Every university instructor would be thrilled if their students came to their mathematics classes with the ability to make viable arguments and to critique the reasoning of others; if their inclination were

- to persevere through difficult problems,
- to look for and make use of mathematical structures, and
- to strategically use tools in their mathematical toolbox.

But how do students develop these mathematical practices? The foundation is laid during a student’s 13 years of mathematics classes in K-12 – learning from their teachers and engaging in mathematics with their peers. The eight Mathematical Practice Standards that are an integral part of the Common Core State Standards (CCSS) for Mathematics, have elevated the importance and visibility of productive mathematical habits of mind in K-12 education. It is now an expectation and not a bonus. But are teachers equipped to help their students develop the practices until they become habits? Do teachers even have productive mathematical habits of minds themselves?

We actually know quite a bit about pre-service teachers’ habits of mind from research (Karen King: Because I love mathematics, Mathfest 2012, Madison). For example, pre-service teachers who hold mathematics degrees have an inclination to first state rules (Floden & Maniketti, 2005). They are not in the habit to seek meaning, which is such an important mathematical habit of mind. We can think of habits as acquired actions that we have practiced so much, that we eventually do them without thinking. At first, they are deliberately chosen but at some point they become automatic.

This has important implications for teaching at the university level, especially for pre-service teachers. Many professors and policy makers assume that completing a major in mathematics builds some kind of maturity. Undergraduate courses should be an opportunity to further refine productive mathematical habits of mind. Instead, this coursework often appears to reinforce unproductive habits of mind for engaging in mathematical practice. So I think we college/university faculty should take a serious look at what we are doing in our classes—not just in specific classes for future teachers, but in all our math classes. Mathematics faculty have a tendency to assign responsibility for K-12 math teacher quality to math education courses. But let’s think about that for a moment. In California, future high school teachers take 4 credit hours of math methods courses in their credential program. If they are lucky, they take at most a handful of courses as part of a math major specifically designed for future teachers, maybe 6 more credit hours. And they complete about 40 credit hours of mathematics content courses that are part of the normal mathematics degree programs. If they don’t learn productive mathematical habits of mind from their professors in their 10 or more college math courses, then who is responsible for this?

This is our responsibility and our opportunity! Pre-service teachers come to college with already formed ideas of what mathematics is and how the game of mathematics is played. They have already developed mathematical habits of mind—for good or for bad. It is up to us to help them replace unhelpful habits and develop productive habits, and we have approximately 4 years to do it.

When we are trying to change habits and practices, we often focus on directly changing actions and we hope this will lead to better results. In this case, we want teachers to change their teaching practice so that all students will develop productive mathematical habits of mind. But actions are affected by beliefs and beliefs are based on experiences. So it would be much more productive for us to provide pre-service teachers (and all students) with a series of compelling and positive experiences to change their beliefs. This, in turn, will lead to more coherent, consistent, grounded, and therefore stable results.

In my work with in-service teachers around transitioning to the CCSS, we have explored a variety of productive pedagogical ideas that provide students with experiences where they engage in mathematical practices. I have adopted several into my college classroom to better prepare my students for their work as teachers but also because I think this is simply good teaching for everybody. I’ll give two examples that focus on “Make a viable argument and critique the reasoning of others”.

**Gallery Walks**

In many of our courses, students write proofs; this is a mathematician’s idea of a viable argument. How do students learn how to write a proof? What are characteristics of a good proof? How do you critique other people’s arguments? On the first day of my combinatorics and graph theory class we worked on the following problem:

Students first collaborated on the problem in groups of 3—4. After students solved the problem, they made a poster to explain how they found their solution and how they knew that they had found all solutions. We then did a gallery walk: With a stack of sticky-notes in hand, students studied each poster. They asked questions about parts that they did not understand and they made suggestions when they found something that could be improved. They also pointed out aspects of the posters they found helpful in understanding the argument.

(sample posters with sticky note feedback)

Next, students went back to their own posters and studied the feedback they had received. They discussed revisions, and for homework each student individually wrote up an improved version of their proofs.

Before we finished the class, we had a discussion about the purpose of this activity. Students were surprised about the variety of proofs they had seen. After reading each other’s solutions, they were able to decide if there were gaps in arguments and describe what made a proof easy to read. They saw that there are a variety of ways to structure the argument, that a complete proof is not necessarily a good proof, and that a “proof by example” is not a proof but could possibly be revised into a general proof. They recognized the value of their peers’ feedback; and that they did not need the instructor to validate their proofs—rather, they possessed the mathematical authority to do so themselves.

You may ask: Our students write proofs and have to show their work all the time, why is this activity useful? In this case, it set the tone for the semester, and it made expectations clear to the students. Aside from seeing that they would be expected to actively work with their peers in class, they also experienced giving feedback and then using feedback to revise their work. They learned that an important goal of mathematics is communicating solutions, not just getting answers, and for the future teachers in the room, they saw a pedagogical structure they can use at any grade level and in any subject.

I do variations of the gallery walk in most of my classes a few times each semester. It works with modeling problems in calculus just as well as with proofs in real analysis.

**Re-engagement Lessons**

Every instructor knows the following situation very well: Students have done a task. You assess it. There are major gaps. What do you do? You could

- Re-teach the topic or do more examples.
- Offer review sessions or office hours for students with gaps and work with them separately.
- Ignore the gap, go on, and hope the students will pick the content up later.

I want to describe another option: re-engage the students with the task and the concepts, using their responses to move everybody forward.

While learning how to write proofs involving the algebra of sets in my “Intro to Proofs” class, students did the following standard problem on a homework assignment: Given sets *A* and *B,* prove that *A* U (*B* – *A*) = *A* U *B.* While grading the homework, I found myself writing the same comments over and over again: “Pick a point,” “double set inclusion,” etc. I decided to use the proofs that students had written as the basis for the next day’s activity. To prepare, I compiled a collection of students’ proofs. In class, I handed out copies of these proofs to pairs of students. I asked them to discuss:

- What is good about each proof?
- Are there actual mistakes? Gaps?
- What makes a proof easy to understand? Hard to understand?
- Fill in gaps, correct mistakes.

Then we had a whole class discussion, keeping track on a document camera of changes students suggested.

Why was this activity better than just going over the proof again on the board or doing a similar problem, which would certainly have been faster?

By using a compilation of actual student work, students were invested in the exercise from the start. They already had engaged with this problem, so even if they had not written a perfect proof, they had a basis to build on. The examples I chose included good and bad features of proofs. The contrast and repetition allowed the students to transfer ideas from one to the other. The setup of the activity allowed students at every level to engage and benefit. One of my top students told me after the class that he had learned a lot about reading and critiquing others’ work. Finally, by contrasting several proofs, we had an excellent discussion about the structure of proofs, not just small details.

Research is compelling that students learn more from making and then confronting mistakes than from avoiding them (Boaler, 2016). My goal as a teacher is shifting from providing clear explanations so students don’t make mistakes, to creating situations, which are likely to produce important mistakes, and then helping the entire class confront and learn from those mistakes. Re-engagement lessons are a great method for this confrontation.

This is just one example of a re-engagement lesson. David Foster from the Silicon Valley Math Project contrasts re-teaching and re-engagement:

Re-teaching |
Re-engagement |

Teach the unit again. | Revisit student thinking. |

Address basic skills that are missing. | Address conceptual understanding. |

Do the same or similar problems over. | Examine task from different perspective. |

Practice more to make sure student learn the procedures | Critique student approaches/solutions to make connections |

Focus mostly on underachievers. | The entire class is engaged in the math. |

Cognitive level is usually lower. | Cognitive level is usually higher. |

(Foster & Poppers, 2009)

I offer the two classroom activities as examples to help us start talking about changing the mathematics culture in our classrooms and schools so that all students, including future teachers, have experiences that support them in forming productive mathematical habits of mind.

To educate our students to become mathematicians and teachers we have to do more than role-model mathematical practices, we have to create the environment where students engage in them, and we have to talk more about what we are doing and why. We have 4 years to help our students replace bad mathematical habits (speed, answer-getting, anxiety) with productive ones (sense-making, perseverance, use of tools and structure). This is our responsibility, but maybe even more importantly, this is our opportunity.

**References:**

Boaler, J., & Dweck, C. S. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching.

Connors, R., & Smith, T. (2012). Change the culture, change the game: The breakthrough strategy for energizing your organization and creating accountability for results. [Also https://www.partnersinleadership.com/insights-publications/changing-your-culture/]

Common Core State Standards: http://www.corestandards.org/Math/Practice/

Floden, R., and Meniketti, M. (2005). Research on the effects of coursework in the arts and sciences and in the foundations of education. In M. Cochran-Smith and K. Zeichner (Eds.), Studying teacher education: The report of the AERA panel on research and teacher education. Mahwah, NJ: Lawrence Erlbaum Associates

Foster, D. and Poppers, A. (2009). Using Formative Assessment to Drive Learning: http://www.svmimac.org/images/Using_Formative_Assessment_to_Drive_Learning_Reduced.pdf

]]>“I am so glad you made that mistake,” I’ve come to realize, is one of the most important things I say to my students.

When I first started using inquiry-based learning (IBL) teaching methods, I had a tough time creating an atmosphere where students felt comfortable getting up in front of class and presenting their work. It is a natural human instinct to not want to expose your weaknesses in front of others. Making a mistake while presenting the solution to a problem at the board is a huge potential source of embarrassment and shame, and hence also anxiety. So how do we—as educators who understand the critical importance in the learning process of making and learning from mistakes—diminish the fear of public failure in our students? For me, the answer involves persistent encouragement. It also relies on setting the right tone on the first day of class.

To prepare my students on Day One of class, I talk about the importance of making and learning from mistakes. I often refer to one of my favorite books on this subject, *The Talent Code *by Daniel Coyle [1]. Coyle has studied several hotbeds of “genius,” places where an unreasonable number of virtuosos—e.g., world-famous violinists, baseball players, and writers of fiction—emerge. He is interested in discovering just how people like Charlotte Brontë, Pelé, and Michelangelo learn to perform at the top of their fields. The answer involves a simple idea: talented people are those who have made far more mistakes than others and who have deliberately learned from those mistakes. For my students, the takeaway is that the most accomplished people have made many more mistakes than the average person. Consequently, it is of high value for us to make our mistakes public and discover how to correct them together. (As a side note, Francis Su employs the same strategy in his article “The Value of Struggle” [2].)

After the first day of class, whether I am teaching Quantitative Reasoning, Calculus, or a more advanced course such as Introduction to Knot Theory, nearly every class period begins with presentations of homework problems by student volunteers. Students have homework due each day, and they are required to present problems a certain number of times during the term. The number of problems we do depends on how long the class period is, how complex the problems are, and what I need to teach in the remainder of class. In a course like Introduction to Knot Theory, we might spend 45 minutes or an hour on student presentations, while we will spend 20-30 minutes on calculus homework presentations in an 85-minute class period. This general structure could be modified to fit shorter class periods or weekly recitation sections at universities with larger lecture courses. For instance, we used to teach calculus classes four days a week in 50-minute blocks at Seattle University. Within this structure, I had a weekly “Problem Day” for my calculus classes instead of having daily student presentations of homework. After students volunteer to present problems at the board on a typical class day, all students who are chosen to present simultaneously write up problem solutions while their classmates review the homework or work on another activity. Once all solutions have been written up, we reconvene; one by one, students come to the board to walk us through their solution. *This is where supportive facilitation becomes critical.*

Encouraging students to make mistakes in the abstract—as I do one Day One—is one thing, but helping students accept their mistakes in front of class is quite another. This is where my new catch phrase comes in. Let’s say, for example, a student is computing the derivative of \(y=x^2\sin x\) at the board and writes \(y’=2x\cos x\). I might say, “I am so glad you made that mistake! You’ve just made one of the most common mistakes I’ve seen on this type of problem, so it’s worth us spending some time talking about. Can anyone point out what the mistake is?” If someone in the class comments that the presenter should have used the Product Rule, I might follow up with, “That’s a good idea. How can we see that this function is a product? Let’s work together to break the problem down into pieces.” Going forward, I facilitate the process of the class coming up with their collective correction of the mistake. Collaboratively working to correct mistakes like this tends to help students observe more subtle differences between different types of problems while building a more sophisticated mental problem-solving framework.

Making and correcting mistakes together can also help address more basic misconceptions. Suppose a student—let’s call them Riley—writes, in the middle of a calculus problem, a line like the following.

\(1/(x+x^2) = 1/x + 1/x^2\)

This mistake will most likely lead to an incorrect final answer. Many of the presenter’s classmates will discover the final answer is wrong, and some will even be able to pinpoint where the computation went awry. How would I address this? Once a classmate has identified the problem, I might say, “Riley, I’m so glad you made that mistake! This is one of the most common algebraic mistakes students make in calculus—I’m willing to bet others in the class made this same exact mistake, so it’ll be really helpful for us to talk about it together. This is a question for anyone in the class: How can we prove that this equality doesn’t hold, in general?” Suppose a student, Dana, in the audience suggests we try plugging in some numbers to see what happens. I’d follow up with, “Riley, could you be a scribe for this part of the discussion? Please write up Dana’s suggestion beside your work. Dana, can you tell Riley exactly what to write?” Once we’ve cleared up the confusion with Riley’s algebra, I might ask them to work through the rest of their problem again at the board, fixing their work accordingly. On the other hand, if Riley appears to be too shaken or confused to fix the rest of the problem or if the actual problem was much more complex than the one that resulted from the algebraic error, I might ask the class to collectively help Riley figure out what to write each step of the way. A third option I frequently use is the “phone a friend” option. I could see if Riley wants to “phone a friend” in the class to dictate a correct answer.

Mistakes can be common in class presentations, but I occasionally have a class that is so risk-averse that very few people offer to present their work unless they know it’s perfect. If I have too many correct solutions presented, but I know some in the class are struggling, I might follow up with a comment like: “That was perfect! Too bad there were no mistakes in your work for us to learn more from. I’d like to hear from someone who tried a method for solving this problem that *didn’t* work out so well. Would anyone be willing to share something they tried with the class?” At this point, someone may come forward with another (incorrect, or partially correct) way to attempt the problem. If nobody comes forward, I could offer a common wrong way to do the problem and ask my students to identify the misunderstanding revealed by my “solution.” I might even tell a little white lie and say something like, “When I first learned this concept, I had a lot of trouble understanding it. I made the following mistake all the time before I figured out why I was confused.” Alternatively, I could mention, “The last time I taught this class, someone made the following mistake. What’s wrong with this approach to solving this problem?”

Now, let’s say one of my students has just presented a problem at the board. Perhaps they made a mistake, or perhaps they did everything perfectly. What happens next? I will ask the class, “Any questions, comments, or *compliments*?” The request for compliments is one of the most important parts of this solicitation of feedback. It is so important that, during the first several weeks of class, I make my students give each presenter at least one compliment. Some of the best compliments I’ve heard from students follow some of the worst presentations. For instance, after a disastrous presentation where the presenter appeared clueless and needed their peers to help them complete all parts of a problem, a student of mine once observed, “That took a lot of guts to get up there and make mistakes. I thought you did a great job fixing the solution and taking constructive criticism from us!” If nobody offers up such a supportive compliment after a bad presentation, I might give this feedback myself to publicly recognize the presenter’s courage. What’s more, if a student appears shaken by the experience of messing up so thoroughly, I’ll follow up again after class, reinforcing my appreciation for their bravery. Over time, this strategy helps build a supportive classroom environment.

Looking back on how my classes have evolved, I can see that it is difficult to convince students to be vulnerable in a math class without the three following elements:

(1) setting the stage by sharing my expectations of students making mistakes and being clear about the *reasons* for these expectations,

(2) encouraging students to help each other come to the right answer while recognizing the benefits of making specific mistakes, and

(3) acknowledging students’ willingness to make mistakes both publicly and privately.

We’ve been primarily focused on *how* to encourage students to make mistakes, but let’s turn our attention to *why* it might be important in our math classes. One thing that I found to be particularly striking when I started teaching this way was my students’ exam performance. I typically ask a mixture of conceptual and computational questions on exams. I was surprised to see how much more sophisticated students’ responses were to conceptual questions in courses where I spent a great deal of class time on student presentations. At first, this was surprising to me since we spent quite a lot of time in class working through computational problems. The more I reflected on this phenomenon, though, the more it made sense. The repairing of computational mistakes in class often led to a discussion of the more conceptual mathematics underlying the computations. What’s more, these discussions were sparked by students grappling with problems that they cared about—problems they had spent time outside of class trying to solve—and not simply problems they had just been introduced to in the course of a lecture. Discussion that takes place during a homework presentation session seems to stick with students in a way that a “discussion” (where the instructor is doing much of the talking) during a lecture does not.

There are myriad other benefits I’ve observed, including development of a tight-knit classroom community, increased student self-confidence, and more engaged student participation in all aspects of class. In short, I’m convinced. I’m all in. The benefits of teaching this way far outweigh the costs of redistributing precious class time, making room for students to publicly make and collaboratively fix their delightful mathematical mistakes.

**References**

[1] Coyle, Daniel. *The Talent Code: Greatest Isn’t Born, It’s Grown, Here’s how*. Bantam, 2009.

[2] Su, Francis. The Value of Struggle. *MAA FOCUS.* June/July 2016.

What happens to the data from your teaching evaluations? Who sees the data? Are your numbers compared with other data? What interpretations or conclusions result? How well informed is everyone, including you, about the limitations of this data, and conditions that should be satisfied before it is used in evaluating teaching?

Despite many shortcomings of student ratings of teaching (SRT), some of which I mention below, their use is likely to continue indefinitely because the data is easy to collect, and gathering it requires little time on the part of students or faculty. I refer to them as student ratings, not evaluations, because “evaluation” indicates that a judgment of value or worth has been made (by the students), while “ratings” denote data that need interpretation (by the faculty member, colleagues, or administrators) (Benton & Cashin, 2011).

Readers may be asked to interpret the data from their SRT on their annual reviews or in their applications for tenure or promotion. They may even find themselves on committees charged with reviewing the overall teaching evaluation process or the particular form that students use at their institutions, as I did. For these reasons, I thought it might be helpful to discuss some general issues concerning SRT and then present a few practical guidelines for using and interpreting SRT data.

My career as a mathematics professor spanned four decades (1973-2013) at Loyola Marymount University, a comprehensive private institution in Los Angeles. During that time, my teaching was assessed each semester by student “evaluations.” For nearly all of those 40 years this was the only method used on a regular basis. If there were student complaints, a classroom observation by a senior faculty member might take place, which happened to me once as an untenured faculty member. Later on, as a senior faculty member, I myself was called upon to perform a few classroom observations.

During 2006–2011, I also directed a number of faculty development programs on campus, including the Center for Teaching Excellence. In that role, I served as a resource person to a Faculty Senate committee appointed in 2010 to develop a comprehensive system for evaluating teaching. Prior to that, I had participated in a successful faculty-led effort to revise the form students used to rate our teaching, and I worked to develop and disseminate guidelines about how that data should be interpreted. During that two-year process (2007-2009), I discovered that my colleagues and I, and even faculty developers on other campuses, had a lot to learn about the limitations of this data (Dewar, 2011).

Because teaching is such a complex and multi-faceted task, its evaluation requires the use of multiple measures. Classroom observations, peer review of teaching materials (syllabus, exams, assignments, etc.), course portfolios, student interviews (group or individual), and alumni surveys are other measures that could be employed (Arreola, 2007; Chism, 2007; Seldin, 2004). In practice, SRT are the most commonly used measure (Seldin, 1999) and, frequently, the primary measure (Ellis, Deshler, & Speer, 2016; Loeher, 2006). Even worse, “many institutions reduce their assessment of the complex task of teaching to data from one or two questions” (Fink, 2008, p. 4).

The use of SRT has garnered many critics (e.g., Stark & Freishtat, 2014) and supporters (e.g., Benton & Cashin, 2011; Benton & Ryalls, 2016) of their reliability and validity. Back-and-forth discussions about SRT occur frequently on the listserve maintained by the professional society for faculty developers known as the POD (for Professional and Organizational Development ) Network (see http://podnetwork.org). Earlier this month, in just one 24-hour period, there were 18 postings by 12 individuals on the topic (see https://groups.google.com/a/podnetwork.org/forum/#!topic/discussion/pBpkkck_xEk)

The advent of online courses has provided new opportunities to investigate gender bias in SRT, leading to new calls for banishing them from use in personnel decisions (MacNell, Driscoll, & Hunt, 2015; Boring, Ottoboni, & Stark, 2016). Still, as noted above, experts continue to argue their merits.

Setting aside questions of bias, readers should be aware of many factors that can affect the reliability and validity of SRTs. These include the content and wording of the items that are on the form and how the data are reported.

Some issues related to the items on the form are:

- They must address qualities that students are capable of rating (e.g., students would not be qualified to judge an instructor’s knowledge of the subject matter).
- The students’ interpretation of the wording should be the same as the intended meaning (e.g., students and instructors may have very different understandings of words like “fair” and “challenging”).
- The wording of items should not privilege or be more applicable to certain types of instruction than others (e.g., references to the instructor’s “presentations” or the “classroom” may inadvertently favor traditional lecture over pedagogies such as IBL, cooperative learning in small groups, flipped classrooms, or community-based learning).
- The items should follow the principles of good survey design (e.g., no item should be “double-barreled,” that is, ask for a rating of two distinct factors, such as
*The instructor provided timely and useful feedback.*See Berk (2006) for more practical and entertaining advice.) - Inclusion of global items, such as
*Rate this course as a learning experience*, maybe be efficient for personnel committees, but data obtained from such items provide no insight into specific aspects of teaching and can be misleading (Stark & Freishtat, 2014).

Regarding how the data are reported:

#1.* Sufficient Response Ratio*

There must be an appropriately high response ratio. If the response rate is low, the data cannot be considered representative of the class as a whole. For classes with 5 to 20 students enrolled, 80% is recommended; for classes with between 21 and 50 students, 75% is recommended. For still larger classes, 50% is acceptable. Data should not be considered in personnel decisions if the response rate falls below these levels (Stark & Freishtat, 2014; Theall & Franklin, 1991, p. 89). (NOTE: Items left blank or marked Not Applicable should not be included in the count of the number of responses. Therefore, the response ratio for an individual instructor may vary from item to item.)

#2. *Appropriate Comparisons*

Because students tend to give higher ratings to courses in their majors or to electives than they do to courses required for graduation, the most appropriate comparisons are made between courses of a similar nature (Pallet, 2006). For example, the average across all courses in a College of Arts and Sciences or even across all mathematics department courses would *not* be a valid comparison for a quantitative literacy course.

#3. *When Good Teaching is the Average*

When interpreting an instructor’s rating on a particular item, it is more appropriate to look at the descriptor corresponding to the rating, or the rating’s location along the scale, instead of comparing it to an average of ratings (Pallet, 2006). In other words, a good rating is still good, even when the numerical value falls below the average (for example, getting a 4.0 on a scale of 5, when the average is 4.2). Stark and Freishtat (2014) go even farther, recommending reporting the distribution of scores, the number of responders, and the response rate, but not averages.

#4. *Written Comments*

Narrative comments are often given great consideration by administrators, but this practice is problematic. Only about 10% of students write comments (unless there is an extreme situation), and the first guideline recommends a minimum 50% response threshold. Thus decisions should not rest on a 10% sample just because the comments were written rather than given in numerical form! Student comments can be valuable for the insights they provide into classroom practice and they can guide further investigation or be used along with other data, but they should not be used by themselves to make decisions (Theall & Franklin, 1991, pp. 87-88).

#5. *Other Considerations*

- Class-size can affect ratings. Students tend to rank instructors teaching small classes (less than 10 or 15) most highly, followed by those with 16 to 35 and then those with over 100 students. Thus, the least favorably rated are classes with 35 to 100 students (Theall & Franklin, 1991, p. 91).
- There are disciplinary differences in ratings. Humanities courses tend to be rated more highly than those in the physical sciences (Theall & Franklin, 1991, p. 91).

Many basic, and difficult, issues related to the use of SRT for evaluating teaching effectiveness have not been addressed here, such as how to *define* “teaching effectiveness.” I hope even this limited discussion has helped make readers more aware of issues surrounding the use of SRT, and that they will sample the resources and links provided.

**References**

Arreola, R. (2007). *Developing a comprehensive faculty evaluation system: A handbook for college faculty and administrators on designing and operating a comprehensive faculty evaluation system* (3rd ed.). San Francisco: Anker Publishing.

Berk, R. A. (2006). *Thirteen strategies to measure college teaching*. Sterling, VA: Stylus.

Benton, S. L., & Cashin, W. E. (2011). *IDEA Paper No. 50: Student ratings of teaching: A summary of research and literature.* Manhattan, KS: The IDEA Center. Retrieved from: http://ideaedu.org/wp-content/uploads/2014/11/idea-paper_50.pdf

Benton, S. L., & Ryalls, K. R. (2016). *IDEA Paper #58: Challenging misconceptions about student ratings of instruction. *Manhattan, KS: The IDEA Center. Retrieved from http://www.ideaedu.org/Portals/0/Uploads/Documents/IDEA%20Papers/IDEA%20Papers/PaperIDEA_58.pdf

Boring, A., Ottoboni, K., & Stark, P.B. (2016). Student evaluations of teaching (mostly) do not measure teaching effectiveness. *Science Open Research*. DOI: 10.14293/S2199-1006.1.SOR-EDU.AETBZC.v1

Chism, N. (2007). *Peer review of teaching: A sourcebook* (2nd ed). Bolton, MA: Anker.

Dewar, J. (2011). Helping stakeholders understand the limitations of SRT data: Are we doing enough? *Journal of Faculty Development, 25*(3), 40-44.

Ellis, J., Deshler, J., & Speer, N. (2016, August). How do mathematics departments evaluate their graduate teaching assistant professional development programs? Paper presented at the 40^{th} Conference of the International Group for the Psychology of Mathematics Education, Szeged, Hungary.

Fink, L. D. (2008). Evaluating teaching: A new approach to an old problem. In D. Robertson & L. Nilson (Eds.), *To improve the academy: Vol. 26 *(pp. 3-21). San Francisco, CA: Jossey-Bass.

Loeher, L. (2006, October). *An examination of research university faculty evaluations policies and practices. *Paper presented at the 31^{st} annual meeting of the Professional and Organizational Development Network in Higher Education, Portland, OR.

MacNell, L., Driscoll, A. & Hunt, A.N. (2015). What’s in a name: Exposing gender bias in student ratings of teaching. *I**nnovative Higher Education, 40*(4), 291-303. DOI:10.1007/s10755-014-9313-4

McKeachie, W. J. (2007). Good teaching makes a difference—and we know what it is. In R. P. Perry and J.C. Smart, (Eds.), *The scholarship of teaching and learning in higher education: An evidence-based approach *(pp. 457-474). New York, NY: Springer.

Pallett, W. (2006). Uses and abuses of student ratings. In P. Seldin (Ed.), *Evaluating faculty performance: A practical guide to assessing teaching*, *research, and service*. Bolton, MA: Anker Publishing.

Seldin, P. (Ed.). (1999). *Changing practices in evaluating teaching*. Bolton, MA: Anker Publishing.

Seldin, P. (2007). *The teaching portfolio: A practical guide to improved performance and promotion/tenure decisions* (3rd ed). Bolton, MA: Anker Publishing.

Stark, P. B., & Freishtat, R. (2014). An evaluation of course evaluations. *ScienceOpen Research*. DOI: 10.14293/S2199-1006.1.SOR-EDU.AOFRQA.v1

Theall, M., & Franklin, J. (Eds.). (1991). *New directions for teaching and learning: No. 48. Effective practices for improving teaching.* San Francisco, CA: Jossey-Bass.

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Over the years I have been asked the questions: Why do you direct undergraduate research? How do you pick a research problem for your students? How do you manage a research group? In this blog post I would like to present my personal points of view regarding these questions.

I have been involved in research with undergraduates since 2001. I have worked with students as part of REU programs at large research universities, at mostly undergraduate state universities, and at programs in mathematics institutes. I have also worked with small groups of local students. In 2001, I was a graduate TA at the Summer Institute of Mathematics for Undergraduates (SIMU), an REU program hosted at the University of Puerto Rico – Humacao that received the first ever Mathematics Programs that Make a Difference award from the AMS in 2006. This program fundamentally shaped my view regarding working on research with undergraduate students.

There are several resources for students to learn about undergraduate research programs. There are webpages with lists of active REU programs, some maintained by the AMS, NSF, and the Math Alliance and also some other independent websites. The webpages of many of these programs also relate the points of view of previous students and what makes these programs successful and special. There is an MAA website that, while outdated, does a nice job at answering the question: Is an REU for You? There are also more recent articles and blogs describing the relevance of undergraduate research from the students’ own point of view.

In contrast, there is only a limited amount of information targeting faculty who are interested in leading an REU but lack the experience to work with undergraduate students on research. Currently, the American Institute of Mathematics (AIM) and The Institute of Computational and Experimental Research in Mathematics (ICERM) offer a one-week workshop on Research Experiences for Undergraduate Faculty (REUF). The goal of this workshop is to equip faculty at primarily undergraduate institutions with tools to engage in research with undergraduate students. In 2014, Leslie Hogben and Ulrica Wilson published an article in Involve detailing this program. The REUF program has been very successful and I encourage anyone who wants to start advising undergraduate students in research to apply. As another example, for several years the Center for Undergraduate Research in Mathematics (CURM) has promoted academic year undergraduate research in mathematics. This program, located at Brigham Young University, has provided training and funds to professors to establish undergraduate research groups. Some of the materials developed in this program are accessible online. For those who are interested in reading more about leading REUs, the journal PRIMUS published in 2013 a Special Issue on Undergraduate Research in the Mathematical Sciences. All the articles in this issue are great reads for anyone interested in the topic. In the rest of this article, I will provide some personal reflections about leading REUs.

**Why do you direct undergraduate research?**

In my opinion, the job of a mathematician consists in learning, discovering, and disseminating mathematical knowledge. More than 60% of undergraduate mathematics degrees are awarded by colleges and universities that do not have doctoral programs. At these institutions, students may not necessarily get enough training focused on discovering and presenting mathematical knowledge. So, working with students on research is an essential complement to their undergraduate education.

Some mathematicians feel that students interested in research should simply continue to graduate programs and do “real research” there. First, I think that undergraduate research can be real research, and I will talk more about this later. But as a general response to this perspective, I offer an analogy. When a child starts to learn how to ride a bike, they go in stages. First, they have training wheels, then the wheels are removed but the very concerned adults remain jogging right next to the kids to catch them in case they are about to fall, and finally the stage arrives when the kids ride free and unassisted. The second stage is very short but important in building confidence and self-assurance. Undergraduate research plays the role of this second stage. In our classes, students learn mathematics using training wheels. The problems are not too difficult and they have all the tools that they need to solve the problem. Undergraduate research is a short experience where the safety net is removed, students explore their capabilities, but the faculty is nearby to make sure that students do not fall or, if they do, to encourage them to get up and continue working.

I also believe that it is important to provide mentoring of students from a diverse range of backgrounds and demographics. Doing research with students that are ready and prepared to do research is very exciting. Doing research with students that will greatly benefit from having companionship, guidance, and mentoring is similarly extremely fulfilling. For this reason, I have focused my attention on first generation and other underrepresented groups in the STEM sciences. At the end of the day, these groups of students have produced mathematics that are just as beautiful and significant as any other “top students”. But this group usually lacks knowledge about graduate school and the diverse jobs that exist outside of academia which require advanced degrees, and they tend to be more aware of their perceived mathematical limitations. They also have to fight against some preconceived notions in their family and society about their future careers.

**How do you pick a research problem? **

A research problem must truly be carefully selected to simultaneously provide an honest “real research” experience for the student but also an experience that is meaningful and productive. I do not think that the ultimate product should necessarily be a research paper. But at the end of the program, a student should be able to pinpoint some specific contribution to the subject that is entirely their own. I have always been open to both concrete problems and open-ended investigations that start without a clear target. The main thing is to pick a problem that is flexible enough to get adjusted according to students’ needs so that there can be a successful outcome at the end of the experience. Partial results, conjectures, and databases of non-trivial computations, or even a detailed report regarding the pitfalls in a certain approach are great examples of positive outcomes.

Yet how does one find such problems? It is usually difficult to find problems that satisfy the above constraints and that are also in the faculty’s research program. So one has to be willing to expand the search. In my opinion, there are three main sources: articles, talks, and conversations. Read undergraduate research journals like Involve, Principia, Rose-Hulman undergraduate math journal, SIURO and the Minnesota Journal of Undergraduate Mathematics. CURM has a more complete list of undergraduate math journals. Travel to conferences like the Joint Math Meetings, Math Fest, Field of Dreams, SACNAS or the NSF Mathematics Institutes’ Modern Math Workshop (at SACNAS). Most of these conferences have poster and/or talk sessions devoted to undergraduate research. Talk to colleagues or presenters at conferences or workshops.

Through the years, I have done a combination of the three activities detailed above. I would usually write a note on the main area of research, a certain open problem, and some references. I would then read a couple of introductory papers, write a short introduction to the problem and perform some computations. At the end, I have a 4-5 page self-contained note that I can use to remember the problem or give to a group of students. Some of these notes get refined over the years as students work on some aspects of the problem or discover new avenues to pursuit.

**How do you manage a research group?**

First, I always start a collaboration with a crash course on the subject. My goal is to provide all the necessary information that the students need to understand the given problem and be able to do some experiments. I do this to cover the background material in the shortest possible time but also to establish a relationship with the students. Once students start working on their problem, I meet with them every day. In a short 15-20 minute meeting, each group presents the advances and challenges of the previous day. Only one student in the group presents on a given day and he or she must discuss the advances of the entire group. Students rotate through the week and at the end of the week they give a beamer presentation on their weekly advances. After this presentation, I discuss the goals for the weekend and the following week but also improvements on their presentation and their report. I have found that at the end of the program, it is much easier to compile the partial reports into the final report and the weekly presentations into a poster or a final talk.

As I have mentioned above, the experience must be a real research experience. So I listen to the students, point them to some useful references. I also give them suggestions. But mostly I act as a cheerleader. Using the bike analogy as before, I am not as close as to catch them when they fall, but I am always right there to cheer them up and continue looking at the road ahead.

**Final Comments**

Undergraduate research is usually a short time event. But faculty involvement is a long race. It takes time to find good problems. It also takes time to learn how to interact with students to improve their abilities and confidence while making sure that students retain ownership of their own work. It takes time to successfully find funding to support this activity. And despite all the time it takes, one only affects very few students every year. But even then, for me it has been one of the most rewarding activities that I have been involved in.

]]>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:

- 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.
- 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.

**Mathematical mindsets**

In her research, Carol Dweck describes implicit theories of intellectual and social traits that influence how and whether people choose to invest effort in developing skills (see, for example, Dweck, 2008). Dweck uses the term *entity theory *to refer to the idea that traits such as mathematical skill are innate, and that adversity and failure are indications that one does not possess these traits. She uses the term *incremental theory* to refer to the idea that traits such as mathematical skill are malleable and can be developed through sustained effort. Students who have an entity theory of mathematical intelligence often demonstrate a “fixed mindset” in mathematics classes, interpreting challenges as opportunities to display their innate abilities in mathematics, or as threats to their mathematical identity. On the other hand, students who have an incremental theory often demonstrate a “growth mindset,” embracing challenging and open-ended tasks as opportunities to discover and develop new ideas.

Jo Boaler’s recent book *Mathematical Mindsets* (2015) provides a wealth of advice on how to structure mathematics instruction to promote the development of growth mindsets. She recommends practices that recent research has proven successful, such as the use of low-threshold-high-ceiling problems that are accessible to all students but require extended effort to solve completely, and strategies for managing groupwork that are consistent with Complex Instruction (Cohen *et al.*, 1999).

**Integrating the growth mindset into assessment and grading**

I have taken some steps to reframe assessment and grading in my courses as a way of stimulating growth and providing guidance for learning, rather than rewarding success or punishing failure.

*Specifications grading.* Specifications (specs) grading (Nilson, 2015) is a system in which students earn course grades by meeting a set of clearly defined criteria rather than by achieving a certain weighted average across exams, homework, and other assignments. I now use specs grading in all of my courses; in most cases, to earn a grade of A, students must pass exams with specified scores, give a successful presentation in class, and earn a passing score on homework problem sets. I also include class attendance and participation in my specs grading scheme; since I started doing this, I have had over 90% attendance in my classes. I have an “exception clause” in which students who fall short of a standard in one category can compensate by exceeding standards in another category. This provides flexibility and sets the tone that there are many ways to demonstrate mastery. In November 2015, Kate Owens wrote about a similar system called standards-based grading (SBG); while SBG is generally organized around learning goals rather than assignment types, both systems have the essential feature of providing opportunities for students to deepen their own mastery of course content.

I’ve found that specs grading provides greater clarity for students; at the end of the semester, there is little mystery about what students must do in order to earn a desired grade. The grading scheme also allows me to be serious about things that matter: since I’ve adopted this system, I’ve never had to award a course grade of B to a student who did extremely well on homework (by getting external assistance) but did not demonstrate any mastery of the course material on exams.

*Revision policy.* I knew that if I implemented a specifications grading scheme and did nothing else, I would only end up being stingier with grades. I wanted to reshape my course policy into one that embraces mistakes as opportunities for growth and learning. Therefore, I have the policy that any written homework assignment in my course can be revised. Students get constructive feedback on problem sets; if they read the feedback and submit revisions, I replace the old grade with a better one. I used to impose a nominal penalty (say, one point out of ten) on revisions, but I stopped doing this because I could no longer defend a practice that punished students for making mistakes. The revision policy allows me to be much more consistent in holding students accountable for producing high-quality work. This does not cause too great an increase in my overall grading load, because students’ revised work is usually of higher quality and therefore easier to grade.

*Exam scores as “work in progress.”* Exams have a way of bringing students’ sense of non-belonging in mathematics into sharp relief. I try to manage exams in my classes in ways that encourage growth and do not position students as competing with one another. First, I set cut scores for each exam based on the difficulty of the test itself, not based on a “curve” nor in response to student performance on the test. I don’t use a 90-80-70-60 scale to interpret exam scores; there is nothing mathematically natural about this scale (Reeves, 2006), and it offers little hope for students who earn a score in the 20s or 30s on a test.

I make it clear that exam scores are “in progress” until the end of the semester, as each student earns a number of “extra lives” that can be used to retry exam questions at the end of the course. Students earn “extra lives” by doing things that will help them succeed in the course, such as completing the homework, doing practice problems, and completing short “Lesson Launch” assignments in which they watch video examples prior to class and write summaries (as in some “flipped” instruction models). On the last day of class, students take a customized test with questions covering topics on which they didn’t demonstrate mastery during the midterm exams. Their scores on these questions replace their old midterm question scores.

Finally, I try to make sure my own messaging about exams is consistent. After each exam, I send a brief e-mail summarizing a few places on the exam where I thought the class as a whole did well, and reinforcing the “growth mindset” message. My most recent e-mail contained the following:

I don’t make it a practice to give class-wide statistics from the exam … My purpose in giving exams (as with all the other work) is to give you opportunities to discover where your knowledge is already strong, and where you still have room to grow. I’d sooner see you spend your energy and effort on learning the material you haven’t mastered yet, rather than positioning yourself with respect to your classmates. My view is that this class has 19 terrific mathematical thinkers, and your current score on this exam is an indicator of your current level of mastery of this material, not of how smart you are in mathematics. I say “current score” because as you know, under the Extra Life system your score on this exam may well improve at the end of the semester if you do a good job of learning the material you haven’t mastered yet.

**Impact on students**

Students in my courses seem to appreciate the various opportunities to revisit and improve their work. In a typical semester, I will receive homework revisions from about 75% of my students, with some students submitting revisions for as many as 50% of the problems. The revised solutions that students submit are usually substantial improvements; the majority of revisions earn at least two additional points on a five-point scale.

I asked students in my Fall 2016 capstone course for preservice secondary teachers for feedback on how the course policy influenced their learning and their identity as mathematics learners. One student responded,

A huge benefit was that we could correct our assessments after being graded. It made us go back and actually think about every problem and how we could correct it. You gave great feedback and showed that you were willing to help us.

Another student commented not only on the revision policy, but on the overall tone of collaboration and personal growth it set:

Feedback – Fantastic. No other course has allowed me to continuously correct my work. Although grading must be time consuming, it’s greatly appreciated.

Engagement/Involvement/Interpersonal Connection – I felt a unique atmosphere of collaboration in MAT 4303. The trichotomy of student, instructor, and group motivated me to consistently work to the best of my ability. This is especially true in homework. In most courses, if I have an imperfect solution to a problem that I can’t resolve, it remains imperfect. This was not true for MAT 4303.

Above all else, MAT 4303 helped me to mature as a student and take a collaborative, selfless approach to courses. I enjoyed helping other students learn just as much as I enjoyed learning.

**What’s next?**

In the future, I also hope to research the use of growth mindset assessment practices and investigate their effects on students’ learning and mathematical identity. Every program in which I have taught has wanted students to be more confident in their potential to solve challenging problems and more motivated to pursue learning opportunities on their own. I believe that an indispensable step in this direction is to help students develop the mindset that even when initial efforts fail, their hard work will result in powerful and long-lasting intellectual growth.

**Acknowledgement: **I would like to thank my colleague Dr. Priya V. Prasad, who uses a similar system in her courses at UTSA and whose feedback led to improvements in my own implementation.

**References**

Aronson, J., Fried, C. B., & Good, C. (2002). Reducing the effects of stereotype threat on African American college students by shaping theories of intelligence. *Journal of Experimental Social Psychology*, *38*(2), 113-125.

Blackwell, L. S., Trzesniewski, K. H., & Dweck, C. S. (2007). Implicit theories of intelligence predict achievement across an adolescent transition: A longitudinal study and an intervention. *Child development*, *78*(1), 246-263.

Boaler, J. (2015). *Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching*. John Wiley & Sons.

Butler, R. (1987). Task-involving and ego-involving properties of evaluation: Effects of different feedback conditions on motivational perceptions, interest, and performance. *Journal of educational psychology*, *79*(4), 474.

Cohen, E. G., Lotan, R. A., Scarloss, B. A., & Arellano, A. R. (1999). Complex instruction: Equity in cooperative learning classrooms. *Theory into practice*, *38*(2), 80-86.

Dweck, C. S. (2008). *Mindset: The new psychology of success*. Random House Digital, Inc.

Good, C., Rattan, A., & Dweck, C. S. (2012). Why do women opt out? Sense of belonging and women’s representation in mathematics. *Journal of personality and social psychology*, *102*(4), 700.

Leslie, S. J., Cimpian, A., Meyer, M., & Freeland, E. (2015). Expectations of brilliance underlie gender distributions across academic disciplines. *Science*, *347*(6219), 262-265.

Nilson, L. (2015). *Specifications grading: Restoring rigor, motivating students, and saving faculty time*. Stylus Publishing, LLC.

Reeves, D. B. (2006). *The learning leader: How to focus school improvement for better results*. ASCD.

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:

- Use students’ interest in contextualized tasks
- Expose students to a diverse group of mathematicians
- Design assessments and assignments with a variety of response types
- Use systematic grading and participation methods
- Consider your course logistics
- 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.

**1. Use students’ interest in contextualized tasks. **

What communities and interests are represented in the problems you assign students? How do these backgrounds align with those of your students? Researchers have shown that students are more motivated in material when it is applicable to their own interests and communities (Carlone & Johnson, 2007; Jones, Howe, & Rua, 2000). In order to identify the interests of our students, consider giving your students a survey to ask for their hobbies, motivations for taking the course, and career goals (for example, a form I made for my own students is available here). Use what you learn about your students while framing the mathematical tasks and problems. Be sure to consider if the tasks you assign represent *all* of those interests in your classroom and which students might be left out.

In a traditional calculus course, for example, a common topic is related rates. Some typical related rates problems involve falling ladders, hemispherical reservoirs, and cars and trucks and things that go. These applications may be very appropriate for students with these interests or interests in certain types of engineering. However, exposing students exclusively to such applications signals to students who do not have such interests that mathematics is not relevant for them. Consider diversifying such tasks and (depending on the interests of your students) include applications to medicine, biology, conservation, music, baking, etc. Here are a few suggestions I have used with my own students.

*Chris and Jake are ***cooking*** pancakes. Jake ladles the pancake batter into the fry pan. While the pancake cooks, the radius of the circular pancake formed increases at a rate of 1 cm per minute. How fast is the circumference changing when the radius is 7 cm?*

*At a ***conservation*** site in the Amazon rainforest, a hyacinth macaw ***parrot*** is spotted flying horizontally 37 feet above a research site. The parrot is flying at 20 ft/sec. How fast is the distance from the parrot to the research site changing when the bird is 35 feet away?*

*The ***velocity of blood*** in a human’s blood vessels is related to the radius R of the blood vessel and the radius r of the layer of blood in the blood vessel. This relationship known as Poiseuille’s law and is given by** *\(v=375(R^2-r^2)\)*. Assume the radius of the layer of blood r is constant but cold weather causes the radius of the blood vessel R to contract at a rate of 0.01mm per minute. What is the velocity of blood flow when the radius R of the blood vessel has contracted to 0.03mm?*

**2. Expose students to a diverse group of mathematicians.**

Who are the mathematicians you tell your students about? Are they white, male, and introverted? These common stereotypes make students who do not identify with such qualities feel they do not belong in mathematics (Carlone & Johnson, 2007; Cheryan & Plaut, 2010; Good, Aronson, & Harder, 2008; Thoman, Arizaga, Smith, Story, & Soncuya, 2014). Diversify your students’ image of mathematicians by highlighting mathematicians who do not fit the typical stereotype. Describe mathematicians as multidimensional individuals with struggles, hobbies, and families. Communicating short biographies to students and showing students pictures of mathematicians from underrepresented groups are great ways to do this. If students are able to see mathematicians as genuine individuals, they are more able to identify with them and see themselves in mathematics. For resources to increase your own exposure to individuals in the mathematics community, consider perusing books and websites that highlight important contributions from women or individuals from underrepresented ethnicities in the field. For example, check out the Mathematically Gifted and Black website, recent articles such as Lathisms: Latin@s and Hispanics in Mathematical Sciences and The Black Female Mathematicians Who Sent Astronauts to Space, or books such as Kenschaft, 2005 and Murray, 2000.

Communicate stories of mathematicians to students while engaging in mathematical contributions the individuals have made. For example, in a calculus course, consider discussing with students the mathematics related to the curve known as the “Witch of Agnesi” (a mistranslation from *averisera* meaning “turned sine curve”) given in Figure 1 from Weiqing Gu’s website. This curve was studied in the calculus textbook *Analytical Institutions* written by the Italian mathematician Maria Gaetana Agnesi (1718-1799). Agnesi published this text at the age of thirty; she began writing the book at age twenty, originally writing the text as a resource for her brothers (Osen, 1975). The curve can be constructed by tracing the points P obtained from the \(x\) (horizontal) and \(y\) (vertical) coordinate of the points \(A\) and \(Q\) (respectively) in Figure 1 below. The curve can be given parametrically as \(x(t)=2a \cot(t)\) and \(y(t)=a[1-\cos(2t)]\) (for \(0 \leq t \leq \pi\) and a suitable positive constant \(a\)). Activities for students related to this curve could have students construct the parametric equation (from a more suitable description of the curve) or deriving an equation of the tangent line at any point P on the curve (see MathForum for a construction of the curve).

**3. Design assessments and assignments with a variety of response types. **

We as mathematics instructors have been successful mathematics students. Thus, many of us have likely found success with traditional mathematics assessments in traditional settings. However, not all students succeed in such environments. Create and structure assignments to include a variety of types of problems as well as settings. For example, consider including problems that ask students to write long responses to explain their thinking or draw a visual to demonstrate an argument. Vary the test environment by allowing students to work in groups or give a take-home assessment in order to give students flexibility in the amount of time for completion. Consider allowing students to retest. This strategy has been shown to provide students who experience math anxiety with a mental “safety net” that can help alleviate some of the pressures involved in testing and improve their test performance (Juhler, Rech, From, & Brogan, 1998). If you are not able to vary the assessments in your course (possibly due to departmental or other constraints), consider using these suggestions in class assignments or quizzes.

In my own courses, I encourage students to express and develop their thinking outside of class through something I call “Try it” opportunities. Try it opportunities have included responding to open-ended questions I post on our course discussion board, posting practice test solutions, or responding to fellow students’ practice test solutions. I also encourage students to tweet class summaries to my professional Twitter account, bring to class a picture or news article of a math concept we have discussed in class that they see in their own lives, or take a photo of their math study group. The following is an example of an open-ended question I asked students in a recent geometry for teachers course on our course discussion board as part of a try it opportunity.

*Discuss the following two statements: “two triangles put together always make a square” and “a square cut in half always forms two triangles.” Are these statements true? Why might a student think this? What would you tell this student?*

The question provided students an opportunity to think about their own conceptions of triangles and shapes before we formally discussed the topic in course. The setting of the discussion board environment allowed students the flexibility in their timing of response and opportunity to reflect on other students’ thinking.

**4. Use systematic grading and participation methods.**

Who are the students that you expect to succeed in your course? Who are the students whose contributions you encourage in class? Teachers often have expectations and judgments of different groups of students based on student identity (Anderson-Clark, Green, & Henley, 2008; Riegle-Crumb & Humphries, 2012; Van den Bergh, Denessen, Hornstra, Voeten, & Holland, 2010). It has also been reported that teachers provide a “warmer” academic climate to students for whom they hold higher expectations, in the form of in-class interactions and assignment feedback (Rosenthal, 2002). Such treatment has positive effects on student performance.

Attempt to hold all of your students to the same high standard. Consider implementing systematic ways of getting student participation and methods of grading. Keep a record of which students participate in your class and make an effort to elicit contributions from all students. While grading, create a rubric to evaluate student work. After grading, look over the comments and feedback you give your students. Do all students have similar depth and specificity of feedback? Consider having a colleague who is unfamiliar with the identities of your students look over a sample of the work you have graded and provide *you* feedback on the types of responses you give to your students.

**5. Consider your course logistics.**

*Office Hours. *What time do you host office hours? Are they immediately after class when a student might have to rush off to work in the campus cafeteria? Or early in the morning when a student might be commuting into campus? Another useful item for a pre-semester survey is a question about the best times for office hours.

*Deadlines. *When are your assignments due? What obligations do your students have outside of your course? Requiring students to turn in a homework set to your office door by 5PM might not be doable for a student who has to work until 6PM. Having an online homework set due on Sunday evening might not be feasible for a student without access to a computer on the weekends.

*Technology. *What technologies do your assignments require? If your department requires online quizzes or homework, is there technology on campus that students can use to complete these assignments? Know when such resources are available to students and be sure your students know as well.

**6. Encourage students to embrace a growth mindset.**

Carol Dweck’s popular work shows that individuals’ mind-sets regarding intelligence can influence their academic motivation and performance (Dweck, 2008). Dweck describes students with a fixed mindset as having a static view of intelligence and students with a growth mindset believing intelligence can be developed, the latter mindset being able to persist in the face of challenges and setbacks and grow in the process.

Remind students that mistakes are an essential part of learning and a vehicle for growth. Provide feedback on students’ strategies and reasoning, rather than just their answers. Celebrate students’ effort and persistence and avoid praising a student for getting an answer quickly. Treat exams as an opportunity for students to demonstrate their effort and understanding rather than their intelligence and ability. Allow students to engage in productive failure by providing limited scaffolding and challenging students to collaborate with each other (Kapur & Bielaczyc, 2012).

In my own classes, I begin the semester by assigning my students the task to watch and give a short reflection on the TED talk by Eduardo Briceno on “Mindsets and Success.” My students’ responses reflect encouragement from the talk; many express a shift from believing they are “not good at mathematics” to believing they are “not good at mathematics *yet*.” I then usually emphasize to students that they *are* already mathematical thinkers but with persistence and effort they can feel success in our mathematics course.

I hope these strategies invite you to reflect on your teaching practices and consider the influence we can have in creating inclusive classrooms that support diversity. As a final recommendation, I hope this post can start or continue conversations with colleagues on the topic of diversity and inclusion in mathematics. Having a community to discuss and develop the ways we teach and interact with students is essential for making such efforts lasting and productive.

**Acknowledgements:** I would like to express my gratitude to Laura Provolt, Debbie R. Hale, and Dr. Kecia M. Thomas for their helpful feedback and many insightful discussions.

References

Anderson-Clark, T. N., Green, R. J., & Henley, T. B. (2008). The relationship between first names and teacher expectations for achievement motivation. *Journal of Language and Social Psychology, 27*(1), 94-99.

Carlone, H. B., & Johnson, A. (2007). Understanding the science experiences of successful women of color: Science identity as an analytic lens. *Journal of research in science teaching, 44*(8), 1187-1218.

Cheryan, S., & Plaut, V. C. (2010). Explaining underrepresentation: A theory of precluded interest. *Sex roles, 63*(7-8), 475-488.

Dweck, C. S. (2008). *Mindset: The new psychology of success*: Random House Digital, Inc.

Good, C., Aronson, J., & Harder, J. A. (2008). Problems in the pipeline: Stereotype threat and women’s achievement in high-level math courses. *Journal of Applied Developmental Psychology, 29*(1), 17-28.

Jones, M. G., Howe, A., & Rua, M. J. (2000). Gender differences in students’ experiences, interests, and attitudes toward science and scientists. *Science education, 84*(2), 180-192.

Juhler, S. M., Rech, J. F., From, S. G., & Brogan, M. M. (1998). The effect of optional retesting on college students’ achievement in an individualized algebra course. *The Journal of experimental education, 66*(2), 125-137.

Kapur, M., & Bielaczyc, K. (2012). Designing for Productive Failure. *Journal of the Learning Sciences, *21(1), 45-83.

Kenschaft, P. C. (2005). *Change is possible: Stories of women and minorities in mathematics*: American Mathematical Soc.

Lopez, A. D., Sosa, G., Langarica, A. P., & Harris, P. E. (2016). Lathisms: Latin@ s and Hispanics in the Mathematical Sciences. *Notices of the American Mathematical Society, 63*(9), 1019-1022.

Murray, M. A. M. (2000). *Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America*: MIT Press Cambridge MA.

Osen, L. M. (1975). *Women in mathematics*: Mit Press.

Riegle-Crumb, C., & Humphries, M. (2012). Exploring bias in math teachers’ perceptions of students’ ability by gender and race/ethnicity. *Gender & Society, 26*(2), 290-322.

Rosenthal, R. (2002). Covert communication in classrooms, clinics, courtrooms, and cubicles. *American Psychologist, 57*(11), 839.

Thoman, D. B., Arizaga, J. A., Smith, J. L., Story, T. S., & Soncuya, G. (2014). The Grass Is Greener in Non-Science, Technology, Engineering, and Math Classes Examining the Role of Competing Belonging to Undergraduate Women’s Vulnerability to Being Pulled Away From Science. *Psychology of Women Quarterly, 38*(2), 246-258.

Van den Bergh, L., Denessen, E., Hornstra, L., Voeten, M., & Holland, R. W. (2010). The implicit prejudiced attitudes of teachers: Relations to teacher expectations and the ethnic achievement gap. *American Educational Research Journal, 47*(2), 497-527.

Whitney, A. K. (2015). The Black Female Mathematicians Who Sent Astronauts to Space. 2017, from __http://mentalfloss.com/article/71576/black-female-mathematicians-who-sent-astronauts-space__

*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:

*Collaboration:*the importance of collaboration, friendships, and networking*The Nature of Mathematics:*an appreciation or gained understanding of the nature of mathematics and mathematical research*Self-Beliefs and Agency:*heightened awareness and/or insight about oneself as a learner or person in general*Back to the Classroom:*the positive impact of REUs on subsequent coursework*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.

**Collaboration**

In the interview, a repeating theme was the value of collaboration for students faced with the task of engaging with challenging mathematics at a new level of depth, starting from the first day of the REU.

- The first two days were just lecture and these were all new ideas to me, words I… hadn’t heard before. Much less know how to compute things using them and that kind of thing… The comfort came in the fact that the other students felt the same way.
- There’re really solid friendships that come out of that struggle.
- What I think I’ll remember the most is the friendships.

The students’ friendships with each other were valuable because they found other people from different places and different backgrounds, in some cases finding a mathematical community for the first time.

- I appreciate the REU because it introduced me to a lot of people really serious about math and that’s something I don’t get too much at my college. There are a couple of people interested in grad school but not nearly as many as being at an REU and… now when I come to conferences I know a lot more people and it’s, we’ve kind of like supported each other through the application process of applying to grad school. So I guess, I mean like a lot of people I see myself being friends with pretty much for the rest of my life, being that we plan on being in the same field and I really, I really do appreciate that.
- It’s very easy to make good friends because you drop the pretense in some sense so it’s just a lot of fun. But yeah, definitely walking in ‘cause the different backgrounds you’re kind of able to collaborate in a different way from class … So you’re trusting someone else’s background, their knowledge, their expertise in these kinds of things and it allows you to work better as a group but also simultaneously it makes it easier to pull back.
- I think meeting different people from different places is the other thing you get out of REU and that should stick with you.

Eventually, in some cases, students formed efficient teams, creating “magical” moments.

- A lot of what I learned really happened outside of the hours in which we did research back at someone’s apartment asking, “Hey what does this mean again?” or “Can you explain this to me?” or “This came up today and at the time I didn’t know what it really meant,” and so I think that’s just a rewarding experience that I’m sure I’ll be able to take with me for the rest of my life.
- Working on the actual problem… there was little bit more of that give and take that [another student] spoke about. One was an applied math major I believe and the other a statistics major and I’m just a regular math major. So I guess I kind of contributed helping put stuff in “math speak”, and I guess that they contributed in other ways: ‘cause we were looking for patterns so one of the people had taken a number theory course and so he helped me and the other person in other areas.
- Attending the REU really kind of opened my eyes into the magic that can happen when you have a whole group of people who are interested in the same topic… I saw that change happen this past semester coming back from the REU, just my ability and my willingness to kind of invite that collaboration and to search for it in a greater way than I ever had before
- There are seven other people on this planet that have seen me at my most frustrated moment and at my happiest moment. You know you come to care about these people a lot… it’s just such a rewarding thing to know there are other people out there who have had the same passions as you and who will always kind of be that support team for you and who are going through the same, the same sort of I guess struggles that happen when you’re an undergrad looking toward a career in mathematics

**The Nature of Mathematics**

At their REUs, students learned a lot of new mathematics, delved deeply into the process of doing mathematics, and discovered how research is different than classwork.

- I think because these programs are well-organized [by] people who are really interested in helping and working with students, it [was] a little bit easier… I think for the most part trying to understand this background and the context with… the end goal of genuine research is much different than, I don’t know, someone sitting in a class and proving theorems that have been proved, I don’t know, hundreds of years before or something.
- If you’re just used to taking math classes the problems that you have experience solving are more straightforward and you have an idea of what tools you’re supposed to be using and what you’re drawing on. But then once you get into math research it’s like, you have everything anyone’s ever done to draw on and that can feel very overwhelming but also very exciting… Progress is not linear necessarily so it might feel like you’re making no progress at all for like several weeks and then suddenly you have a breakthrough and everything comes together.
- Our group had a few people who were much more experienced and then a couple of juniors and we just hadn’t seen a lot of things. So there were some things that came up all the time and we just didn’t know what they were and so getting to finally learn that stuff was so exciting.

Many students discovered that the world of mathematics is bigger than they had realized.

- Getting introduced to the math world and what math research is, is like almost bigger than learning the math than you need for your project.
- One thing I’ll definitely remember is a trip that we took to MathFest that summer and that was my first math conference and that was such an eye opener, because the REU was actually also at my college. So I had only, you know, really experienced math through this very small community at my school. And to suddenly be opened up to this world of thousands of mathematicians and all these different areas of math research that was all super exciting and something I’ll definitely remember.
- I felt, after my REU, that a lot about math not just as a subject but also culturally as a community in a lot of ways became demystified… for example, like, reading math research or something like that or reading math papers and this kind of thing just feels like something far off on the horizon or something like this and then you just sit down and do it.
- There’s so much math out there so it’s kind of, you know the saying “you don’t know what you don’t know”… I just learned that there’s so much more out there involving math that I had no idea existed. Which is really exciting for me so I guess what will affect me the most is it introduced me [to] something that I think will motivate me for the rest of my life.

**Self-Beliefs and Agency**

A key feature of these REUs was that students gained confidence and independence, became more comfortable admitting what they don’t know, and learned new ideas.

**You go to an REU and now you’re working with people from all universities. Everyone has a sort of different knowledge base, some people are more interested in things that you’ve never heard of and so what I found was this past semester these terms coming up, “oh hey I remember talking about that at the REU and I didn’t know what it meant then but now I have this motivation to learn what it means” because those conversations kind of made you want to learn these things that these other students knew.**- You find this sense of independence and you start learning these things and it is a very rewarding experience to look back after those two months time. You know, I didn’t even know what this word meant two months ago and now I like to talk to people about it kind of thing.
- I wish I had been a little more comfortable not knowing every last bit of the topic we were working on… trusting that you’re not going to spearhead every aspect of your project. There are other people in the group who are going to understand some things better than you.
- You can’t really have an ego, it’s very obvious what you know and what you don’t know.

In particular, students had to learn when and how to ask for help.

- We had very different levels of experience [with] the area that we were working in and so I think one of the difficulties was trying not [to be] discouraged by that. Because a lot of the time it can feel like you’re not contributing equally but just because you’re not, you know, the one like coming up with something new every single day, that doesn’t mean that you don’t have a role to play. And one other thing that I sort of learned over that summer was how to be assertive and make sure I knew what was going on and be willing to ask my group members to explain something in more detail so that we could all be on the same page. And that’s something they’re always willing to do but you sometimes have to ask for it, ‘cause they’re not always going to realize that they’re moving really fast and not everyone’s following.
- The REU experience, first off, is that it’s a very humbling experience, right. You find out what you know and what you don’t know very quickly… the most progress I think is when you finally admit that to yourself … It can be an uncomfortable thing but I think once you allow yourself to admit what you know and what you don’t know it opens the door, it opens the door to learn a lot more.
- It’s ok to ask people for help and there’s always going to be someone who knows something you don’t know. And you don’t have to look at that as any sort of discouragement towards yourself. It’s just an opportunity to learn something new. So I think that’s a very valuable group dynamic.

**Back to the Classroom**

When students returned to their home institutions following the REU, their summer experience influenced their work in subsequent mathematics courses.

- It was just very motivating and I found a lot of tie-ins to the course work I did the semester after the REU.
- [The REU] improve[d] my mathematical maturity so when I went to take analysis… it was much easier to read the analysis book than it had been to read the math stats book last year.
- I think it was from the first time I ever had to read some very difficult books. The books we had to read from were like really really tough to crack and they were using kind of a language I wasn’t very comfortable in… this made it easier I think to go back into the classroom and maybe read books.
- For me it kind of made it harder to go take classes because I enjoyed doing research so much.

**Graduate School**

The REU experience motivated students to consider whether or not to go to graduate school.

- It got me more motivated to go to graduate school
- Some of us have already decided “I don’t want to do research”… I’m really glad I did that [REU] so I found that out.
- Going into my REU I was sort of thinking of it as something, like, tell me whether or not I would like math research and whether or not I should think about going into [it]… and I think now, a year and a half later, I don’t think [this is] at all how you should approach your REU. I mean if you really really like it that’s definitely a good sign (chuckles) but if there are some things that you don’t like, that doesn’t necessarily mean that research is not for you.
- There are just so many factors going on in REU that I think even if it doesn’t go exactly how you would of wanted it to, you should still think about giving research another chance.

**Overall Observations**

Students agreed that one should not judge an REU by the first few days, and that the total REU experience was worthwhile and rewarding.

- One thing I wish I had known is that just as much as the details of your project, one thing that can determine like how the REU goes is your advisor’s advising style.
**Your experience the first two days is not representative of the whole experience.**- I don’t think you’re going to get the chance to work with professors who are so patient, so knowledgeable and so interested in their fields, I mean the amount of patience that it takes on the part of REU by the advisors is just entirely astounding to me. (laughing) It’s, to get someone from not knowing the definition of the basic object that you’re working on and getting them to produce a result I just I don’t even know what that takes.
- You know what everyone is facing and so it’s a really, it’s just a rewarding experience and I think that persevering past those initial discomforts is key to success.
- The graph of the REU experience is like the graph of the sine function: you’re going to have highs and lows and I promise you those positive parts are going to stick with you at the end.

**Interview Participant Bios **

David Burton completed his undergraduate degree at East Tennessee State University during the fall of 2016 and has just started as a graduate student there. He participated in an REU at the University of Connecticut Health Center for Quantitative Medicine during the summer of 2016, where he worked on an ongoing project entitled “Functional data analysis of copy number alterations in bladder cancer tumor chromosomes.” Next year, he hopes to enter a Ph.D. program in statistics.

Kelly Emmrich is a junior at the University of Wisconsin, La Crosse. She participated in the REU at California State University, Fresno, during the summer of 2016, where she worked on a project in complex analysis called “Sufficient conditions for a linear operator on R[x] to be monotone.” After graduation, she plans to enter a Ph.D. program in mathematics.

Micah Henson is a senior at Spelman College. She participated in the Mathematical Sciences Research Institute Undergraduate Program during the summer of 2016, where she worked on a problem in algebra called “The sandpile group of thick cycle graphs.” After graduation, she plans to pursue a Ph.D. in mathematics.

Andres Mejia is a junior at Bard College. He participated in the WADE Into Research REU at Wake Forest University during the summer of 2016, where he worked on a project in number theory called “Classically integral quadratic forms excepting at most two values.” After graduation, he plans to enter a graduate program in mathematics.

Nina Pande is a senior at Williams College. She participated in the SMALL REU at Williams College during the summer of 2015 and an REU at the University of Michigan during the summer of 2016. At Williams she worked on a problem in commutative algebra called “Controlling the dimensions of formal fibers,” and at Michigan she worked on another problem in commutative algebra called “First Koszul homology over a local Artinian ring.” After graduation she will teach high school mathematics in North Carolina before entering a graduate program in mathematics.

]]>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.

**1. Excellence is possible, perfection is not**

While perfection is impossible, excellence as a teacher is achievable, though elusive. The most difficult part of teaching for me is that I deeply want *all* of my students to succeed. In reality, this rarely happens, for many different reasons. Nevertheless, it is possible to reach excellence in teaching and learning, even if that doesn’t translate into an idealized outcome for every student in every course. By holding ourselves to the standard of excellence rather than perfection, it also becomes easier to hold our students to more reasonable standards of excellence as well. I have found that it is easy for me to slip into a mode where I am disappointed when my students don’t reach what I feel is their full potential, but by doing so I can also miss the opportunity to recognize the successes that they do achieve. To seek excellence rather than perfection, in ourselves and others, allows us to maintain our ideals while accepting the challenge of our reality.

**2. All human systems have flaws**

Colleges and universities are complex institutions, many of which serve diverse communities of students and employ faculty in a broad range of positions. Like many of my friends and colleagues, I have at times become focused on specific institutional flaws that are impossible to effectively address, often at the expense of turning my energy toward reaching more tractable goals. It is common to hear people say “pick your battles,” but at an institutional level I prefer to phrase this as choosing to engage with certain challenges and to yield to other challenges. There are many times when yielding to a challenge can provide significantly more freedom than fighting “the good fight.”

Consider Richard Tapia, a mathematician at Rice University, who received the National Science Board’s 2014 Vannevar Bush Award for “his extraordinary leadership, inspiration, and advocacy to increase opportunities for underrepresented minorities in science; distinguished public service leadership in science and engineering; and exceptional contributions to mathematics in the area of computational optimization.” In a video produced by the NSF, Tapia states:

When I started, I was so naive I thought I could change my colleagues, OK. You don’t change colleagues. You get them to maybe tolerate things you’re doing. You know, “Richard Tapia does ‘this’.” But where I see things changing and things going on is through my students. Without even directly telling them “you have to do ‘this’ and ‘this’,” they see it by example. And so I am really satisfied when I see how many students of mine are doing exactly what I was doing.

What Tapia describes is not a direct confrontation with the cultural norms and reward structures that influence his colleagues, but rather a yielding to these forces and a redirection of his energy in more effective directions. While there are certainly times when we must directly challenge flawed systems, we must also recognize that for many institutional problems, we create a higher impact by yielding to them in the short term and making progress through a different approach.

**3. Maximize student learning within a set of constraints**

In addition to large-scale institutional flaws, there are many additional constraints on our teaching. It is important to remember that our goal is not to have perfect learning from every student, but rather to maximize student learning given these constraints. The way we do this will vary dramatically given our situations, but there is a core principle that we can and should always rely on: *focus on the experiences of our students*. Here is an example of what I mean.

At the University of Kentucky, our first-year courses in Calculus (for students majoring in engineering, mathematics, and the physical sciences) are taught in large lectures of ~150-180 students with ~32-student recitation sections led by graduate teaching assistants. The first three times I taught these courses, the outcome was mediocre at best. My original strategy with large lectures was to import the best methods I had developed for small-scale teaching into the large lectures, but they were not effective. The constraints for teaching large lectures are completely different from those for my small courses, and the solutions I had used to maximize student learning in my small courses were not optimal solutions for the large courses.

The fourth time I taught a large-lecture calculus course, I completely yielded to these constraints. I was not excited about this, and was not expecting the course to be enjoyable for anyone, myself or my students. I could not have been more wrong — this was one of the most memorable courses I have ever taught, and my students were both successful and happy with their experiences. In hindsight, I realized that yielding to hard constraints had led me to a profound change in my perspective about large lectures: my primary focus should be to identify positive aspects of the large-lecture environment *from the perspective of my students* and take advantage of these as much as possible. Previously, I had focused almost exclusively on the negative aspects of large courses *from my personal perspective as a teacher*. This caused me to overlook most of these potential positive aspects, such as the effectiveness of a well-organized teaching team, the vibrancy of student excitement in a large class, and the broader range of peer interactions students can have among a large group.

A concrete example of an in-class change I made is my method of presentation. Like most mathematicians, I prefer to use the chalkboard when I teach; with large lectures, this was never as effective as it is when I teach 20-30 students. I also never wanted to use a microphone, as a personal preference. I finally gave up on all of these teacher-focused preferences. I now use Crayola markers to write on blank white paper projected using a document camera, use desmos.com for all my graphing, and use the lapel microphone. With the microphone, the students hear me clearly and the class is more relaxed since I am not straining my voice. By using markers and desmos, students can see better, I can scan my actual in-class notes and post them after class, and the lectures are literally colorful. I had dramatically underestimated the impact of these simple changes — my student evaluations are now consistently full of positive comments about how my use of colorful markers and dynamic graphics are uplifting in a drab room and help students pay attention.

Do these things make every student learn perfectly? Of course not. However, by thinking more purposefully about working within constraints to maximize student learning, leaving some of my own personal preferences aside, I have developed an approach to teaching large lectures that is more successful, and which my students and I feel reasonably positive about.

**4. Students can have meaningful mathematical experiences without us**

In my early teaching, I had bought into a false idea that student contributions were most meaningful when I could provide feedback about them. This was one reason for my preference for whole-class inquiry-based learning courses, and my distaste for large lectures. As with many other things in life, sometimes less is more in this regard. In my courses for first-year graduate students and in my large-lecture calculus courses, students are engaged and report positive experiences when I give 7-10 minute lectures followed by a 2-3 minute pause where students can discuss any points of confusion with their neighbors. The most effective prompt that I have found is to tell the students to turn to their neighbor and ask “do you have a question, yes or no? If no, why does this make sense?” It actually does not seem to matter whether or not I hear these conversations, what matters is that the students are talking about mathematics, struggling with the ideas, and are regularly engaged in conversation about what we are doing.

Similarly, in my small courses, I am less concerned than I used to be with having every small group report on their work, or check with me. This does increase the risk that students might have a misconception that is not immediately addressed, but it gives students more agency and authority in their own learning. It also recognizes the reality that students can achieve excellence in their learning without being perfect, and have meaningful experiences in mathematics without me being intellectually present at every moment.

At a deeper level, when we recognize that students can have meaningful mathematical experiences without us, we allow ourselves to embrace our most important task, to guide and inspire students, rather than to seek a false sense of control over their learning. Our most fundamental role as teachers is not to transmit truths to our students, but to create and sustain supportive environments in which students deeply learn, to create opportunities for students to engage with mathematics at a fundamental and profound level. We balance the tension between the aspirations and ideals that Su and Ardila-Mantilla offer and the reality of teaching by honoring this fundamental role we play, while simultaneously allowing students the choice of whether or not to take advantage of the opportunities they have. This leads to my final mantra.

**5. Do not be afraid of honest failure**

This has been the most important mantra for me. Like most people, I want to reach my goals. I want my students to succeed in my courses. However, the dichotomy of “success versus failure” is not sufficient when we set challenging goals, and goals in the context of mathematics are almost always challenging! Instead, we should strive to succeed or fail honestly. It is debilitating to have dishonest failure, where we fail because we choose not to put in our best work, where we fail because we do not risk anything. It is also a waste to have dishonest success, through cheating or gaming the system. If we succeed in our teaching, if our students succeed in their learning, these successes are most meaningful when they are honestly earned. If we fail in our teaching, or if our students fail in a given course, that is still a meaningful experience as long as the failure is honest.

I have been fortunate that I have not yet encountered epic failures in my mathematical life. However, this is not true of my life overall; whether in mathematics or something else, each of us has stories to tell of when things went awry despite our best efforts. If our students are doing what we hope they will, are pursuing challenging goals, chasing after dreams, learning beyond what they thought they were capable of, there will be honest failure along the way. We must honor those failures, and value them, and make sure our students know this. If we as teachers are striving to realize our aspirations and ideals, we will have honest failures as well.

**A final thought**

These mantras and my approach to teaching have been influenced by the concept of strengths-based practice in social work, by my interest in mindfulness practice, and by my readings about history and politics. At a fundamental level, all of these are about the challenge of resolving the tension between ideals and reality. I am far from unique in having had a significant influence on my teaching come from non-mathematical sources. For example, in his essay mentioned at the beginning of this article, Francis Su describes how his teaching has been informed by the ancient Greek idea of *eudaimonia*, and Federico Ardila-Mantilla’s essay describes how his work with students has been informed by research in social science and psychology. While it is worthwhile and meaningful for us to look inward and see how the strengths of mathematics and our community can be used in the practical pursuit of our ideals, we should remain open to inspiration from all aspects of our lives.

**Acknowledgements**

Thanks to my colleague Serge Ochanine for his insightful comments about Francis Su’s article, which inspired me to put these thoughts into coherent form. Thanks to my father James Braun for introducing me to strengths-based practice in social work. Thanks to the other editors of this blog for many thought-provoking conversations and their helpful comments on a previous version of this essay.

]]>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

]]>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.