## Theory into Practice: Growth Mindset and Assessment

By Cody L. Patterson, University of Texas at San Antonio

Several years ago, I took up running. At first, I wasn’t particularly good at it, but I persisted: about two or three times each week, I would go for a jog, increasing my pace or distance in small increments. This measurable growth in my running ability and physical fitness was a great motivator for me, and I increased the frequency of my workouts. After about a year, I was able to complete a local 5K race; this remains among the proudest achievements of my life to date. This was the most authentic experience I’ve had of putting sustained effort into a domain in which I had little natural ability, observing my own growth, and working toward a specific, achievable goal. I attribute my success to two factors:

1. I didn’t measure my own performance against others’. I knew that many people were more accomplished at running than I was when I got started. I set this thought aside and enjoyed the fresh air and the feel of the pavement under my feet.
2. I took notice of any growth in my distance or speed, no matter how small. I took pleasure in being able to observe so many improvements in such a short time.

I have often wondered how I can create a similar experience for students in my mathematics classes, especially for those students who lack confidence in their mathematical knowledge and skills. These are the students who are in danger of developing the mindset that the sustained effort they need to master challenging topics indicates that they are not qualified for advanced study in mathematics. Therefore, one goal of every class I teach is to help students let go of concerns about how they are performing relative to their peers, and enjoy observing their own growth and learning. In his September 2015 article in this blog, Benjamin Braun described some of the mindset interventions he uses to help focus students’ attention on their mathematical growth.   In this article, I’ll describe how the recent work on growth mindset has influenced assessment practices in my own courses. Continue reading

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## Six Ways Mathematics Instructors Can Support Diversity and Inclusion

By Natalie LF Hobson, Graduate Student, University of Georgia

What teaching practices support a diverse student body in your mathematics classroom? In this post, I suggest six concrete teaching practices you can implement today to help make your classroom a more inclusive environment for your students:

1. Use students’ interest in contextualized tasks
2. Expose students to a diverse group of mathematicians
3. Design assessments and assignments with a variety of response types
4. Use systematic grading and participation methods
5. Consider your course logistics
6. Encourage students to embrace a growth mindset

I hope these strategies can spark conversation with colleagues on how we, as educators, can support a diverse and inclusive mathematics classroom. Continue reading

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## Students’ Views of REUs: a “Magical Place of Thinking”

By the Editorial Board, based on an interview at the 2017 Joint Mathematics Meeting with REU students David Burton, Kelly Emmrich, Micah Henson, Andres Mejia, and Nina Pande.

Editor’s note: The editors thank David, Kelly, Micah, Andres, and Nina for taking the time to share their thoughts and insights with us.  Biographical information for each of these students is included at the end of this article.

“Something that I’ll remember the most is there were a couple epiphany moments where we just all of sudden we seemed like we just stumbled into this, you know magical place of thinking of something we never would have thought or come up with that was really important for our project and the reason I think that I’ll also remember that for a long time is that it gave me a lot of confidence that I could do research because being able to come up with a creative way forward is sometimes I think one of those important parts.”
— REU student

Why should students participate in a summer Research Experience for Undergraduates (REU)?  What do undergrads gain from such programs?  What has driven their growth and popularity over the past several decades?  In this post, we share highlights of a conversation that the editors had with five undergraduates at the 2017 Joint Mathematics Meetings about their experiences at five different REUs (described in the final section).  If you are a faculty member, we hope this inspires you to share information about REUs with your students.  If you are an undergraduate student, we hope this inspires you to apply for an REU! (Lists of REUs can be found here and here.)

In our conversation, five major themes emerged regarding the students’ REU experiences:

1. Collaboration: the importance of collaboration, friendships, and networking
2. The Nature of Mathematics: an appreciation or gained understanding of the nature of mathematics and mathematical research
3. Self-Beliefs and Agency: heightened awareness and/or insight about oneself as a learner or person in general
4. Back to the Classroom: the positive impact of REUs on subsequent coursework
5. Graduate School: an increased or decreased interest in graduate school or insight into graduate school

While there were some additional comments off these themes, which we include below, in this article we hope to tell a story of the impact of REUs on undergraduates through the students’ own words.  Note that all student quotes in this article have been lightly edited for clarity. Continue reading

## Aspirations and Ideals, Struggles and Reality

By Benjamin Braun, Editor-in-Chief

Two of my favorite pieces of mathematical writing are recent essays: Francis Su’s January 2017 MAA Retiring Presidential Address “Mathematics for Human Flourishing”, and Federico Ardila-Mantilla’s November 2016 AMS Notices article “Todos Cuentan: Cultivating Diversity in Combinatorics”.  If you have not yet read these, stop everything you are doing and give them your undivided attention.  In response to the question “Why do mathematics?”, Su argues that mathematics helps people flourish through engagement with five human desires that should influence our teaching: play, beauty, truth, justice, and love. In a similar spirit, Ardila-Mantilla lists the following four axioms upon which his educational work is built:

Axiom 1. Mathematical talent is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.

Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.

Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.

Axiom 4. Every student deserves to be treated with dignity and respect.

These essays are two of my favorites because they provide visions of teaching and learning mathematics that are rich with humanity and culture, visions that welcome and invite everyone to join our community.

The ideals and aspirations offered by Su and Ardila-Mantilla are inspiring, emotional, and profound, yet also fragile — for many mathematicians, it can be difficult to balance these with the sometimes harsh reality of our classes.  An unfortunate fact is that for many of us, a significant part of teaching mathematics consists of the struggle to support students who are uninterested, frustrated, inattentive, or completely absent.  We are regularly faced with the reality that large percentages of our students fail or withdraw from our courses, despite our best efforts, and often despite genuine effort on the part of our students as well.  How does a concerned, thoughtful teacher navigate this conflict between the truth of the tremendous potential for our mathematical community and the truth of our honest struggle, our reality?

In my practice of teaching, I have found that the only way to resolve this conflict is to simultaneously accept both truths.  This has not been, and still is not, an easy resolution to manage.  In this essay, I want to share and discuss some of the mantras that I have found most helpful in my reflections on these truths. Continue reading

## 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

## 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.

## 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

## 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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