Those of us who teach mathematics know that students struggle writing the symbolism of mathematics even through they can articulate some of the concepts behind the symbolism. Those of us who interact with children know that they struggle articulating their thoughts even though they can convey their thoughts through gesture. For example, children point to indicate what they want and touch items or use their fingers as they learn to count. It is through such bodily action that children learn to recognize three objects as the quantity three without simultaneously touching and counting one, two, three. Athletes and musicians also apply bodily actions to master their sport or instrument respectively. For example, how many times have you have seen a basketball player shoot an imaginary ball into an imaginary hoop? Consider how a piano teacher places a student’s hand on top of the teacher’s hand as the teacher plays the piano. These are just a few ways in which we use our body to learn, so why not use it purposefully to promote the learning of mathematics?
In layman’s terms the philosophy of embodied cognition argues that learning is a result of interactions with our environment [1]. There are broad interpretations of this philosophy, but like others I interpret it to mean that we learn through bodily movements. Various mathematics education researchers adopt embodied cognition as a theoretical lens because it allows them to use gesture as a source of evidence as students learn linear algebra, differential equations, complex variables, etc. [2,3,4]. I too document research participants’ gestures as they tackle tasks related to my research on the teaching and learning of complex analysis, but embodied cognition also informs my teaching. In this blog, I describe how I highlight students’ gestures to help them articulate their thinking and I illustrate embodied activities designed to elucidate mathematical concepts via bodily movement.
As a first illustration, consider how I expose preservice elementary teachers to Euclidean transformations. I commence by asking students to define a translation, reflection, rotation, and dilation. Similar to children, students tend to gesture when they are unable to articulate their thoughts. For example, after asking students to define a rotation, Sammy (pseudonym) raised his hand and the following dialogue occurred. Although Sammy was unable to provide a definition, he turned his right hand back and forth as though turning a doorknob.
Sammy: Well it’s a, it’s kind of like, well you know, well I don’t know.
Me: Sammy I am going to repeat what you said and regesture your gestures.
I repeated Sammy’s words and emulated his gestures, which quickly prompted him to say, “Oh, it’s like a turn, you are just turning the object.” This exemplifies neuroscientists’ belief that if a person is “not engaged in an intentional action, or watching another person engage in an intentional action” then there is no expectation for neurons in the premotor cortex brain area to activate [5, p. 13]. These neurons are believed to be responsible for helping us interpret other’s actions as well as our own actions. In Sammy’s case, attention to his gestures facilitated creating working definitions based on the gesture characteristics. For example, the students commented that Sammy’s wrist could be perceived as the center of rotation. We have similar conversations when students alternate between turning their palm faceup and facedown as they convey their reflection definition.
After creating working definitions, the students collaboratively complete worksheets using manipulatives (see Table 1 for sample tasks). The purpose of these worksheets is for students to determine the image of a figure under a given transformation, to work backwards, and to make discoveries about the properties of Euclidean transformations. After completing the worksheets, the students present their work to the class. After this, we proceed with some embodied activities, which reinforce classroom work.
Table 1. Sample Worksheet Questions 
Suppose the point (x,y) was translated in the direction of (5,2) to obtain the image (4,8). What is the preimage point (x,y)?

In the following figure determine the line of reflection and explain your reasoning.

Consider the figure below.
1. Reflect triangle ABC about line m and label it as triangle A’B’C’. 2. Reflect triangle A’B’C’ about line n and label it as triangle A”B”C”. 3. Construct the circle with center O and radius \(\overline{OA}\). Do the same for radius \(\overline{OB}\) and \(\overline{OC}\). 4. Describe all the points that pass through each circle and explain why this happens. Use mathematical transformation ideas for your explanation. 
As part of the embodied activities, students act out many of the worksheet tasks on a giant grid where the students are the points and rope serves as segments. Figure 1 depicts students as image points after translating in the direction of (1,2). As a result of this activity the students realized that not moving at the “same rate” causes the rope to become loose. With some probing, they connect the notion of the “same rate,” to an equal slope, and to the worksheet discovery that under a given translation, the segments connecting a preimage point and its corresponding image point are parallel and congruent. It is during this activity that the students also use language alluding to rigid motions of the plane. That is, they realize that under a transformation every point on the preimage transforms simultaneously rather than one point at a time as they performed it on the worksheet.
Figure 1. Embodied Translations
Figure 2 illustrates the students determining their image point under the given line of reflection. The combination of the rope and large right triangle helped them make meaning of the fact that a line of reflection is the perpendicular bisector of the segment connecting a preimage point and its image point. While working on the worksheet some students generally forget about the perpendicular aspect or the bisector aspect of the definition, but somehow using the rope facilitated attending to both facets of the definition. This could be because they are able to simply fold the rope over the line of reflection. It is not uncommon to hear comments such as “Oh that’s what perpendicular bisector means.” Similar comments are made with the rotation task. As the students rotate about the center of rotation (another student) while holding the rope they remark, “So a rotation means you are traveling in a circular fashion.”
Figure 2. Embodied Reflections
I now highlight an example where students unconsciously engage in bodily movement. As part of a second semester geometry course for prospective secondary teachers, the students performed stereographic projections using Lénárt spheres (Figure 3).
Figure 3. Stereographic Projection
One group of students was not satisfied with their image sketches because the string should go through the sphere. Thus, they relied on their eyes (another example of embodied cognition) to determine the image of the circles (Figure 4). The group progressed quite rapidly through the tasks until they arrived at the great circle that passes through the North Pole. At this point one of the students, Neil (pseudonym) got up and pointed both of his arms up to denote the North Pole (Figure 5a). While engaged in bodily motion he mentioned that one half of the great circle would be projected down (Figure 5b) and the other half would get projected in the opposite direction (Figure 5c). During this action both he and his teammember remarked, “So it will map to a line and there is a break at the North Pole.” This dynamic engagement did not go unnoticed by the other students and I asked Neil to regesture his discovery. Furthermore, when we started the unit on inversions the students quickly recalled Neil’s bodily action as they hypothesized about the image of a circle that passes through the center (O) of the circle of inversion. They knew the circle would break at point O and map to a line.
Figure 4. Using eyes for stereographic projection
Figure 5a  Figure 5b  Figure 5c 
Figure 5. Stereographic projection of a great circle 
Currently, there is much buzz about active learning of mathematics [6] but the definition of active learning is vague and sometimes it is difficult to determine if students are truly engaged in learning. By paying attention to students’ gestures instructors can hypothesize about students’ mathematical reasoning and ask probing questions that help students convey their mathematical reasoning. It is also an effective technique for assessing mathematical misconceptions, but this can only occur if instructors are attuned to gesture. Furthermore, tasks such as embodied activities help bring to life the mathematics where students are actively learning and have “aha” moments – these aha moments are the best part of teaching.
References:
[1] Anderson, M. L. (2003). Embodied cognition: A field guide. Artificial Intelligence, 149, 91130.
[2] Nemirovky, R., Rasmussen, C., Sweeney, G., & Wawro, M. (2012). When the classroom floor becomes the complex plane: Addition and multiplication as ways of bodily navigation. Journal for the Learning Sciences, 21(2), 287323.
[3] Rasmussen, C., Stephan, M., & Allen, K. (2004). Classroom mathematical practices and gesturing. Journal of Mathematical Behavior, 23, 301323.
[4] Tabaghi S. G. & Sinclair, N. (2013). Using dynamic geometry to explore eigenvectors: The emergence of dynamicsyntheticgeometric thinking. Technology, Knowledge and Learning, (18), 149164.
[5] Gallagher, S. (2014). Phenomenology in embodied cognition. In L. Shapiro (Editor), The Routledge Handbook of Embodied Cognition (918). London: Routledge Taylor and Francis Group.
[6] Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., and Wenderoth, M. P. (2015) Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences, 11(23), 84108415.
]]>I had what seemed the perfect first fulltime teaching position, in that much of the planning for Calculus had already been done when I arrived. The department chair handed me the textbook and the syllabus, essentially a daybyday schedule of book sections and homework assignments. This being the United States Naval Academy at Annapolis, where every student takes Calculus, a lot of wisdom had gone into the schedule. I now look back at that syllabus with a mixture of gratitude for the jump start and recognition that much has changed. What’s in your syllabus? What does your institution require, and what is most important to you? What is decidedly not in your syllabus? Do you hand out a paper copy on the first day, or is it all online? How well does the syllabus reflect what you want your course to be?
For some time after my move to Middlebury College, my syllabi followed that basic first model. Eventually, however, I felt constrained by detailed plans that had made sense in August but didn’t fit the October reality. After too many classes that ended in a rush to cover specific content, I began to offer a general outline of topics for the semester, along with exam dates, reminders about the honor code, and only the first assignment. Putting together assignments a week at a time allowed me to be more responsive to what I was seeing and hearing in the classroom, while honoring my commitment to engage particular concepts. Looking back, I see that shift in the syllabus as an early sign of my disillusionment with a strict lecture format.
Once I started bringing students into InquiryBased Learning territory, a syllabus needed to include a description of what we were doing and why. As I tried to articulate a rationale for more active class sessions, I was compelled to consider what my compositions in the syllabus genre communicated to students, intentionally or not. For example, apart from locking us all into a rigid timetable, devoting most of the typing to a list of homework exercises risks putting the textbook at the center, while I want to put learning at the center. A reproduction of the catalog course description, including terms that novices don’t yet understand, isn’t exactly an invitation into a captivating intellectual experience.
Hence my current syllabus starts with a short description of the course content for the nonexpert. Next comes my contact information, followed by my learning objectives, again with as little technical language as possible. Only then does the reader come to the schedule of topics and important due dates, at the bottom of the first page.
One of the challenges of writing a syllabus is that, as the first document I present to my students, it serves multiple purposes. As much as I would prefer to stick to a conversation about learning, external pressures intrude. In the interest of minimizing disputes, I’ve learned to state clearly my policies on absence and late homework. Middlebury’s honor code necessitates an explicit description of what kind of sharing is acceptable and what is not. More generally, many colleges and universities have explicit requirements for what is included in a syllabus. Also, the audience is not just students; internal and external committees routinely collect syllabi in their reviews of faculty members, departments, and institutions.
Most important to many students, it seems, is information about how I will assess their work. Some tell me bluntly that they allocate their study time in direct proportion to the percentages in my grading policy, which can send me into an internal rant about how The System has driven them to focus on extrinsic rewards rather than the intrinsic satisfactions of intellectual growth.
Actually, though, these seemingly invasive topics are connected to learning, and I try to be explicit about those connections in the syllabus. In an active learning environment, I tell students, it is especially important to arrive to class having put in a good faith effort on the homework due that day. To establish and maintain a learning community, everyone should come to every class on time, barring illness. You must engage your own brain in order to learn, and this is an institution devoted to learning, which is why we have an honor code.
The possibility of a website for each course introduces both flexibility and complexity into syllabus composition. (Did I mention that I started at Middlebury in 1984?) My course sites have included, in addition to daily assignments, a rubric for evaluating proofs (adapted from Keith Devlin’s), a a statement on InquiryBased Learning, and samples of written work. This helps me meet my goal of keeping the actual syllabus to two sides of one sheet of paper, a valuable exercise. My hope is that the shorter length increases the chance that students will actually read it.
Do they read it? Thinking deeply about what is or is not on the syllabus leads quickly into a broader consideration of the nature of communication between me and my students. Should I ensure that they read it by giving a quiz, in the interest of their mathematical progress, or should I allow them to learn from experience that they are responsible for that progress? Given that I’ve decided to do less, and more judicious, telling about the mathematics, could I be more creative about conveying my expectations and teaching philosophy? One thing is clear: the syllabus can only do so much. A technique I learned from a colleague in the humanities is to start the first class not with handing out the syllabus or taking attendance, but rather with the actual material of the course. “Let’s start with some math” seems the best introduction of all.
A careful consideration of the syllabus leads to all kinds of questions about contemporary higher education. In addition to the coddling vs. building character debate, there’s the question of grade inflation. If I value class participation and reward it with a significant portion of the grade, then will the resulting grade distribution be objectionable to some colleagues? Does the need to explicit about late work policies and academic honesty, along with some institutions’ exacting requirements for syllabi, say anything about legalism on campus? I recently took a required online course concerning Title IX regulations on harassment; it ended with “… you can inform students by including information in your syllabus on resources, reporting options, and student rights. Check with your Title IX Coordinator to see if your school has approved language regarding Title IX reporting and resources to include in syllabi.” I’m ready to include such language on the course website, but would including it on the actual syllabus help or hinder my efforts to create a small community centered on learning mathematics?
Evidently the changes in my own syllabi reflect transitions in my attitudes toward my students. As a nervous young lecturer trying to stay on schedule, I prepared my notes on a pad; three pages was usually just about right for a 50minute session. Small classes meant I learned names quickly and thought I knew who was struggling, but I’m afraid I only interacted with those who raised hands in class or came to office hours. My job was to present information clearly; their job was to absorb, and the syllabus contained the map through the textbook.
Now the syllabus lays out what I find most important: the learning goals, which concern both content and practice; a framework for meeting those goals; and the principles, based on what I know about learning, connecting the goals with the framework. The syllabus is only the beginning, and I try to reinforce and elaborate on the goals and principles repeatedly in class and on the website. I now see my job as providing a structure in which students can practice mathematics on the content of the course, and responding to their efforts appropriately along the way.
]]>We want to begin this post with thanks to all of our readers and contributors — we appreciate your feedback and ideas through your writing, social media comments, and inperson conversations at mathematical meetings and events. Inperson conversations have been on the minds of the editors recently because we had our firstever inperson meeting as an editorial board at the 2016 Joint Meetings in Seattle. This was great fun and gave us a chance to seriously reflect on our blog, its role in the mathematical community, and what we want to do over the next year or two. In this post, we give a brief update about a change to the structure of our blog, followed by some highlights of our experiences attending the joint meetings.
Update
As regular readers of our blog know, since we began in June 2014 we have been publishing articles on or around the 1st, 10th, and 20th of each month. Starting with this post, we will be changing to a biweekly publication schedule. Starting on Monday, January 25th, new articles will appear every two weeks. If you want to receive notice when new blog posts appear, please subscribe to our email distribution list on the righthand sidebar of the blog or subscribe to our RSS feed.
JMM Highlights
With so many excellent activities going on at the Joint Meetings, even with five of us we could only attend a small fraction of the offerings. Given that, here are some of the highlights from our experiences.
Diana:
It amazes me how much my focus at conferences has gradually shifted over time as I have progressed through the academic ranks. As a graduate student and postdoctoral fellow, I almost exclusively attended talks and chitchatted with colleagues and friends during breaks. Now, with many projects in progress with folks from around the country, and with several leadership roles, I feel lucky if I can choose one talk per day to attend! I spend most of my time in meetings, but I enjoy this greatly. I find it wonderful to connect in person with colleagues on various projects and committees, work and connect intensely for a bit, and then resume working remotely. How wonderful this combination of modern technology and in person meeting can be when used well!
One of my main foci at the Joint Math Meetings was in my capacity as Director of the National Association of Math Circles (NAMC), one of the main outreach activities of the Mathematical Sciences Research Institute. It’s great to see Math Circles growing and becoming more wellknown across the country. I strongly encourage you to become involved with an existing one, or to start a new one. At the NAMC happy hour, it was particularly wonderful to meet PoShen Loh, coach of the U.S. International Math Olympiad team that won the Gold Medal in 2015. I’m particularly excited about his new Expii website that is posting a wonderful weekly problem set to engage kids of all backgrounds and experiences with mathematics.
Another major focus for my time at the JMM consisted of events surrounding the development of the Mathematical Association of America’s Instructional Practices Guide. I’m honored and excited to be a part of developing this as an accompaniment to the decadal MAA Curriculum Guide. At the JMM, we conducted focus groups, held meetings of the steering committee and advisory board, hosted a panel, and had several leadership team meetings. The 23 year project is off to a great start!
Art:
The entire AMS Special Session on Essential Mathematical Structures and Practices in K–12 Mathematics featured mathematicians and mathematics educators talking with (not at!) each other about treating the mathematical topics that arise in K12 with the rigor that they deserve. Expect a post about this soon, focusing on ratio and proportion as an illustrative example, from some of the speakers. The panelists at the AMS & AWM Committees on Education Panel Discussion Work in Mathematics Education in Departments of Mathematical Sciences showed the different ways that mathematicians can be involved in mathematics education. Yvonne Lai presented research she and her collaborators conducted on the mathematical knowledge for teaching at the high school level, and they will also be contributing a post here in the future. I also learned more about Illustrative Mathematics, a community of educators collecting quality classroom resources that encourage mathematical understanding; their materials are easy to browse at their website.
Priscilla:
The first JMM stop for me was a panel discussion on “Creating a meaningful Calculus I experience for students entering with high school calculus.” David Bressoud (Macalester) suggested that such students “don’t need a course of techniques — they need big ideas,” and presented an outline of his department’s course. He also recommended Michael Oerhtman’s Clear Calculus materials for a laboratory approach. Robin Pemantle (UPenn) described students’ backgrounds as “porous,” so that he and his colleagues offer a course which is ⅓ filling in holes in calculus, algebra, and even arithmetic, ⅓ topics (probability densities, differential equations) suggested by faculty in client departments, and ⅓ multivariable calculus. Uri Treisman encouraged us to accept the fact that “it’s in the culture that you’re supposed to repeat calculus” after taking it in high school. His response is to offer an introduction to university mathematics, rather than simply an introduction to calculus; he wants to “startle students” with the power of the subject and to emphasize proof and precision. “Don’t make high school calculus the enemy,” Treisman insists; instead we should “work collaboratively with the high school teacher community.”
The premiere showing of “Navajo Math Circles” was a treat. I’ve bought a copy for my department so that we can show it on campus. The “extra features” page has some fascinating short videos that go beyond the film content.
Elise:
A highlight for me was a talk by Chris Rasmussen about teaching called “Advances in inquiryoriented instruction at the postsecondary level: Student success and instructor practices.” This was in Alan Schoenfeld’s MAA Invited Paper Session on What Do We Know about University Mathematics Teaching, and How Can It Help Us? (which provided a number of insightful and thoughtprovoking talks). In his talk, Chris highlighted a few papers and projects that demonstrate the progress that is being made at the postsecondary level. It was exciting to see that there are quite a few new advances, and he cited some recent and upcoming papers that speak to this issue. I appreciate that there are opportunities for productive discussions about instruction at the postsecondary level.
Ben:
I had two highlights from the 2016 JMM. First, I greatly enjoyed giving two talks about teaching, one as part of a Project NExT panel on broadening assessment in postsecondary education and another on the topic of growth mindset interventions to support IBL pedagogies in the MAA contributed paper session on inquirybased learning. It was rewarding to have excellent questions and interactions with audience members, both in person and through email, following my talks. Second, I attended Alan Schoenfeld’s talk about improving K16 mathematics, in which he described the Teaching for Robust Understanding (TRU) Math framework. The five dimensions of this framework strongly resonated with me, and I encourage all of our readers to visit the TRU Math website for more information. One of the comments that Schoenfeld made stuck with me: at this point in mathematics education research, we know what to do to teach right, and it is hard work to implement our knowledge. This led me to reflect on my belief that half of our challenge at the postsecondary level is to broaden awareness and understanding about what constitutes powerful mathematics classrooms — it is impossible to implement change for the better if we don’t recognize and value the complexities of our classrooms and our students’ learning.
]]>Like many mathematicians, the only formal training I have received as a teacher was in graduate school. After a one semester seminar on teaching, I was set loose on three recitation sections of unsuspecting calculus students and expected to improve my teaching primarily by trial and error, discussion with peers and mentors, and feedback from students and classroom observations. While I still use all of these to improve my teaching, I have found that social media has become an indispensable tool to helping me improve as a teacher. I use “social media” in a broad sense here — I would include any quasipublic interactive online discussion in my definition. This includes platforms like Facebook and Twitter that most people associate with the term “social media”, but also things like a discussion in the comments section of a blog, or a discussion boardbased online community. Further, the key value of social media is not in the availability of information, but the interactions and discussions that are generated. In conjunction with trial and error, I have learned more about teaching through social media than I have through any other method.
The primary way you can use social media to improve in teaching is simply by using it to expand your network of peers and mentors who can discuss and offer feedback on your ideas and techniques. As someone who works at a large public flagship university with a very traditional student population, social media allows me to connect with people teaching at a variety of institutions — smaller universities, liberal arts colleges, and schools with a large portion of nontraditional students. Each of these settings presents unique challenges to teachers, and connecting with diverse groups of people and seeing how they overcome these challenges offers various insights I can take back to my own classroom. For example, I can safely assume that 99% of my calculus students have seen college algebra/precalculus in the last two or three years. Reading about the teaching experiences of those who often teach a larger percentage of nontraditional students has helped me to recalibrate my expectations of what a student is likely to retain from prerequisite courses five or ten years earlier, allowing me to more effectively teach the few nontraditional students I do have.
One particular group deserves special mention here, namely middle and high school math teachers. There is a particularly lively group of teachers loosely organized in the “Math Twitter Blogosphere,” with the primary medium of discourse being blog postings (and ensuing comments), and Twitter. I find reading their discussions valuable for several reasons. First, it gives me a glimpse of what is going on in K12 mathematics classrooms. My own personal recollection of what and how high school students learn is becoming more and more dated, so I find reading what secondary teachers are doing helps me to understand the mathematical background of my calculus students better. Second, I have found that reading about the challenges these teachers face has altered my perceptions about what I should be emphasizing to the mathematics education majors I teach. One example here is a discussion among high school teachers about how to answer a student’s question, “Why isn’t zero the least common multiple of every number?” The confusion mostly seemed to stem to from not having clear definitions of terms like “least common multiple” and “greatest common divisor”. Our mathematics education majors learn these topics in an abstract algebra course, but now I place a greater emphasis on discussing these definitions, why they are the correct ones, and what would happen if we defined them in a different manner.
A more concrete way that social media has influenced my teaching is by exposing me to various pedagogical techniques I would otherwise be unaware of. One particular example is that I have switched to using a grading methodology called Standards Based Grading in my calculus courses. I have not yet met in person a fellow mathematician who uses this methodology, but fortunately several people who use it have shared their experiences in various blog posts. Moreover, I’ve found a number of people on Twitter more than happy to answer my questions as I tried this new grading system, and more recently a Google Plus group was formed where people can discuss their use of SBG.
I would like to reemphasize that the true value of social media derives from the interactive nature of these discussions. Reading a single blog post is not that different than reading an article in a print medium. It most often offers some sort of summative snapshot of the topic. However, by learning about SBG on social media, I could read a series of blog posts, offering insight into the rationale behind the choices the author made in how to implement this in his or her classroom, and even watch their use of SBG evolve from semester to semester. The public nature of social media also allowed me to “eavesdrop” on discussions between various people using this in their classes, giving me a great deal of insight into the kinds of challenges I would face when implementing this, and how other people overcame them. And then, when I finally decided to try it out, I was able to bounce ideas and questions off of people who had already run into many of the same issues. These discussions gave me the confidence to introduce a technique into my classroom that seemed, at the time, like a radical change, especially in light of the fact that none of my colleagues in my department were doing anything like it.
Another important way I have grown as a teacher from using social media is that it has given me a greater understanding about the diversity issues facing the mathematics community. When I first started teaching I gave little thought to the topic of diversity in my classroom. Given no evidence to the contrary, I simply assumed my classroom was a welcoming environment to all. Since then, I have read (most often on social media) a number of experiences where mathematics students feel marginalized by wellmeaning but perhaps oblivious instructors. This has allowed me critically assess my own behaviors in the classroom, trying to make sure I am not unwittingly engaging in any of those behaviors. For example, as a fledgling teacher, I gave little thought to the process of learning students’ names. I simply gradually learned them in an ad hoc manner as the semester progressed, paying little attention to which sorts of names I learned easier than others. After reading a discussion on the topic, I realized I was learning names that were familiar to me more quickly than others; now I make an extra effort to deliberately learn all students’ names as best I can.
I find people are often skeptical of the value of social media for professional growth, most often concerned that any useful content is difficult to find among the noise. However, I have found that following fellow mathematicians on Twitter or Google Plus avoids this problem; in fact, I find this is the best way to become aware of interesting articles and blog postings on various teaching and mathematics topics. Moreover, there is usually an interesting discussion that follows, which is often as worthwhile to read as the original article. The nature of these services allows you to passively read and observe, and only chime in to a discussion if and when desired. I would encourage every mathematician to explore how social media can help improve your teaching.
]]>Introduction
Solving counting problems is one of my favorite things to do. I love the challenge of making sense of the problem, the work of correctly modeling what I am trying to count, and the fact that I get to reason about astonishingly large numbers. I did not always feel this way about solving counting problems, though. For much of my mathematical career, counting was a mystery – a jumble of poorly understood formulas and equations that just made me miserable. As an undergraduate, I struggled to grasp the difference between order mattering or not mattering, what the respective factorials represented in confusinglooking formulas, and why I should care about how many full houses could be chosen from a deck of cards. My teachers at the time may have shared the sentiment nicely captured by Annin and Lai: “Mathematics teachers are often asked, ‘What is the most difficult topic to teach?’ Our answer is teaching students to count” (2010, p. 403).
At some point during graduate school (thanks to an influential professor who loved counting), I turned the corner and became more interested in understanding counting. Through lots of practice, I began to improve in my ability to solve counting problems. Since that time I have committed my research interests to learning everything I can about undergraduate students’ counting – what they do when they approach counting problems, why they struggle, and how we might help them solve such problems more effectively.
A Model of Students’ Combinatorial Thinking
In this post, I introduce a model of students’ combinatorial thinking that has been helpful for me to make sense of students’ counting activity. I also offer examples that illustrate aspects of the model, and I conclude with specific recommendations for teaching students to solve counting problems.
The model (initially introduced in Lockwood, 2013, and then refined in Lockwood, Swinyard, & Caughman, 2015) consists of three components – formulas/expressions, counting processes, and sets of outcomes – as well as relationships between these components (see Figure 1 below).
Figure 1 – A Model of Students’ Combinatorial Thinking
The formulas/expressions component refers to mathematical expressions that may be evaluated numerically; these are often considered to be the answer to a counting problem. They may be inherently combinatorial (like a binomial coefficient \({10 \choose 3}\)) or may simply involve numbers and operations (such as \(3^4+2^4)\). Counting processes refer to the actual stepbystep procedures in which someone engages (mentally or physically) as they solve a counting problem. This might involve applying the multiplication principle or implementing a case breakdown. The set of outcomes for a given problem refers to desirable outcomes of that problem – the items that are actually being counted. These outcomes may be encoded in some way (as strings of numbers or letters, for instance), and this component could consist of various ways of encoding and structuring outcomes.
As a simple example to elaborate these components and to highlight the relationships between formulas/expressions, counting processes, and sets of outcomes, consider the following problem: How many 3letter sequences made from the letters a, b, c, d, e, f can be formed if repetition is not allowed and we must include the letter e? (This problem was introduced in Tucker, 2002).
Before solving the problem, we can consider what the outcomes would look like. They are threeletter sequences (so, abe is different from eab) that contain the letter e where repetition is not allowed (outcomes like abc or aaa are not allowed). To count all such sequences, a threestage counting process to solve the problem would be first to choose a position in which the e should go (there are 3 options), and then to choose which of the 5 remaining letters (a, b, c, d, f) can go in the next available position, and then to choose which of the 4 remaining letters (the four that were not previously chosen) can go in the last available position. This process yields the formula/expression of \(3 \cdot 5\cdot 4\), which equals 60.
It is important to note that the counting process described above structures the outcomes in a particular way. Specifically, it groups according to where the e is positioned, as the following list of outcomes in Figure 2 suggests:
Figure 2 – One organization of the set of outcomes
It is worth noting that different counting processes might lead to different structures of the set of outcomes. For example, even if it might not be as elegant a solution, another process might be to organize outcomes according to the first letter. Specifically, we begin with the choice of which letter is first, and if it is a none we consider two cases: putting an e in the second position (then cycling through the remaining letters in the third position) and then putting an e in the third position (then cycling through the remaining letters in the second position). If e is the first letter, we cycle through each of the remaining letters for the second and third positions. So, we could consider sequences with different respective first letters, yielding the outcomes being structured as in Figure 3. An expression that reflects this process is \(5 \cdot 2 \cdot 4 + 1\cdot 5 \cdot 4\), which also equals 60. The point of this example is that different ways of structuring or organizing the set of outcomes can reflect different respective counting processes, and, conversely, different counting process can result in different ways of organizing the outcomes.
Figure 3 – An alternative organization of the set of outcomes
So, why might we care about this model and these components? I contend that there is much value in having students focus on the set of outcomes (Lockwood, 2014), and in particular on thinking about the relationship between counting processes and sets of outcomes. There are a couple of reasons why this relationship is so important to understand. For one thing, if students are not attuned to the set of outcomes (and instead focus primarily on counting processes and formulas/expressions), then rules about which formula to use become harder to parse. Counting can become an exercise in simply manipulating formulas and not clearly understanding what is being counted. Also, certain common pitfalls, like overcounting, are difficult to discover and fix without a solid sense of outcomes and how counting processes relate to those outcomes. As an example of why this relationship is important, consider the following, similar problem involving 3letter sequences (also found in Tucker, 2002): How many 3letter sequences made from the letters a, b, c, d, e, f can be formed if we must include the letter e, and repetition of letters IS allowed?
Here, a common counting process is the following: First, place an e in one of three positions. Then, for each placement of the e we can argue that since repetition is allowed, we are now free to put any of the remaining 6 letters in the remaining two positions. This process suggests a formula/expression of \(3 \cdot 6 \cdot 6\). This counting process seems to make sense, and indeed it is a very common response among students. However, this is an incorrect answer, as this process counts some outcomes too many times. To see this, we must carefully consider the outcomes, and in particular how the outcomes are generated and organized by this counting process.
Let us think about the following: Suppose in the first stage of the process, we placed an e in the third position, and then in the remaining stages of selecting one of 6 letters to be in the remaining 2 positions, we chose an e and then an a. This gives a password of eae. However, consider another way of completing this process: we could have first placed an e in the first position, and then in the 6 \(\cdot\) 6 stage we could have selected an a and then an e. This, too, generates the password eae. The ways to complete the process are not in ontoone correspondence with the number of desirable outcomes, and this process thus causes an overcount.
If a student does not understand that there is a relationship between counting processes and outcomes in this way, how might he or she be aware of when they overcount, let alone be prepared to address and fix that overcount? Counting problems are difficult, and there can be many reasonablesounding processes. Without grounding that in the set of outcomes, it can be difficult to tell whether (and why) a given process may overcount.
Practical Implications
This issue of overcounting is but one example of why all three components of the model are important, and why we should encourage students to consider sets of outcomes and how they relate to counting processes. In light of this, below I offer some advice about teaching counting problems (particularly to undergraduates, although the same advice might apply to teaching counting at any level).
1) Have students focus on sets of outcomes. The main finding of my research so far is that there is value in having students focus on sets of outcomes. There are a couple of practical ways to do this. First, broadly, teachers should strive to frame counting as an activity of determining the cardinality of a set – specifically, the set of desirable outcomes specified in the counting problem. While this seems rudimentary, we have evidence that students do not always view counting in this way – instead, counting can be a matter of matching formulas, blindly guessing at problem types, etc. For example, when asked about an issue of order in a problem, one of my students once said, “I don’t know, I kind of go off of my gut for it on the ones that don’t specifically say order matters or it doesn’t matter.” Ironically, if students were more attuned to sets of outcomes, I believe this would help them in their pursuits toward finding a given problem type or in applying formulas. Perhaps if we always encouraged students to articulate the nature of what they are counting – by asking questions like What are you trying to count? Are your outcomes more appropriately modeled as sets of things or arrangements of things? – students may be more prone to view counting as an activity involving outcomes. This setoriented perspective on counting is outlined in Lockwood (2014) and is also mentioned in Hadar & Hadass (1981), Mamona Downs & Downs (2004), and Batanero, NavarroPelayo, & Godino (1997).
Second, there is another way in which students might actively engage with outcomes: to have students create partial lists of outcomes. There is mounting evidence that actually engaging in listing activities is a beneficial strategy for students (English, 1991; Halani, 2012), and statistical significance has even been demonstrated (Lockwood & Gibson, in press). Thus, a practical tip would be to have students engage in listing activity.
In some cases, listing can actually offer an answer to a counting problem, and, even more, this solidifies for students what an outcome is. For example, consider the Domino problem, which states: A Domino is a small, thin rectangular tile that has dots on one of its broad faces. That face is split into two halves, and there can be 0 through 6 dots on each of those halves. Suppose you want to make a set of dominos (i.e., one of every possible domino). How many distinguishable dominos would you make for a complete set? On this problem, I have seen students, prior to listing, offer an answer like \(7! \cdot 7!\), which does not make much sense in the context of the problem. By engaging in a problem like this, students can understand the nature of an outcomes, what we want to consider as distinguishable. Also, it is worth noting that partial listing is also helpful, especially on a problem in which the outcomes might be difficult to see/recognize as similar. Partial listing helps because, again, it can orient students to the nature of what is being counting, and often students may extrapolate a more general solution or strategy even from a partial list.
2) Emphasize the relationship between counting processes and sets of outcomes. As students engage in listing, the idea that there is a link between their counting processes and their set of outcomes should be reinforced. As we have noted above, this relationship can be key in helping students detect and deal with issues of order and overcounting, which are two common difficulties that students face. It is also important to note that the expressions and formulas should not necessarily be deemphasized, because they are important aspects of counting that provide streamlined ways of efficiently solving problems. The issue is that we tend to overemphasize them, and students view them simply as a formula to memorize, not as a generalization and/or formalization of a counting process that makes sense and that actually structures the set of outcomes in some way. Practically, emphasizing this relationship may involve giving students tasks that involve listing, or giving them problems that do not involve a simple application of a formula. The Domino problem and the 3letter sequences problems are good examples, as is the following Language Book problem (also adapted from Tucker, 2002): You have 5 different Spanish books, 6 different French books, and 4 different Japanese books. In how many ways can you select two books that are not of the same language?
3) Remind students that counting problems are fun and are excellent opportunities for critical thinking. Because there are not clearly prescribed algorithms for solving every problem (unlike solving a page full of integration by parts calculus problems), students can find counting to be frustrating. However, teachers should try to share the perspective that counting is, in fact, an intellectual challenge and can be quite enjoyable. A wonderful example of this is recent videos of students in a middle school math class trying to solve a counting problem (https://www.youtube.com/watch?v=SrWt_XvWLUk). These kids are not stressed about formulas or getting the right answer – they are engaged in critical thinking and problem solving and seem to be having fun doing it!
For undergrads, consider a problem like the following: Suppose you want to put 8 identical white socks and 8 identical red shoes on your pet octopus, who has 8 distinguishable legs. You can do this in any order as long as, for any given leg, the sock goes on before the shoe. In how many different ways can you put shoes and socks on your octopus? If students are given time and space to think about and explore counting as a fun opportunity to solve problems, students may be more comfortable with engaging in outcomes and not simply trying to apply a memorized but not well understood formula.
References:
Annin, S. A., & Lai, K. S. (2010). Common errors in counting problems. Mathematics Teacher, 103(6), 402409.
Batanero, C., NavarroPelayo, V., & Godino, J. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32, 181199.
English, L. D. (1991). Young children’s combinatorics strategies. Educational Studies in Mathematics, 22, 45147.
Hadar, N., & Hadass, R. (1981). The road to solve combinatorial problems is strewn with pitfalls. Educational Studies in Mathematics, 12, 435443.
Halani, A. (2012). Students’ ways of thinking about enumerative combinatorics solution sets: The odometer category. In the Electronic Proceedings for the Fifteenth Special Interest Group of the MAA on Research on Undergraduate Mathematics Education. (pp. 5968) Portland, OR: Portland State University.
Lockwood, E. (2013). A model of students’ combinatorial thinking. Journal of Mathematical Behavior, 32, 251265. Doi: 10.1016/j.jmathb.2013.02.008.
Lockwood, E. (2014). A setoriented perspective on solving counting problems. For the Learning of Mathematics, 34(2), 3137.
Lockwood, E., & Gibson, B. (In press). Combinatorial tasks and outcome listing: Examining productive listing among undergraduate students. To appear in Educational Studies in Mathematics.
Lockwood, E., Swinyard, C. A., & Caughman, J. S. (2015). Patterns, sets of outcomes, and combinatorial justification: Two students’ reinvention of counting formulas. International Journal of Research in Undergraduate Mathematics Education, 1(1), 2762. Doi: 10.1007/s4075301500012.
Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.). (2011). Combinatorics and Reasoning: Representing, Justifying, and Building Isomorphisms. New York: Springer.
MamonaDowns, J. & Downs, M. (2004). Realization of techniques in problem solving: the construction of bijections for enumeration tasks. Educational Studies in Mathematics, 56, 235253.
Tucker, A. (2002). Applied Combinatorics (4th ed.). New York: John Wiley & Sons.
]]>Mathematicians often consider knowledge of how algebraic structure informs the nature of solving equations, simplifying expressions, and multiplying polynomials as crucial knowledge for a teacher to possess, and thus expect that all high school teachers have taken an introductory course in abstract algebra as part of a bachelor’s degree. This is far from reality, however, as many high school teachers do not have a degree in mathematics (or even mathematics education) and have pursued alternative pathways to meet content requirements of certification. Moreover, the mathematics education community knows that more mathematics preparation does not necessarily improve instruction (DarlingHammond, 2000; Monk, 1994). In fact, some research has shown that more mathematics preparation may hinder a person’s ability to predict student difficulties with mathematics (Nathan & Petrosino, 2003; Nathan & Koedinger, 2000). Nevertheless, the requirements for traditional certification to teach secondary mathematics across the country continue to include an undergraduate major in the subject, and many mathematicians and mathematics educators still regard such advanced mathematics knowledge as potentially important for teachers.
Given this, it is important that, as a field, we investigate the nature of the present mathematics content courses offered (and required) of prospective secondary mathematics teachers to gain a better understanding of which concepts and topics positively impact teachers’ instructional practice. That is, we need to explore links not just between abstract algebra and the content of secondary mathematics, but also to the teaching of that content (e.g., see Wasserman, 2015). In November 2015, a group of mathematicians and mathematics educators met as a working group around this topic at the annual meeting of the North American Chapter of the Psychology of Mathematics Education. We began to probe the impact understanding connections such as those described above might have on teachers’ instructional choices. For example, how does understanding the group axioms shift teacher instruction around solving equations? How does understanding integral domains shift teacher instruction around factoring? Through answering questions such as these, mathematicians and mathematics educators can better support teachers to connect advanced mathematical understanding to school mathematics in meaningful ways that enhance the quality of instruction.
In the remainder of this blog post, we explain and discuss three frequently cited examples of connections between abstract algebra and high school mathematics.
Example 1: Solving equations
Solving equations and simplifying expressions is a technique used in multiple settings within mathematics. It uses the precise axioms of a group, but this is often not made transparent to students
What would you do to solve this “onestep” equation? Many students are taught to subtract 5 from both sides to isolate the variable x, and they might write something like this (crossing out the 5s on the left hand side):
x + 5 = 12
5 5
x = 7
However, on closer inspection, a variety of algebraic properties come to bear that the above work suppresses. (See Wasserman [2014] for a more complete elaboration and discussion.) An expanded version might look like this, with justifications for each step.
(x + 5) + 5 = (12) + 5 (Additive Equivalence)
x + (5 + 5) = 12 + 5 (Associativity of addition)
x + 0 = 12 + 5 (Additive Inverse)
x = 12 + 5 (Identity Element for addition)
x = 7 (Closure under addition)
Similarly, if attention is given to algebraic properties used to solve equations, the solution to an equation of the form 5x=12 might appear as follows:
⅕*(5x)= ⅕*12 (Equivalence)
(⅕*5)x = ⅕*12 (Associativity of multiplication)
1*x = ⅕*12 (Multiplicative Inverse)
x = ⅕*12 (Identity Element for multiplication)
x = 12/5 (Closure)
These solution techniques can be related to students’ learning of matrix algebra in a course on linear algebra. Specifically, students learn, under appropriate conditions, to solve matrix equations of the form AX = B using these same steps.
In each case above, the last four steps being used – the ones “hidden” from view in the onestep cancellation process – are the precise axioms for a group. In the first case, we’re working on the additive group of integers, in the second on the nonzero multiplicative group of rational numbers, and in the last under the group of n by n square matrices with nonzero determinant (i.e., invertible) under matrix multiplication. Thus, these are three a priori separate problems, all united by the same algebraic structure of a group – and that structure becomes evident in the algebraic solution process. Wasserman and Stockton (2013) discuss one vignette for how such knowledge might be incorporated into secondary instruction.
Example 2: Simplifying expressions
As a related example, consider the following two samples of student work:
In each case, clearly a form of “cancellation” is being attempted. But what, technically, results in “cancellation”? And what remains after the cancellation is complete? Do sin and sin1 make “1”? Is the “x” still an exponent? While we recognize this “cancellation” as attending to both the inverse elements and the meaning of the identity element in the group of invertible functions, these are subtle issues that are often not clear to students, and they are often taught in isolation, without the underlying structure being made apparent.
In using the above two examples to illustrate, we do not intend to imply that teachers should require students to make explicit each and every use of a mathematical property when they solve equations. Rather, we aim to draw attention to the importance of recognizing the consistency going on across all of these examples of solving equations. Moreover, it is the collective power of individual properties – as they form the group (or ring/field) axioms – that allow for algebraic solution approaches and also help reconcile the meaning of “cancellation” in these different contexts as an interaction of both inverse and identity elements.
Example 3: Polynomials and Factoring
As another example of the connection between abstract algebra and secondary mathematics, we consider the problem of multiplying two polynomials. (See Baldinger [2013, 2014] for additional examples of this type.) In high school, students learn that the degree of the product of two nonzero polynomials is the sum of the degrees of the factors. Yet this does not hold in all types of algebraic settings. Consider, for example, the product of the following two polynomials when working modulo 7 versus modulo 8.
As mathematicians, we of course recognize that the the degree of the product of two polynomials is the sum of the degrees of the factors — when the coefficients are elements of an integral domain, but that this relationship need not hold in other settings. Students, however, may be mystified when they first encounter an example like this in modular arithmetic, as their prior conceptions and understandings are being challenged, and they are thus being asked to deepen their understanding of the underlying structures that permit a result to hold in one setting, but break down in another.
This example also ties directly into student misconceptions. For example, we teach students in high school that if the product of two polynomials is zero, then to solve we set each one separately equal to zero. Yet this does not hold with nonzero numbers. For example, working in polynomials with real coefficients, we know that f(x) * g(x)=0 implies either f(x) = 0 or g(x) = 0. Yet it is not the case that if f(x) * g(x) = 4, then either f(x) = 2 or g(x) = 2.
The three above examples represent just a few of the many connections between abstract algebra and secondary mathematics. There has been a longstanding debate in the mathematics and mathematics education communities concerning the knowledge secondary mathematics teachers need to provide effective instruction. Central to this debate is what content knowledge secondary teachers should have in order to communicate mathematics to their students, assess student thinking, and make curricular and instructional decisions. This debate has already led to many fruitful projects (e.g., Connecting Middle School and College Mathematics [(CM)2] (Papick, n.d.); Mathematics Education for Teachers I (2001) and II (2012); Mathematical Understanding for Secondary Teaching: A Framework and ClassroomBased Situations (Heid, Wilson, & Blume, in press). A common thread in these projects is the belief that mathematics teachers should have a strong mathematical foundation along with the knowledge of how advanced mathematics is connected to secondary mathematics (Papick, 2011). But questions remain regarding what secondary content stems from connections to advanced mathematics, which connections are important, and how might knowledge of such connections influence practice. Our working group hopes to continue to explore these connections and contribute to our collective understanding of teacher education.
References
Baldinger, E. (2013). Connecting abstract algebra to high school algebra. In Martinez, M. & Castro Superfine, A. (Eds.). Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 733–736). Chicago, IL: University of Illinois at Chicago.
Baldinger, E. (2014). Studying abstract algebra to teach high school algebra: Investigating future teachers’ development of mathematical knowledge for teaching (Unpublished doctoral dissertation). Stanford University, Stanford, CA.
Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers (Issues in Mathematics Education, Vol. 11). Providence, RI: American Mathematical Society.
Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II (Issues in Mathematics Education, Vol. 17). Providence, RI: American Mathematical Society.
DarlingHammond, L. (2000). Teacher quality and student achievement: A review of state policy evidence. Educational Policy Analysis Archives, 8(1). Retrieved from http://epaa.asu.edu
Heid, M. K., Wilson, P., & Blume, G. W. (in press). Mathematical Understanding for Secondary Teaching: A Framework and ClassroomBased Situations. Charlotte, NC: Information Age Publishing.
Monk, D. H. (1994). Subject matter preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125–145.
Nathan, M. J. & Koedinger, K. R. (2000). An investigation of teachers’ beliefs of students’ algebra development. Cognition and Instruction, 18, 209–237.
Nathan, M. J. & Petrosino, A. (2003). Expert blind spot among preservice teachers. American Education Research Journal, 40, 905–928.
Papick, I. (n.d.) Connecting Middle School and College Mathematics Project. Retrieved March 7, 2015 from http://www.teachmathmissouri.org/
Papick, I. J. (2011). Strengthening the mathematical content knowledge of middle and secondary mathematics teachers. Notices of the AMS, 58(3), 389392.
Wasserman, N. (2014). Introducing algebraic structures through solving equations: Vertical content knowledge for K12 mathematics teachers. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 24, 191–214.
Wasserman, N. (2015). Abstract algebra for algebra teaching: Influencing school mathematics instruction. Canadian Journal of Science, Mathematics and Technology Education (online first). DOI: 10.1080/14926156.2015.1093200
Wasserman, N. & Stockton, J. (2013). Horizon content knowledge in the work of teaching: A focus on planning. For the Learning of Mathematics, 33(3), pp. 20–22.
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This fall, the Mathematical Association of America released a fiveyear 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 mathloving 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 fulltime 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 prealgebra 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 prealgebra to precalculus, 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 precalculus 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 precalculus. 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 oneonone 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 stepbystep 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 stepbystep 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 numberreliant 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 standardsbased 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 inclass 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 inclass 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 04 scale), roughly correlated with a GPA or lettergrade 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 reattempt up to one course standard per week. Usually these reattempts occurred during office hours and were in the form of a one or twoquestion 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:
I adapted this system from one Joshua Bowman used. I like it because it captures my feeling that an “Alevel” student is a student who shows mastery of nearly all concepts and shows good progress toward mastery on the others; meanwhile, a “Blevel” student is one who consistently does Blevel 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 reattempt 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 buyin 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 emails 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 jumpstart your journey:
Online SBG Resources
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 nowlandmark 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 largescale Canadian study in 2010 [2], which found that early math skills were stronger predictors of general academic success than either reading skills or socialemotional 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 socioeconomic 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 latergrade 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 “pottytrained” 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. Sitandlisten 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 lesslimited 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 studentdirected/teacherfacilitated 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 readaloud 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 inservice 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, BrooksGunn J, Sexton H, Duckworth K, Japel C. “School readiness and later achievement.” Dev Psychol. 2007 Nov;43(6):142846.
[2] Pagani, Linda S. et al. “School Readiness and Later Achievement: A French Canadian Replication and Extension.” Developmental Psychology. Vol. 46(5): 984994. 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 PostSecondary 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 buyin from individual mathematicians who are in the classroom daily, working facetoface 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 earlycareer 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 NSFFunded 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 largelecture size). In departments such as the University of Michigan, University of Illinois UrbanaChampaign, 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 reconceive these activities as examples of active learning, then it becomes easier to see how one might try to incorporate some smallgroup 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 K12 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 extracurricular K12 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 earlycareer mathematicians
The training and mentoring of earlycareer 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 multiyear 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 activelearning teaching style. The MAA also offers 4hour minicourses 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 wellestablished, 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 earlycareer faculty members do not receive sufficient mentoring and training, suggests that continued efforts are needed to improve the overall landscape of pedagogical training for earlycareer faculty.
Further, depending on departmental and institutional culture, junior faculty often have reasonable concerns regarding earning tenure or ensuring that a shortterm 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 earlycareer 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 clearlycommunicated ways to support earlycareer 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 earlycareer 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 proofcentered discourse does not always translate well to the classroom. Lowerlevel students may have weak backgrounds, including scant practice with the logic that now comes naturally to us. Even upperlevel 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, realworld 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/isalgebranecessary.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/arecollegelecturesunfair.html
[3] Lecture Me. Really. Molly Worthen. New York Times, October 18, 2015. http://www.nytimes.com/2015/10/18/opinion/sunday/lecturemereally.html
[4] Engage to Excel. PCAST report, January 2012. https://www.whitehouse.gov/sites/default/files/microsites/ostp/pcastengagetoexcelfinal_feb.pdf
[5] Beisiegel, M. & Simmt, E. (2012). Formation of mathematics graduate students’ mathematicianasteacher identity. For the Learning of Mathematics, 32(1), 3439.
[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. 916). 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. 644648. 2001.
[8] Nathan, Mitchell J., and Anthony Petrosino. “Expert blind spot among preservice teachers.” American educational research journal 40.4 (2003): 905928.
[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.
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