When I first started incorporating active learning in the classroom, I struggled with getting my students to buy into being active. I made worksheets, put the students in groups, and excitedly set them off to discover and play with mathematical ideas. Despite this, many students were inclined to sit silently in a group of four and work on the problems on their own.
But really, who can blame them? First, this propensity towards solitude can be explained by basic human nature: specifically, the fear of being wrong. We don’t want to be wrong. At least, we don’t want to be wrong in front of other people. From that perspective, working alone is safe and comfortable. We should view our job as teachers as one of helping our students overcome this basic human inclination, as opposed to viewing it as a failure or shortcoming on their part.
Beyond this, the desire to work alone can be attributed to culture and expectations. Many students’ formative educational years have been spent sitting silently in desks passively absorbing lectures. If they feel this is what is expected of a math class, then it is natural for them to continue to sit silently, even if the environment is meant to be collaborative. Of course, it is not my intention to imply that this is an issue that is entirely the students’ fault – maybe my questions weren’t sufficiently open-ended, maybe I wasn’t doing a good enough job at “selling it,” maybe the students just like working alone, maybe, maybe, maybe… The list goes on.
I tried some of my standard tricks to foster communication among the students. I would prepare impassioned pep talks about the benefits of working with your peers. This technique flopped for obvious reasons – no one wants to listen to what they are told is good for them. Otherwise, cigarette companies and fast food restaurants would go bankrupt and I would be much more diligent about flossing. I’d try to lighten the mood, saying “this isn’t a library, you’re welcome to talk to one another.” I’d give a difficult problem and leave the room to get a drink of water, forcing the students to rely on one another. These strategies helped, but never served to create the classroom of my dreams – one where students discuss math problems at such a frenzied pace that time ceases to exist; one that causes passersby to wonder whether we are having a math class or developing some bizarre scientific improv comedy troupe.
Over time, I continued to reflect on my own teaching and sought advice from more experienced practitioners of active learning. As a result, I have developed a few strategies that have been effective in my classrooms. One of the most effective strategies for me has come from eliminating those pesky desks that keep getting in the way of my students’ learning.
One day after watching Back to the Future for the umpteenth time, Doc Brown gave me a spark of inspiration: “Roads? Where we’re going we don’t need roads!” But now replace “roads” with “desks.” Maybe the desks were the problem.
In the spring of 2014 I was teaching a Graph Theory class in my favorite classroom on campus. It has chalkboards on three of the walls, and the fourth wall is a bank of windows that looks out onto our campus quad, which is very pretty with a fountain and trees. It’s sort of a mathematician’s paradise. The most important aspect of this room is the board space – there’s enough room for 24 people to work on the board at the same time. I decided to test this desk-free learning idea: instead of sitting in their desks, what if my students spent most of their class time at the board? This had a noticeable impact on the quality of conversations and engagement in the classroom. I’ve found success in implementing this strategy in different ways in different classes. It has led to more dynamic exam review sessions for lower-level calculus and linear algebra classes, deeper learning in my introduction to proofs classes, and hotbeds of mathematical ideas in my graph theory classes. I’d like to offer some tips and tricks on how this can be done in a variety of settings.
I typically prepare a worksheet with problems for each class that are meant to guide the students through a certain topic or idea. If necessary, I start class with a short lecture introducing some new ideas or definitions and then set the students to work on the problems I have prepared, telling them that I want them to get up and work at the board in groups with their peers.
From a practical teaching perspective, there are a number of benefits to doing this. Because I can see what everyone is doing from just about anywhere (as long as the room is convex), I can easily assess each group’s progress from a vantage point in the center of the room and quickly determine which group needs the most immediate attention. If everyone seems to be making a common mistake, I can pull the group back together and add clarification or facilitate a large group discussion. Similarly, if one of the groups has something interesting to share – for example, an interesting example/counterexample or a solution that is different from what everyone else has done – it is easy to bring the whole class to their work area and let them share their ideas.
This practice can be particularly effective in a graph theory class where dynamic problem solving is so important. For example, solving problems about planar graphs on paper can be frustrating if you have to keep redrawing the same graph until you find a nice planar drawing. On the board, it is much easier to just erase the problematic edges without having start from scratch. In a different activity, I give the students pseudocode for an algorithm (for example, Dijkstra’s Algorithm), without telling them what the algorithm does. They decipher the pseudocode and run through the algorithm on an example graph to try to figure out what the algorithm does. After that, we discuss what the algorithm does and prove that it works as a group. This is easier once the students already have a solid intuitive understanding of what the algorithm does because it separates the difficulty of such proofs into more manageable pieces: first, understand what the algorithm does, and second understand techniques for proving an algorithm works.
Most importantly, being at the board helps students overcome the fear of being wrong. Because work on the board is inherently temporary, students don’t have the same reservations about writing down some ideas and sharing their thoughts, even if they are incomplete and especially if they may be wrong. I call this the “Bob Ross approach to mathematics.” On The Joy of Painting, Bob Ross taught us “We don’t make mistakes. We just make happy little accidents.” Overcoming their initial fear of making mistakes helps students get to the point where real learning happens.
Unlike working on a piece of paper, which carries a natural expectation that you will start writing at the top and finish writing at the bottom, work at the chalkboard can flow more organically. Students come to see that solutions often take serpentine paths through different parts of the problem and various examples until you figure out how to put all the pieces together. Then, once they find a solution, they can write out a clean version of the solution in their notebooks. In many cases, they just take pictures on their phones and transcribe their notes later.
Having students stand at the board also makes for a more social environment that naturally fosters collaboration and does seem to create a more active classroom. Standing helps everyone be more engaged, more physically active and, as a result, more mentally active. In a course evaluation, one student commented “Using the chalkboards in class was a great way to get our blood flowing and keep focused during class.” When you are standing, it is more reasonable to expect that you should be talking with the people around you.
As a result of spending more time collaborating with their peers, students come to see they are not alone in their confusion or struggles. They learn to ask questions, which can be as simple as “I didn’t get what you just said, can you say it again?” When I was in grad school I started asking that question, perhaps to the extent that people got tired of hearing it. It has been hugely beneficial to me and my students. When working together, students see different approaches and different ways of thinking about problems. As a result of having to answer questions posed by their peers, they reflect more deeply on the way they approach problems.
Heuristically, this also seems to help the students develop better problem solving skills. They realize that in order to solve a problem, it helps to write something, anything, to get your brain wrapped around the problem. Students learn to explore small examples, reflecting on their observations, and thinking about how to generalize those observations. By the end of the quarter, students who had initial reservations saying “do we have to work at the board?” have changed their attitude, with some saying “do we get to work at the board today?”
The tips I’ve discussed here are applicable beyond a course in Graph Theory and can be used beyond classes at a small liberal arts school with small classes. Having students work together to solve practice problems at the board without their notes or books can be valuable in helping them prepare for an exam. Teaching Assistants can implement similar practices in recitation sections with smaller groups of students, even if they are part of a large lecture course. At Seattle University, we don’t have a graduate program, but we have undergraduate Learning Assistants who facilitate study groups for our lower-level math courses. We train our LAs to facilitate group work in this way so that students are actively engaged during their study groups.
In mathematics, we often pride ourselves on the fact that our research can be done wherever there’s a chalkboard. We should strive to include this in the way we help our students learn!
]]>Summer 2017 brought the third anniversary of On Teaching and Learning Mathematics and with it our annual review of the articles we have published since our previous year in review article. Over the past year, our articles have covered a range of topics and ideas, and I have loosely collated them by the following topics: active learning, K-12 education, summer experiences, assessment, diversity and inclusion, curricular issues, and mathematical culture. As we begin a new academic year, we hope you will take some time to read them (or read them again!) and be inspired.
Active Learning
Active learning was a major topic for us again this year. Henrich, Blanco, and Klee shared ideas for supporting productive collaboration and conversation. LaRose argued that effective teaching is essentially inefficient, and that active learning is a prime example of this. Ellis Hagman interviewed several colleagues to find out how their use of active learning impacts students from marginalized populations. Bremser reflected on a broadly-used form of active learning many of us overlook: tutoring.
K-12 Education
It is impossible to discuss postsecondary mathematics education without considering K-12 education as well. Lai, Howell, and Lahme discussed effective pre-service teacher education. Wilson, Adamson, Cox, and O’Bryan made the case that our standard method for teaching inverse functions is counterproductive. Schanzer outlined the challenges that exist for mathematics due to the growing movement to teach computer science at the K-12 level. Beck and Wiegers shared their experiences directing an NSF-funded program connecting graduate students with K-12 students.
Summer Experiences
Both K-12 and undergraduate education take place beyond the constraints of classrooms; summer programs are frequently a source of inspiration for students. Through an interview with REU students, members of the editorial board explored their impact on five current undergraduates. García Puente provided a faculty perspective on leading undergraduate research projects. Duval reflected on his own profound experience as a high school student in a summer program that inspired a lifetime of mathematics.
Assessment
Along with the responsibility of creating meaningful classroom experiences, mathematics faculty have the responsibility of assessing students in a meaningful way. Bagley, Gleason, Rice, Thomas, and White investigated the efficacy of the Calculus Concept Inventory as a means to assess student conceptual understanding. Patterson discussed the influence of growth mindset research on his classroom assessment techniques. Dewar turned the focus around with a thorough consideration of what instructors should know about student ratings of teaching.
Diversity and Inclusion
A deep and important challenge for the mathematics community is to find ways to increase our diversity and meaningfully include every mathematics student. Pons discussed her experience at ECCO 2016, a research conference that excelled in this mission. Hobson provided six ideas for instructors seeking ways to support diversity and inclusion. Katz reflected on the impact of implicit messages in our teaching, providing frameworks through which instructors can evaluate their impact on students.
Curricular Issues
Curricular issues are a perennial concern for mathematicians and mathematics departments. Armstrong made the case for an expanded presence of linear algebra in standard undergraduate coursework. Pudwell described her experience teaching courses on experimental mathematics and the role this course offers within the standard undergraduate curriculum.
Mathematical Culture
Our final three articles this year dealt in different ways with mathematical culture. Braun wrote about the challenge of balancing our ideals and our reality in the realm of teaching. Ellis Hagman wrote about the cultural differences between mathematics research and mathematics education research, and the questions she often gets from colleagues about her work as an educational researcher. Buckmire, Murphy, Haddock, Richardson, and Driscoll described several of the mathematics education projects funded by the National Science Foundation, and invited readers to contact them with ideas for proposals and projects.
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Since we began On Teaching and Learning Mathematics in June 2014, I have had an amazing experience working with fantastic editorial board members and contributing authors. I believe that the commitment to excellence by our board members and contributors is the reason our blog attracts over 5,000 unique page views every month. For anyone who is interested in deepening their knowledge regarding the teaching and learning of mathematics, and in contributing to the improvement of postsecondary mathematics education, I encourage you to apply for this position.
]]>On a recent trip to Santa Fe, New Mexico, I met a really cool woman named Anna Sale who runs a podcast called Death, Sex, and Money (check it out). In this podcast she interviews people about things she is curious about. We talked about how her work is similar to research (come up with something you want to know more about, then go learn about it), except much less rigorous and you get answers much more quickly.
One thing I am very curious about is how students from marginalized populations experience active-learning classes. I believe deeply in teaching in a more active way, and I also believe deeply in teaching so that all of my students have the best opportunity to succeed, and sometimes I wonder if all my active-learning moves are enough to support all of my students. So, taking inspiration from Anna, I decided to interview some experts. (I am also working on a grant proposal to look at this in a much more rigorous/slow way).
The interviewees: I chose four people to interview who I have either talked to about active learning and issues of equity or who were recommended as someone good to talk to about these things. This led me to: Brian Katz, Gail Tang, Darryl Yong, and Christine von Renesse. At the end of this post I introduce them in more depth, and I will admit that I didn’t realize I had asked only people from small, teaching-focused colleges (Augustana College, University of La Verne, Harvey Mudd College, and Westfield State University, respectively). If this were a research project, this may be an issue. However, for this blog-post, I take this as indicative of who is spending time thinking critically about their teaching and developing reputations as thoughtful teachers of undergraduate mathematics. Either way, they provided different and interesting perspectives, and answered everything so well I no longer need to submit my grant proposal looking into these questions a larger scale (just kidding!).
The interview: After BK, CvR, GT, and DY agreed to participate in my interview, I sent them each a link to a Google doc containing the questions and space for each to answer. We did it this way so that each person could react to the others’ responses, which led to fun and interesting exchanges. For this post, I will present a summary of some of the main themes to each answer and share some especially illuminating excerpts. Questions 1-3 provide some background on the interviewees approaches to active learning, while questions 4-6 get to the meat of my interests. Before I move on, I must say: Interviewing people who are experts in something you care about is SO FUN and I highly recommend it!
Questions 1-3 asked what classes they have used active-learning approach to teach, why they use active learning, and how they typically implement active-learning.
What: Between the four of them, my interviewees listed literally every course taught in a typical undergraduate mathematics department, from general education courses such as “Math in Society” and “Quantitative Reasoning” to upper-level mathematics courses like Numerical Analysis and Algebraic Geometry.
Why: Four main reasons for implementing active learning stood out to me in the responses. Active learning:
CvR referenced a comprehensive list of benefits of active learning for students on the Art of Mathematics website.
How: The interviewees were hesitant to describe a typical day because their teaching is so responsive – they say it depends on the content (“Everything about the active learning depends on the learning objectives for the course as a whole and for the specific class session.” – DY) and on the students (“I come up with a loose plan to do small-group work (let’s say). But then as they’re working in groups, I may have students present depending on what happens during that activity. Sometimes we’ll break in whole-class discussion, also depending on what happens during the activity… It all depends on what is happening in the class!” – GT). As a whole, the interviewees utilize small groups (purposefully structured, will say more later about this), whole-class discussion, student presentations, partner work, and individual work followed by sharing with others. BK and CvR also referenced the importance of the work they have students do before and after class, emphasizing that the active-learning class does not live only within the time and space of class.
Here is where things get meaty:
All four interviewees said that this resonated with their experiences. GT has noticed that in her department women tend to get better grades in the active learning courses compared to traditional courses, and BK has had students from underrepresented groups (both women and students of color) tell him that they feel able to talk in his active learning classes when they haven’t felt comfortable in other classes. CvR noted that she has seen particular benefits for women in her classes, provided that she “find[s] a good group for them.”
She also noted that with active learning, she can more easily differentiate instruction to support students who are lacking prior knowledge and students who are ahead in their knowledge and so has seen benefits for “typically low achievers.” [Note: A few of the interviewees took slight issue with this phrasing in my question. I took it directly from the Laursen et al. paper, which grouped students by previous mathematics GPAs.] DY said “Active learning is awesome and I have seen it create a more equitable playing field for everyone,” but qualified this statement a bit: “I just feel like we shouldn’t assume it to be an automatic thing.” (Which perfectly set up my next question – thanks for the smooth transition DY!)
The interviewees shared that this did resonate with their experiences, and shared specific instances where a student from a marginalized population shared with them that they were not comfortable either in their groups or contributing to their whole-class discussion.
DY summarized a few of their ideas well: “I really think it is important to give students some instructions and models for how to work well with each other. We cannot assume that they know how to work well with each other and in fact we need to expect that students will bring with them all of their prejudices and unconscious biases to class (just like we do) and that we need to set up the working environment to mitigate these things…I am also now more conscious about making sure my instructions to students are crystal clear and that I give them enough individual think time before they have to answer or work together. I do this because our introverted students and students whose first language is not English need a little more time to put together their thoughts. And, I don’t want students to have to read my mind about what exactly I’m asking them to do—that privileges students who might have had more experience working in more active classrooms and I want to create a more level playing field for my students.”
My interviewees shared specific ways that they structure groups, and emphasized how important this was to their class functioning equitably:
I hope you enjoyed this interview as much as I did. I am thrilled to know equitable active learning is becoming a topic of more discussion – during this years’ IBL workshops we had some great discussions related to this post, and recently discussed related ideas at this year’s MathFest in Chicago.
However, there is still much more for us to learn. As mentioned briefly above, in the area of K-12 mathematics researchers use the term “Complex Instruction” to describe instruction that supports equitable experiences in active learning mathematics courses. As college educators, we can absolutely learn from this work, but there are also reasons why not all of this work is easily transferrable. For instance, typically K-12 teachers spend much longer with their students than we do as college teachers, providing more time to develop a classroom culture that supports the practices they encourage. So – while more discussion is happening related to equity and active learning, as a community there is still much for us to learn. While we learn more, here are some resources for some things we do know:
Charles Duhigg’s Chapter about Groupwork in Smarter, Better, Faster
Elizabeth Cohen’s Designing Groupwork: Strategies for the Heterogeneous Classroom
Laura Rendón Sentipensante (Sensing/Thinking) Pedagogy: Educating for Wholeness, Social Justice and Liberation
Discovering the Art of Mathematics website
Tim Erickson and Rose Craig’s United We Solve: 116 Math Problems for Groups, Grades 5-10, (DY: even though this is for younger kids, this is still a great book to use for undergraduates because they get your students to cooperate in productive ways)
Biographical Information:
Brian Katz BK is an associate professor of mathematics and computer science at Augustana College, with a PhD in mathematics from UT Austin. His research interests include proof, the evolution of shared meaning, epistemology, and equity/justice in the context of inquiry-based mathematics classrooms. For BK, all teaching is political and hence should be about justice. He is currently the Chair of the IBL SIGMAA and coeditor of the AMS Blog on equity and social justice, called inclusion/exclusion.
Christine von Renesse CvR is an associate professor of mathematics at Westfield State University with a Ph.D. in Mathematics at the University of Massachusetts. CR is originally from Germany, and has a Master’s Degree in Elementary Education, a Minor in Music and a Master’s Degree in Mathematics from the Technical University Berlin, Germany. She is an author and principle investigator for the Art of Mathematics project, and a passionate teacher, who loves teaching at all levels — from elementary school through College. In her free time Christine loves to explore nature, sing in harmony and go dancing, especially with her daughter. To learn more about CvR and the Art of Mathematics, see the website.
Gail Tang Passionate about broadening the participation of women in mathematics, GT relies upon asset-based mentorship and teaching. Her research interests include equity in mathematics and fostering mathematical creativity. GT is an Assistant Professor of Mathematics at the University of La Verne. Additionally, she is the Mathematics Curriculum Lead for Guided Pathways to STEM Success, a DoE-funded grant aimed at providing inclusive opportunities for more students in STEM. In her spare time, GT loves to spend time with her partner, their two dogs, three cats, and five chickens. Frequently she can be found in their garden among the bees, ladybugs, and butterflies, watching plants grow.
Darryl Yong DY is a Professor of Mathematics at Harvey Mudd College. This year, he is on leave from Harvey Mudd to serve as the Director of the Claremont Colleges Center for Teaching and Learning. Previously, he served as Associate Dean for Diversity at Mudd from 2011-2016 and has served as associate chair of the mathematics department. He received his PhD in applied mathematics at the University of Washington. His scholarly activities focus on the retention and professional development of secondary school mathematics teachers and improving undergraduate mathematics education.
]]>I just returned from an all-years reunion of the Hampshire College Summer Studies in Mathematics (HCSSiM) program, a six-week program I attended during the summer between my sophomore and junior years of high school. It has been run by David C. Kelly, whom everyone refers to just as Kelly, since he started it in 1971. There are several other summer high school math programs around the country (a good start is this list from the AMS), which likely share some characteristics with Hampshire, but since Hampshire is the one I have personal experience with, this is the one I am compelled to talk about. And while several people and experiences were instrumental in my path to becoming a mathematician, Hampshire is the one that stands out most prominently in my mind, the one mathematical encounter that changed my life. And from talking to other people at the reunion last weekend, I know that many other program alumni feel the same way.
There are other accounts written about Hampshire. The AMS has a nice commentary on Hampshire and other similar programs, and Jim Propp has a very nice blog post about it, from the perspective of someone who has been a student, and has taught at the program as junior staff and senior staff. I was only a student one summer, and never taught there, but the program had such a profound effect on me that I want to share my personal reflection on the experience. Now over 34 years later I am a professor at UTEP, and I hope that the benefit of my looking back with the hindsight of years of learning and teaching math will outweigh the loss of some details through those same years. But, as with any transformative experience, some details remain crystal clear.
I’d loved math from an early age, but when I showed up at Hampshire as a rising high school junior, the only proofs I’d seen were the two-column proofs in high school geometry that previous year. From the outset, it was clear that this program was going to be … different from high school. The first day of class (we met in 4 classes with about 17 students for 4 hours in the morning, 6 days a week; each class with different instructors, but discussing similar topics), Kelly started immediately with a problem, which I still remember: If we have a (three-dimensional) hunk of cheese, and we slice it with some number of cuts with a knife, how many regions will we have? This was not at all like my math classes in school, and a little like some of the puzzles I read about in mathematical puzzle books on my own, but it was somehow a little deeper than those puzzles, and it was an entire class spent exploring the problem together.
We spent almost all morning (four hours!) on just this one problem. It served as a vehicle to develop for ourselves (with guiding questions from the staff, to be sure) mathematical problem-solving strategies we were to return to all summer: Work examples and gather data; formulate your assumptions carefully (early idea, soon discarded: What if all the cuts are parallel? That’s not an interesting problem anymore); don’t assume that patterns always continue (in this case, 1,2,4,8 was followed not by 16, but by 15); when problems are too hard, try a simpler problem first (we moved from 3-dimensions to 2-dimensions); draw pictures; make conjectures; try to prove your conjectures, or make new ones if necessary.
The rest of the summer proceeded similarly. Topics were introduced by way of problems, some of them imaginative, which we discussed and refined, with lots of student input. Methods of proof were woven into discussions about how to verify our conjectures. Classes were engaging, and even fun, because of the interesting problems and because of the interaction among students and staff. Looking back at it now I would describe it as active learning, with an additional ingredient: It never felt like anyone (student or staff) was doing something because they had to, or for a grade, or really for any reason other than that it was inherently interesting. But at the time, I only knew that I really liked it. If this was math, I could spend all day doing math!
Evenings were devoted to 3-hour problem sessions. Problem sets ranged in difficulty from working examples to writing proofs. Problems reviewed that day’s material, expanded on ideas, or let us play with ideas that had come up during the day. Sometimes they previewed upcoming material. During the problem sessions, we could work on whichever problems struck our fancy, and, when we solved something we were proud of, we could turn it in for constructive feedback. In a change from my previous learning/student experience, nothing was ever awarded a grade. (Instead, at the end of summer, Kelly wrote a detailed descriptive letter of recommendation for each student.) When I looked back many years later at some of the proofs I’d written, they looked very rudimentary to my more experienced eyes, but I know that by the end of the summer I really had learned at least the basics of writing proofs, and that I loved it.
Another remarkable difference from my previous experiences was that we were strongly encouraged to work in groups on problems. In fact, throughout the program there was a strong sense of cooperation instead of competition. I quickly grew to embrace this cooperative view of mathematics and of education, and never turned back.
Afternoons were free time, but I spent most of my afternoons working with other students on the weekly “program journal”, hanging out in the room where we put it together. Like many aspects of the program, it was almost entirely student-run. We wrote summaries of the week’s activities in the classes and of the daily “Prime Time Theorem” lectures (self-contained hour-long talks given by visiting mathematicians or by the staff), and ran a problems section (pose problems, solicit solutions, print solutions the next week). Of course, we also had some less serious features, such as reviews of the weekly math movies, cartoons, and silly math songs. I wrote some of the Prime Time Theorem write-ups, and I distinctly remember noting that by the final week of the program I was paying more attention to the precision I had to use to get the details correct.
Eventually we saw many different topics, none requiring much prerequisites beyond some high school algebra and, more importantly, an intense curiosity and a willingness to experiment and learn. I don’t recall precisely all the topics, but we certainly covered a lot of number theory, the basics of group theory, and some combinatorics and probability, some topology, some notions of infinity. It very much felt like that any subject might come up on any day. Halfway through the program, we finished the overview class, and each student could pick a class focused on one of four specific topics; I picked the class on large prime numbers and factoring large numbers, but later wished I’d picked the class on group theory centered around Rubik’s Cube (a few months before the Cube became wildly popular in the United States). I got to see lots of other things I would later take for granted (the geometry of how complex numbers multiply; how to think of GCD in terms of buying postage stamps; etc.). It took me some time to realize that not all math students learned these things in high school!
As with any good educational experience, I also learned a lot from my fellow students. Many were attending selective high schools in New York City and elsewhere (I was attending the public high school in my suburban town), and they generally had much higher expectations than I’d even thought about. They went to national math competitions, and did well. They planned to go to Ivy League colleges. Being around them raised my own expectations of college and my future. I entered Hampshire thinking I would be a meteorologist (because I liked looking at clouds), and left thinking I would be, well, if not a mathematician, at least an engineer. But I was also certain I would take any math class I could. (Which I did, and then eventually switched to math.) And I also believed I could go to the best programs in the country to pursue further education.
Even though we were studying serious and advanced mathematics (even without having taken calculus!), everything was infused with a sense of playfulness. From Kelly on down, staff conveyed the idea that what we were doing was inherently interesting, and that it was fun to just play around with ideas, and problems, and explore ideas. Though there were jokes and kidding around, it was the wonder of the mathematics that always took center stage. One of the ways in which this playfulness was transmitted was through the program’s adopted mascot and number.
You probably were expecting me to get to this part if you have heard of Hampshire before. We quickly found out that Kelly, and everyone else at the program, had a thing about yellow pigs and the number 17. Yellow pigs were everywhere, including on our t-shirts once Yellow Pigs Day happened on July 17, when Kelly gave his talk on the mathematical and social history of 17. (For instance, a regular 17-sided polygon can be constructed with ruler and compass because 17 is a Fermat prime, \(17=2^{2^n}+1\) with \(n=2\).) Soon all the students found and used YP’s and 17’s everywhere (for instance, look again carefully at the first sentence of this paragraph, especially the first two words and the number of words).
Of course, one purpose of yellow pigs and 17 was for a program identity and cohesion (and for alumni to be able to recognize one another), but 17 had another useful purpose. If you wanted to give a proof, but start with an example, you could pick 17 as the value of a variable such as \(n\). Everyone (at Hampshire) would recognize you were using it just as a placeholder, and the next step, replacing all the 17’s by \(n\)’s might not be too hard. (Kelly also showed us how this transition could be achieved typographically; see below illustration.)
Some years after I’d been teaching at the university level, I realized that almost every innovation I tried to implement was based on some aspect of Hampshire. Well, I didn’t try to use yellow pigs, but I do use 17 in examples in class when I can. (And I do reflexively look for 17’s everywhere.) More than that, the Hampshire way of thinking about mathematics remains at the core of how I approach mathematics and education. It remains remarkable to me that just six weeks could change my life so profoundly, but I remain eternally grateful to Kelly and HCSSiM that it did.
Let us know in the comments what your most significant mathematical experience was, and what effect it had on your life.
]]>“Can you recommend a good math tutor?” I hear this question from friends with children in local schools, academic support staff at my institution, and my own students. Once or twice I’ve even heard it from a student on the first day of class. Although tutoring has much in common with other educational settings, it presents its own opportunities and challenges. In this post, I explore why one-on-one instruction is so appealing as a supplement to classroom instruction, and how effective tutors make the most of tutoring sessions.
As Lepper and Woolverton point out in setting the stage for “The Wisdom of Practice: Lessons Learned from the Study of Highly Effective Tutors” [2, p. 138], “tutorials provide a venue for learning that is inherently more individualized, more immediate, and more interactive than most common school settings.” Specifically, individualization ensures more focused attention from both tutor and tutee. Immediacy allows for instantaneous feedback. Interactivity means that the tutor can make real-time decisions and adjustments as the student’s comprehension level and emotional state become more clear.
The authors go on to identify specific practices of expert tutors. While the overview is limited to studies of tutors for elementary school students studying mathematics, many of the effective practices it describes are also applicable to secondary and college mathematics settings. For example, “our best tutors seem to prefer a Socratic to a more didactic approach” [2, p. 146]. Naturally this approach involves asking questions and providing hints rather than providing quick answers. It also includes making a distinction between “productive” and “nonproductive” errors [p. 147] and responding accordingly. A productive error is one that the student can self-correct, with the long-term learning benefits that ensue, while a nonproductive error is best corrected immediately by the tutor.
Readers of How People Learn and related works will recognize a metacognition theme in this observation of Lepper and Woolverton: “more effective tutors are more likely to ask students to articulate what they are learning, to explain their reasoning and their answers, and to generalize or relate their work in the tutoring session to other contexts and problems.” An expert tutor, then, guides the interaction not only for strong communication with the tutee, but also to strengthen and reinforce learning.
The question of tutor-student communication is a complex one. In a research review, Graesser et al. [1, p.418] point out five “illusions” that tutors may hold. These are the illusions of grounding, feedback accuracy, discourse alignment, student mastery, and knowledge transfer. They categorize the misunderstandings that tutors often have about their students’ thinking. Have you ever been asked whether you understood something, and said “yes” even though you weren’t sure? You were giving inaccurate feedback, and your questioner may not have caught on. Of course even a sincere “yes, I understand” may be inaccurate, as “it is the knowledgeable students who tend to say ‘No, I don’t understand.’ This result suggests that deeper learners have higher standards of comprehension” (p. 414).
For an example of poor discourse alignment, note that “tutors sometimes give hints, but the students do not realize they are hints” [p. 418]. Now that is a reality check. In our previous post, Jess Ellis Hagman wrote, “Mathematics education research is the systematic study of the teaching and learning of mathematics.” Sometimes a seemingly small detail emerging from such study can have profound implications.
More from Graessner et al.: “A good tutor is sufficiently skeptical of the student’s level of understanding. … A good tutor assumes that the student understands very little of what the tutor says and that knowledge transfer approaches zero … (E)xpert tutors are more likely to verify that the student understands what the tutor expresses by asking follow-up questions or giving follow-up troubleshooting problems” [1, p. 419]. I recall working with an algebra student who insisted that he understood the relationship between the graphs of $y = x^2$ and $y = x^2 +2$, even though his graphs intersected. Rather than pointing at the intersection and explaining my concern, I should have suggested that he add the graph of $y = x^2 + 1$ and tell me what he noticed.
Given recent research on the effects of students’ emotions and mindsets on learning, how do good tutors attend to those factors? For one thing, while they are supportive and kind, they are sparing with praise. When these tutors do offer compliments, they refer to the work, not the person. The compliment might be an indirect one, such as a simple, “That was a hard problem you just did.” Good tutors also find ways to turn control over to their students by, for example, letting the tutee choose between two equally challenging problems [2].
Many of the above observations about effective tutoring, and potential pitfalls, are relevant to considerations of classroom instruction, especially active learning environments in which instructors have frequent, though short, interactions with individual students and small groups. In addition, faculty office hours are often sequences of tutoring sessions. Occasionally I’ve had the sense that a meeting with a student didn’t go well because I said too much or corrected an interesting mistake too soon. The research seems to confirm my impressions.
Still, tutoring is different from classroom instruction in significant ways. Most obviously, perhaps, tutoring usually happens when someone determines that special intervention is required. A student is struggling, or not doing as well as expected. Perhaps the student’s parents see tutoring as a way to improve grades or test scores for college applications. Under these conditions, it is especially important for the tutor to attend to the student’s affective state.
Additionally, although the appeal of tutoring as a remedy springs from the one-on-one nature of tutoring sessions, there are usually other people on the periphery. There’s the classroom instructor, who may have recommended tutoring, or may not know that it is happening. Perhaps the student’s parents are involved. Many school districts coordinate tutoring programs in cooperation with local organizations. It seems reasonable to conclude that communication challenges come along with those added relationships.
For one thing, the tutor may not know or understand the instructor’s learning objectives for the student. A peer tutor for my Calculus I students may have taken AP Calculus in high school, which can be a very different course from mine. A volunteer tutor in a public school might remember shortcuts for working with fractions, while the teacher wants to Nix the Tricks.
Further, the tutor might not have a deep understanding of the relevant mathematical content. As a sophomore in college, I signed up to be a peer tutor. A junior came to me for help with multivariable calculus. She was baffled by parametric curves, which hadn’t been covered in my multivariable course the previous year. At the time I was mortified, feeling somehow that I’d failed personally. But she and I tried to work through that section of the textbook together, which (I now recognize) was probably good for both of us. According to [1], “tutors in …same-age and cross-age collaborations tend to learn more than the tutees” (p. 412). It’s probably important that I knew what I didn’t know about parametric curves. In contrast, a colleague once overheard a peer tutor say, “the individual terms of the series go to zero, so it has to converge” in our department common room. Fortunately, our peer tutors now undergo appropriate training before they start.
Can I recommend a good math tutor? Yes, but I would want that tutor to get training first. It wouldn’t hurt to also read [1] and [2]! (Other resource suggestions are welcome in the comments.) Good tutors know that showing and telling should be used sparingly, and only after careful listening.
Thanks to Steve Klee for directing me to [2].
REFERENCES
[1] Graesser, A. C., D’Mello, S., & Cade, W. (2011). Instruction based on tutoring. Handbook of research on learning and instruction, 408-426.
[2] Lepper, M. R., & Woolverton, M. (2002). The wisdom of practice: Lessons learned from the study of highly effective tutors. Improving academic achievement: Impact of psychological factors on education, 135-158.
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I’ve recently finished my third year as an assistant professor in the mathematics department at Colorado State University. Since my research area is mathematics education, I am often asked what it is like to be a math-ed researcher in a math department. Such curiosity points to a cultural difference between mathematicians and mathematics-education researchers, and alludes to a specific culture where it may be difficult to be an education researcher in a mathematics department. To me, this question sometimes feels akin to being asked what it is like to work at Hogwarts as a Muggle, surrounded by real witches and wizards. Certainly, this comparison carries with it some information about how I perceive the question: that mathematicians are the real researchers, and that as a mathematics-education researcher I am lurking in their world. While this may be how I hear the question, it is very far from my experience in my math department with my colleagues. There are about 30 faculty in my department and three of us are active mathematics-education researchers. I have had overwhelmingly positive interactions in my department and feel valued as a teacher and as a researcher. When asked how I have had such a positive experience in my department (i.e. how I have gained acceptance at Hogwarts by the wizards and witches), my answer is both that my colleagues are just great people and that we have good relationships because we have gotten to know each other and each other’s work through conversations rooted in curiosity. I think it’s been valuable that we respect each other both as people and as researchers. In this blog post, I want to share some of the substance of what I have shared with them about mathematics education research.
Overview of mathematics education research
Mathematics education research is the systematic study of the teaching and learning of mathematics. This means that the types of questions we ask are about how people (students of all ages, non-students of all ages) think about and do mathematics, and about how people teach, why they teach that way, and what students might learn from that instruction. We use quantitative (typically survey based) and qualitative (typically interviews and observations) research methods to answer these questions. By its nature, math-ed research is extremely broad. All sorts of people think about and do mathematics – including school children, college students, graduate students, nurses, research mathematicians, mathematics teachers, food card vendors, etc. So when we ask questions about how people learn mathematics, we can attend to different conceptions of learning (I’ll say more about this below), different populations of people, different types of mathematics, where the people are thinking about or doing the mathematics, how their experience with mathematics relates to other things, and more. When we ask questions about the teaching of mathematics, we can also attend to the components above, as well as different ways that we can teach effectively and how teaching is impacted by internal and external factors.
Flavors of Math-Ed Research
There are a number of ways of categorizing different types of math-ed research. I’ll go over a few, specifically: topic of focus, pure versus applied, and two different strands emphasizing the post-secondary level.
Topic of Focus
The most obvious and common ways to split up different areas within math-ed research are by population of students/learners (elementary, secondary, post-secondary, future teachers, in-service teachers, graduate students, mathematicians, etc.) and by content area (geometry, algebra, calculus, etc.) Thus, it is common to describe a math-ed researcher as: “She studies proof and reasoning of students across age levels” or “He studies how teachers understand proportions and fractions.” While these are overly simplified versions of how we might actually describe these two specific math-ed researchers, it illustrates my point.
Pure and Applied
Just like mathematics researchers differentiate between research conducted without any practical end-use in mind and research conducted in order to solve a specific problem, math-ed research also has pure and applied flavors. While it may be easier to make someone’s pure math-ed research applicable because it is focused on education, there is certainly an abundance of math-ed research done without an intended concrete application. Alan Schoenfeld (2000) delineates pure and applied math-ed research by their goals: Pure math-ed research is done in order “to understand the nature of mathematical thinking, teaching, and learning” and applied math-ed research is done in order “to use such understandings to improve mathematics instruction” (p. 641). Often in math-ed research, pure investigations quickly become relevant and other researchers are able to directly leverage such work in concrete settings.
Pure math-ed research may look at topics such as the cognitive structures people hold and develop surrounding calculus (e.g. Pat Thompson’s work). Applied math-ed research, on the other hand, is more directly focused on how to improve mathematics instruction. I consider much of the MAA Calculus Project’s work to fall under this category – we have focused on investigating what makes a calculus program especially good for students and how to support other mathemaics departments to improve their programs. Often I find that pure math-ed research relies and extends theory (this will be explained more below) much more than applied work.
RUME and SoTL
One subfield within applied math-ed research comes from college mathematics faculty who do scholarly work around their teaching, called Scholarship of Teaching and Learning (SoTL). This contrast the Research in Undergraduate Mathematics (RUME) community, which is the primary academic home for mathematics education researchers (pure and applied) who focus our work on undergraduate mathematics, undergraduate mathematics students, teachers of undergraduate mathematics, or undergraduate mathematics programs. SoTL is a community of academics of different disciplines who are interested in scholarly inquiry into their own teaching of their discipline. In mathematics, this community is primarily populated by mathematicians who engage in scholarship related to college level mathematics. Since both communities use scholarly principals to investigate the teaching and learning of undergraduate mathematics, there are many overlaps between questions of interest. However, there are also some key differences. Curtis Bennett and Jacqueline Dewer, prominent leader in the mathematics SoTL community, describe the differences between “teaching tips”, SoTL, and RUME as follows:
Teaching tips refers to a description of a teaching method or innovation that an instructor reports having tried “successfully” and that the students “liked.” If the instructor begins to systematically gather evidence from students about what, if any, cognitive or affective effects the method had on their learning, she is moving toward scholarship of teaching and learning. When this evidence is sufficient to draw conclusion, and those conclusions are situated in the literature, peer reviewed, and made public, the instructor has produced a piece of SoTL work…. Mathematics education research or RUME is more in line with Boyer’s “scholarship of discovery” wherein research methodologies, theoretical frameworks, empirical studies, and reproducible results would command greater importance. This naturally influences the questions asked or considered worth asking, the methods used to investigate them, and what the community accepts as valid. (Bennet & Dewer, 2012, pp.461).
To carry their progression of the teacher’s description of a good teaching innovation toward her production of SoTL work onto a RUME study, I will put this in the context of teaching proofs. A nice example of a teaching tip related to proofs is found on a blog post called “How to teach someone how to prove something,” where the author describes that when she teaches proof she has asked “each student to give a presentation to the class on some proof they particularly enjoyed, and I sat through a preview of their presentation and gave them extensive advice on board work and eye contact.” She says that though this took a lot of work on her end, it was beneficial for the students, claiming that it “really helped them prepare and also boosted their egos while at the same time increased their sympathy with each other and with me.” The author shares a teaching approach with an (unsubstantiated) claim about how this positively affected her students. Both SoTL scholars and RUME researchers would agree that this claim is unsubstantiated because she did not collect data (either from her classroom or others) to support it.
Suppose this same teacher wanted to provide some evidence for this claim that may convince others that her approach is beneficial. She may survey her students’ mathematical confidence before and after the class, and interview them to understand the role of the classroom presentations on their confidence. She could write a paper describing her approach and her findings, connect her work to other literature, and submit this work to a SoTL outlet (such as PRIMUS). The result may look similar to Robert Talbert’s 2015 PRIMUS publication describing the benefits of inverting the transition to proof class based the author’s personal reflections as the teacher of the course and responses to a questionnaire about the class from about 30 of the 100 students in class. One of the authors’ conclusions from this work was that a student-centered introduction to proof course shows promise for “helping students emerge as competent, confident, self-regulating learners”.
If this teacher then wanted to pursue this work in a way more aligned with RUME work, she would have to identify a specific research question. In RUME, the research question is a necessary component of the work to identify the scope of the question and ensure that the research methods are aligned with the research question, and that the results answer the question. One such question that would explore the role of proof on students’ beliefs could be: “What are undergraduate students’ beliefs about the nature of proof, about themselves as learners of proof, and about the teaching of proof?”, and explore the question on a scale larger than her own classroom. Such a research question partially guided the work of Despina A. Stylianou, Maria L. Blanton, and Ourania Rotou in their 2015 publication in the International Journal of Research in Undergraduate Mathematics Education. To answer their research questions, the authors surveyed 535 early undergraduate students from six universities and then conducted follow up written test and interviews with a subset of the students to better understand the survey results. One of the findings from this work was a strong positive relationship between students’ beliefs about the role of proof and with their views of themselves as learners.
The claims made by teaching tip, SoTL work, and RUME work all shed light on the positive relationship with engaging in mathematical proofs and students’ beliefs about themselves. The difference is the audience of the claims, the degree to which the argument may convince others of the claims’ validity, and the role of theory in the arguments.
Role of Theory
Since mathematics education researchers are concerned with the teaching and learning of mathematics, math-ed research draws on theories of learning, often from psychology. In pure math-ed research, this theory is often made very explicit, such as in Pat Thompson’s work where he draws very explicitly on Jean Piaget’s constructivist perspective. In applied math-ed research, such as the work through the MAA’s Calculus projects (that I am involved with), the theory may be more implicit in the work, meaning that it is not at the forefront of the work but that there is an underlying theory guiding the work. In SoTL work, there is often no implicitly or explicitly mentioned guiding theory of learning. This is mostly due to a combination of differences in expectations and goals of SoTL versus mathematics education research. To wrap up this post, I will give a (very!) brief overview this idea.
A theory of learning is an explanation of how people learn – observe that this is subtle, as other ways of phrasing this sentence carry with them different assumptions about learning.
Based on which explanation of how people learn a researcher subscribes to, they will ask different research questions and answer these questions using different approaches. For instance, suppose a researcher were interested in exploring student learning of derivatives. Taking the first approach to learning (i.e. drawing on an acquisition metaphor), a math-ed researcher may create a research study to investigate “How much do students learn about derivative in teaching approach A?” To answer this question, the researcher could develop a test with a number of derivative questions and administer the same test to students at the beginning and end of the class and compare the results. This approach assumes that what students have learned in the setting of the classroom is carried with them into a testing situation and how they do on the exam is indicative of what they know. If, instead, the researcher draws on the second approach to learning (a constructivist approach), then she may ask the question “What are different student conceptions of derivative?”. To answer this question, she may create a think-aloud interview where the students are filmed or recorded working on various problems about derivative, and asked to explain how they are thinking about the problems. This approach assumes that an interview setting can recreate a situation where students can access and share how they make sense of derivative. Lastly, if the researcher ascribes to the third perspective of learning (a participation-oriented approach), then the question asked may be “How do students use their understanding of derivative in mechanical engineering classes?”. This question could be answered by observing student interactions in the classroom as they work in groups on problems that rely on the derivative. This approach assumes that it does not make sense to decontextualize student thinking from the real learning environment.
Conclusions
This brief introduction to mathematics education research was written to shed some light on the aspects of math-ed research that are often of interest to mathematics researchers, from my perspective as a Muggle in a Math department. If you have more questions about what we do – ask us! For more information, check out the SIGMAA on RUME page which has more information on publication venues and conferences related to RUME. This book about SoTL has more information about the community, and much more information can be found online. Lastly, both RUME and SoTL sessions appear at the Joint Meetings, which are great ways to get a small taste of this work. In closing, please enjoy these musings about Muggles:
]]>“Muggles have garden gnomes, too, you know,” Harry told Ron as they crossed the lawn… “Yeah, I’ve seen those things they think are gnomes,” said Ron, bent double with his head in a peony bush, “like fat little Santa Clauses with fishing rods…”
― J.K. Rowling, Harry Potter and the Chamber of Secrets“The wizards represent all that the true ‘muggle’ most fears: They are plainly outcasts and comfortable with being so. Nothing is more unnerving to the truly conventional than the unashamed misfit!”
― J.K. Rowling, Salon 1999
For several years I’ve been incorporating active-learning and inquiry-based learning activities in my teaching. There is ample documented evidence of the benefits of these approaches for students, but equally as important, they make teaching and learning more fun! Shifting class time from lecturing to having students work on problems, present their solutions to the class, and explain answers to each other has a dramatic effect: students become more engaged, learn communication skills, and gain confidence. These soft skills are in high demand in the job market. In this article, I will describe my use of these approaches and my experience teaching in a classroom designed for collaborative learning.
So far I’ve mostly been doing these active learning activities in traditional classrooms, but for smaller classes of about 25 students I’ve used collaborative classrooms with great success. The main difference between a “traditional classroom” and a “collaborative classroom” are (A) the seating arrangement; and (B) the presence of integrated technology. In a collaborative classroom, students usually sit around tables, often facing each other, which facilitates working in small groups. Many collaborative classrooms do not have an obvious “central location” where the instructor can stand, so teaching in such as classroom requires getting used to (see picture below). The main hesitation I had with using a collaborative classroom is this lack of a central location from which to lecture. I normally don’t use slides when lecturing so I wanted a way of emulating writing on a blackboard. I used a tablet computer with writing software to project what I would write on overhead screens. It ended up working very well. Students took notes as I wrote them, and I made the notes available to them after class. As can be seen from the pictures, collaborative classrooms tend to have many screens so students can see at least one of them easily.
As instructors, we are aware that “traditional classrooms” can come with different seating arrangements. Some have individual desks that one can move around, some have tables that are fixed and chairs that can be moved, some have multiple tables and chairs that cannot be moved, and some have a typical auditorium setting. I have taught in all of these types of classrooms and I have tried to incorporate active learning techniques with different degrees of success. It is significantly harder to have students work on a problem collaboratively if they can’t really face each other in a natural way. Collaborative classrooms, on the other hand, are designed to foster discussion by having multiple tables where one can move chairs as needed. For the particular classroom I was using, the tables were distributed in such a way that it made it easy for the instructor and the teaching staff (composed mostly of undergraduate students who had done well in the class in prior semesters) to circulate in the room to answer questions and address students.
Many of these collaborative classrooms also have multiple screens where the instructor can project information in a way that all students can see easily, without rearranging the way they are seated. So, a collaborative classroom accomplishes two goals: it allows students to work in groups, thus allowing the teaching staff easy access to every student, and allows for multiple displays so that the entire class has an easy view of what the instructor is projecting. There is no need to rearrange the seating every time one transitions from group-work time to “instruction” time and back.
This past summer I had the opportunity to teach in a collaborative classroom for a larger class of 59 students. This class was a proof-based introductory discrete mathematics course that emphasized logic, proof techniques, and both oral and written communication of mathematical ideas. The class did better overall than the same class in the regular semester. I was happy about how things went, and I decided to share my experience in case other instructors are considering utilizing more collaborative approaches to teaching. To take advantage of the collaborative space, I incorporated the following components.
Course staff helped me answer questions while students worked during class. To make this process work, it was important to have more than one teaching staff member in the classroom. To accomplish this, I recruited a few undergraduate students who had taken the class previously and had done exceptionally well. When it was time to work on the worksheet problems, we had about five people walking around (one instructor and four undergraduate instructors), answering questions, and talking to the students about the class material. These undergraduate instructors also held office hours, so we ended up having about 13 office hours every week.
Choosing the right undergraduate instructors is extremely important. I selected students who I knew could do the job, understood the material reasonably well, and were able to express mathematical ideas. Seeing them work with students was also a rewarding experience, as I was able to notice a significant improvement in their mathematical ability since they had taken the class. There is no better way to learn a topic than to teach it! We also had graduate assistants, who were in charge of grading homework, but in my experience undergraduate instructors do an excellent job understanding student questions, even if they are not perfectly formulated. There is something about talking to a peer that makes everyone, student and teaching assistant, more comfortable.
Reading quizzes, both individual and team-based. The idea of these reading quizzes comes from team-based learning (TBL), where instructors assign a reading before class, and at the beginning of the class they give an individual quiz (referred to as an individual readiness assessment test, or iRAT) and a team-based quiz (referred to as team readiness assessment test, or tRAT). Both the iRAT and tRAT for a given day have the same questions. At the beginning of the term, students were placed in teams according to a brief survey asking them about their level of comfort with teamwork as well as with logical and mathematical thinking. Then groups of four were formed according to their answers in such a way as to have “balanced teams.” These teams were used for the team quizzes and in-class work. For the reading quizzes, I assigned a specific section from the textbook for each class, and then gave a quiz on that section before it was officially covered in class. For many students, the idea of being asked questions before seeing a topic in class is preposterous. Nonetheless, reading comprehension is an important skill to develop. So that it wouldn’t greatly affect the students’ grades, the topics were carefully chosen and these quizzes didn’t count for a large portion of the final grade (but did count for something, as otherwise students might not be motivated enough to do the reading). The students would first do the quiz individually, and then would get together in their teams and work on the team quiz. Not surprisingly, students did better in the team version of the quiz than in the individual version. I witnessed many spirited discussions as members of the same team were choosing their answers: students were indeed teaching each other!
Worksheets containing a summary of the major concepts for a given class, along with problems to test student knowledge. I prepared a worksheet for every class that included the basic definitions, and then several problems for students to work on. Students were given time to work on the problems while the teaching staff walked around, answered questions, and discussed the problems with students (without giving them the answers). Most of lecture time was spent clarifying concepts from the reading, and providing examples that would inevitably bring more questions. But I tried to avoid talking continuously for more than 10 minutes and would provide several “breaks” where students would work on the problems provided in the worksheets.
Opportunities for students to explain their work to others. After several students had worked on a problem, we selected someone to present the solution to the rest of the class. We utilized a document camera to project students’ work on an overhead screen, and had the student walk us through their solutions. Sometimes the instructor or other students would ask questions. I would often compare the work of multiple students, which was a great way to highlight the fact that there are multiple correct ways of solving a problem or proving a proposition. I would also show work that wasn’t quite complete and correct, but without revealing the student who had made the work, and I would ask the class how to fix the mistake or how to complete the problem.
Overall, teaching in a collaborative classroom was a great experience. I will be politely requesting these kinds of rooms to the powers-that-be for all my future classes!
]]>Every university instructor would be thrilled if their students came to their mathematics classes with the ability to make viable arguments and to critique the reasoning of others; if their inclination were
But how do students develop these mathematical practices? The foundation is laid during a student’s 13 years of mathematics classes in K-12 – learning from their teachers and engaging in mathematics with their peers. The eight Mathematical Practice Standards that are an integral part of the Common Core State Standards (CCSS) for Mathematics, have elevated the importance and visibility of productive mathematical habits of mind in K-12 education. It is now an expectation and not a bonus. But are teachers equipped to help their students develop the practices until they become habits? Do teachers even have productive mathematical habits of minds themselves?
We actually know quite a bit about pre-service teachers’ habits of mind from research (Karen King: Because I love mathematics, Mathfest 2012, Madison). For example, pre-service teachers who hold mathematics degrees have an inclination to first state rules (Floden & Maniketti, 2005). They are not in the habit to seek meaning, which is such an important mathematical habit of mind. We can think of habits as acquired actions that we have practiced so much, that we eventually do them without thinking. At first, they are deliberately chosen but at some point they become automatic.
This has important implications for teaching at the university level, especially for pre-service teachers. Many professors and policy makers assume that completing a major in mathematics builds some kind of maturity. Undergraduate courses should be an opportunity to further refine productive mathematical habits of mind. Instead, this coursework often appears to reinforce unproductive habits of mind for engaging in mathematical practice. So I think we college/university faculty should take a serious look at what we are doing in our classes—not just in specific classes for future teachers, but in all our math classes. Mathematics faculty have a tendency to assign responsibility for K-12 math teacher quality to math education courses. But let’s think about that for a moment. In California, future high school teachers take 4 credit hours of math methods courses in their credential program. If they are lucky, they take at most a handful of courses as part of a math major specifically designed for future teachers, maybe 6 more credit hours. And they complete about 40 credit hours of mathematics content courses that are part of the normal mathematics degree programs. If they don’t learn productive mathematical habits of mind from their professors in their 10 or more college math courses, then who is responsible for this?
This is our responsibility and our opportunity! Pre-service teachers come to college with already formed ideas of what mathematics is and how the game of mathematics is played. They have already developed mathematical habits of mind—for good or for bad. It is up to us to help them replace unhelpful habits and develop productive habits, and we have approximately 4 years to do it.
When we are trying to change habits and practices, we often focus on directly changing actions and we hope this will lead to better results. In this case, we want teachers to change their teaching practice so that all students will develop productive mathematical habits of mind. But actions are affected by beliefs and beliefs are based on experiences. So it would be much more productive for us to provide pre-service teachers (and all students) with a series of compelling and positive experiences to change their beliefs. This, in turn, will lead to more coherent, consistent, grounded, and therefore stable results.
In my work with in-service teachers around transitioning to the CCSS, we have explored a variety of productive pedagogical ideas that provide students with experiences where they engage in mathematical practices. I have adopted several into my college classroom to better prepare my students for their work as teachers but also because I think this is simply good teaching for everybody. I’ll give two examples that focus on “Make a viable argument and critique the reasoning of others”.
Gallery Walks
In many of our courses, students write proofs; this is a mathematician’s idea of a viable argument. How do students learn how to write a proof? What are characteristics of a good proof? How do you critique other people’s arguments? On the first day of my combinatorics and graph theory class we worked on the following problem:
Students first collaborated on the problem in groups of 3—4. After students solved the problem, they made a poster to explain how they found their solution and how they knew that they had found all solutions. We then did a gallery walk: With a stack of sticky-notes in hand, students studied each poster. They asked questions about parts that they did not understand and they made suggestions when they found something that could be improved. They also pointed out aspects of the posters they found helpful in understanding the argument.
(sample posters with sticky note feedback)
Next, students went back to their own posters and studied the feedback they had received. They discussed revisions, and for homework each student individually wrote up an improved version of their proofs.
Before we finished the class, we had a discussion about the purpose of this activity. Students were surprised about the variety of proofs they had seen. After reading each other’s solutions, they were able to decide if there were gaps in arguments and describe what made a proof easy to read. They saw that there are a variety of ways to structure the argument, that a complete proof is not necessarily a good proof, and that a “proof by example” is not a proof but could possibly be revised into a general proof. They recognized the value of their peers’ feedback; and that they did not need the instructor to validate their proofs—rather, they possessed the mathematical authority to do so themselves.
You may ask: Our students write proofs and have to show their work all the time, why is this activity useful? In this case, it set the tone for the semester, and it made expectations clear to the students. Aside from seeing that they would be expected to actively work with their peers in class, they also experienced giving feedback and then using feedback to revise their work. They learned that an important goal of mathematics is communicating solutions, not just getting answers, and for the future teachers in the room, they saw a pedagogical structure they can use at any grade level and in any subject.
I do variations of the gallery walk in most of my classes a few times each semester. It works with modeling problems in calculus just as well as with proofs in real analysis.
Re-engagement Lessons
Every instructor knows the following situation very well: Students have done a task. You assess it. There are major gaps. What do you do? You could
I want to describe another option: re-engage the students with the task and the concepts, using their responses to move everybody forward.
While learning how to write proofs involving the algebra of sets in my “Intro to Proofs” class, students did the following standard problem on a homework assignment: Given sets A and B, prove that A U (B – A) = A U B. While grading the homework, I found myself writing the same comments over and over again: “Pick a point,” “double set inclusion,” etc. I decided to use the proofs that students had written as the basis for the next day’s activity. To prepare, I compiled a collection of students’ proofs. In class, I handed out copies of these proofs to pairs of students. I asked them to discuss:
Then we had a whole class discussion, keeping track on a document camera of changes students suggested.
Why was this activity better than just going over the proof again on the board or doing a similar problem, which would certainly have been faster?
By using a compilation of actual student work, students were invested in the exercise from the start. They already had engaged with this problem, so even if they had not written a perfect proof, they had a basis to build on. The examples I chose included good and bad features of proofs. The contrast and repetition allowed the students to transfer ideas from one to the other. The setup of the activity allowed students at every level to engage and benefit. One of my top students told me after the class that he had learned a lot about reading and critiquing others’ work. Finally, by contrasting several proofs, we had an excellent discussion about the structure of proofs, not just small details.
Research is compelling that students learn more from making and then confronting mistakes than from avoiding them (Boaler, 2016). My goal as a teacher is shifting from providing clear explanations so students don’t make mistakes, to creating situations, which are likely to produce important mistakes, and then helping the entire class confront and learn from those mistakes. Re-engagement lessons are a great method for this confrontation.
This is just one example of a re-engagement lesson. David Foster from the Silicon Valley Math Project contrasts re-teaching and re-engagement:
Re-teaching | Re-engagement |
Teach the unit again. | Revisit student thinking. |
Address basic skills that are missing. | Address conceptual understanding. |
Do the same or similar problems over. | Examine task from different perspective. |
Practice more to make sure student learn the procedures | Critique student approaches/solutions to make connections |
Focus mostly on underachievers. | The entire class is engaged in the math. |
Cognitive level is usually lower. | Cognitive level is usually higher. |
(Foster & Poppers, 2009)
I offer the two classroom activities as examples to help us start talking about changing the mathematics culture in our classrooms and schools so that all students, including future teachers, have experiences that support them in forming productive mathematical habits of mind.
To educate our students to become mathematicians and teachers we have to do more than role-model mathematical practices, we have to create the environment where students engage in them, and we have to talk more about what we are doing and why. We have 4 years to help our students replace bad mathematical habits (speed, answer-getting, anxiety) with productive ones (sense-making, perseverance, use of tools and structure). This is our responsibility, but maybe even more importantly, this is our opportunity.
References:
Boaler, J., & Dweck, C. S. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching.
Connors, R., & Smith, T. (2012). Change the culture, change the game: The breakthrough strategy for energizing your organization and creating accountability for results. [Also https://www.partnersinleadership.com/insights-publications/changing-your-culture/]
Common Core State Standards: http://www.corestandards.org/Math/Practice/
Floden, R., and Meniketti, M. (2005). Research on the effects of coursework in the arts and sciences and in the foundations of education. In M. Cochran-Smith and K. Zeichner (Eds.), Studying teacher education: The report of the AERA panel on research and teacher education. Mahwah, NJ: Lawrence Erlbaum Associates
Foster, D. and Poppers, A. (2009). Using Formative Assessment to Drive Learning: http://www.svmimac.org/images/Using_Formative_Assessment_to_Drive_Learning_Reduced.pdf
]]>“I am so glad you made that mistake,” I’ve come to realize, is one of the most important things I say to my students.
When I first started using inquiry-based learning (IBL) teaching methods, I had a tough time creating an atmosphere where students felt comfortable getting up in front of class and presenting their work. It is a natural human instinct to not want to expose your weaknesses in front of others. Making a mistake while presenting the solution to a problem at the board is a huge potential source of embarrassment and shame, and hence also anxiety. So how do we—as educators who understand the critical importance in the learning process of making and learning from mistakes—diminish the fear of public failure in our students? For me, the answer involves persistent encouragement. It also relies on setting the right tone on the first day of class.
To prepare my students on Day One of class, I talk about the importance of making and learning from mistakes. I often refer to one of my favorite books on this subject, The Talent Code by Daniel Coyle [1]. Coyle has studied several hotbeds of “genius,” places where an unreasonable number of virtuosos—e.g., world-famous violinists, baseball players, and writers of fiction—emerge. He is interested in discovering just how people like Charlotte Brontë, Pelé, and Michelangelo learn to perform at the top of their fields. The answer involves a simple idea: talented people are those who have made far more mistakes than others and who have deliberately learned from those mistakes. For my students, the takeaway is that the most accomplished people have made many more mistakes than the average person. Consequently, it is of high value for us to make our mistakes public and discover how to correct them together. (As a side note, Francis Su employs the same strategy in his article “The Value of Struggle” [2].)
After the first day of class, whether I am teaching Quantitative Reasoning, Calculus, or a more advanced course such as Introduction to Knot Theory, nearly every class period begins with presentations of homework problems by student volunteers. Students have homework due each day, and they are required to present problems a certain number of times during the term. The number of problems we do depends on how long the class period is, how complex the problems are, and what I need to teach in the remainder of class. In a course like Introduction to Knot Theory, we might spend 45 minutes or an hour on student presentations, while we will spend 20-30 minutes on calculus homework presentations in an 85-minute class period. This general structure could be modified to fit shorter class periods or weekly recitation sections at universities with larger lecture courses. For instance, we used to teach calculus classes four days a week in 50-minute blocks at Seattle University. Within this structure, I had a weekly “Problem Day” for my calculus classes instead of having daily student presentations of homework. After students volunteer to present problems at the board on a typical class day, all students who are chosen to present simultaneously write up problem solutions while their classmates review the homework or work on another activity. Once all solutions have been written up, we reconvene; one by one, students come to the board to walk us through their solution. This is where supportive facilitation becomes critical.
Encouraging students to make mistakes in the abstract—as I do one Day One—is one thing, but helping students accept their mistakes in front of class is quite another. This is where my new catch phrase comes in. Let’s say, for example, a student is computing the derivative of \(y=x^2\sin x\) at the board and writes \(y’=2x\cos x\). I might say, “I am so glad you made that mistake! You’ve just made one of the most common mistakes I’ve seen on this type of problem, so it’s worth us spending some time talking about. Can anyone point out what the mistake is?” If someone in the class comments that the presenter should have used the Product Rule, I might follow up with, “That’s a good idea. How can we see that this function is a product? Let’s work together to break the problem down into pieces.” Going forward, I facilitate the process of the class coming up with their collective correction of the mistake. Collaboratively working to correct mistakes like this tends to help students observe more subtle differences between different types of problems while building a more sophisticated mental problem-solving framework.
Making and correcting mistakes together can also help address more basic misconceptions. Suppose a student—let’s call them Riley—writes, in the middle of a calculus problem, a line like the following.
\(1/(x+x^2) = 1/x + 1/x^2\)
This mistake will most likely lead to an incorrect final answer. Many of the presenter’s classmates will discover the final answer is wrong, and some will even be able to pinpoint where the computation went awry. How would I address this? Once a classmate has identified the problem, I might say, “Riley, I’m so glad you made that mistake! This is one of the most common algebraic mistakes students make in calculus—I’m willing to bet others in the class made this same exact mistake, so it’ll be really helpful for us to talk about it together. This is a question for anyone in the class: How can we prove that this equality doesn’t hold, in general?” Suppose a student, Dana, in the audience suggests we try plugging in some numbers to see what happens. I’d follow up with, “Riley, could you be a scribe for this part of the discussion? Please write up Dana’s suggestion beside your work. Dana, can you tell Riley exactly what to write?” Once we’ve cleared up the confusion with Riley’s algebra, I might ask them to work through the rest of their problem again at the board, fixing their work accordingly. On the other hand, if Riley appears to be too shaken or confused to fix the rest of the problem or if the actual problem was much more complex than the one that resulted from the algebraic error, I might ask the class to collectively help Riley figure out what to write each step of the way. A third option I frequently use is the “phone a friend” option. I could see if Riley wants to “phone a friend” in the class to dictate a correct answer.
Mistakes can be common in class presentations, but I occasionally have a class that is so risk-averse that very few people offer to present their work unless they know it’s perfect. If I have too many correct solutions presented, but I know some in the class are struggling, I might follow up with a comment like: “That was perfect! Too bad there were no mistakes in your work for us to learn more from. I’d like to hear from someone who tried a method for solving this problem that didn’t work out so well. Would anyone be willing to share something they tried with the class?” At this point, someone may come forward with another (incorrect, or partially correct) way to attempt the problem. If nobody comes forward, I could offer a common wrong way to do the problem and ask my students to identify the misunderstanding revealed by my “solution.” I might even tell a little white lie and say something like, “When I first learned this concept, I had a lot of trouble understanding it. I made the following mistake all the time before I figured out why I was confused.” Alternatively, I could mention, “The last time I taught this class, someone made the following mistake. What’s wrong with this approach to solving this problem?”
Now, let’s say one of my students has just presented a problem at the board. Perhaps they made a mistake, or perhaps they did everything perfectly. What happens next? I will ask the class, “Any questions, comments, or compliments?” The request for compliments is one of the most important parts of this solicitation of feedback. It is so important that, during the first several weeks of class, I make my students give each presenter at least one compliment. Some of the best compliments I’ve heard from students follow some of the worst presentations. For instance, after a disastrous presentation where the presenter appeared clueless and needed their peers to help them complete all parts of a problem, a student of mine once observed, “That took a lot of guts to get up there and make mistakes. I thought you did a great job fixing the solution and taking constructive criticism from us!” If nobody offers up such a supportive compliment after a bad presentation, I might give this feedback myself to publicly recognize the presenter’s courage. What’s more, if a student appears shaken by the experience of messing up so thoroughly, I’ll follow up again after class, reinforcing my appreciation for their bravery. Over time, this strategy helps build a supportive classroom environment.
Looking back on how my classes have evolved, I can see that it is difficult to convince students to be vulnerable in a math class without the three following elements:
(1) setting the stage by sharing my expectations of students making mistakes and being clear about the reasons for these expectations,
(2) encouraging students to help each other come to the right answer while recognizing the benefits of making specific mistakes, and
(3) acknowledging students’ willingness to make mistakes both publicly and privately.
We’ve been primarily focused on how to encourage students to make mistakes, but let’s turn our attention to why it might be important in our math classes. One thing that I found to be particularly striking when I started teaching this way was my students’ exam performance. I typically ask a mixture of conceptual and computational questions on exams. I was surprised to see how much more sophisticated students’ responses were to conceptual questions in courses where I spent a great deal of class time on student presentations. At first, this was surprising to me since we spent quite a lot of time in class working through computational problems. The more I reflected on this phenomenon, though, the more it made sense. The repairing of computational mistakes in class often led to a discussion of the more conceptual mathematics underlying the computations. What’s more, these discussions were sparked by students grappling with problems that they cared about—problems they had spent time outside of class trying to solve—and not simply problems they had just been introduced to in the course of a lecture. Discussion that takes place during a homework presentation session seems to stick with students in a way that a “discussion” (where the instructor is doing much of the talking) during a lecture does not.
There are myriad other benefits I’ve observed, including development of a tight-knit classroom community, increased student self-confidence, and more engaged student participation in all aspects of class. In short, I’m convinced. I’m all in. The benefits of teaching this way far outweigh the costs of redistributing precious class time, making room for students to publicly make and collaboratively fix their delightful mathematical mistakes.
References
[1] Coyle, Daniel. The Talent Code: Greatest Isn’t Born, It’s Grown, Here’s how. Bantam, 2009.
[2] Su, Francis. The Value of Struggle. MAA FOCUS. June/July 2016.
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