This fall, the Mathematical Association of America released a five-year study on college calculus that showed that, no matter how elite their learning institution may be, far too many students lose confidence in their math abilities after Calculus 1. As someone who recently spent a lot of time in calculus classrooms, I understand how that can happen.

Between 2012 and 2013, I enrolled in four different Calculus I courses. This may seem excessive even to the math-loving crowd reading this blog, but let me explain. Of the four, I dropped two, failed one and passed one. Of the four, two were in a community college classroom (the dropped and the failed), while two were Massive Open Online Courses, or MOOCs (one dropped, one passed, the latter with an 89.3 percent).

To be honest, I never set out to take this many calculus courses. Ideally, it would have been one and done. Some quick context: I am a print journalist with 20 years of experience in print and online. While always interested in science, I gave up on math at age 12. I spent the next 26 years as an avowed word person and math phobe, until leaving my full-time newsroom job to go freelance. Suddenly having so much time to think (the freelance career took a while to get going) made me question my youthful decision, and since I was already taking a computer class, I gave a remedial pre-algebra class a try. This turned into the Mathochism Project, where I was determined to revisit high school math as an adult, and write a blog about the joys and terrors of the experience.

To my surprise, there were mostly joys. From pre-algebra to pre-calculus, I did very well, and became delighted not just with math as a subject but also with my ability to understand it, getting mostly As and high Bs. I finished pre-calculus with a high B, and a strong level of confidence. Then the terror began, though I didn’t realize it at first.

**If at first you don’t succeed…**

In my first calculus course, the lectures were crystal clear. The homework, while not super easy, wasn’t hard. But the red flags were there: The instructor was not personable and seemed unwilling to answer questions. There was a lot of information, it came at breakneck speed, yet there was very little depth to it. Surely there was more to this, I thought as I went through limits, delta epsilon proofs and the squeeze theorem. Calculus can’t be that easy?

Then we had our first test, and all hell broke loose. Most questions barely resembled what I’d been working so hard on for weeks; others introduced new material, such as applying calculus to trigonometry. This was the homework if it had been on steroids, and it frustrated me. Why had we been wasting our time on simple limits involving rational equations, when the limits we were expected to do required much fancier algebraic footwork? Why not give us something meatier to practice on? True, they were limits I could have correctly computed, but they required deeper thinking and more time, and the sudden leap in expectations on the exam was unnerving, and threw me off my game. In previous courses, doing all the homework, attending all classes, studying and reviewing hard was enough. But apparently not now.

For the first time since I started Mathochism, I failed spectacularly, with a 33 percent. So did most of the rest of the class. No one got As. A select few got Bs. After telling us he didn’t think of us as test scores, the instructor announced that those who got those lower scores really should drop the class, since it would only get worse. This was disheartening, yet I decided to stay. I was no quitter.

But after a few more weeks, and another test (which while less difficult was still at a much higher level than the provided study material) I quit. I might have soldiered through, but the instructor lied about the test score thing. I discovered this when he refused to help me with a concept I was stuck on when I consulted him during office hours. As it turns out, my confusion was over something very simple, but I was 33 percent, and therefore not worth his time. He was not interested in further communication.

My calculus adventure could have ended there, but in spite of the awful experience, I found the subject fascinating and wanted to keep going. So I hit a dud instructor! I would try again!

**Try, try again**

My second instructor more personable, but she always seemed stressed because we had such a lot of material to get through and very little time to cover it. In previous courses, my teachers had made an effort to answer student questions and solve the occasional homework problem when most of the class was stuck on it, but that never happened now. Occasionally, we were asked to solve a problem during the lecture, but had less than a minute to do it, which was frustrating when the concept was particularly complicated, and I needed time to let it sink in.

Then we had our first test. I failed, though with a more respectable 56 percent. What did me in this time was root functions and absolute values. Again, not impossible, but the problem sets I’d been given focused heavily on quadratics and cubics. As before, I needed practice, but as before, barely had enough to practice on, with only two problems out of more than 60 tackling roots.

This time, though, since the instructor was encouraging and we had rapport, I stuck with it. In the next three months, I did everything I could to meet her expectations, though she was terrible at communicating them. I quickly figured out that she would test us on more obscure concepts, and did my best to practice those, supplementing problems since the book invariably failed to provide them. I bought or borrowed other calculus texts and consulted online and real life tutors. But it was never enough. The pace was simply too punishing, and I never caught up, and my confidence and energy were gone by the end of the semester. Ironically, I understood the material just fine, and she knew it.

“I know you understand calculus,” she told me. “You’re just not good at taking my tests.”

Unfortunately, since those tests were the entire grade, I failed the class. And my confidence was shredded.

**Whither confidence in college calculus?**

Or was it? What was really shredded was my confidence that I could have a good calculus experience at that college, which is a shame, since I deliberately took all my courses there in hopes that their system wouldn’t fail me.

But it did. The system failed me in that, in spite of doing well in the courses leading up to calculus, they were clearly not preparing me for the rigors of that course. Had I known I needed to seek out supplemental problems to train on, since the book wasn’t enough, I would have done that from the beginning. Had I known that the professors would be so busy getting through material that they barely had time (or in the first case, inclination) to go into any depth, or tell students that depth was even expected – well, I would probably just have stopped at pre-calculus. And then I would have bought myself a bunch of calculus textbooks, hired a tutor and homeschooled.

But my calculus story has a happy ending. After considering an extension course at a nearby university, I turned to massive open online courses, or MOOCs. They didn’t have the one-on-one element, true, but they were free, and Coursera was offering one by UPenn professor Robert Ghrist that was getting raves. Those raves were deserved. Unfortunately, I had to drop the course, because it was more advanced than Calculus 1.

I got my second chance in fall 2013, when Ohio State University offered “Mooculus,” again through Coursera. Taught by professors Bart Snapp and Jim Fowler, this was my dream experience. They were both great, personable lecturers, always available for questions on the online forums (and if not them, multiple TAs were available), and I loved their attitude that calculus was not impossible to learn, even if you stumbled at first.

Although it was two weeks shorter than the community college courses, this MOOC packed in way more material, including log and inverse trig calculus, and handy techniques like L’Hopital’s Rule. And yet, I never felt rushed, probably because I wasn’t spending time commuting to and from school or sitting in class waiting for lectures to begin. I could have this class any time I wanted, and even repeat video lectures over and over again when I missed something.

But best of all, they understood how important it was to have enough problems to practice on! They offered those problems in two formats. The easier ones were interactive, and offered step-by-step solutions if you got stuck. The software acted a lot like a video game; if you showed that you really understood a concept, it leveled you up and asked you tougher questions. If you were having trouble, it gave you as many problems as you needed to get it right before allowing you to the next level.

The hardest questions were in the course’s pdf textbook. Like any textbook, they first explained the concepts you needed, then gave you problems applying those concepts. But unlike in my community college text, most of these problems were at a high level, and once you solved them (no step-by-step solution available here, though you did get an answer, like \(\sin x\) or 2), you really felt a sense of accomplishment and that you had gone deeper.

Once you had tackled the problems in a particular section, it was time to take a quiz. The quizzes were all graded, as were the midterm and final. You could take a quiz any time of day or night you wanted (though by a certain deadline). They were also untimed. What really made the difference was that, if you had done both the interactive and pdf problems, the exams contained problems that reflected your prior work, even if they were more challenging.

A year earlier, I had finished Calculus 1 at a community college depressed and exhausted over having failed in spite of having understood the material. I finished Mooculus online exhilarated and exhausted, with my grade finally resembling the ones I had gotten in previous courses. And I had learned amazing things, like \(e^x\) is its own derivative.

**Where Do We Go From Here?**

In October, the Mathematical Association of America released its study. Financed in part by the National Science Foundation, it surveyed 213 colleges and universities, 502 instructors and more than 14,000 students. Not only did students report less confidence after Calc 1, they also reported lower levels of enjoyment, worries about readiness for future courses and about their ability to understand future material. Women were way more affected than men.

“What can be done?” the study authors asked, adding that such attitudes did not bode well for getting more people, particularly women, into number-reliant STEM careers.

To which I answer, even as I concede I am not an aspiring scientist or engineer (at least not this year): Take a cue from the guys at OSU.

Yes, it is possible to teach calculus effectively! No, you don’t have to offer untimed quizzes, and I understand why that is not doable. But interactive homework is doable. So are more challenging problem sets, a more vigorous curriculum that includes logs, and most of all, professors who don’t refuse to help students who struggle, whether it is by disdaining them or by not communicating expectations effectively because they are so stressed out by the pace of the course.

It is possible to get through a calculus course and still feel confident in one’s abilities. I have the faith, and lived experience, to say we should try. Don’t we owe our students that?

]]>In the past, I was frustrated with grades. Usually they told me very little about what a student did or didn’t know. Also, my students didn’t always know what topics they understood and on what topics they needed more work. Aside from wanting to do well on a cumulative final exam, students had very little incentive to look back on older topics. Through many conversations on Twitter, I learned about Standards Based Grading (SBG) and I implemented an SBG system in several consecutive semesters of Calculus II.

The goal of SBG is to shift the focus of grades from a weighted average of scores earned on various assignments to a measure of mastery of individual learning targets related to the content of the course. Instead of informing a student of their grade on a particular assignment, a standards-based grade aims to reflect that student’s level of understanding of key concepts or standards. Additionally, students are invited to improve their course standing by demonstrating growth in their skills or understanding as they see fit. In this article I will explain the way I implemented SBG and describe some benefits and some drawbacks of this method of assessment.

*Image: **The Integrity of the Grade, courtesy of Dr. Justin Tarte, @justintarte*

I chose Calculus II to try an SBG approach because it was my first time teaching the course, so I could build my materials from the ground up. Also, unlike several other courses I teach, the student count remains low — approximately 25 per section. Before the start of the semester, I created a list of thirty course “standards” or learning goals. Roughly, each goal corresponded to one section of the textbook. I organized the thirty standards around six Big Questions that I felt were the heart of the course material. One Big Question was, “What does it mean to add together infinitely many numbers?” The list of standards served as answers to these Big Questions. The list of standards and a description of the grading system were distributed to the students on the first day of class. During the semester, students were given in-class assessments in the form of weekly quizzes, monthly examinations, and a cumulative final examination. The assignments themselves were similar to those found in courses using a traditional grading scheme, but they were assessed differently. Rather than track a student’s total percentage on each particular assignment, for every problem I examined each student’s response and then assigned a score to one or more associated course standards. I provided suggested homework problems both from the textbook and using an online homework platform, but homework did not factor directly into a student’s grade. Instead, if I noticed a student needed more practice at a particular sort of problem, I would direct her to the associated homework problems for additional practice.

During in-class assessments, a single quiz or exam question asking a student to determine if an infinite series converged might also require the student to demonstrate knowledge of (a) “The Integral Test,” a strategy for determining if a series converges or diverges; (b) “Improper Integrals,” the process used to evaluate integrals over an infinite interval; (c) some method of integration, such as “Integration by Parts,” and (d) some prior knowledge about how to evaluate limits learned earlier in Calculus I. For each of these concepts, I assign a different score (on a 0-4 scale), roughly correlated with a GPA or letter-grade system. During the semester, I tracked how well each student did on each of the thirty standards.

Since some standards appeared in a multitude of questions throughout the semester, a student’s current score on a standard was computed as the average of the student’s most recent two attempts. Outside of class, each student could re-attempt up to one course standard per week. Usually these re-attempts occurred during office hours and were in the form of a one- or two-question quiz. My rationale for continually updating student scores is that I want grades to reflect a *current* level of understanding since I want students to aim for a continued mastery of course topics. Over the course of the semester, their scores on standards can move up or down several times. Students are motivated to continue reviewing old material since they know that they might be assessed on those ideas again and their previous grades could go in either direction.

At the end of the term, each student had scores on approximately thirty course standards. To determine a student’s letter grade, I used the following system:

- To guarantee a grade of “A”, a student must earn 4s on 90% of standards, and have no scores below a 3.
- To guarantee a grade of “B” or higher, a student must earn 3s or higher on 80% of course standards, and have no scores below a 2.
- To guarantee a grade of “C” or higher, a student must earn 2s or higher on at least 80% of course standards.

I adapted this system from one Joshua Bowman used. I like it because it captures my feeling that an “A-level” student is a student who shows mastery of nearly all concepts and shows good progress toward mastery on the others; meanwhile, a “B-level” student is one who consistently does B-level work. Also, this system requires students earn at least a passing grade on each course topic. In a traditional system, a student might do very well in some parts of the course, very poorly in others, and earn an “above average” grade. In the system I used, for a student to earn an “above average” grade, they must display at least a passing level of understanding of all course concepts. While students aren’t initially thrilled with this requirement, most are happy once I explain they can re-attempt concepts often (within some specific boundaries) and so the only limit on improving performance is their motivation to do so.

There are three major advantages of tracking scores on standards. First, I can quickly assess student performance:

Second, I can give meaningful advice to students:

Third, I can determine what topics are in need of review or additional instruction:

Students have noted that SBG has several benefits for them as well. They aren’t limited by past performance and can always improve their standing in the course. Many students who describe themselves as “not math people” or those who say they suffer from test anxiety appreciate that their grades can continue to improve, thereby lowering the stakes on any particular assessment. In my office, conversations are almost always about mathematical topics instead of partial credit, why they lost points here or there, or what grade they need on the next test to bring their course average above some threshold. The change in types of conversations during my office hours has been amazing, and for this reason alone I will stick with SBG in the future. Students review old material without prompting, they feel less stress over any individual assignment, we don’t have conversations about partial credit or lost points, and they are able to diagnose their own weaknesses.

With that said, the SBG system also has some disadvantages. First, it takes a thorough and careful explanation to students about the way the system works, why it was chosen, and why I believe it is to their benefit. Student buy-in is critical and it isn’t always easy to attain. I have found that spending a few minutes of class time discussing SBG every day for the first one or two weeks is more helpful than giving a lot of explanation on any particular day. Students need some time to think about what questions and concerns they have, and I encourage them to voice these in class whenever they like. Initially, students think that this system will be too much work for them, or that their course grades will suffer since past strong performance could be wiped out in the future. (In contrast, by the end of the semester, almost all students say they really appreciated this method and felt they learned more calculus than they would have in a traditionally graded course.) Second, several students complained that their grades were not available through our online learning management system; I still haven’t found a way to convince our online gradebook to work in an SBG framework. Instead, students must come to my office to review their scores with me outside of class time. Third, choosing both the correct number of course standards as well as a thorough description of each standard has been challenging. It’s difficult to balance wanting each standard to be as specific as possible while keeping the total number of standards workable from both my viewpoint and that of the students.

After several semesters of using an SBG framework, I believe the benefits to the students outweigh the disadvantages. At this point, I don’t have any firm data about student learning outcomes, but I do have some anecdotal evidence. The feedback from my students about this method of grading and, in particular, the details of my implementation has been very positive. I have received several e-mails from former students who, even semesters later, realize how much SBG changed their perspective on the learning process, or who wished their new instructors would switch to an SBG system. Comments on my student evaluations have mentioned that they feel their grade accurately reflects how much calculus they know, rather than how well they performed on a particular assignment, or how much they were punished from making arithmetic mistakes. As one student noted, “this class was not about how well you could take a test or quiz or do homework online that sucked. It was about the amount of calculus you understood and your effort to be better at it.” As a calculus instructor, this describes my exact goal for my course.

If you are interested in trying an SBG approach in your own courses, here are four questions to jump-start your journey:

- What are the core ideas of your course? What concepts or ideas do you want students to master?
- How many standards do you think you can track? You need them to be specific enough that students can understand exactly what each one means, but you also need to have few enough that your grading workload is manageable. I have 30 for a 16-week semester.
- Will you allow re-attempts? What kinds of limits will you set, if any? I found that limiting students to re-attempting only one standard per week was essential in cutting down my grading workload. This limit also gave students the opportunity to focus on one topic at a time, rather than re-attempting several at once just to see what would stick.
- How will a final assessment, project, or exam count? In my course, a student’s course score on each standard is a weighted average: 80% comes from their pre-final exam score and the remaining 20% comes from the score earned on the final itself. In this way, the final exam contributes about 20% to the student’s letter grade in the class, a figure in line with what is commonly used in my department.
- How will you convert all the scores on standards into a letter grade?

**Online SBG Resources**

- Twitter hastags: #sbg, #sbgchat, #sblchat
- http://tinyurl.com/SBGLiterature, Scholarly articles related to SBG (list maintained by Matt Townsley)
- http://thalestriangles.blogspot.com/search/label/sbg, SBG blog posts by Joshua Bowman (@Thalesdisciple)
- http://shawncornally.com/wordpress/?p=673, Standards-Based Grading FAQ by Shawn Cornally
- http://blogs.cofc.edu/owensks/tag/sbg/, my own blog posts about SBG
- https://plus.google.com/communities/117099673102877564377, a newly formed Google Plus community for anyone interested in conversations about standards-based or specifications-based grading

Effective early childhood math teaching is much more challenging than most people anticipate. Because the math is foundational, many people assume it takes little understanding to teach it, and unfortunately this is distinctly not the case. In fact, the most foundational math ideas — about what quantity is, about how hierarchical inclusion makes our number system work, about the things that all different shapes and sizes of triangles have in common — are highly abstract ones. While we should not expect or encourage young children to formally recite these ideas, they are perfectly capable of grappling with them. Further, they need to do so to develop the kind of robust understanding that will not crumble under the necessary memorization of number words and symbols that is to come in kindergarten. In preschool, before there is really any opportunity for “procedural” math, it is important that we give children ample opportunity to think about math conceptually. In this essay I will discuss several profound ideas from early childhood mathematics, including examples of effective early math classrooms. Along the way I will share some of the resources that my colleagues and I have developed to help early childhood educators develop as skillful teachers of early mathematics.

**About Early Math**

As a doctoral student, I first got interested in early mathematics by way of cognitive science. I fell in love with the precise and thoughtful cognitive developmental work that built on what Jean Piaget had begun. Through clever experimental designs and a careful parsing of concepts over the last 40 years, developmental psychologists have made enormous strides in understanding how the mind develops during childhood. Many of their findings have profound implications for mathematics, and since my degree was to be in applied child development, early math education provided a way to make studying cognitive development useful to me.

As it turned out, early math was a useful place to put energy for far more important reasons. In a now-landmark study in 2007 [1] using six longitudinal data sets, Duncan et al. found that math concept understanding at kindergarten entry predicted not only later math achievement, but also later reading achievement; reading at kindergarten entry, however, did not predict later math. This finding was replicated in a large-scale Canadian study in 2010 [2], which found that early math skills were stronger predictors of general academic success than either reading skills or social-emotional skills at school entry. We don’t yet know for certain why this association is so strong, but it is at least clear that early math is important. It is also true that the differences we observe in math achievement at kindergarten entry tend to fall along socio-economic lines, so alleviating those differences relates to issues of educational equity.

Early math was also a useful focus because of the pronounced need (in the U.S. especially) for improved instruction in mathematics in preschool and early elementary settings. Years after the seminal work by Deborah Ball [4] on the need for improved pedagogical content knowledge, and by Liping Ma [3] on the lack of a “profound understanding of fundamental mathematics” among later-grade elementary teachers, math educators turned their lens to those teaching our youngest students. It turns out that students of teacher education who “love kids but hate math” are commonly directed by faculty to teach in the younger grades. This has left us with a preponderance of preschool and primary teachers who are both underconfident and underprepared in mathematics teaching.

**Teaching Early Math**

So what does mathematics teaching look like in a preschool classroom? Recall first that preschool means children between the ages of 3 and 5, and that their range of normative development is exceedingly wide. In this group of kids there will be children who are not “potty-trained” alongside children who have begun to read, so teachers have to cast a very wide net. Further, for this age group, “teaching” is something that is often done only when all the heavy lifting of being sure everyone is comfortable, rested, and not in tears is complete. Sit-and-listen techniques are effective only when the content is exceedingly entertaining — as in a story is being read — and the children have very limited capacity for absorbing information directly from text, and less-limited but still primitive abilities to communicate their own ideas.

For these reasons, learning in early childhood classrooms consists almost entirely of “active learning.” In fact, early childhood has a long and proud connection to the type of teaching that emphasizes student-directed/teacher-facilitated activities. Child choices and the use of prepared “centers” are favored, with limited time spent on whole group activities of any kind (“circle time” being the exception), and small groups being occasionally led by a teacher. This is not an environment that is amenable to worksheets, and for that, early childhood teachers are generally extremely grateful. It also means, however, that whatever content is introduced comes fairly directly from the intentions and understandings of the teacher, who designs and facilitates experiences that lead children to construct new thinking.

**Some Useful Interventions**

Given this learning environment, my colleagues and I decided to focus our work on improving teachers’ understanding of the early math content they should be working into their interactions with young children. By studying the cognitive developmental and early math education literatures, we developed 26 Big Ideas that we wanted to be sure early childhood teachers understood well and knew how to address. One example is the idea that “any collection of objects can always be sorted in more than one way.” While this is not a conventional mathematical idea, it is foundational to the types of thinking that underlie our experience of sets (there are 6 pieces of fruit; there are 2 apples, 2 lemons, and 2 bananas; there are 2 red pieces of fruit and 4 yellow pieces of fruit) and therefore an important understanding for young children to see, explore, and experience. It has generative implications for understanding number and algebra in later life, and helps children flex and develop their logical thinking skills.

To help teachers make such an idea come to life, we developed what we call “Research Lessons.” These are skeleton lesson plans for activities teachers can use over the period of a month or more (through slightly altered iterations). For the Big Idea above, we ask teachers to conduct a read-aloud of a beautifully illustrated children’s book called *Five Creatures* by Emily Jenkins. In the book, a family of two adults, one child, and two cats is described differently from page to page, as in “In my family, there are five creatures…three who like milk, one who does not, and one who only drinks it in coffee…three with orange hair (child, one adult, one cat), one with gray hair, and one with stripes…” This book is read several times over a period of days, with lots of discussion. At some point, the teacher introduces two large circles, drawn out on the rug with tape: half the class are the “creatures” and half are the audience. Together, teacher and audience sort the “creatures” using binary (A/B) sorting to place them inside the circles, as in “the creatures with long hair and the creatures with short hair” or “the creatures with white in their shirts and the creatures without white in their shirts.” This leads to useful discussions about shared definitions for categories and sometimes generates the (exciting!) need for a third circle.

**Conclusion**

While it often goes unrecognized, the need for strong early math skills among children and early childhood educators is strong. Early math is highly abstract, and is a key indicator of later school success. What happens in preschool and early elementary classrooms has a direct impact on students for the rest of their educational experiences, from elementary school through postsecondary work. Our early childhood teachers need better preparation and in-service training to understand their crucial role in mathematics education. We will best be able to rise to the challenges of early math education through collaborative efforts involving teachers, teacher educators, and mathematical scientists.

**References**

[1] Duncan GJ, Dowsett CJ, Claessens A, Magnuson K, Huston AC, Klebanov P, Pagani LS, Feinstein L, Engel M, Brooks-Gunn J, Sexton H, Duckworth K, Japel C. “School readiness and later achievement.” *Dev Psychol*. 2007 Nov;43(6):1428-46.

[2] Pagani, Linda S. et al. “School Readiness and Later Achievement: A French Canadian Replication and Extension.” *Developmental Psychology*. Vol. 46(5): 984-994. September 2010.

[3] Ma, Liping. *Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States*. Routledge, 1999.

[4] Ball, D.L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University, Lansing.

]]>*Editor’s note: This is the sixth and final article in a series devoted to active learning in mathematics courses. The other articles in the series can be found **here**.*

How are mathematicians trained as teachers, what are the effects of this training, and what can we do to improve the quality of this training? We feel these questions are particularly important at this time, as a clamor of recent calls for the dramatic improvement of postsecondary education, made from both inside and outside of the mathematical community, has not abated. From the outside, we hear this call in venues ranging from opinion pieces in major newspapers [1,2,3] to federal advisory reports to the President of the United States [4] and beyond. The message has also been clearly conveyed by leadership from professional societies in the mathematical community: in early 2014, an article titled “Meeting the Challenges of Improved Post-Secondary Education in the Mathematical Sciences” was published in the *AMS Notices*, *MAA Focus*, *SIAM News*, and *AMSTAT News* through a coordinated effort by the professional societies — we urge any readers who have not already done so to read this statement.

Yet in order to be effective and achieve meaningful change, any actions taken by our professional societies and other leadership in the mathematics community must get buy-in from individual mathematicians who are in the classroom daily, working face-to-face with students. From our training in both mathematics and the teaching of mathematics, we each carry disciplinary habits, ways of thinking, biases, and strengths, many of which occur subconsciously as part of our mathematical culture, and all of which impact our teaching. In order to improve mathematical teaching and learning on a large scale, we must all work to better understand how mathematicians grow and develop as teachers, so that we may more thoughtfully respond to the educational challenges of our time. In this article, we focus our discussion on the topic of pedagogical training and development for graduate students and early-career faculty, with a view toward active learning.

**Mathematical training in graduate school**

To earn a graduate degree in mathematics, one must master a body of mathematics content through coursework, demonstrate a deep understanding of this through oral and written examinations, and (for doctoral degrees) complete a dissertation demonstrating original research. Thus, the primary focus of graduate students is on mastering advanced mathematical ideas and producing new mathematical results, and these qualities of graduate education are consistent across programs at different institutions, though specific program details can vary.

Less consistent across the graduate school spectrum is the preparation of students for positions where they will have teaching responsibilities. Duties for teaching assistantships in graduate school vary greatly, from providing grading support to professors, to working in a tutoring lab, to leading recitation sections, to having full responsibility to lead a course. Preparation for these duties is equally various, but an increasing number of graduate schools have explicitly attended to preparing their students for teaching by instituting or enhancing teaching programs for their TAs. More broadly, a look at the program for the 2016 Joint Mathematics Meetings reveals some promising developments, including a panel called *Improving the Preparation of Graduate Students to Teach Mathematics: An NSF-Funded Project*. While these are positive developments, it is not uncommon for graduate teaching assistants to be supervised by faculty members who are only familiar with a small set of teaching techniques, or who have had frustrating prior experiences with active learning methods, and for graduate students to receive little formal training as teachers.

Further, though this is slowly changing, many graduate students in mathematics have not personally experienced teaching environments that include active learning components. Thus, for many mathematicians and current graduate students, their first experience with active learning techniques will be as teachers rather than students. A consequence of this is that we cannot expect students to emerge from graduate programs prepared to be guided by their own classroom experiences where active learning is concerned. If we want graduate students to consider using active learning in undergraduate courses, we must provide them with some experiences, either as students or teachers, to help inform their practice.

There are many reasons to be optimistic that this can be accomplished. Researchers in mathematics education have begun to study training of teaching assistants in mathematics, which should lead to better information about effective practices [5,6]. Further, the gap between the reality of graduate school and the goal of producing graduate students who have a reasonable level of training as teachers, including some exposure to active learning methods, is not as wide as it might appear. One key is to recognize and promote the aspects of graduate programs that already have active learning embedded in them. Here are some examples.

Many universities with doctoral programs rely on their TAs to serve as recitation leaders, or to teach small sections of courses, roles which can easily incorporate active learning methods (avoiding, for example, the issue of scaling things up to large-lecture size). In departments such as the University of Michigan, University of Illinois Urbana-Champaign, University of Kentucky, and *many* others, graduate students lead recitations that are based on having students work in small groups through an activity built from a sequence of problems. In such settings, graduate students are already leading a class setting based on active learning, but they may or may not be receiving explicit training regarding how to effectively structure small group work, how to lead a discussion without directly providing an answer, etc. With a small amount of effort, course coordinators and TA supervisors can provide training in these areas for TAs, as long as they themselves are aware of how to do this effectively.

As another example, after initial coursework, many graduate students participate in formal or informal seminars where participants read through a paper or book and gather to discuss problem sets, sticking points in the reading, and general questions about the topic. This practice, which mathematicians would call “doing mathematics,” is active learning at the core. Most senior graduate students and mathematicians can reflect on times when they have struggled with an idea or topic, only to have it clarified through helpful conversation with others. If mathematics faculty and graduate students re-conceive these activities as examples of active learning, then it becomes easier to see how one might try to incorporate some small-group discussion in classes. In both the recitation model above, and in this example from the experience of many mathematicians, we recognize the benefits of conversing and communicating, doing mathematics with others. Active learning methods seek to bring this into the classroom, and it would be helpful for graduate students to be trained to make this connection explicitly.

As a final example, there are many outreach programs for K-12 students that are operated by mathematics departments with graduate programs, including Math Students’ Circles, Math Teachers’ Circles, math days, math camps, and more. Many of these programs are strongly based on active learning methods. Graduate students who serve as assistants for such programs might not make an explicit connection between these programs and their own teaching, though certainly many students do see connections. In any event, it would be a positive step forward if graduate students serving as assistants in these programs were explicitly encouraged as a part of their assistantship to consider and discuss ways in which effective techniques in extra-curricular K-12 outreach programs might be transferred into their own courses.

Even though most mathematicians consider graduate school the foundation of a mathematical career, it isn’t clear how much responsibility should be placed on the shoulders of doctoral programs for teacher training. It is unreasonable to expect that every doctoral student in math will emerge as an expert teacher, given the many demands of graduate school and the need for students to develop and defend a research dissertation. Yet it is clear that we can do more than we are at present, and that we have many strengths on which faculty and students can immediately build by being more explicit on the issue of effective teaching.

**Training as early-career mathematicians**

The training and mentoring of early-career faculty has long been recognized as important by the mathematics community, which has responded with a variety of efforts. As perhaps the largest single effort to date by a professional organization, since 1994 the Mathematical Association of America (MAA), through Project NExT (New Experiences in Teaching), has provided multi-year intensive mentorship and support for over 1500 early career faculty in the mathematical sciences. Other opportunities now abound as well. The Academy of Inquiry Based Learning provides weeklong summer training workshops, mentorship programs, and small grants to assist faculty with transitioning to an active-learning teaching style. The MAA also offers 4-hour mini-courses at each of the Joint Math Meetings and Mathfest, which vary greatly in topic as illustrated by the 2016 JMM offerings. Thus, at the national level there are many opportunities for professional development regarding teaching, a large number of which are focused on active learning methods; however, issues of access certainly exist for faculty at institutions with limited funds available to support participation in these programs.

At the local level, it is common for departments and colleges to have faculty mentoring programs, though as with graduate training, these programs vary widely across institutions. Unfortunately, at some institutions new instructors and assistant professors may go several terms before receiving feedback about their teaching (if they receive any at all). Other institutions do have forms of mentoring in place, such as regular classroom observations or meetings with a “master teacher” in the department. Still others have well-established, formal mentoring programs in which new instructors are paired with more experienced faculty. However, as we noted before regarding graduate school, these mentors may not have much experience with active learning techniques. The ways in which teaching is assessed also vary, with some departments emphasizing student evaluations and some prioritizing classroom observations. The variety of mentoring opportunities for new faculty, and the reality that many early-career faculty members do not receive sufficient mentoring and training, suggests that continued efforts are needed to improve the overall landscape of pedagogical training for early-career faculty.

Further, depending on departmental and institutional culture, junior faculty often have reasonable concerns regarding earning tenure or ensuring that a short-term contract is renewed. This can cause them to hold back from trying unfamiliar teaching methods for fear of negative student responses or of a classroom observation that is negative because the observer does not agree with the teaching method. This can sometimes cause early-career faculty to delay gaining experience with active learning methods that have been shown to have positive impacts on students. It is particularly important for department leadership and higher administrators to find clearly-communicated ways to support early-career faculty who wish to pilot the use of unfamiliar teaching methods, especially those active learning methods that have evidence supporting their effectiveness.

Even when quality support is available to early-career faculty, there remain inherent challenges for developing as a teacher. The term “expert blind spot” probably rings true for anyone who has tried to teach anything to a relative novice. It describes situations in which instructors’ advanced knowledge of content interferes with their ability to understand their students’ learning processes [7,8]. Within mathematics, our custom of proof-centered discourse does not always translate well to the classroom. Lower-level students may have weak backgrounds, including scant practice with the logic that now comes naturally to us. Even upper-level mathematics majors are not always ready for the presentations that have become second nature to their instructors with doctoral degrees. This is related to our previous post about telling in teaching mathematics; sometimes we as mathematicians can insist upon telling students facts over and over — \((a^2 + b^2)\) does not equal \((a+b)^2\) — facts which may be obvious to us, without fully acknowledging or accepting a student’s struggle to learn such facts.

This is not to say that the graduate school experience has nothing to offer an instructor. Having wrestled with an open problem in preparing a thesis, a PhD mathematician surely understands the value of struggle and occasional failure — recognized in current public discussions of education [9] — to the learning process. At the same time, experts tend to see the classroom as a place to organize ideas, while novices are better served if it is also an environment for discovery, error, and invention [10]. Mathematicians who are teaching should consider the broader vision in order to reach the particular novices in their classes. This is not a trivial exercise.

**Conclusion**

Having come to the end of our series on active learning methods in mathematics, we wish to emphasize one last point. There is a fundamental way in which our training as mathematicians can help us develop as teachers: mathematicians are expert problem solvers. As a community of mathematicians and academics, we are in the process of solving the problem of how best to teach mathematics, and we are jointly working together toward that end. As with all complex, real-world problems, the challenge for us is that there is not an exact solution, but rather a collection of approximate solutions. Nevertheless, our mathematical training has prepared us as problem solvers to hone our intelligence, our diligence, our spirit of curiosity, and our love of learning in order to develop meaningful and effective ways of teaching. These qualities are directly related to who we are as mathematicians, and it gives us hope for success in our continued endeavor of improving mathematics teaching and learning for all.

**References**

[1] Is Algebra Necessary? Andrew Hacker. New York Times, July 29, 2012. http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=0

[2] Are College Lectures Unfair? Annie Murphy Paul. New York Times, September 13, 2015. http://www.nytimes.com/2015/09/13/opinion/sunday/are-college-lectures-unfair.html

[3] Lecture Me. Really. Molly Worthen. New York Times, October 18, 2015. http://www.nytimes.com/2015/10/18/opinion/sunday/lecture-me-really.html

[4] *Engage to Excel.* PCAST report, January 2012. https://www.whitehouse.gov/sites/default/files/microsites/ostp/pcast-engage-to-excel-final_feb.pdf

[5] Beisiegel, M. & Simmt, E. (2012). Formation of mathematics graduate students’ mathematician-as-teacher identity. *For the Learning of Mathematics, 32*(1), 34-39.

[6] Ellis, J. (2014). Preparing Future Professors: Highlighting The Importance Of Graduate Student Professional Development Programs In Calculus Instruction. Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3 (pp. 9-16). Vancouver, British Columbia: PME.

[7] Nathan, Mitchell J., Kenneth R. Koedinger, and Martha W. Alibali. “Expert blind spot: When content knowledge eclipses pedagogical content knowledge.” In *Proceedings of the Third International Conference on Cognitive Science*, pp. 644-648. 2001.

[8] Nathan, Mitchell J., and Anthony Petrosino. “Expert blind spot among preservice teachers.” *American educational research journal* 40.4 (2003): 905-928.

[9] Lahey, Jessica. *The Gift of Failure*. Short Books, 2015.

[10] Bransford, John D., Ann L. Brown, and Rodney R. Cocking. *How people learn: Brain, mind, experience, and school*. National Academy Press, 1999.

*Editor’s note: This is the fifth article in a series devoted to active learning in mathematics courses. The other articles in the series can be found here.*

Facts, methods, and insights all are essential to all of us, all enter all our subjects, and our principal job as teachers is to sort out the what, the how, and the why, point the student in the right direction, and then, especially when it comes to the why, stay out of his way so that he may proceed full steam ahead.

— Paul Halmos (Halmos, pp. 848-854)

Because of our passion and love for our subject, mathematicians want to share with students the joy, excitement, and beauty of doing mathematics. Our natural human impulse is to do so by telling students about the ways we have come to understand our discipline, to shed light on the subtleties that surround most mathematical ideas, and to explain the fundamental insights of our field. As we have discussed in our previous articles in this series, there is strong evidence that these goals of inspiring students and helping them deeply learn mathematics are often most effectively reached through the use of active learning techniques. Yet there are some good reasons why we might choose to tell students about mathematics when the time is right. In this article we will explore the act of instructor “telling” and discuss some of the roles that telling can play in active learning environments. We seek to balance our inclination to tell students about math, which is inherently passive for the students, with our desire to foster students’ active engagement with mathematical ideas. By doing so, we can simultaneously acknowledge the value of telling while challenging the idea that traditional telling is the best or only way to communicate mathematics with students.

**Telling to Transmit Information**

The way in which one thinks about instruction is shaped by how one believes students learn. For example, if one believes that learning occurs as a result of direct transmission of information from instructor to student, and that students learn by a process of directly taking in bits of information that their instructors say or write, then telling is the natural mechanism to help students learn. However, learning is not this simple: almost every teacher has experienced telling a student a certain mathematical fact, during class or during office hours — \((a^2+b^2)\) does not equal \((a+b)^2\) is a classic example — only to have them demonstrate on a test that they have not learned it. This kind of experience, which happens all too often, suggests that it is not enough for students simply to be told information if we want to produce deep and meaningful learning. As a result of such experiences, many mathematics educators and mathematicians now draw upon learning philosophies in which students construct (mathematical) meanings based on their own experiences (e.g., Glasersfeld, 1995; Piaget, 1954; Steffe & Thompson, 2000). These theories of learning align more closely with instructional techniques that reinforce active learning, as discussed in Part II of this series.

However, a part of teaching mathematics involves injecting new ideas and concepts into the classroom. We cannot and do not expect students to come up with brand new formulas, conventions, and concepts without any guidance. Therefore, at times instructors must introduce new bits of information. The question of how best to do this, though, is a tough one. If we want to be able to share new ideas and yet also help students actively engage with material and learn, how might we best introduce new material? One possible answer to this question is through judicious telling, a notion introduced by Smith (1996) in which teachers try to minimize the amount of direct telling they do, encouraging active engagement among students, and yet still occasionally tell when they deem it necessary. This perspective acknowledges the fact that we want to facilitate active engagement when we can, but that there may be aspects of mathematics that students simply need to be told (such as useful terminology, ways of representing mathematical ideas, etc.). This underscores the idea that the goal is not for an instructor never to tell, but rather to avoid telling when students might otherwise be able to actively engage with and understand an idea.

Even in a classroom in which active student engagement is valued and fostered, there are situations in which judicious telling may be particularly productive. One such time is directly following student activities in which students have had the opportunity to engage actively, e.g. through group problems, classroom voting, clickers, or other techniques referred to in Part III of this series. By first engaging students in tasks that stimulate their curiosity and help them to think about the mathematics, we are creating intellectual need in students, where by intellectual need we mean when a student encounters an intrinsic problem that they genuinely understand, appreciate, and are curious about. In his framework of DNR-based instruction, Harel (2007) places the intellectual need of a student in equal importance to the integrity of the content being taught. After establishing this intellectual need through a student activity, it can be effective to use focused telling to solidify resulting ideas and insights.

Another time that may be particularly appropriate for judicious telling is during a wrap up of key ideas. A potential problem with active learning environments is that students can be in situations in which there is rarely closure or a sense of completion for a task or idea. Students may be encouraged to engage, ask questions, think critically, etc., but if there is never any resolution for them, this can be problematic. Telling after students have extended engagement with material could help to solidify, clarify, and confirm ideas they have been developing. Indeed, failure to (judiciously) tell can lead to confusion, and there is something to be said about instruction that fosters resolution of ideas for students. As previously discussed in Part III of this series, the MAA Calculus study (Bressoud et al., 2015) found that a factor in determining whether students persisted in calculus is that “good” teaching can be more important than “ambitious” teaching. Thus, there is value in consistent instruction, and instruction that fosters active engagement is perhaps more effective when other foundational aspects of teaching (such as wrapping up or resolving mathematical discussions or questions) are in place.

**Gaining a Sense of Efficacy Through Telling**

A well-crafted mathematical lecture goes beyond a simple sequencing of facts. Recent research from the perspective of discourse analysis and instructor pedagogical moves (Artemeva and Fox, 2011; Fukawa-Connelly, 2012) shows that when given by an expert practitioner, a mathematical “chalk talk” is quite complex and relies on practices that are surprisingly stable across social and cultural boundaries. The creation of a complex mathematical lecture, where ideas are carefully introduced and developed so as to build upon each other in meaningful and insightful ways, is one source through which mathematicians provide themselves with evidence of their own mathematical knowledge, understanding, and development. Thus, the crafting of rich lectures contributes to mathematicians’ feelings of efficacy in their discipline.

This positive influence of instructor telling on sense of efficacy has been observed at the K-12 level as well. Smith (1996) points out that the act of telling provides teachers with a sense of efficacy, and he discusses the fact that reforms that move away from lecturing and telling can leave teachers without a clear source of efficacy. He astutely points out that “Telling mathematics allows teachers to build a sense of efficacy by (a) defining a manageable mathematical content that they have studied extensively and (b) providing clear prescriptions for what they must do with that content to affect student learning. The current reform [in the 1990’s at the K-12 level] removes both supports (p. 388).” We can apply this line of thought to our discussion of active learning in university classrooms. The active learning reforms in which many university instructors are engaging can be similarly disorienting, for exactly the reasons Smith lists. In considering when to tell, then, we must consider that instructors’ sense of self-efficacy might be affected by changes in their teaching practices.

The key observation in this context is that while the creation and delivery of a rich lecture contributes positively to instructors’ sense of self-efficacy, this contribution only involves students in a marginal way, if at all. John Mason describes the difference between the student experience of instructor telling and the instructors’ experience of telling as follows:

Yves Chevallard introduced the term didactic transposition to describe the way in which the intuitions and experiences of an expert are trimmed and edited for teaching purposes, so that what learners encounter is often little more than refined formal definitions, proofs of theorems, and examples of applications of techniques. Expert awareness is transposed or transformed into training of behavior. The result is that no appeal is made to learner’s emotions, learners’ powers are not called upon, and mathematical themes remain implicit. The pleasure and insight achieved by the expert in organizing the topic and ‘making sense’ leaks away and is lost to the learner, who experiences merely behavior training. (Mason, p. 259)

When considering active learning techniques and the role telling plays in them, we must be attentive to the need for teaching techniques that are both a positive influence on instructors’ sense of efficacy and a positive influence on student learning. Judicious telling is one example that would allow for instructors to maintain some sense of efficacy while still generally trying to limit direct telling focused on information transfer. There are a number of other ways in which teachers may gain efficacy even while reducing the amount of traditional telling in their classrooms in favor of more active learning techniques. Smith provides alternatives for sources of efficacy, which he calls “new moorings of efficacy” (p. 396). He suggests that, rather than finding their efficacy in traditional telling, teachers could learn to find efficacy in other kinds of classroom activities. These include activities such as choosing problems, predicting student reasoning, generating and directing discourse, pushing for more explanation, asking for questions that extend, and getting immediate feedback from students on what was just told. A high level of expertise and care is needed to construct meaningful sequences of student tasks, see for example the Journal of Inquiry-Based Learning and textbooks such as Active Calculus.

Lobato, Clarke, and Ellis (2005) have built on Smith’s (1996) work by arguing for a reformulation of traditional telling, noting that some key elements of telling can be captured in a way that allows students to be active participants in their mathematical learning. These authors reformulate telling as initiating, which they define as “the set of teaching actions that serve the function of stimulating students’ mathematical constructions via the introduction of new mathematical ideas into a classroom conversation” (p. 110), and as eliciting, “an action intended to ascertain how students interpret the information introduced by the teacher” (p. 111). They contend that some of the pedagogical results that telling accomplishes (like introducing new material into a classroom or communicating mathematical ideas with students) can still be accomplished by alternative means. By considering reformulations of telling such as asking probing questions, implementing carefully designed tasks, and encouraging students to share ideas, teachers can still accomplish goals of traditional telling without sacrificing a commitment to active learning.

**Telling to Achieve Coverage**

Another reason that instructors might traditionally tell is related to the issue of coverage. Covering all of the necessary course material in an allotted time is a perennial concern for instructors at all levels in most disciplines, and there is no denying that telling can be an efficient way of getting through material. If we fall a day behind in our syllabus, a natural response is to try to transmit the material quickly and efficiently so the students can at least be exposed to what they need to see. However, the example of students believing that \((a+b)^2=a^2+b^2\) should not be far from our minds — telling in order to cover more material is not always effective for students.

By exclusively considering course content coverage, and responding to content coverage with telling, we risk forgetting the many other elements of student learning that active learning addresses. For example, in the 2004 and 2015 CUPM Curriculum Guides, content goals only form one portion of the overall set of goals for mathematics students and programs — equally important are goals such as (from the 2015 guide) “assess the correctness of solutions, create and explore examples, carry out mathematical experiments, devise and test conjectures… [and] approach mathematical problems with curiosity and creativity and persist in the face of difficulties.” By recognizing that “coverage” should refer to both content and mathematical processes and practices, we can alleviate our tendency to tell to achieve coverage. A number of experienced teachers and programs have reached effective balances in this regard. For example, the IBL community has established the Journal of Inquiry-Based Learning to serve as a source of refereed course notes for IBL classes. The calculus program at the University of Michigan has been carefully developed by course coordinators with a strong conceptual focus; this also supports new instructors, including graduate students and postdocs, who are trained to quickly become proficient using active learning techniques.

In addition to the recognition that content topics are not the exclusive subject of coverage, recent research suggests that coverage of material is actually less important for student persistence and achievement in mathematics than the effective use of active learning techniques. In a study that found generally positive long-term impacts of inquiry-based courses on undergraduates in mathematics, Kogan and Laursen (2014) state that “‘covering’ less material in inquiry-based sections had no negative effect on students’ later performance in the major. Evidence for increased persistence is seen among the high-achieving students whom many faculty members would most like to recruit and retain in their department.” Thus, although we acknowledge the difficulties that issues of coverage present, and while judicious telling might be appropriate when coverage is important, we would argue that active engagement can be more effective for student learning than resorting to traditional telling.

**Conclusion**

It is difficult to measure student learning, especially if we want to assess high-cognitive demand tasks. Hence instructors often depend on proxies: scores on exams that may be testing lower-level tasks, student course evaluations, and the perceived quality of a lecture, among others. Given the evidence supporting active learning, we must move beyond information transmission, instructor’s sense of efficacy, and achieving coverage as proxies for student learning. By expanding our vision of instructor telling, we can develop methods of telling that are better aligned with student learning, and we can train ourselves to look for signs that students have been helped in richer ways. A common theme in the reflections from Part IV of our series, unintended yet clearly present in hindsight, is that the use of active learning techniques has increased our own sense of satisfaction and efficacy as teachers. By re-conceptualizing telling, and moving toward more mature forms of telling, we can find richer ways of recognizing how to help students succeed.

**References**

Artemeva, N., & Fox, J. (2011). The writing’s on the board: the global and the local in teaching undergraduate mathematics through chalk talk. *Written Communication*, 28(4), 345-379.

Bressoud, D., V. Mesa, C. Rasmussen. *Insights and Recommendations from the MAA National Study of College Calculus*. MAA Press, 2015.

Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okorafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics. *Proceedings of the National Academy of Sciences*, 111(23), 8410-8415.

Fukawa-Connelly, T. P. (2012). A case study of one instructor’s lecture-based teaching of proof in abstract algebra: making sense of her pedagogical moves. *Educational Studies in Mathematics*, 81(3), 325-345.

Glasersfeld, E. v. (1995). *Radical Constructivism: A Way of Knowing and Learning*. New York: Routledge-Falmer.

Halmos, Paul. What is teaching?, *American Mathematical Monthly*, 101(9), November 1994.

Harel, G. (2007). The DNR System as a Conceptual Framework for Curriculum Development and Instruction, In R. Lesh, J. Kaput, E. Hamilton (Eds.), *Foundations for the Future in Mathematics Education*, Erlbaum

Kogan, M., & Laursen, S. L. (2014). Assessing long-term effects of inquiry-based learning: A case study from college mathematics. *Innovative Higher Education*, 39(3), 183-199

Lobato, J., Clarke, D., & Ellis, A. B. (2005). Initiating and eliciting in teaching: A reformulation of telling. *Journal for Research in Mathematics Education*, 36(2), 101-136.

Mason, John. From Concept Images to Pedagogic Structure for a Mathematical Topic, in *Making the Connection: Research and Teaching in Undergraduate Mathematics Education*, edited by Marilyn Carlson and Chris Rasmussen. MAA Notes #73, Mathematical Association of America, 2008.

Piaget, J. (1954). *The Construction of Reality in the Child*. New York: Basic Books Publishing.

Smith, J. P. III. (1996). Efficacy and teaching mathematics by telling: A challenge for reform. *Journal for Research in Mathematics Education*, 27(4), 387-402.

Steffe, L. P. & Thompson, P. W. (2000). *Radical Constructivism in Action: Building on the Pioneering Work of Ernst von Glasersfeld.* New York: Routledge.

*Editor’s note: This is the fourth article in a series devoted to active learning in mathematics courses. The other articles in the series can be found **here**.*

In contrast to our first three articles in this series on active learning, in this article we take a more personal approach to the subject. Below, the contributing editors for this blog share aspects of our journeys into active learning, including the fundamental reasons we began using active learning methods, why we have persisted in using them, and some of our most visceral responses to our own experiences with these methods, both positive and negative. As is clear from these reflections, mathematicians begin using active learning techniques for many different reasons, from personal experiences as students (both good and bad) to the influence of colleagues, conferences, and workshops. The path to active learning is not always a smooth one, and is almost always a winding road.

Because of this, we believe it is important for mathematics teachers to share their own experiences, both positive and negative, in the search for more meaningful student engagement and learning. *We invite all our readers to share their own stories in the comments at the end of this post.* We also recognize that many other mathematicians have shared their experiences in other venues, so at the end of this article we provide a collection of links to essays, blog posts, and book chapters that we have found inspirational.

There is one more implicit message contained in the reflections below that we want to highlight. All mathematics teachers, even those using the most ambitious student-centered methods, use a range of teaching techniques combined in different ways. In our next post, we will dig deeper into the idea of instructor “telling” to gain a better understanding of how an effective balance can be found between the process of student discovery and the act of faculty sharing their expertise and experience.

**Priscilla Bremser:**

I began using active learning methods for several reasons, but two interconnected ones come to mind. First, Middlebury College requires all departments to contribute to the First-Year Seminar program, which places every incoming student into a small writing-intensive class. The topic is chosen by the instructor, while guidelines for writing instruction apply to all seminars. As I have developed and taught my seminars over the years, I’ve become convinced that students learn better when they are required to express themselves clearly and precisely, rather than simply listening or reading. At some point it became obvious that the same principle applies in my other courses as well, and hence I was ready to try some of the active learning approaches I’d been hearing about at American Mathematical Society meetings and reading about in journals.

Second, I got a few student comments on course evaluations, especially for Calculus courses, that suggested I was more helpful in office hours than in lecture. Thinking it through, I realized that in office hours, I routinely and repeatedly ask students about their own thinking, whereas in lecture, I was constantly making assumptions about student thinking, and relying on their responses to “Any questions?” for guidance, which didn’t elicit enough information to address the misunderstandings around the room. One way to make class more like office hours is to put students into small groups. I then set ground rules for participation and ask for a single set of problem solutions from each group. This encourages everyone to speak some mathematics in each class session, and to ask for clarity and precision from classmates. Because I’m joining each conversation for a while, I get a more accurate perception of students’ comprehension levels.

This semester I’m teaching Mathematics for Teachers, using an IBL textbook by Matthew Jones. I’ve already seen several students throw fists up in the air, saying “I get it now! That’s so cool!” How well I remember having that response to my first Number Theory course; it’s why I went into teaching at this level in the first place. On the other hand, a Linear Algebra student who insists that “I learn better from reading a traditional textbook” leaves me feeling rather deflated. It seems that I’ve failed to convey why I direct the course the way that I do, or at least I haven’t yet succeeded. The truth is, though, that I used to feel the same way. I regarded mathematics as a solitary pursuit, in which checking in with classmates was a sign of weakness. Had I been required to discuss my thinking regularly during class and encouraged to do so between sessions, I would have developed a more solid foundation for my later learning. Remembering this inspires me to be intentional with students, and explain repeatedly why I direct my courses the way that I do. Most of them come around eventually.

**Elise Lockwood:**

I have a strong memory of being an undergraduate in a discrete mathematics course, trying desperately to understand the formulas for permutations, combinations, and the differences between the two. The instructor had presented the material, perhaps providing an example or two, but she had not provided an opportunity for us to actively explore and understand why the formulas might make sense. By the time I was working on homework, I simply tried (and often failed) to apply the formulas I had been given. I strongly disliked and feared counting problems for years after that experience. It wasn’t until much later that I took a combinatorics course as a master’s student. Here, the counting material was brought to life as we were given opportunities to work through problems during class, to unpack formulas, and to come to understand the subtlety and wonder of counting. The teacher did not simply present a formula and move on, assuming we understood it. Rather, he persisted by challenging us to make sense of what was going on in the problems we solved.

For example, we once were discussing a counting problem in class (I can’t recall if it was an in-class problem or a problem that had been assigned for homework). During this discussion, it became clear that students had answered the problem in two different ways — both of them seemed to make sense logically, but they did not yield the same numerical result. The instructor did not just tell us which answer was right, but he used the opportunity to have us consider both answers, facilitating a (friendly) debate among the class about which approach was correct. We had to defend whichever answer we thought was correct and critique the one we thought was incorrect. This had the effect not only of engaging us and piquing our curiosity about a correct solution, but it made us think more carefully and deeply about the subtleties of the problem.

Now, studying how students solve counting problems is the primary focus of my research in mathematics education. My passion for the teaching and learning of counting was probably in large part formed by the frustrations I felt as an undergraduate and the elation I later experienced when I actually understood some of the fundamental ideas.

When I have been given the opportunity to teach counting over the years (in discrete mathematics or combinatorics classes, or in courses for pre-service teachers), I have tried my hardest to facilitate my students’ active engagement with the material during class. This has not taken an inordinate amount of time or effort: instead of just giving students the formulas off the bat, I give them a series of counting problems that both introduce counting as a problem solving activity and motivate (and build up to) some key counting formulas. For example, students are given problems in which they list some outcomes and appreciate the difference between permutations and combinations firsthand. I have found that a number of important issues and ideas (concerns about order, errors of overcounting, key binomial identities) can emerge on their own through the students’ activity, making any subsequent discussion or lecture much more meaningful for students. When I incorporate these kinds of activities for my students, I am consistently impressed at the meaning they are able to make of complex and notoriously tricky ideas.

More broadly, these pedagogical decisions I make are also based on my belief about the nature of mathematics and the nature of what it means to learn mathematics. Through my own experiences as a student, a teacher, and a researcher, I have become convinced that providing students with opportunities to actively engage with and think about mathematical concepts — during class, and not just on their own time — is a beneficial practice. My experience with the topic of counting (something near and dear to my heart) is but one example of the powerful ways in which student engagement can be leverage for deep and meaningful mathematical understanding.

**Diana White:**

What stands out most to me as I reflect upon my journey into active learning is not so much how or why I got involved, but the struggles that I faced during my first few years as a tenure-track faculty member as I tried to switch from being a good “lecturer” to all out inquiry-based learning. I was enthusiastic and ambitious, but lacking in the skills to genuinely teach in the manner in which I wanted.

As a junior faculty member, I was already sold on the value of inquiry-based learning and student-centered teaching. I had worked in various ways with teachers as a graduate student at the University of Nebraska and as a post-doc at the University of South Carolina, including teaching math content courses for elementary teachers and assisting with summer professional development courses for teachers. Then, the summer before I started my current position, I attended both the annual Legacy of R.L. Moore conference and a weeklong workshop on teaching number theory with IBL through the MAA PREP program. The enthusiasm and passion at both of these was contagious.

However, upon starting my tenure track position, I jumped straight in, with extremely ambitious goals for my courses and my students, ones for which I did not have the skills to implement yet. In hindsight, it was too much for me to try to both switch from being a good “lecturer” to doing full out IBL and running an intensely student centered classroom, all while teaching new courses in a new place. I tried to do way too much too soon, and in many ways that was not healthy for either me or the students, as evidenced by low student evaluations and frustrations on both sides.

Figuring out specifically what was going wrong was a challenge, though. Those who came to observe, both from my department and our Center for Faculty Development, did not find anything specific that was major, and student comments were somewhat generic – frustration that they felt the class was disorganized and that they were having to teach themselves the material.

I thus backtracked to more in the center of the spectrum, using an interactive lecture Things smoothed out and students became happier. What I am not at all convinced of, though, is that this decision was best for student learning. Despite the unhappiness on both our ends when I was at the far end of the active learning spectrum, I had ample evidence (both from assessments and from direct observation of their thought processes in class) that students were both learning how to think mathematically and building a sense of community outside the classroom. To this day, I feel torn, like I made a decision that was best for student satisfaction, as well as for how my colleagues within my department perceive me. Yet I remain convinced that my students are now learning less, and that there are students who are not passing my classes who would have passed had I taught using more active learning. (It was impossible to “hide” with my earlier classes, due to the natural accountability built into the process, so struggling students had to confront their weaknesses much sooner.)

It is hard for me to look back with regrets, as the lessons learned have been quite powerful and no doubt shaped who I am today. However, I would offer some thoughts, aimed primarily at junior faculty.

Don’t be afraid to start slow. Even if it’s not where you want to end up, just getting started is still an important first step. Negative perceptions from students and colleagues are incredibly hard to overcome.

Don’t underestimate the importance of student buy-in, or of faculty buy-in. I found many faculty feel like coverage and exposure are essential, and believe strongly that performance on traditional exams is an indicator of depth of knowledge or ability to think mathematically.

Don’t be afraid to politely request to decline teaching assignments. When I was asked to teach the history of mathematics, a course for which I had no knowledge of or background in, I wasn’t comfortable asking to teach something else instead. While it has proved really beneficial to my career (I’m now part of an NSF grant related to the use of primary source projects in the undergraduate mathematics classroom), I was in no way qualified to take that on as a first course at a new university.

I have personally gained a tremendous amount from my participation in the IBL community, perhaps most importantly a sense of community with others who believe strongly in active learning.

**Art Duval:**

My first experience with active learning in mathematics was as a student at the Hampshire College Summer Studies in Mathematics program during high school. Although I’d had good math teachers in junior high and high school, this was nothing like I’d seen before: The first day of class, we spent several hours discussing one problem (the number of regions formed in 3-dimensional space by drawing \(n\) planes), drawing pictures and making conjectures; the rest of the summer was similar. The six-week experience made such an impression on me, that (as I realized some years later) most of the educational innovations I have tried as a teacher have been an attempt to recreate that experience in some way for my own students.

When I was an undergraduate, I noticed that classes where all I did was furiously take notes to try to keep up with the instructor were not nearly as successful for me as those where I had to *do* something. Early in my teaching career, I got a big push towards using active learning course structures from teaching “reform calculus” and courses for future elementary school teachers. In each case, this was greatly facilitated by my sitting in on another instructor’s section that already incorporated these structures. Later I learned, through my participation in a K-16 mathematics alignment initiative, the importance of conceptual understanding among the levels of cognitive demand, and this helped me find the language to describe what I was trying to achieve.

Over time, I noticed that students in my courses with more active learning seemed to stay after class more often to discuss mathematics with me or with their peers, and to provide me with more feedback about the course. This sort of engagement, in addition to being good for the students, is very addictive to me. My end-of-semester course ratings didn’t seem to be noticeably different, but the written comments students submitted were more in-depth, and indicated the course was more rewarding in fundamental ways. As with many habits, after I’d done this for a while, it became hard *not* to incorporate at least little bits of interactivity (think-pair-share, student presentation of homework problems), even in courses where external forces keep me from incorporating more radical active learning structures.

Of course, there are always challenges to overcome. The biggest difficulty I face with including any sort of active learning is how much more time it takes to get students to realize something than it takes to simply tell them. I also still find it hard to figure out the right sort of scaffolding to help students see their way to a new concept or the solution to a problem. Still, I keep including as much active learning as I can in each course. The parts of classes I took as a student (going back to junior high school) that I remember most vividly, and the lessons I learned most thoroughly, whether in mathematics or in other subjects, were the activities, not the lectures. Along the same lines, I occasionally run into former students who took my courses many years ago, and it’s the students who took the courses with extensive active learning, much more than those who took more traditional courses, who still remember all these years later details of the course and how much they learned from it.

**Other Essays and Reflections:**

Benjamin Braun, The Secret Question (Are We Actually Good at Math?), http://blogs.ams.org/matheducation/2015/09/01/the-secret-question-are-we-actually-good-at-math/

David Bressoud, Personal Thoughts on Mature Teaching, in *How to Teach Mathematics, 2nd Edition*, by Steven Krantz, American Mathematical Society, 1999. Google books preview

Jerry Dwyer, Transformation of a Math Professor’s Teaching, http://blogs.ams.org/matheducation/2014/06/01/transformation-of-a-math-professors-teaching/

Oscar E. Fernandez, Helping All Students Experience the Magic of Mathematics, http://blogs.ams.org/matheducation/2014/10/10/helping-all-students-experience-the-magic-of-mathematics/

Ellie Kennedy, A First-timer’s Experience With IBL, http://maamathedmatters.blogspot.com/2014/09/a-first-timers-experience-with-ibl.html

Bob Klein, Knowing What to Do is not Doing, http://maamathedmatters.blogspot.com/2015/07/knowing-what-to-do-is-not-doing.html

Evelyn Lamb, Blogs for an IBL Novice, http://blogs.ams.org/blogonmathblogs/2015/09/21/blogs-for-an-ibl-novice/

Carl Lee, The Place of Mathematics and the Mathematics of Place, http://blogs.ams.org/matheducation/2014/10/01/the-place-of-mathematics-and-the-mathematics-of-place/

Steven Strogatz, Teaching Through Inquiry: A Beginner’s Perspectives, Parts I and II, http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-1, http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-2

Francis Su, The Lesson of Grace in Teaching, http://mathyawp.blogspot.com/2013/01/the-lesson-of-grace-in-teaching.html

]]>*Editor’s note: This is the third article in a series devoted to active learning in mathematics courses. The other articles in the series can be found here.*

It is common in the mathematical community for the phrases “active learning” and “inquiry-based learning” (IBL) to be associated with a particular teaching technique that emphasizes having students independently work and present to their peers in a classroom environment with little-to-no lecturing done on the part of the instructor. Yet it is counterproductive for this method to be a dominant cultural interpretation of “active learning,” as it does not represent the range of teaching styles and techniques that fall along the active learning and IBL spectrums as considered by mathematicians who use these pedagogies, mathematics education researchers, federal and private funding agencies, and professional societies such as the AMS, MAA, SIAM, ASA, AMATYC, and NCTM. In this article we will provide multiple examples of active learning techniques and environments that arise at institutions with different needs and constraints. We begin by reflecting on general qualities of classroom environments that support student learning.

**Student-Centered Classroom Environments**

In his decade-long study of highly-effective college teachers [1], Ken Bain found that such teachers establish in their courses a *natural critical learning environment* in which…

…people learn by confronting intriguing, beautiful, or important problems, authentic tasks that will challenge them to grapple with ideas, rethink their assumptions, and examine their mental models of reality. These are challenging yet supportive conditions in which learners feel a sense of control over their education; work collaboratively with others; believe that their work will be considered fairly and honestly; and try, fail, and receive feedback from expert learners in advance of and separate from any summative judgment of their effort. — Ken Bain,

What the Best College Teachers Do

From this description is it clear that these environments engage students with tasks at all levels of cognitive demand, a concept described in Part II of this series. While these environments require effort and diligence to establish and maintain, Bain makes it clear that these environments arise in every conceivable teaching environment, including small discussion-focused courses in the humanities, large-lecture style courses in the sciences, practicum-based courses in medical fields, field-based courses in the social sciences, and more. These diverse teaching environments have been the catalyst for the development of many successful models of active learning that support student engagement; this is one source of the challenge behind defining the phrase “active learning,” as we discussed in Part I of this series. Bain’s work complements and reinforces the explicit consideration in the meta-analysis by Freeman et al. [2] of varied active learning techniques.

Acknowledging the effectiveness of a range of active learning techniques across diverse settings is particularly important in the context of postsecondary mathematics teaching and learning. In contemporary college and university courses, lecturing remains the dominant teaching technique used by mathematics faculty. Many faculty view the use of either active learning (with the stereotypical interpretation mentioned previously) or lecture as an exclusive choice with two diametrically opposed options, yet nothing could be farther from the truth. Marrongelle and Rasmussen [6] have described a spectrum of teaching that ranges from “all telling” to “all student discovery.” Mathematics education researchers have invested significant effort toward understanding teaching and learning across this spectrum, including recent efforts to better understand the pedagogical moves of mathematicians who use traditional lecture as their instructional practice [4], [5]; we will investigate this topic in more depth in a forthcoming article in this series. The most important aspect of this for mathematics teaching is that there are opportunities to use active learning techniques at all points on this spectrum, the single exception being the extreme end of “instructor lecture only — no questions or comments allowed by students,” which we believe is extremely rare in practice.

In the rest of this article we will describe techniques and environments that we include as active learning, using our definition from Part I of this series. We will begin with techniques that fall closer to the “all telling” end of the spectrum and end with techniques closer to the “all student discovery” end. It is important to discuss techniques that can be used across this spectrum because there are many high-quality, concerned teachers who, while not wanting to make the jump to all-student-discovery, are deeply interested in increasing student learning and engagement. These responsible, committed teachers are valuable members of the mathematical community. Indeed, in the case of Calculus, a recent report by the MAA regarding successful calculus programs [3] found that the most important aspect regarding student persistence from first- to second-semester calculus is the presence of three factors: classroom interactions that acknowledge students; encouragement and availability on the part of the instructor; and the use of fair assessments. These are among the qualities that the report uses to describe “good” teaching, and these qualities afford ample opportunities for the introduction of effective active learning techniques. The MAA report uses the term “ambitious teaching” to describe the use of more sophisticated and complex active learning techniques by teachers moving beyond the qualities of good teaching, which represents a shift further along the spectrum discussed by Marrongelle and Rasmussen.

An important observation is that the words “active” and “interactive” are not synonymous in our discussion. For example, a lecture in which an instructor tells jokes that elicit laughter from students, or asks students to fact-check an elementary arithmetic calculation with their calculator, is interactive. These actions acknowledge the presence of students, hence fall under “good” teaching. However, those techniques do not provide tasks in which students directly engage with content during class, thus aren’t within the boundaries of active learning. Similarly, active learning must go beyond asking students to “think hard.” For example, after a particularly complicated example in a calculus class, or upon completion of a proof in an advanced course, an instructor might tell students to “think about what we just did for a minute or two,” then ask if there are any questions. While again this act would fall within the bounds of “good” teaching, the absence of a specific task given to students, with a specific goal, prevents this from being considered an active learning technique.

**Active Learning Techniques for Lectures**

One of the best examples of an active learning technique suitable for use in lectures is “think-pair-share.” In this technique, the instructor provides students with a short task — perhaps a short computational problem, or a step in a proof to complete, or an example for them to create a hypothesis based on. After providing the students with 2-5 minutes of time to independently consider the task, the students are asked to compare their answers with the people sitting around them, or with their partner if they have been placed into explicit work pairs. Finally, some or all of the students are asked to share their answers in some manner, either with the groups next to them or with the entire class. The benefits of using this technique for students are that they have a chance to be energized during lecture, have a chance to pause and consider the content the lecturer has been presenting, and have to explain their thinking to peers. In classes with large numbers of students for whom English is not their first language, students also can discuss the content with peers in a language they might understand more clearly. Instructors benefit as well, as they can gather feedback from students to determine how well content is understood. The main drawback instructors report for this technique is that in the time it takes to complete a think-pair-share, the instructor could have covered more examples or moved on to other content topics more quickly. According to research on activities such as this, for example a study of physics students by Deslauriers et al. [7] that is discussed by Bressoud [8] in the context of mathematics, these benefits far outweigh the drawbacks.

Because this technique is relatively simple, it is applicable in almost every conceivable teaching environment. Even in medium- and large-lecture settings, instructors have used both low- and high-tech feedback response systems for the “share” stage of the technique. For example, many instructors use multiple-choice problems as think-pair-share prompts in conjunction with classroom response systems, i.e. “clickers.” These systems typically come with additional data analysis features that allow instructors to more carefully review student responses over time to detect problematic content areas. Even at institutions where faculty do not have access to sophisticated systems of this type or do not want to deal with the technology, many instructors have successfully had students share their answers by holding up colored pieces of paper, providing a visual representation of their responses. This technique is introduced in Prather and Brissenden [9] (p. 10) as a small part of a larger article about a very focused form of think-pair-share applicable to all disciplines; for a more practical introduction to these “A-B-C-D cards”, with examples from a statistics classroom, see Lesser [26].

In addition to think-pair-share, there are many related examples of “classroom voting” techniques that can be used to increase student engagement during a lecture-based course. An in-depth description of these techniques can be found in the MAA volume *Teaching Mathematics with Classroom Voting: With and Without Clickers* [10].

**Inverted (or “Flipped”) Classes**

In an inverted (or “flipped”) classroom environment, instructor presentations of basic definitions, examples, proofs, and heuristics are provided to students in videos or in assigned readings that are completed prior to attending class. As a result, class time becomes available for tasks that directly engage students. The type of task that instructors use during this time ranges from using complicated think-pair-shares, with complex problems or examples, to having students work in small groups on a sequenced activity worksheet with occasional instructor or teaching assistant feedback. The inverted model of teaching has been used as the structure for entire courses, as an occasional event for handling topics that are less amenable to lecture presentations, as the basis for review sessions or problem solving sessions, and more. While the mere act of inverting a classroom is not inherently active, the structure of the inverted classroom environment is typically used to support in-class tasks with higher levels of cognitive demand, hence our inclusion of this as an active learning environment.

Compared to implementing think-pair-share and classroom voting techniques, creating inverted classroom environments requires both more effort and time on the part of the instructor and significantly more institutional support, especially in the areas of technology and data storage support. Having said that, the inverted classroom model is being explored in many disciplines, and many colleges and universities have experience with this technique even if mathematics faculty do not. This breadth of use across disciplines is reflected in a recent volume on best practices for flipped classrooms [11]. In mathematics, faculty have used combinations of video- and readings-based assignments to invert classes across a surprising range of content areas, including linear algebra [12], [15], calculus [13], [16], math courses for pre-service elementary school teachers [14], statistics [17], and mathematical biology [18]. We refer the interested reader to these references for in-depth discussions regarding the benefits and drawbacks of inverted classroom environments.

**Math Emporium**

The emporium model of teaching, like inverted classrooms, is not a technique but a learning environment that supports active learning techniques. The typical math emporium [23] is centered around a large room filled with computer workstations, in which students progress through self-paced online courses. Unlike inverted classes, many emporium models do not include a lecture component at all. Also unlike inverted classes, most math emporiums have been developed to handle remediation issues and low-level courses such as developmental mathematics and college algebra. An emporium usually has tables at which students can work collaboratively and is staffed by a large number of teaching assistants and tutors. Because the work of students is self-paced, and is driven in some emporium models by adaptive learning systems such as Aleks, students spend most of their time actively engaging with course content, providing opportunities for engagement with a range of tasks. In the emporium environment it is important that tasks be designed with levels of cognitive demand in mind, as there is evidence that some students who are successful in emporium programs are not engaging in high-cognitive work that promotes deep learning [19].

An interesting aspect of the math emporium model is that it was developed and is promoted as a means of both helping students learn and managing the economic reality that many institutions face of increased student enrollment with flat or decreasing instructional resources. The operating costs of an emporium can be lower than that of traditional teaching environments [23], and for this (among other factors) the math emporium model has attracted attention from national news organizations [21]. With a teaching environment that combines significant infrastructure investment at the institutional level and a shift from the traditional economic model on which college classes are built, it is not surprising that the emporium model has been more controversial in the mathematical community than techniques like classroom voting or less comprehensive changes such as inverted classes. Thoughtful discussions and methodological studies, for example Bressoud’s *Launchings* column on this topic [20] and a recent study by Webel et al. [19], are available for readers interested in learning more about the math emporium model.

**Laboratory Courses**

The use of computer technology in math courses does not have to be as dramatic as in emporium models. Since the 1990’s, many mathematics courses have included exercises and computer lab activities using programs such as Mathematica, Maple, and MATLAB. The use of computer algebra systems in postsecondary mathematics courses is now widespread, with a wide range of benefits reported by mathematicians teaching with technological tools, often representing students engagement at higher levels of cognitive demand [22].

The use of technology to teach mathematics can go far beyond simple augmentation of traditional courses, serving as the basis for an environment focused on active learning. For example, in 1989 the mathematics and statistics department at Mount Holyoke College created a new sophomore-level course for their majors that they called the Laboratory in Mathematical Experimentation, or, for short, “the Lab”. The course consisted of six to seven mathematical labs in which students were given a problem to explore, usually with a computer (or calculator) and programs already written by the instructors. Students would use the results of their experiments to make and test conjectures, and then ultimately write arguments to justify some of their conjectures. The course succeeded “beyond any of [the faculty’s] expectations.” Students became more likely to engage with mathematics actively, and did better in their upper-division analysis and algebra courses than students who did not take the course. The labs for this course were eventually distilled into a book, *Laboratories in Mathematical Experimentation* [24], from which the above historical summary was taken. While the original computer code was written in BASIC, mathematicians have adapted the code to other languages such as *Mathematica* (and even improved it on the way). Students typically write up the results of each lab, and this is where they get to practice writing mathematics. In order for students to succeed in this type of course, they are forced to abandon the common misconception that mathematics consists of nothing more than applying formulas the teacher gives you. Another example of a laboratory-style course, influenced by the Mount Holyoke approach, is given by Brown [25] in an article regarding the recent development of a course on experimental mathematics suitable for both mathematics majors and students fulfilling a general education requirement.

**Inquiry-Based Learning**

Arguably the most well-known example of active learning in mathematics is Inquiry-Based Learning (IBL). Recent research studies have found that IBL courses have a positive effect on students, with particularly strong benefits for low-achieving students [31]. In mathematical culture, IBL (sometimes incorrectly identified as synonymous with the “Moore-method”) has its roots in the teaching methods of R.L. Moore, whose teaching methods were extremely beneficial for some students. However, his overt racism and bias in his classroom precluded many students from participating in his classes [27]. This tension has led some mathematicians to be caught between a desire to use and promote IBL methods and a desire to remove any suggestion of acceptance of the negative aspects of Moore’s teaching [28], [29], [30], a situation the mathematical community needs to resolve.

One of the main organizations promoting IBL is the Educational Advancement Foundation (EAF), which holds an annual “Legacy of R.L. Moore Conference” each summer. Despite the tension surrounding this naming, the EAF has been by far the largest promoter of IBL (which is now much more broadly construed) in mathematics. In addition to their summer conference, now a vibrant meeting full of early-career faculty eager to learn and share best-practices related to IBL, they sponsor both large grants and small grant programs through the Academy of Inquiry Based Learning (AIBL). As an example of a classroom environment that falls close to the aforementioned “all student discovery”, AIBL describes a “typical” day in an IBL class:

Class starts. The instructor passes out a signup sheet for students willing to present upcoming problems. The bulk of the time is spent on student presentations of solutions/proofs to problems. Students, who have been selected previously or at the beginning of class, write proofs/solutions on the board. One by one, students present their solutions/proofs to their class. The class as a group (perhaps in pairs) reviews and validates the proofs. Questions are asked and are either dealt with there or the presenter can opt to return with a fix at the next class period. If the solution is approved as correct by the class, then the next student presents his/her solution. This cycle continues until all students have presented. If the class cannot arrive at a consensus on a particular problem or issue, then the instructor and the class devise a plan to settle the issue. Perhaps new problems or subproblems are written on the board, and the class is asked to solve these. Teaching choices include pair work immediately or asking students to work on the new tasks outside of class, with the intention of restarting the discussion the next time. If a new unit of material is started, then a mini lecture and/or some hands-on activities to explore new ideas and definitions could be deployed. If no one has anything to present OR if everyone is stuck on a problem, pair work or group work can be used to help students break down a problem and generate strategies or ways into solving a particularly hard problem.

Note that while this clearly falls toward one end of the active learning spectrum discussed previously, this does not describe a classroom consisting of pure, unguided student discovery. Rather, students are provided direction through a scaffolded series of activities, some independent, some in pairs, some in small groups, and some with the whole class, including mini-lectures as appropriate. Faculty teaching in an IBL environment need to develop facility with a range of teaching strategies, and need to develop familiarity with many “teaching moves” that are not typically used in lecture environments. Opportunities exist for faculty to receive training in these areas, for example through workshops and minicourses at the Joint Mathematics Meetings and MathFest, or through workshops sponsored by the Academy for Inquiry-Based Learning and other organizations.

The other aspect of IBL that requires attention from faculty is the scaffolding of content. Fortunately, many existing resources are available for faculty interested in teaching an IBL course. The Journal of Inquiry-Based Learning in Mathematics contains refereed course notes on a variety of topics, ranging from first-semester calculus to modern algebra to real analysis to mathematics for elementary school teachers. These notes contain sequences of tasks carefully designed to guide students through an area or topic of mathematics. There are also many excellent freely available texts that are suitable for IBL use contained on independent websites, such as the Active Calculus textbook series and Ken Bogart’s guided inquiry combinatorics text. Published textbooks also exist to support IBL courses, e.g. in number theory [32] and algebraic geometry [33].

**Conclusion**

Active learning is hard to define, but at its core is having students work on mathematical tasks of varying levels of cognitive demand during class. As we have seen in this survey, there are multiple teaching environments in which active learning can be used, and multiple active learning techniques through which student tasks can be provided. However, thus far in our series on active learning we have avoided discussion of a fundamental truth: learning to effectively design and use active learning techniques is challenging, and the process of integrating these activities into one’s “teaching toolbox” requires both patience and a willingness to persist through setbacks. In this way, the process of developing and implementing new pedagogical tools is akin to the process of learning and discovering mathematics.

In the three remaining articles in this series on active learning, we will direct our attention to the ways in which personal experiences can shape and affect our development and choices as teachers. In Part IV of this series, we, the authors, will reflect on aspects of our personal experiences as teachers who have struggled to find effective ways to engage students. In Part V, we will explore the role of “telling” in the mathematics classroom and gain a better understanding of the subtle ways in which instructor lecture, student activities, and constructivist educational philosophies can support each other. In Part VI, our sixth and final article on this topic, we will consider the ways in which professional training as a mathematician can be both a benefit and a hindrance to broadening and developing as a teacher of mathematics.

**References**

[1] Scott Freeman, Sarah L. Eddy, Miles McDonough, Michelle K. Smith, Nnadozie Okoroafor, Hannah Jordt, and Mary Pat Wenderoth. Active learning increases student performance in science, engineering, and mathematics. *Proc. Natl. Acad. Sci. U.S.A. *2014, 111, (23) 8410-8415

[2] Bain, Ken. *What the Best College Teachers Do*. Harvard University Press, 2004.

[3] Bressoud, D., V. Mesa, C. Rasmussen. Insights and Recommendations from the MAA National Study of College Calculus. MAA Press, 2015.

[4] Artemeva, N., & Fox, J. (2011). The writing’s on the board: the global and the local in teaching undergraduate mathematics through chalk talk. *Written Communication*, 28(4), 345-379.

[5] Fukawa-Connelly, T. P. (2012). A case study of one instructor’s lecture-based teaching of proof in abstract algebra: making sense of her pedagogical moves. *Educational Studies in Mathematics*, 81(3), 325-345.

[6] Marrongelle, Karen and Rasmussen, Chris. Meeting New Teaching Challenges: Teaching Strategies that Mediate between All Lecture and All Student Discovery. *Making the Connection: Research and Teaching in Undergraduate Mathematics Education.* Carlson, Marilyn P. and Rasmussen, Chris, eds. MAA Notes #73, 2008. pp 167-177.

[7] Deslauriers, L., E. Schelew, and C. Wieman. Improved Learning in a Large-Enrollment Physics Class. *Science*. Vol. 332, 13 May, 2011, 862-864.

[8] Bressoud, David. The Worst Way to Teach. *MAA Launchings Column*, July 2011. https://www.maa.org/external_archive/columns/launchings/launchings_07_11.html

[9] Prather, E., & Brissenden, G. (2008). Development and application of a situated apprenticeship approach to professional development of astronomy instructors, Astronomy Education Review, 7(2), 1-17. http://astronomy101.jpl.nasa.gov/files/Situated%20Apprentice_AER.pdf

[10] Cline, Kelly Slater Cline and Zullo, Holly, (eds). *Teaching Mathematics with Classroom Voting: With and Without Clickers*. MAA Notes #79, 2011

[11] Julee B. Waldrop, Melody A. Bowdon, (eds). *Best Practices for Flipping the College Classroom*. Routledge, 2015.

[12] Robert Talbert (2014) Inverting the Linear Algebra Classroom, *PRIMUS*, 24:5, 361-374.

[13] Jean McGivney-Burelle and Fei Xue (2013) Flipping Calculus, *PRIMUS*, 23:5, 477-486.

[14] Pari Ford (2015) Flipping a Math Content Course for Pre-Service Elementary School Teachers, *PRIMUS*, 25:4, 369-380.

[15] Betty Love, Angie Hodge, Neal Grandgenett & Andrew W. Swift (2014) Student learning and perceptions in a flipped linear algebra course, International Journal of Mathematical Education in Science and Technology, 45:3, 317-324.

[16] Veselin Jungić, Harpreet Kaur, Jamie Mulholland & Cindy Xin. On flipping the classroom in large first year calculus courses. *International Journal of Mathematical Education in Science and Technology*. Volume 46, Issue 4, May 2015, pages 508-520.

[17] Jennifer R. Winquist and Kieth A. Carlson. Flipped Statistics Class Results: Better Performance Than Lecture Over One Year Later. *Journal of Statistics Education*. Volume 22, Number 3 (2014).

[18] Eric Alan Eager, James Peirce & Patrick Barlow (2014) Math Bio or Biomath? Flipping the mathematical biology classroom. *Letters in Biomathematics.* 1:2, 139-155

[19] Corey Webel, Erin Krupa, Jason McManus. Benny goes to college: Is the “Math Emporium” reinventing Individually Prescribed Instruction? *Math*AMATYC* Educator*, May 2015, Vol. 6 Number 3.

[20] Bressoud, David. The Emporium. *MAA Launchings Column*, March 2015. http://launchings.blogspot.com/2015/03/the-emporium.html

[21] Daniel de Vise. “At Virginia Tech, computers help solve a math class problem.” The Washignton Post. April 22, 2012. https://www.washingtonpost.com/local/education/at-virginia-tech-computers-help-solve-a-math-class-problem/2012/04/22/gIQAmAOmaT_story.html

[22] Neil Marshall, Chantal Buteau, Daniel H. Jarvis, Zsolt Lavicza. Do mathematicians integrate computer algebra systems in university teaching? Comparing a literature review to an international survey study. *Computers & Education*, Volume 58, Issue 1, January 2012, Pages 423-434

[23] Barbara L. Robinson and Anne H. Moore. The Math Emporium: Virginia Tech, in *Learning Spaces*, Oblinger, Diana G. (ed). Educause, 2006.

[24] Cobb, G., G. Davidoff, A. Durfee, J. Gifford, D. O’Shea, M. Peterson, Pollatsek, M. Robinson, L. Senechal, R. Weaver, and J. W. Bruce. 1997. *Laboratories in Mathematical Experimentation: A Bridge to Higher Mathematics*. Key College Publishing.

[25] David Brown (2014) Experimental Mathematics for the First Year Student, *PRIMUS*, 24:4, 281-293. http://faculty.ithaca.edu/dabrown/docs/scholar/experimental.pdf

[26] Lesser, L. (2011). Low-Tech, Low-Cost, High-Gain, Real-Time Assessment? It’s all in the cards, easy as ABCD! *Texas Mathematics Teacher*, 58(2), 18-22. http://www.math.utep.edu/Faculty/lesser/LesserABCDcardsTMTpaper.pdf

[27] Reuben Hersh & Vera John-Steiner. *Loving and Hating Mathematics: Challenging the Myths of Mathematical Life*. Princeton University Press, 2011.

[28] Kung, David. Empowering Who? The Challenge of Diversifying the Mathematical Community. Presentation at June 2015 Legacy of R.L. Moore — IBL Conference, Austin, Texas. https://www.youtube.com/watch?v=V03scHu_OJE

[29] Lamb, Evelyn. Promoting Diversity and Respect in the Classroom. *AMS Blog on Math Blogs,* 17 August 2015. http://blogs.ams.org/blogonmathblogs/2015/08/17/promoting-diversity-and-respect-in-the-classroom/

[30] Salerno, Adriana. Talkin’ Bout a Teaching Revolution. AMS PhD+Epsilon Blog, 3 August 2015. http://blogs.ams.org/phdplus/2015/08/03/talkin-bout-a-teaching-revolution/

[31] Kogan, M., & Laursen, S. L. (2014). Assessing long-term effects of inquiry-based learning: A case study from college mathematics. *Innovative Higher Education,* *39*(3), 183-199

[32] David Marshall, Edward Odell, and Michael Starbird, *Number Theory Through Inquiry*. The Mathematical Association of America, 2007.

[33] Thomas Garrity, Richard Belshoff, Lynette Boos, Ryan Brown, Carl Lienert, David Murphy, Junalyn Navarra-Madsen, Pedro Poitevin, Shawn Robinson, Brian Snyder, Caryn Werner. *Algebraic Geometry: A Problem Solving Approach*. American Mathematical Society, Student Mathematical Library, 2013.

*Editor’s note: This is the second article in a series devoted to active learning in mathematics courses. The other articles in the series can be found here.*

Mathematics faculty are well-aware that students face challenges when encountering difficult problems, and it is common to hear instructors remark that successful students have high levels of “mathematical maturity,” or are particularly “creative,” or write “elegant” solutions to problems. To appreciate research results regarding active learning, it is useful to make these ideas more precise. Motivated by research in education, psychology, and sociology, language has been developed that can help mathematicians clarify what we mean when we talk about difficulty levels of problems, and the types of difficulty levels problems can have. This expanded vocabulary is in large part motivated by…

…the “cognitive revolution” [of the 1970’s and 1980’s]… [which] produced a significant reconceptualization of what it means to understand subject matter in different domains. There was a fundamental shift from an exclusive emphasis on knowledge — what does the student know? — to a focus on what students know and can do with their knowledge. The idea was not that knowledge is unimportant. Clearly, the more one knows, the greater the potential for that knowledge to be used. Rather, the idea was that having the knowledge was not enough; being able to use it in the appropriate circumstances is an essential component of proficiency.

— Alan Schoenfeld,

Assessing Mathematical Proficiency[17]

In this article, we will explore the concept and language of “level of cognitive demand” for tasks that students encounter. A primary motivation for our discussion is the important observation in the 2014 Proceedings of the National Academy of Science (PNAS) article “Active learning increases student performance in science, engineering, and mathematics” by Freeman, et al. [8], that active learning has a greater impact on student performance on concept inventories than on instructor-written examinations. Concept inventories are “tests of the most basic conceptual comprehension of foundations of a subject and not of computation skill” and are “quite different from final exams and make no pretense of testing everything in a course” [5]. The Calculus Concept Inventory is the most well-known inventory in mathematics, though compared to disciplines such as physics these inventories are less robust since they are in relatively early stages of development. Freeman et al. state:

Although student achievement was higher under active learning for both [instructor-written course examinations and concept inventories], we hypothesize that the difference in gains for examinations versus concept inventories may be due to the two types of assessments testing qualitatively different cognitive skills. This is consistent with previous research indicating that active learning has a greater impact on student mastery of higher- versus lower-level cognitive skills…

After introducing levels of cognitive demand in this article, our next article in this series will directly connect this topic to active learning techniques that are frequently used and promoted for postsecondary mathematics courses.

**Bloom’s Taxonomy and its Variants**

A well-known and long-established framework in educational psychology is Bloom’s taxonomy [2]. In 1956, Benjamin Bloom and a team of educational psychologists outlined multiple levels of skills in the cognitive domain of learning, increasing from simple to complex. These are often simplified into six skill levels: knowledge, comprehension, application, analysis, synthesis, evaluation. By associating verbal cue words with each level, they categorized test questions over a variety of topics at the college level, and found that *over 95%* of these questions were at the very lowest level, “recall of knowledge” [11, p. 1]. Since these original findings, which were further developed in a second volume published in 1964, the core ideas of Bloom’s taxonomy have been widely used in education across disciplines.

The original taxonomy has been extended and adapted by many researchers in educational psychology. For example, Anderson et al. [1] developed a two-dimensional extension of Bloom’s taxonomy with a cognitive process dimension (remember, understand, apply, analyze, evaluate, create) similar to Bloom’s taxonomy, but also with a knowledge dimension (factual knowledge, conceptual knowledge, procedural knowledge, and metacognitive knowledge) — a taxonomy table encoding this appears below. When categorizing a task by this taxonomy, the cognitive process is represented by the verb used when specifying the task (what the student is doing) and the knowledge process dimension corresponds to the noun (what kind of knowledge the student is working with). In 2002, a special volume of the journal *Theory Into Practice* was devoted to this revised taxonomy; examples of applications of this taxonomy can be found throughout the volume.

Remember | Understand | Apply | Analyze | Evaluate | Create | |

Factual Knowledge | ||||||

Conceptual Knowledge | ||||||

Procedural Knowledge | ||||||

Metacognitive Knowledge |

An important shortcoming of each of these taxonomies for mathematicians is that the specific descriptors used for the different levels aren’t always appropriate for mathematics. For instance, in Bloom’s taxonomy, application comes after comprehension, which does make sense in a general context. But trying to apply this to mathematics, it is too easy to put routine word problems in the “application” category. The idea of “application” in the general sense is to take ideas presented in one context and be able to use them in a somewhat new setting, but in mathematics the word “application” can be used to represent both the development of a mathematical model to fit a situation or data set and the “cookbook” application of a previously-established mathematical model; most word problems in textbooks fit into the latter category.

**Specialized Cognitive Taxonomies and General Student Intellectual Development**

Around the same time as [1], several papers appeared that used taxonomies specialized to mathematics, e.g., [15, 19, 20, 21]. These have the two-dimensional nature of [1], with the columns or verbs replaced by labels that are specific to mathematics, while the rows or nouns simply correspond to different topics in mathematics. In 2006, Andrew Porter [14] explained it this way:

Unfortunately, defining content in terms of topics has proven to be insufficient at least if explaining variance in student achievement is the goal [9]. For example, knowing whether or not a teacher has taught linear equations, while providing some useful information, is insufficient. What about linear equations was taught? Were students taught to distinguish a linear equation from a non-linear equation? Were students taught that a linear equation represents a unique line in a two space and how to graph the line? For every topic, content can further be defined according to categories of cognitive demand. In mathematics cognitive demand might distinguish memorize; perform procedures; communicate understanding; solve non-routine problems; conjecture, generalize, prove.

More details about this taxonomy of levels of cognitive demand can be found in [15]. A comparison of various such taxonomies can be found in [15].

Similarly, several papers of Mary Kay Stein and various co-authors [19, 20, 21] analyze mathematical tasks and how they are implemented, focusing on middle school, using four levels of cognitive demand: Memorization; procedures without connections; procedures with connections; and “doing mathematics”. They identify the first two levels as “low-level”, matching the first two levels of [15]; and they identify the last two levels as “high-level”, matching the last three levels of [15].

There are also broad models for student intellectual development across not only individual topics but their entire college experience. One of the first such models is due to William Perry, and it can be (overly) simplified into the following description. Most college students will begin with the belief that there are right and wrong answers to questions, and that professors hold the knowledge of which these are. As students progress through their studies, they realize that sometimes their teachers are not always aware of the answers to questions, and also that answers can be more subtle than merely “right” or “wrong.” After this realization, students often enter a phase of relativism, where everyone’s opinions are equally valid. In the final stages of intellectual development, students recognize that different areas of intellectual inquiry have different standards and (some students) develop a balance between intellectual independence and commitment to the discipline. The Perry model has been refined and revised by many psychologists to account for diverse student experiences with respect to gender and other factors; an excellent survey of these developments, with pedagogical implications, has been given by Felder and Brent [6,7].

As Thomas Rishel points out [16], students in the early stages of the Perry model or one of its variants often enjoy mathematics precisely because all the answers are perceived as known, and they frequently value mathematical problems that focus on verification of these truths. As these students begin to encounter complicated modeling problems, or as they are first asked to seriously participate in proof-based mathematical reasoning, the cognitive load of such tasks can be much higher than for students who have developed further along this model. Thus, the intellectual stage of development for a given student can impact the level of cognitive demand for various tasks and problems they will encounter in mathematics courses.

**Practical Issues: Level Identification and Task Assessment**

Given these theoretical frameworks for both cognitive engagement and intellectual development, a practical challenge for instructors is to use these frameworks effectively to increase the quality of teaching and learning in the classroom.** **With any of the cognitive taxonomies, it can be hard to assess precisely which level(s) a given student task is hitting. The taxonomy tables discussed in previous sections provide instructors with tools to produce reasonable cognitive demand analysis of the tasks they give students. Engagement with all cognitive levels is necessary for deep learning to take place, so it is important that mathematics faculty identify and provide students with tasks representing a range of levels. Since lower-level tasks are typically already most prevalent, and easiest to assess both in terms of time and resources, faculty have to make the effort to bring in the higher levels. As a result, three challenges for instructors are to identify high-quality mathematical activities for students at higher levels of cognitive demand, to develop methods for assessing student work on such activities, and to create or make use of institutional programs, culture, and resources to support the use of high-quality activities. We will comment on the third issue in our next article in this series.

Some mathematics problems afford a wide range of cognitive engagement. For example, in the K-12 setting Jo Boaler and others have promoted activities described as “low-floor, high-ceiling” (LFHC) [23]. These are activities that can give students practice in lower levels of cognitive demand, but also are open-ended enough to eventually lead to (grade-appropriate) mathematical investigations with high-cognitive demand. Good examples of problems that students can engage with all the way from elementary school procedures to the highest levels of cognitive demand, leading to college-level abstract topics, can be found on the youcubed website, on sites for Math Circles, and on sites for Math Teachers’ Circles. When students are working on LFHC problems, they have flexibility in how they navigate through the problem. Unless explicit guidance is given regarding how students should investigate a LFHC problem, it is possible for them to spend most of their time working inside a small range of cognitive demand. Consequently, it is important for instructors to provide some pathways or scaffolding for students to use when first engaging with such problems.

Though they are not as common as they deserve to be, mathematicians have developed a wide range of techniques for assessing high-cognitive demand tasks, including written assignments, group work, projects, portfolios, presentations, and more [3, 4, 10, 12, 13]. However, task-appropriate techniques for assessing a given high-cognitive demand task can be challenging to identify and put in practice. It is important that the method for assessing specific tasks be selected in the context of overall course assessment. Some mathematicians have been experimenting with grading schemes that more directly support high-cognitive demand assignments, such as specifications grading and standards-based grading. Unfortunately, the fact remains that there is much to be learned about the efficacy of different methods of assessment [17].

**Active Learning and Theories of Learning**

Implicit in our discussion has been an important point that should be made explicit: as Stein et al. state [19], “…cognitive demands are analyzed as the cognitive processes in which students actually engage as they go about working on the task” as opposed to what students witness others doing. Thus, it is not possible to discuss the cognitive level of a mathematical proof itself, though proofs certainly vary in level of sophistication. Rather, one focuses on the cognitive level of what a student is asked to do with the proof: memorize the proof verbatim? construct a concrete example illustrating the proof method? derive a similar result using the same technique? analyze the proof in order to identify the key steps? compare the proof to a different proof of the same result? These tasks are all different from the perspective of cognitive demand, hence they are not interchangeable from the perspective of student learning, yet they exhibit superficial similarities and would each generally be considered valuable for students to complete. It is worth remarking that the verbs used in describing each of the tasks are helpful indicators of level of cognitive demand, as the taxonomies suggest.

This observation brings us back to active learning, which by definition has as a primary goal to engage students through explicit mathematical tasks in the classroom, in view of peers, instructors, and teaching assistants. One major effect of active learning techniques is that the mathematical processes and practices of students, which are tightly interwoven with high-cognitive achievement, are brought into direct confluence with peer and instructor feedback. Thus, active learning techniques complement the shift in emphasis described by Schoenfeld, from received knowledge to committed engagement, of the primary goal of student learning. Active learning techniques are also well-aligned with contemporary theories of learning, for example constructivism, behaviorism, sociocultural theory, and others [18, 22].

As one example of this alignment, constructivism is based on the idea that people construct their own understanding and knowledge through their experiences rather than through the passive transfer of knowledge from one individual to another. This is a prominent theory of learning among mathematics education researchers with many refinements, e.g. radical constructivism remains agnostic about whether there actually is any objective truth/reality, while social constructivism views individual thought and social interaction as inseparable with no model for a socially isolated mind. Generally, constructivism’s emphasis on the actions of the learner reinforces the need to emphasize consideration of the cognitive demands placed on students.

**Conclusion**

As research regarding the teaching and learning of postsecondary mathematics and science matures and becomes more well-known, both inside the mathematics community and beyond, significant evidence is building that active learning techniques have a strong impact on student achievement on high-cognitive demand tasks. We began this article with recognition that mathematicians are fully aware of students’ difficulties with mathematical tasks at all levels, and the observation that the mathematical community has developed language such as “mathematical maturity” and “elegance” which is often applied to distinguish successful from unsuccessful student work. Our main purpose in writing this survey of concepts related to levels of cognitive demand is to introduce mathematicians to the rich and complex set of ideas that have been developed in an attempt to distinguish different types of student activities and actions related to learning. Given the current evidence supporting the positive impact of active learning techniques, mathematics faculty will have an increased need for a refined language with which to discuss both the successes and failures of our students and the efficacy of the large variety of active learning techniques that are available. In our next post in this series, we will discuss the most prominent of these active learning techniques and environments with an eye toward both institutional constraints (as discussed in Part I of this series) and student learning in the context of levels of cognitive demand.

**References**

[1] Anderson, L.W. (Ed.), Krathwohl, D.R. (Ed.), Airasian, P.W., Cruikshank, K.A., Mayer, R.E., Pintrich, P.R., Raths, J., & Wittrock, M.C. *A taxonomy for learning, teaching, and assessing: A revision of Bloom’s Taxonomy of Educational Objectives (Complete edition)*. New York: Longman. 2001

[2] Bloom, Benjamin, et al., eds. *Taxonomy of Educational Objectives: the classification of educational goals. Handbook I: Cognitive domain.* New York: Longmans, Green. 1956

[3] Benjamin Braun. Personal, Expository, Critical, and Creative: Using Writing in Mathematics Courses. *PRIMUS*, 24 (6), 2014, 447-464.

[4] A. Crannell, G. LaRose, and T. Ratliff. *Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go.* Mathematical Association of America, 2004.

[5] J. Epstein. 2013. The Calculus Concept Inventory—Measurement of the Effect of Teaching Methodology in Mathematics. *Notices of the American Mathematical Society.* 60 (8), 1018–1026.

[6] Felder, Richard M. and Brent, Rebecca. The Intellectual Development of Science and Engineering Students. Part 1: Models and Challenges,* Journal of Engineering Education*, Volume 93, Issue 4, October 2004, 269–277.

[7] Felder, Richard M. and Brent, Rebecca. The Intellectual Development of Science and Engineering Students. Part 2: Teaching to Promote Growth,* Journal of Engineering Education*, Volume 93, Issue 4, October 2004, 279–291.

[8] Scott Freeman, Sarah L. Eddy, Miles McDonough, Michelle K. Smith, Nnadozie Okoroafor, Hannah Jordt, and Mary Pat Wenderoth. Active learning increases student performance in science, engineering, and mathematics. *Proc. Natl. Acad. Sci. U.S.A. *2014, 111, (23) 8410-8415

[9] Gamoran, A., Porter, A.C., Smithson, J., & White, P.A. (1997, Winter). Upgrading high school mathematics instruction: Improving learning opportunities for low-achieving, low-income youth. *Educational Evaluation and Policy Analysis, 19*(4), 325-338.

[10] Bonnie Gold, Sandra Z. Keith, William A. Marion, (eds). *Assessment Practices in Undergraduate Mathematics*. Mathematical Association of America Notes #49, 1999.

[11] Karin K. Hess. Exploring Cognitive Demand in Instruction and Assessment. National Center for Assessment, Dover, NH 2008. http://www.nciea.org/publications/DOK_ApplyingWebb_KH08.pdf

[12] Reva Kasman. Critique That! Analytic writing assignments in advanced mathematics courses. *PRIMUS* XVI (2006) 1–15.

[13] John Meier and Thomas Rishel. *Writing in the Teaching and Learning of Mathematics*. MAA Note #48, 1998.

[14] Porter, Andrew. Curriculum Assessment, In J. L. Green, G. Camilli, & P. B. Elmore (Eds.), *Complementary methods for research in education (3rd edition)*. Washington, DC: American Educational Research Association, 2006. http://www.andyporter.org/sites/andyporter.org/files/papers/CurriculumAssessment.pdf

[15] Andrew C. Porter and John L. Smithson. Defining, Developing, and Using Curriculum Indicators. CPRE Research Report Series RR-048, December 2001. Consortium for Policy Research in Education University of Pennsylvania Graduate School of Education. https://secure.wceruw.org/seconline/Reference/rr48.pdf

[16] Rishel, Thomas. *Teaching First: A Guide for New Mathematicians*. MAA Notes #54, 2000

[17] Schoenfeld, Alan H., ed. *Assessing Mathematical Proficiency*. MSRI Book Series, Volume 53, 2007.

[18] B. Sriraman, & L. English (Eds.). *Theories of mathematics education.* New York: Springer, 2010.

[19] Mary Kay Stein, Barbara W. Grover, and Marjorie Henningsen. Building Student Capacity for Mathematical Thinking and Reasoning: An Analysis of Mathematical Tasks Used in Reform Classrooms. *American Educational Research Journal*, Vol. 33, No. 2 (Summer, 1996), pp. 455-488

[20] Stein, Mary Kay and Smith, Margaret Schwan. “Mathematical Tasks as a Framework for Reflection: From Research to Practice.” *Mathematics Teaching in the Middle School,* Vol. 3, No. 4 (January 1998), pp. 268-275

[21] Stein, Mary Kay and Smith, Margaret Schwan. “Reflections on Practice: Selecting and Creating mathematical Tasks: From Research to Practice.” *Mathematics Teaching in the Middle School*, Vol. 3, No. 5 (February 1998), pp. 344- 350

[22] T. Rowland & P. Andrews (Eds.). *Master class in mathematics education: International perspectives on teaching and learning*. London: Continuum Publishers, 2014.

*Editor’s note: This is the first article in a series devoted to active learning in mathematics courses. The other articles in the series can be found here.*

“…if the experiments analyzed here had been conducted as randomized controlled trials of medical interventions, they may have been stopped for benefit.”

So strong is the evidence supporting the positive effects of active learning techniques in postsecondary mathematics and science courses that Freeman, et.al, made the statement above in their 2014 Proceedings of the National Academy of Science (PNAS) article *Active learning increases student performance in science, engineering, and mathematics*. Yet faculty adoption of active learning strategies has become a bottleneck in post-secondary mathematics teaching advancement. Inspired by the aforementioned PNAS article, a landmark meta-analysis of 225 studies regarding the positive effects of active learning, we will devote a series of posts to the topic of active learning in mathematics courses.

An immediate challenge that arises when discussing active learning in mathematics is that the phrase “active learning” is not well-defined. Interpretations by mathematics faculty of this phrase range broadly, from completely unstructured small group work to the occasional use of student response systems (e.g., clicker) in large lectures. In this article we discuss several descriptions from the literature, including what we will take as our working understanding throughout this series of posts, discuss important considerations in the adaptation of such methods, and highlight some important aspects of the PNAS article.

**What is Active Learning?**

The core tenet of active learning is that providing students with opportunities to actively engage with content during their classes leads to positive learning outcomes. In mathematics, the phrases “active learning” and “inquiry-based learning” (IBL) are closely related, though opinions vary regarding the extent to which they are related or overlap. Here are some particularly insightful descriptions of active learning and IBL from the literature.

Active learning is generally defined as any instructional method that engages students in the learning process. In short, active learning requires students to do meaningful learning activities and think about what they are doing. While this definition could include traditional activities such as homework, in practice active learning refers to activities that are introduced into the classroom. The core elements of active learning are student activity and engagement in the learning process. Active learning is often contrasted to the traditional lecture where students passively receive information from the instructor.

— Does Active Learning Work? A Review of the Research, Michael Prince, J. Engr. Education, 93(3), 223-231, 2004In the context of mathematics, IBL approaches engage students in exploring mathematical problems, proposing and testing conjectures, developing proofs or solutions, and explaining their ideas. As students learn new concepts through argumentation, they also come to see mathematics as a creative human endeavor to which they can contribute. Consistent with current socio-constructivist views of learning, IBL methods emphasize individual knowledge construction supported by peer social interactions.

— Assessing Long-Term Effects of Inquiry-Based Learning: A Case Study from College Mathematics, Marina Kogan & Sandra L. Laursen, Innov High Educ (2014) 39:183–199A student-centered instructional approach places less emphasis on transmitting factual information from the instructor, and is consistent with the shift in models of learning from information acquisition (mid-1900s) to knowledge construction (late 1900s). This approach includes

- more time spent engaging students in active learning during class;
- frequent formative assessment to provide feedback to students and the instructor on students’ levels of conceptual understanding; and
- in some cases, attention to students’ metacognitive strategies as they strive to master the course material.

— Discipline-Based Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering, S. R. Singer, N. R. Nielsen, and H. A. Schweingruber (eds.), National Research Council, The National Academies Press, 2012

The PNAS authors do not attempt to define active learning, but instead consider in their analysis “papers representing a wide array of active learning activities, including vaguely defined ‘cooperative group activities in class,’ in-class worksheets, clickers, problem-based learning (PBL), and studio classrooms, with intensities ranging from 10% to 100% of class time.”

In alignment with the broad picture painted by the descriptions just given, *our approach is to use the phrase “active learning” to represent any classroom strategy that provides students with opportunities to directly engage with content during class, whether individually or collaboratively with peers.* Avoiding a singular definition of active learning increases the risk of faculty, administrators, and other stakeholders “speaking past” one another. However, as we discuss next, we believe it is important to emphasize the multiplicity of approaches to increased student engagement and to emphasize the need for clear language when discussing different classroom environments.

It is important to observe that mathematics education researchers have investigated the impact of active learning techniques on mathematical learners for decades, especially at the K-12 level. See the paper “Active Learning in a Constructivist Framework” by Anthony, listed in the references, for an example from the mid-1990’s that contains a nice exposition of different interpretations of “active learning.” We are choosing to emphasize in this article that the effects of active learning transcend disciplines, and that student-centered pedagogical techniques are currently the subject of a broad discussion across the sciences.

**Important Considerations**

The following three fundamental issues must be considered when implementing or supporting active learning strategies. These issues complicate our ability to have a coherent national dialogue regarding postsecondary mathematics teaching, and are a frequent source of confusion among different stakeholders in higher education at the national, regional, and local level.

*Classroom Environment*: Often as a result of factors beyond the control of individual faculty (or even departments), classroom environments vary wildly from institution to institution. “Typical” class sizes can run from fifteen to six hundred, with varying levels of grading support. In environments where courses are often taught in a hybrid fashion, meaning a mix of in-person contact time and online modules, contact time is structured differently than in a traditional three-to-five hour per week class meeting structure. All of these considerations and more impact the choices of active learning strategies available for a course or institution, through both restrictions on the type of direct interactions available and enrichments of the type of technology-driven interactions available.

*Teaching Environment*: Mathematics faculty experience an incredibly diverse range of employment conditions. In contrast to public stereotypes of tenure-stream faculty at research-intensive institutions, postsecondary mathematics teachers include both long-term faculty and part-time or adjunct faculty, tenure-stream and non-tenure stream, with many different administrative job requirements and varying levels of support for pedagogical innovation. This range of faculty profiles creates an equally broad range of needs regarding how pedagogical training and mentoring is delivered, and raises questions such as: how much preparation time do faculty have available? Are the courses under consideration being taught by experienced faculty or those teaching for the first or second time? Do faculty performance evaluations and/or job renewals depend on consistently satisfactory student evaluations? Has the institution in question had any historical focus regarding the teacher-training of new faculty hires?

*Course/Student Goals*: Learning outcomes for courses, and the student expectations accompanying them, vary dramatically among faculty, courses, and institutions. For example, courses that primarily serve as part of a general education or quantitative literacy component will typically have fundamentally different goals and expectations for students than courses that primarily serve as a pathway to STEM majors. Individual faculty often have distinct models of student learning, ranging from a view of teaching/learning as the transfer of knowledge and facts to the view of developing students’ ability to solve new problems and/or grapple with and develop understanding of new ideas. Many faculty have different expectations for students regarding the level of cognitive tasks that they are expected to carry out, and often these expectations are implicit in the way they structure their course rather than explicitly communicated to students and peers.

**Aspects Related to Efficacy and Public Policy**

Several additional issues are directly brought up in the PNAS article. The strength of their results led the PNAS authors to suggest that “STEM instructors may begin to question the continued use of traditional lecturing in everyday practice.” Having said that, they point out that to date, active learning has been implemented primarily by faculty interested in experimenting with new pedagogical strategies. What is not clear is if the high efficacy levels observed so far for active learning techniques would be seen if implemented by almost all mathematics faculty. We feel this is a key question, especially if the use of active learning strategies are mandated without a robust support/reward system and full recognition that transitioning to new pedagogical techniques is never a smooth or effortless process. It would also be worthwhile to compare the efficacy of different active learning techniques in mathematics, in analogy to the work of Prince provided in the references.

Finally, the PNAS authors point out that increased student learning as a result of active learning techniques will lead to increased student success rates, thus resulting in fewer repeats of mathematics and science courses. This has the potential to save students significant amounts of time and tuition. Active learning also has been found to have a disproportionately beneficial effect on members of minoritized groups in STEM fields, revealing that fundamental issues regarding equity are at hand. In addition to the ethical and moral questions these points raise for the mathematics teaching community, these qualities of active learning are drawing the attention of individuals involved in student advocacy, public policy, grant and scholarship funding, and related fields. We believe that increased support and attention to the success of our students from people outside the mathematical teaching community should be welcomed, and that an inclusive discussion of how to best help students learn mathematics at a deep level will lead to a richer teaching and learning experience for all.

**A Preview**

In the remaining articles in this series, we will explore how different tasks carry varying levels of cognitive demand on the part of students, and we will provide examples of active learning techniques that address these levels in various teaching and classroom environments. We will share some of our personal experiences as teachers who have experimented with pedagogical strategies, and consider the difficult issue of finding balance between providing students with time for exploration and providing students with direct feedback and instruction. Finally, we will discuss potential points of tension between scholarly training in research-level mathematics and scholarly development of pedagogical strategies and techniques.

**References**

Glenda Anthony

*Active Learning in a Constructivist Framework
*Educational Studies in Mathematics, Vol. 31, No. 4 (Dec., 1996), pp. 349-369

Scott Freeman, Sarah L. Eddy, Miles McDonough, Michelle K. Smith, Nnadozie Okoroafor, Hannah Jordt, and Mary Pat Wenderoth

*Active learning increases student performance in science, engineering, and mathematics
*

Marina Kogan & Sandra L. Laursen

*Assessing Long-Term Effects of Inquiry-Based Learning: A Case Study from College Mathematics
*Innov High Educ (2014) 39:183–199

Michael Prince

*Does Active Learning Work? A Review of the Research
*Engr. Education, 93(3), 223-231, 2004

R. Singer, N. R. Nielsen, and H. A. Schweingruber (eds.)

*Discipline-Based Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering
*National Research Council, The National Academies Press, 2012

“How many of you feel, deep down in your most private thoughts, that you aren’t actually any good at math? That by some miracle, you’ve managed to fake your way to this point, but you’re always at least a little worried that your secret will be revealed? That you’ll be found out?”

Over half of my students’ hands went into the air in response to my question, some shooting up decisively from eagerness, others hesitantly, gingerly, eyes glancing around to check the responses of their peers before fully extending their reach. Self-conscious chuckling darted through the room from some students, the laughter of relief, while other students whose hands weren’t raised looked around in surprised confusion at the general response.

“I want you to discuss the following question with your groups,” I said. “How is it that so many of you have developed negative feelings about your own abilities, despite the fact that you are all in a mathematics course at a well-respected university?”

If this interaction took place in a math course satisfying a general education requirement, I don’t think anyone would be surprised. Yet this discussion repeats itself semester after semester in my upper-level undergraduate courses, for which the prerequisites are at least two semesters of calculus and in which almost every student is either a mathematics major or minor. I’ve had similar interactions with students taking first-semester calculus, with experienced elementary school teachers in professional development workshops, with doctoral students in pure mathematics research seminars, and with fellow research mathematicians over drinks after dinner. These conversations are about a secret we rarely discuss, an invisible undercurrent of embarrassment and self-doubt that flows through American mathematical culture, shared by many but revealed by few. At every level of achievement, no matter what we’ve done, no matter how much we’ve accomplished, many of us believe that we’re simply not good at math.

**************

I first discovered the work of Carol Dweck from a link on Terry Tao’s blog. Dweck is a psychologist at Stanford whose studies on the relationship between self-beliefs and achievement have had a tremendous impact in education and beyond. Her message is simple: when people, whether students or otherwise, believe that they are capable of improving their abilities through hard work and sustained effort, then they achieve more than when they believe they have innate abilities that will at some point be reached. In other words, if you believe that failure is a natural part of growth and development, then you are more likely to persist through failure and setbacks. On the other hand, if you believe you succeed because you are smart, then when you experience a failure, even a small one, you likely conclude that you are not actually smart and give up as a result. The former belief is referred to as a “growth” mindset, while the latter is called a “fixed” mindset.

Dweck’s work fit naturally in the context of math education research I had been reading regarding the use of active learning, and it also fit with my personal experience. As an undergraduate, I majored in English composition and Mathematics, originally intending to be a high school teacher. While I was interested and active in the math club and math competitions, I wasn’t a particularly strong math student, earning a mixture of A’s, B’s and a C in math major courses. After completing my degree, I took a job at a planetarium as a low-level manager. I had previously considered going on to graduate school in mathematics, so in my spare time I read about math and science. Slowly, through a series of fortunate moments, I came to understand the depth of my lack of mathematical knowledge. I found out that \(x^2+y^2=1\) is the equation for the unit circle because of the Pythagorean theorem; in high school, this had simply been presented as a fact. I was cleaning up the science demo room in the planetarium one day when a NASA video about trigonometric functions came on — I quietly closed the doors and watched for 45 minutes, taken aback by the beautiful connection between sine, cosine, tangent, their graphs, and the unit circle, which I had never seen.

The next year I started graduate school. On a regular basis I told my wife that I was definitely the dumbest person in my complex analysis course, but that I was doing my best anyway. On more than one occasion I sat on the living room floor and burst into tears, overwhelmed by the stress of trying to understand the barrage of ideas one encounters in graduate mathematics courses. I spent a lot of time in the math library that first year in graduate school, reading books like Serge Lang’s *Basic Mathematics* (a high-school text) and Liping Ma’s *Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States* (about elementary school mathematics); it was at this time that I came to understand the importance of the distributive law and its role in the multi-digit multiplication algorithm. It was also at this time that I became deeply aware of how I had been doing math without real understanding, demonstrating high-level mathematical knowledge without substance. Only as time passed did I realize how many students of mathematics, even successful students, operate in this way.

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When I first read Dweck’s work on mindsets, I had already begun using various pedagogical and assessment techniques: active learning through small group work, reflective essays as homework, semester-long individual projects. While students responded well to this, I was never satisfied at the end of the course. There were too many students who didn’t develop their understanding of mathematics, who were hesitant to fully engage in the course. I decided to directly intervene, using Dweck’s work as the basis for an explicit discussion of the role of beliefs in learning and achievement.

Because our first-year calculus courses are taught using a large-lecture/recitation, highly coordinated structure, I only felt free to experiment in my smaller upper-level courses for math majors and minors. On the first day of class, I assigned Dweck’s survey article “Is Math A Gift?: Beliefs That Put Females at Risk”; recently I have instead used her Scientific American article “The Secret to Raising Smart Kids.” I also had students write a one-page autobiographical statement about their previous experiences in math courses. The second day of class was devoted entirely to psychological aspects of mathematics: How do you feel about mathematics? Do you actually like it? Do you feel you are good at it? What are the reasons you have succeeded to this point? At first I tried to have discussions with the entire class sitting in a circle, but found that it is much more effective to assign students to groups and have them talk with their peers — I don’t have to hear everything they say in order for the discussion to be meaningful. I found that starting the class this way completely changed the tone of my courses for the better. It was surprising and refreshing to the students for a math course to start in this manner, and it set the stage for our classroom discussions about mathematics to include both technical and psychological aspects.

The biggest surprise I had, and a challenge I still struggle with as a teacher, is the remarkable ability of students to argue in favor of the dominance of innate talent in mathematics. Cultural conditioning regarding the myth of genius is strong and embedded; for many of my students, this had developed into the false belief that the goal of doing math is to be brilliant, rather than to gain reasonable mastery and improve one’s understanding. Students frequently compared doing math to training to be an elite athlete. In more than one small group, in more than one class, I heard statements such as “no amount of hard work will make someone play basketball like Michael Jordan.” The fact that these same students enjoyed playing basketball for the purpose of honing their skills and enjoying the company of friends, rather than becoming a legendary athlete, usually didn’t occur to them until I raised the point explicitly.

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Ten months ago, I was grading homework that had asked students to work on a particularly challenging open-ended problem. I was struck by the fact that almost a third of the students made a negative remark such as “this is wrong, I’m an idiot” in their solutions. This type of language is so culturally embedded in our mathematical discourse that we often don’t realize the level of negative self-talk we use. I frequently hear mathematicians make comments like “Oh, I see that now, I should have realized that before” when the reality is that we earn our realizations through effort and persistence, because real mathematical understanding requires time. The day after I graded that homework assignment, I implemented the following course policy, which is now part of every course I teach.

Students are not allowed to make disparaging comments about themselves or their mathematical ability, at any time, for any reason. Here are example statements that are now banned, along with acceptable replacement phrases.

- I can’t do this –> I am still learning how to do this
- That was stupid –> That was a productive mistake
- This is impossible –> There is something interesting and subtle in this problem
- I’m an idiot –> This is going to take careful thought
- I’ll never understand this –> This might take me a long time and a lot of work to figure out
- This is terrible –> I think I’ve done something incorrectly, let me check it again
Please keep in mind the article we read by Carol Dweck. The banned phrases represent having a fixed view of your own intelligence, which does not reflect the reality that you are all capable of dynamic, continued learning. The suggested replacement phrases support and represent having a growth mindset regarding your abilities and your capacity for improvement.

In my most recent courses, I introduced this policy on the second day of class, following our small group discussions of Dweck’s article, and I subsequently enforced it vigorously. Doing so has revealed even further for me the depth of the challenge math teachers face — everything operates against our goal of student learning, even the words and phrases we are subconsciously trained to use. How can I hope to have my students believe in their own abilities, when their default descriptions of their work are derogatory?

I frequently teach courses for pre-service teachers, and one remarkable aspect of building a classroom environment around growth mindsets is the connection to the Standards for Mathematical Practice in the Common Core State Standards for Mathematics. At some point in time during every course that serves preservice teachers, I show students these standards — their typical response is to be shocked that these are required of K-12 students, and also to feel uncertain of how to interpret some of them. These standards both implicitly and explicitly reflect the fact that authentically doing mathematics involves trying, failing, trying again, making mistakes, correcting, and shifting perspective. I have students work on a tough problem in small groups, one I don’t expect them to solve during class or at all, and stop every few minutes to reflect on which of the practice standards they have used, and whether or not there were missed opportunities to bring others into play. I insist that the students not criticize themselves for their missed opportunities, simply acknowledge them and, from that recognition, improve.

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It is reasonable to wonder if any of these activities have a meaningful effect on students. Mindset interventions, as they are often called, are being included as part of academic orientations at various universities. I often reflect on the ability my students have demonstrated to resist these messages regarding mathematics, even though various studies provide strong evidence that such interventions improve academic achievement. I wonder: while students’ performance improves after a brief mindset intervention, how much does it change what they believe about the nature of mathematical ability?

I’ve found that the hardest questions to ask students are the ones I most want to know the answer to: *“What are you really thinking? What do you truly understand? What do you believe you are capable of accomplishing?”* To obtain reasonably deep answers to these questions, I decided early in my teaching career that I need to have students write reflective essays in my courses. Here are excerpts from end-of-semester essays that four of my students have allowed me to share.

**Student 1:** I had always conceived of mathematics — and, by extension, science and engineering — as a field advanced by sheer brilliance. Yes, I realized that these fields were more parts failure than success, but nothing has contributed to cementing in my mind that anyone can succeed in any field through hard work and dedication than the Dweck article presented near the beginning of the semester. I have made this an integral message in my private chemistry tutoring; no regular client of mine this semester has managed to escape my spiel about how they can’t allow their fears and lack of confidence to hold them back from working hard to succeed.

**Student 2: **Speaking of teaching, the [Dweck] article that we read at the very beginning of the semester has stuck with me this whole time and is something I want to be sure I keep in mind when I have a classroom of my own. While I would still say that most of the mathematicians we have read about are super geniuses, they did work diligently towards what they wanted to achieve. It stood out to me that when you present these geniuses as people who worked really hard it influenced the students’ thought processes in a positive way, making them more likely to try, whereas when presented with material that said they were geniuses the students took a more negative approach of “I don’t have the gift therefore I can’t do the math.” It will definitely influence the way that I present material in my own future classroom, making sure to focus on working hard rather than “just being good at math.”

**Student 3:** One of the misconceptions I held when I came into this class at the beginning of the semester was that if I had to spend a large amount of time on a problem that meant I was dumb. I don’t believe that I ever voiced that opinion to anyone, but I now know that it was there. And it was because of this and my other math courses this semester, that I only slightly have that thought. I learned the very hard way that good mathematics takes time. I can no longer just plug and chug like I could with calculus or matrix; now I actually have to think about what I’m doing. It was, and still is, a very frustrating feeling, but underneath that feeling is the understanding that sometimes this is just what solving problems is: it’s time, and frustration, and sometimes having a tantrum before the problem can be solved.

**Student 4:** I had always thought that mathematics was a gift. You were either good at math or you were not. It was this reasoning that caused me to believe that some people were born to be mathematicians, while others were doomed to always struggle in mathematics. However, I have seen several people (including myself) in this class go from struggling in math class to having an impressive mathematical skill set. I now see mathematics as more like athletics. While some people are more naturally gifted than others, hard work will pay off in the end. This does not mean that I will not have more challenges. It does mean that I can face those challenges, and that in most cases I can learn the mathematics in order to do what is required.

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For reasons that I don’t fully understand, our mathematical culture encourages us to define our mathematical ability by what we don’t know, what we aren’t able to do, rather than by what we do know and have learned how to do. The power of culture is strong, with deep roots — I don’t truly believe that the ripple effect from my teaching will spread very far. Yet I cannot help but think of all the students who persist in mathematics. In spite of so many unspoken doubts, so many negative influences, these students have made their way through the doors to our classrooms. And I cannot help but think of the many thoughtful, capable students who turn away from mathematics and give up hope. We are surrounded by potential, by possibility, by self-inspiration yearning for a spark. I believe that the brightest sparks come from people rather than mathematics. That our thoughts, emotions, and beliefs are the gateway toward a more diverse, equitable, proficient, and beautiful mathematical culture. The key is allowing time for these alongside technical mathematics in our classrooms; real mathematical understanding requires time.

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