have now been teaching undergraduate mathematics here for over ten years. These experiences suggest to me a change that would improve college mathematics education in the US. It won’t solve every problem, but it is something concrete that we can do right now.

Suggestion: Replace the typical one-semester “introduction to linear algebra” course with a two-semester linear algebra sequence. This would be taken in the first year of college, in parallel with calculus. It would not have calculus as a pre-requisite.

In effect, this would place linear algebra and calculus side-by-side as the twin pillars of undergraduate mathematics. I believe this would have several immediate benefits for the curriculum. In this blog post I’ll describe three of these benefits and then I’ll explain how my experience as a student in Canada and as a professor in the US has brought me to this position. Continue reading

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The Inefficiency of Teaching

By Gavin LaRose, University of Michigan

It could be the punchline of a joke that at any given college or university, at some point, the administration will lean on departments to be more “efficient” by teaching classes in larger sections, or online, or with some technology or another. By the metric of student credit hour to faculty work hour, of course, large lectures are tremendously efficient, and scale admirably. One may argue that there is little difference between an instructor lecturing to 100 or to 200 students, and little difference between an instructor rendered small by the distance to the front of a large lecture hall and one rendered small in the pixels of a video screen. This is the Massive Open Online Course (MOOC) model, which extends this efficiency of scale from 200 to 20,000. Anecdotally, the MOOC tide seems to be receding, but the pressures that argue for this efficiency are not going away. Many departments are being asked to teach, with fewer resources and greater accountability, more students whose mathematical preparation is weaker than in the past [2].

The difficulty here is that the student credit hour metric is easy to measure, while student learning is not. Research says that our efficient passive lecture does not result in the student learning gains we can see with more active teaching techniques [6,7,8]. Indeed, through the Conference Board of the Mathematical Sciences, the presidents of fifteen professional societies in the mathematical sciences have recognized this conclusion and endorsed the use of active teaching methods [4]. But neither the research nor the endorsement provide us with a simple, usable measure by which to demonstrate the effectiveness of these techniques. So our endeavor of teaching remains by easily applied metrics an inefficient one, and I increasingly think one that is inevitably inefficient by even more measures.

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Announcement Regarding White House Office of Science and Technology Policy Call to Action on Active Learning in STEM Education

By Benjamin Braun, Editor-in-Chief, University of Kentucky

This is an announcement to the mathematical community that the White House Office of Science and Technology Policy (OSTP) has issued a Call to Action for incorporating Active Learning in K-12 and higher education STEM courses. Their call includes a submission form where they ask: “What new (i.e., not yet public) activities or actions is your organization undertaking to respond to the Call to Action to improve STEM teaching and learning through the use of active learning strategies?” The deadline for submission of responses is Sept 23, 2016.  It would be excellent for the mathematics community to be well-represented among the submissions, so I encourage our readers to submit their activities and to share this information with others.

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An Inclusive Maths Conference: ECCO 2016

By Viviane Pons, maitre de conferences (Assistant Professor), Université Paris-Sud

Editor’s Note: An expanded version of this article previously appeared at http://openpyviv.com/2016/07/12/ECCO/.

Being one of the few women in the men’s world of mathematics and computer science has led me to look around and spot our flaws when inclusivity is concerned. Let’s not fool ourselves: even though we think of us as being purely objective beyond bias, the maths world is not an inclusive paradise.  The academic world I personally live in is made of mostly white men, mostly from western countries (Europe, US, Canada, and just a little bit of Asia). If you look even closer, you’ll see that most of us come from well-off educated families. Except for the fact that I am a woman, I check all the other boxes myself and I am well-aware of it. Considering the multiple causes of this situation, what can be done? What can I do as a single individual in this world, when I’m busy fighting my own fights earning my right to stay around? Well, I’m not going to answer that just now, but I will share a very good experience I just had. I went to a CIMPA maths summer school in Colombia that was different: ECCO 2016. For the first time, I felt it was indeed inclusive in the best possible way. And, it was excellent maths too, so I was really happy.

First, a little bit of context.  As opposed to a classical conference where most presentations are short ones to announce new results, a summer school is usually made of mini-courses on a certain topic. At ECCO 2016, the main audience was made of students (masters students, PhD students, and undergrads) but some postdocs and even professors participated as well, as we are always keen on learning new things. It was in Colombia and the topic was combinatorics, which happens to be my field. ECCO runs every two years, and began in 2003 as a small event organized by Federico Ardila. He is a Colombian mathematician based in the US and we (the academic world) owe him thanks for many great researchers in combinatorics. I had noticed before that the number of Colombian people among researchers in combinatorics was astonishingly high, but before I met Federico I had no idea why. Most of this very active Colombian community is now organizing the conference. Over the years, ECCO has become quite a big event in combinatorics with a very positive, well-earned, reputation. This year, for the first time, it was a CIMPA school and there were over 100 participants.  So why was this conference so good?

IMG_6350_blog

Photo by Federico Ardila

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From the Editors: Changes for the Editorial Board

by Benjamin Braun, Editor-in-Chief, University of Kentucky

To start, I want to thank all of our readers, subscribers, and contributors — we appreciate your feedback and ideas through your writing, social media comments, and in-person conversations at mathematical meetings and events.  Since launching our blog in June of 2014, our articles have received over 189,000 unique page views!  We will continue to strive to provide high-quality articles on a broad range of topics related to post-secondary mathematics, and we welcome your feedback and suggestions.

I have a few changes to announce regarding the editorial board for On Teaching and Learning Mathematics.  Following two years of service as a founding Contributing Editor for our blog, Elise Lockwood is leaving our board to join the editorial board of the new International Journal on Research in Undergraduate Mathematics Education (IJRUME).  Many thanks to Elise for her excellent contributions that helped the blog have a great start over the past two years!  I know that we can look forward to hearing more from Elise in the future as a contributing author.

For 2016-2017, I am happy to welcome three new Contributing Editors to the board:

  • Luis David García Puente, Sam Houston State University
  • Jess Ellis, Colorado State University
  • Steven Klee, Seattle University

Luis, Jess, and Steve bring with them a wealth of expertise in teaching, research, and mentoring, and I am excited that they will be sharing their expertise with our readers.

For those of you who are regular readers, we will continue to publish articles roughly every two weeks, with a target goal of publishing 24 articles per year. Our next post is scheduled for August 22, 2016, where we will hear from Viviane Pons about her experience at a math summer school that was “inclusive in the best possible way.” Stay tuned!

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