By the end of every workshop and conference session on Inquiry-Based Learning that I’ve attended, someone has raised a hand to ask about coverage. “Don’t you have to sacrifice coverage if you teach this way?” Of course coverage took center stage for many of my professional conversations long before I tested the IBL waters; it’s important. But an equally important question is this: What do we sacrifice when coverage dominates? It may well be conceptual understanding; it’s possible to cover more ground, albeit thinly, if we settle for procedural understanding instead. More than once I’ve settled for even less, delivering a quick lecture just so that my students will have “seen” a particular idea. How do we strike a balance between coverage and other considerations when we are so practiced at reducing a course description to a list of topics?
Strong arguments for striking that balance have been made elsewhere. For example, Stan Yoshinobu and Matthew Jones offer a close examination of the “price of coverage”. “Coverage versus depth” is a “false dichotomy,” they say; racing through material makes for a passive student experience, which affects student understanding of what it is to learn mathematics. “Implied messages are sent to students through classroom experiences,” and some of those messages may have unproductive consequences, including overreliance on mimicking the instructor and memorization, and significant difficulties with non-routine problems.
Is there, on the other hand, a price of demoting coverage? Does a more comprehensive view of student learning get in the way of content knowledge? Recent research done by Marina Kogan and Sandra Laursen, brought to my attention by Yoshinobu and David Bressoud, suggests that students don’t necessarily suffer, and may be helped, from a holistic approach. From the conclusion to the Kogan and Laursen paper:
College instructors using student-centered methods in the classroom are often called upon to provide evidence in support of the educational benefits of their approach—an irony, given that traditional lecture approaches have seldom undergone similar evidence-based scrutiny. Our study indicates that the benefits of active learning experiences may be lasting and significant for some student groups, with no harm done to others. Importantly, “covering” less material in inquiry-based sections had no negative effect on students’ later performance in the major. Evidence for increased persistence is seen among the high-achieving students whom many faculty members would most like to recruit and retain in their department.
Still, it’s often difficult to prevent concerns about coverage from hijacking day-to-day teaching practice, regardless of course format. Here are some approaches I am using to keep coverage in perspective.
Regard conceptual understanding, mathematical writing and speaking, and other learning goals as integral parts of the “coverage” list, on an equal par with specific topics. Yoshinobu points out that we have a “systemic” issue, in that our institutions define coverage as no more than the list of topics. Hence I have to make a conscious effort, in planning each course, to weave all of the goals together, and to recognize that procedural skills won’t last without conceptual understanding, which in turn won’t happen if students don’t routinely speak and write mathematics.
Include learning objectives, not just a topics list, on the syllabus. Whether or not all of my students read the syllabus, it’s my way of formalizing my intentions and expectations. It’s also an invitation to consider the course in its entirety. This is especially important in mathematics, where students don’t understand many of the terms in a catalog description until after they’ve taken the course.
Have conversations with students, early and often, about the learning goals for the course. On the first day of linear algebra this semester, I devoted the entire hour to a class activity adapted from a model offered by Dana Ernst. The students’ responses to “What are the goals of a liberal arts education?” included “critical thinking” and “to experience the freedom to explore.” To “What can you reasonably expect to remember from your courses in 20 years?” I heard, “NOT details or the stuff you’re tested on,” but rather “how to figure out what’s relevant.” My own students understand the big picture; surely I can keep it in mind!
Halfway through the term, I had my students read this blog post from Ben Orlin and then fill out a survey online. I asked: to what extent are you practicing in the Church of Learning, as opposed to the Church of the Right Answer? Once again, the students reinforced my choices. Many of them also noted that their pre-college experiences, especially Advanced Placement Calculus, leaned heavily toward the Right Answer doctrine. In at least some cases, I’m working against students’ most recent experience of mathematics learning, so I need to be persistently transparent.
Gather data frequently on student understanding. Formative assessment isn’t just for elementary school teachers. I’m fortunate to teach small classes, so I can learn a lot just from classroom conversations. In an earlier post, I explained how recent research on learning has influenced my teaching. If I hear someone struggling to use “linearly independent” accurately during small group work, I can offer corrective feedback immediately. My students often show their work using a document projector. Anonymous surveys are useful as well; it only takes a few minutes for students to write down what’s puzzling them at the moment. I’ve never used clickers, but I’m intrigued by Eric Mazur’s methods. Most importantly, I try to design homework assignments that ask for deeper understanding. (It takes several weeks to convince students that homework is for formative, not summative, assessment, and that the graders’ job is to give constructive feedback.)
Bring student graders and teaching assistants in on the plan. I handpicked my graders this term, and made it clear that I want homework solutions to be clear and well-written, not just correct. They know that I’ve encouraged the students to show their attempts and partial solutions to more challenging problems. They let me know what misconceptions they see. The student tutors are also aware of my intentions.
It may be that I am especially sensitive to questions about coverage because my semester includes only twelve weeks of classes. My department colleagues and I agree that this poses a particularly vexing challenge in multivariable calculus. Getting to Green’s Theorem is challenging enough, and a thorough treatment of Stokes’ Theorem, which would add coherence to the entire semester, seems a worthy goal. Yet even here, I remind myself, what’s important is not only what I cover; it’s also what the students can retain.
]]>Making fundamental changes to the way you teach is a difficult task. However, with a growing number of students leaving STEM majors, instructors’ dissatisfaction with student learning outcomes, and research indicating positive avenues for improving undergraduate mathematics instruction, some instructors are ready and eager to try something new. In this post, we describe some promising research-based curricular materials, briefly identify specific challenges associated with implementing these materials, and describe a recently funded NSF project aimed at addressing those challenges.
Teaching Inquiry-Oriented Mathematics: Establishing Supports (TIMES) is an NSF-funded project (NFS Awards: #143195, #1431641, #1431393) designed to study how we can support undergraduate instructors as they implement changes in their instruction. A pilot is currently being conducted with a small group of instructors. In the next two years, approximately 35 math instructors will be named TIMES fellows and will participate in the project as they change their teaching of differential equations, linear algebra, or abstract algebra. As project leaders, we will study how to best support these instructors, as well as how their instructional change affects student learning. More details about the project follow later in this blog post.
Inquiry-Oriented Instruction
The curricula we utilize in the project are each examples of inquiry-oriented instructional materials. Inquiry-oriented instruction is a specific type of student-centered instruction. Not surprisingly, different communities characterize inquiry in slightly different ways. In the inquiry-oriented approach we describe here, we adopt Rasmussen and Kwon’s (2007) characterization of inquiry, which applies to both student activity and to instructor activity. In this approach, students learn new mathematics by: engaging in cognitively demanding tasks that prompt exploration of important mathematical relationships and concepts; engaging in mathematical discussions; developing and testing conjectures; and explaining and justifying their thinking. Student inquiry serves two primary functions: (1) it enables students to learn new mathematics through engagement in genuine exploration and argumentation, and (2) it serves to empower learners to see themselves as capable of reinventing important mathematical ideas.
The goal of instructor inquiry into student thinking goes beyond merely assessing student’s answers as correct or incorrect. Instead, instructor inquiry seeks to reveal students’ intuitive and informal ways of reasoning, especially those that can serve as building blocks for more formal ways of reasoning. In order to support students, instructors routinely inquire into how their students are thinking about the concepts and procedures being developed. As instructors inquire into students’ emerging ideas, they facilitate and support the growth of students’ self-generated mathematical ideas and representations toward more formal or conventional ones. The instructor’s role is to guide and direct the mathematical activity of the students as they work on tasks by listening to students and using their reasoning to support the development of new conceptions. Additionally, instructors provide connections between students’ informal reasoning and more formal mathematics.
With an inquiry-oriented instructional approach, instructors use mathematically rich task sequences, small group work, and whole class discussions in order to elicit student thinking, build on student thinking, develop a shared understanding, and introduce formal language and notation.
Curricular Materials for Undergraduate Mathematics Education
The TIMES project is organized around three sets of post-calculus, research-based, inquiry-oriented curricular materials.
· Inquiry-Oriented Abstract Algebra (IOAA), developed by Sean Larsen under the NSF grant Teaching Abstract Algebra for Understanding (#0737299), http://www.web.pdx.edu/~slarsen/TAAFU/ (User:AMSBlog; Password:teacher). These materials are designed for an introductory group theory course and include units on groups and subgroups, isomorphisms, and quotient groups. Supplementary materials for rings/fields are available upon request.
· Inquiry-Oriented Linear Algebra (IOLA), developed by Megan Wawro, Michelle Zandieh, Chris Rasmussen, and colleagues under NSF grant numbers 0634074/0634099 and 1245673/1245796/1246083, http://iola.math.vt.edu (must request login & password). These materials are designed for an introductory linear algebra course and include four units on span, linear dependence and independence, transformations, and eigenvalues, eigenvectors, and change of basis. Tasks for determinants and systems are also available upon request.
· Inquiry-Oriented Differential Equations (IODE), developed by Chris Rasmussen and colleagues under NSF grant number 9875388, website coming soon. These materials are designed for a first course in differential equations and include the following topics: solving ODEs; numerical, analytic and graphical solution methods; solutions and spaces of solutions; linear systems; linearization; qualitative analysis of both ODEs and linear systems of ODEs; and structures of solution spaces.
For each of these three curricular innovations, the student materials have been developed through iterative stages of research and design supported by grants from the NSF. In the early stages of these respective projects, the developers carried out small-scale teaching experiments focused on uncovering students’ ways of reasoning and developing tasks that evoke and leverage productive ways of reasoning. Instructional tasks then went through additional cycles of implementing, testing, and refining over a series of whole class teaching experiments. In the last stages of research and design, instructors who were not involved in the development implemented the materials and provided feedback.
Over the course of the last 10+ years, these extensive and ongoing research projects have produced many results, including: instructional sequences comprised of rich problem-solving tasks, instructor support materials, research showing positive conceptual learning gains (e.g., Kwon, Rasmussen, & Allen, 2005; Larsen, Johnson, & Bartlo, 2013), insights into how students think about these concepts (e.g., Larsen, 2009; Wawro, 2014; Keene, 2007) and the identification of specific challenges that instructors face as they implemented these materials. Some of the difficulties experienced by instructors implementing the materials include: making sense of student thinking, planning for and leading productive whole class discussions, and building on students’ solution strategies and contributions (e.g., Johnson & Larsen, 2012; Speer & Wagner, 2009; Wagner, Speer, & Rossa, 2007).
TIMES Project
The TIMES grant will allow us to better understand how to support instructors as they work to implement these three inquiry-oriented curricula materials. We have a three-pronged instructional support model, consisting of:
(1) Curricular support materials – These materials, created by the researchers who developed the three curricular innovations, include: student materials (e.g., task sequences, handouts, problem banks) and instructor support materials (e.g., learning goals and rationales for the tasks, examples of student work, implementation notes).
(2) Summer workshops – The summer workshops last 2-3 days and have three main goals, 1) building familiarity with the curricula materials, including an understanding of the learning trajectories of the lessons; and 2) developing an understanding of the intent of the curricula in particular and inquiry-oriented instruction in general.
(3) Online instructor work groups – The online instructor work groups have between 4 and 6 participants, each currently implementing the same curricular materials. Each group meets for one hour a week and works on selected lessons from the curricular materials. For each of the focal lessons, we discuss the mathematics and plan for implementation. Then, after instructors have taught the lesson, the group watches video clips of instruction with a focus on student thinking. The goal is to help instructors develop their ability to interpret and respond to student thinking in ways that support student learning. Every meeting also has time dedicated to address specific and immediate needs of the participants (e.g., difficulty with managing small group work, a particularly challenging task, strategies for getting students to share ideas).
Over the course of this three-year grant, we will offer these supports and investigate their impact. Our research will focus on the relationships and interactions among the supports, the instructors, and their instructional practices. In addition to assessing the impact of the support model, project data will be analyzed to identify aspects of the supports and instruction that have a positive impact on students’ learning.
We hope that this post provided a useful description of the inquiry-oriented instructional approach that can help instructors think about how they might (or already do) incorporate some of these ideas into their teaching. For instance, regardless of how you currently teach, really inquiring into your students’ thinking (not just their answers) can provide you with very valuable insights. We also hope that, after reading this post, you will be encouraged to see that some tangible, practical steps are being taken toward scaling up and supporting inquiry-oriented instruction.
If you are interested in learning more about the curricular materials or this project please visit http://times.math.vt.edu/. If you are interested in learning more about becoming a TIMES fellow, please contact Estrella Johnson (strej@vt.edu) for Abstract Algebra, Christy Andrews-Larsen (cjlarson@fsu.edu) for Linear Algebra, or Karen Keene (kakeene@ncsu.edu) for Differential Equations. We are the principal investigators on the project and would be glad to hear from you if you are interested in learning more.
References
Johnson, E. M. S., & Larsen, S. (2012). Teacher listening: The role of knowledge of content and students. Journal of Mathematical Behavior, 31, 117 – 129.
Keene, K. A. (2007). A characterization of dynamic reasoning: Reasoning with time as parameter. The Journal of Mathematical Behavior, 26(3), 230-246.
Kwon, O. N., Rasmussen, C., & Allen, K. (2005). Students’ retention of mathematical knowledge and skills in differential equations. School Science and Mathematics, 105(5), 227-239.
Larsen, S. (2009). Reinventing the concepts of group and isomorphism: The case of Jessica and Sandra. The Journal of Mathematical Behavior, 28(2), 119-137.
Larsen, S., Johnson, E., & Bartlo, J. (2013). Designing and scaling up an innovation in abstract algebra. The Journal of Mathematical Behavior.
Rasmussen, C., & Kwon, O. N. (2007). An inquiry-oriented approach to undergraduate mathematics. The Journal of Mathematical Behavior, 26(3), 189-194.
Wawro, M. (2014). Student reasoning about the invertible matrix theorem in linear algebra. ZDM, 46(3), 389-406.
Speer, N. M., & Wagner, J. F. (2009). Knowledge Needed by a Teacher to Provide Analytic Scaffolding During Undergraduate Mathematics Classroom Discussions. Journal for Research in Mathematics Education, 40(5), 530-562.
Wagner, J. F., Speer, N. M., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician’s knowledge needed for teaching an inquiry-oriented differential equations course. The Journal of Mathematical Behavior,26(3), 247-266.
]]>When I first started teaching, I was mystified (and, frankly, at times panicked) at the thought of having undergraduates work with me on research. I realized this was part of the job, part of my institution’s mission, but I just couldn’t figure out how it would be effective. Sure, these students were bright, eager and motivated to learn, but how much could they contribute with such limited time? A typical research experience might be 8-10 weeks during the summer (full time) or 10 hours a week during a semester; best case, I might find a student who would work with me for a couple years in this way. I had just finished six years in grad school and still felt like I knew nothing. On top of that, my research is at the intersection of computer science and math with applications in the domains of engineering and biology – would I be able to find students with experience in even two of these fields? As it turns out, I would soon discover how powerful research with undergraduates can be, and I’d like to share some of the lessons I’ve learned over the years.
I remember thinking I should come up with a list of very specific problems, solvable with limited time and background, before trying to find students. Looking back, I think I was trying to mirror the familiar classroom experience, where a careful syllabus provides clear expectations to students with specified prerequisites. It turns out that specific problems, while useful in giving students an idea of the research area, almost never provide the direction we end up moving toward. I suppose I should have seen that coming, as research never does go as planned. It can’t be clean and predictable just because undergraduates are involved. I’ve learned to embrace the prospect of the unknown, instead looking for students with more broadly defined interests, such as computational biology or robotics.
Once I’ve found students, the most successful approach comes from guiding them along paths that suit their own passions and interests. Many times, they don’t really know these in advance, so I view the first part of my time with them as a chance to let them play with different types of problems. This may mean coming up with examples that fit a given set of combinatorial properties, reading and presenting a research paper on an algorithm we hope to generalize, or building modules of Mathematica code to explore properties of certain matrices. One summer, my students built little robots with microcontrollers and old VCR boxes; that activity resulted in one student determined to continue working on hardware and another determined to work only on software (a surprise to her). This “discovery period” can be truly transformative for some, and the reward of knowing that I helped a student find out a little more about herself is one of the main reasons I became a faculty member.
After identifying her interests, the student begins to get a glimpse of how research feels by facing the energizing and terrifying prospect of defining her own problems and pathway. I have the students pitch their own projects and timelines (which are always too ambitious) and work with them to create several milestones along the way. I usually let them start off on their overly optimistic timeline, but know they generally won’t make it past the first milestone. The students track their own progress by maintaining a website with blog updates on their work. This serves two purposes: (1) it helps me understand what they have done, what they understand and where they are stuck, and (2) it gives them something to reflect on at the end of the experience. Throughout this time, I am very conscious of each student’s confidence level. For some, the unfamiliarity of not having lectures, assignments and a textbook can cause them to doubt their own ability. Explicitly telling them that research is coupled with a feeling of the unknown and relating imposter syndrome stories of my own and of other researchers often gets them back on track. This is one of the things I enjoy the most, mentoring students who are excited to work on problems related to my research and helping them find the confidence to jump-start their own research careers. It is an amazing feeling when they tell me years later that it was that seed of a research experience which grow into their passion, whether it is pursuing a graduate degree with an NSF fellowship or becoming a teacher who will inspire new generations or working on cutting edge technology at an industry leader.
I used to worry that working with students on problems that interest them might be a distraction from my own research. I had, in earlier years, been asked by a local roboticist to help advise students on a project of his. I had no experience in robotics, but saw the excitement on the students’ faces. As the only faculty member positioned to co-advise, I knew that my saying “no” would crush their hopes. As I became more involved in their projects, helping to build a 3D printer from a kit, I became enamored with microcontrollers and the “maker” movement. At that point, it was just fun to build stuff and create an interactive project with a few lines of code; in my mind, it was completely decoupled from research. Then, two years ago, I began thinking about applying for an NSF grant, but was stumped as to what exciting research pathway I could propose. Serendipitously, a roboticist, whom I’d met through this robotics work, sent me a link to a TED talk with quadcopters cooperating to catch a ball in a net. A light bulb went off, and I saw a connection to the theoretical core of my research. I took a risk and proposed this robotics-based research program. To my delight and surprise, my proposal was funded!
This was not the only time a surprising connection came from working with a student. In fact, my first undergraduate researcher impacted my career in a way she may not even know. She sought me out one day as she was double-majoring in computer science and mathematics and had been told that my research straddled both. I felt completely unprepared as I had no list of specific problems; instead, I described my research on the fly, and my work in computational biology piqued her interest. She has since become a co-author with another undergraduate and two biochemists, but perhaps her most unexpected gift to me was a new collaboration, which I value deeply. As part of writing up her thesis, this student wanted to provide background on Lie groups, and she sought out the expertise of a mathematics professor. This professor saw a connection between the thesis work and the research area of another math professor. She encouraged us to start talking and thus began a collaboration for which I will be forever grateful.
As faculty at a research liberal arts institution, involving undergraduates in research is a core part of what we do. These budding researchers may not always be able to produce significant original contributions, but I can genuinely say my research path has been dramatically transformed for the better because of them. This is the biggest lesson I have learned: don’t underestimate the power of undergraduate researchers. They might directly contribute to your research, becoming co-authors on your next publication, or provide context and intuitions from things you’ve never thought about. And one day, those student interactions just might result in a connection that will transform who you are as a researcher. That connection could lead to a fantastic new collaborator or even a successful grant proposal. And, to top it all off, you get the amazing reward of knowing you played a small role in helping them in their own journey of discovering where to go next.
]]>I serve as chair of the Mathematical Association of America’s (MAA’s) Committee on the Undergraduate Program in Mathematics (CUPM). Approximately every ten years, CUPM publishes a new curriculum guide, with the primary goal of assisting mathematics departments with their undergraduate offerings. Over five years in the making, the 2015 Curriculum Guide to Majors in the Mathematical Sciences encourages departments to engage in a process of review and renewal, by examining their own beliefs, interests, resources, mission, and particularly their own students in designing or revising a major in mathematics or, more generally, in the mathematical sciences. In the remainder of this blog post, we discuss the history, development, process, and key characteristics and recommendations of the 2015 Guide.
CUPM has a long and distinguished history, going back to curriculum reports in the 1950’s. The CUPM recommendations of 1991 were limited to the mathematics major and can be found in the report, Heeding the Call for Change. The CUPM Curriculum Guide 2004 addressed the entire undergraduate mathematics curriculum, with a particular emphasis on service courses of the first two years and pedagogical methods. In 2011, when we began thinking about preparing this guide, we decided to revisit the undergraduate mathematics major, as almost 25 years has passed since the MAA has done an in-depth study of the mathematics major. The 2004 Guide included important basic principles that apply to all mathematics courses. However, the applications of mathematics have expanded so much of late that CUPM felt it important to address the full scope of majors in the mathematical sciences.
At national meetings over the last five years, CUPM has held numerous focus groups, presented ideas to groups of chairs and coordinators of undergraduate mathematics programs, and solicited advice from directors of graduate programs, at both Master’s and doctoral levels. Several years of these discussions of content and cognitive goals have led to a consensus of what might be considered the fundamental components of any mathematical sciences bachelor’s level program. The draft of the 2015 Guide was submitted to the member organizations of CBMS, and many CBMS members created Association Response Groups that sent us valuable suggestions and comments. We were pleased with the response and have incorporated almost all of their advice.
The recommendations and principles in the Curriculum Guide are meant to drive a process of intentional design within a framework of basic cognitive and content goals that have been agreed upon by the mathematical community across many types of institutions. As the title reflects, this report is intended as a guide, not as a prescription. It is not an accreditation document. The members of CUPM and the Curriculum Guide Steering Committee found fundamental agreement as to what should be the core of any mathematical sciences major. From the many applied fields to the doctoral programs in pure mathematics, we heard much the same basic ideas. Although we spent quite a lot of time refining the words in the recommendations, there was very little disagreement on the principles. Nevertheless, CUPM emphasizes that each department is expected to examine its own program(s), mission, students, resources, and interests as it considers its own goals and pathways. At its August meeting in 2014, the Board of Governors of the MAA approved the Cognitive and Content Recommendations discussed in the Overview of the 2015 Guide and also reaffirmed the Principles of the 2004 Guide.
The real planning and intentional structure of the mathematics major derives from the cognitive and content goals and from recognizing that a department must design courses in the core to advance students toward the mathematical maturity that is the sum of the stated goals.
We believe that a successful major program offers a program of courses to gradually and intentionally lead students from basic to advanced levels of critical and analytical thinking, while encouraging creativity and excitement about mathematics. This requires that students be expected to do more each semester to develop and grow so that they achieve the goals eventually. This is the essence of the generic mathematics major. Each department should incorporate the core goals by offering their own intentional design of appropriate experiences and electives for their students. The Overview section of the 2015 Guide is the closest we get to “defining” the major.
There are many parts to the current Guide. There is a downloadable brochure, which briefly explains the contents and can be used to discuss the main points with administrators and others. A printed abbreviated version of the 2015 Curriculum Guide will be sent to all department chairs. The bulk of the details on all the course areas and program areas will appear online only. Carol Schumacher of Kenyon College co-chaired the Curriculum Guide Steering Committee and is co-PI with me and Michael Pearson of the MAA on the NSF grant [DUE-1228636] and the Educational Advancement Foundation grants that made this project possible. Paul Zorn is the editor, and has generously offered wise counsel to Carol and me in preparing all of the disparate parts for distribution.
The Introduction and the Overview are essential reading. They describe the context and essential components of the report and the recommended goals for all mathematical sciences. Although this Guide makes recommendations about a great variety of programs that center on new and developing applications of mathematics, CUPM urges departments to continue to offer an option for a major in mathematics that is perhaps considered traditional in scope and purpose. Some departments call this the “pure” mathematics track; some call it the preparation for doctoral programs in mathematics. Such programs should rely on educational processes that lead to mathematical maturity, should cultivate appreciation for mathematics for its own sake, and, while incorporating some applications of the theory, they are centered on the theoretical nature of the subject. All undergraduate major programs in mathematics/mathematical sciences have common elements however, and the cognitive and content goals reflect those common threads.
Focus groups and MAA SIGMAAs (MAA’s special interest groups) generated many ideas for an in-depth view of undergraduate course areas, and of course, provided a modern look at the many applied areas that beckon mathematicians. The Mathematical Sciences in 2025, a publication of the Board of Mathematical Sciences and Its Applications of the National Research Council, provided CUPM with many examples of fast-growing career opportunities for mathematicians. The INGenIOus Report in 2014 added to the call for career-readiness in college graduates. These resources and the members of the Curriculum Guide Steering Committee were instrumental in the selection of the many Course Areas and Program Areas featured in the report. Invitations to the members of the nearly 40 Study Groups were informed by the Steering Committee, as well.
Some MAA committees provided a natural resource as, for example, the American Statistical Association-MAA Joint Committee on Statistics or the CUPM Subcommittee on Mathematics Across the Disciplines. Of particular importance are the reports on the Preparation of High School and Middle School Teachers, which are reports from the MAA’s Committee on the Mathematical Education of Teachers. CUPM is pleased that more than 200 individuals have contributed to these group reports. The willingness of the mathematical community and many of our colleagues in allied fields to commit their time to this project has been very gratifying. I urge everyone to read these reports. They represent the views of a large and varied group of mathematicians and scientists who teach undergraduates.
Because a successful department is more than just its course and program offerings, the Guide contains a large section on matters “beyond the curriculum.” These are essential departmental and institutional policies that advance and enhance undergraduate major programs. These include articles on recruitment and retention, assessment, and placement, for example.
In some ways, the most interesting CUPM discussions were focused on Review and Renewal, the “how” to go about making change, particularly focused on meeting our cognitive and content goals. We believe that a healthy mathematical sciences program should incorporate intentional evolution and continual improvement. Every mathematical sciences department should have and follow a strategic plan that acknowledges local conditions and resources, but is also informed by recommendations from the greater mathematical community. The process of planning and renewal should be guided by consultation both within the department and with outside stakeholders at the institution and beyond. Departments should assess their progress in meeting cognitive and content goals through systematic collection and evaluation of evidence. The process of program review and departmental responsibilities in renewal are discussed in our report.
The most radical feature of this CUPM Guide is that it is meant to be dynamic in that we expect CUPM will add resource material to the Course and Program Areas regularly and do a thorough review of several course and program areas each year. Rather than trying to write a guide to the undergraduate curriculum and the major every decade, CUPM hopes to maintain an on-going revision and renewal process of its own that is directed at all of the course and program areas. We invite members of the mathematical community to submit ideas such as projects, resources, and videos, so that they can be posted online appropriately. We are preparing a submission procedure at this time. CUPM’s mission is to provide a current, meaningful, and useful resource for the design and improvement of the undergraduate major, and we welcome your input.
Feel free to contact me with your comments and suggestions at msiegel@towson.edu.
]]>Why are fractions hard to learn for so many people? There are many reasons for this, but I like to think about one in particular, a mathematical idea hiding in plain sight, from elementary school to college: equivalence relations. Consider the fraction sum 2/3 + 1/5, which we of course compute by using 2/3=10/15 and 1/5=3/15, arriving at an answer of 13/15. This raises a whole host of fundamental questions about equality: If 2/3 equals 10/15, why can we use one but not the other in evaluating the sum? Does this mean something is wrong with our idea of “equals”? Could we have used something else besides 10/15; or, in the other extreme, should we always use 10/15? This shows that often when we say “equals”, what we really mean is “equivalent”. Equivalence introduces a number of useful mathematical connections, but we must be careful in how we handle it with our students who just want to know, for instance, how to add two fractions.
We’ll see in a bit how these same ideas arise in mathematical topics from elementary school through college, and what potential difficulties they may cause. I’ll conclude by sharing some specific examples of what I do in my classes to confront these issues.
But first, a very quick primer for readers who haven’t seen how to define rational numbers with mathematical rigor: Declare two fractions a/b and c/d (a,b,c,d integers, b, d not 0) to be equivalent if ad=bc. We then verify this relation really is an equivalence relation (a relation satisfying the familiar reflexive, symmetric, and transitive properties of equality), which also means this relation partitions all the fractions into equivalence classes. It’s those equivalence classes that are the rational numbers. So the rational number 2/3 is actually the equivalence class of all fractions that are equivalent to the fraction 2/3, by our usual method of reducing to lowest common denominator.
We should also verify that this relation behaves nicely with respect to operations such as addition (defined by the formula we get from using common denominators: a/b + c/d = (ad + bc)/bd) so that anything equivalent to 2/3 plus anything equivalent to 1/5 gives something equivalent to 13/15.
All of this is more meaningful (and the proofs are much more straightforward) if we define two fractions to be equivalent if they correspond to the same length on a number line. Indeed, this more natural way to define the equivalence is why we care about it in the first place. But that doesn’t help when we want to do exact calculations, or when our fractions contain algebraic expressions. Then we really are confronted with issues of equivalence:
One more issue with some equivalence relations doesn’t show up with fractions, but does in geometry: there may be different equivalence relations on the same set. Sometimes, we want to consider geometric objects equivalent if they are congruent, or sometimes even just similar. But other times we need a finer partition that distinguishes between congruent objects. For instance, a left-hand glove is congruent to a right-hand glove, but not equivalent in most real-world settings, because they differ by a reflection; similarly, a triangle pointing up may be congruent to a triangle pointing to the left, but they convey different information, because they differ by a rotation.
Once you start looking, you can find equivalence relations in many mathematical ideas. Here is a sampling, showing that this notion of equivalence spans a wide variety of mathematical topics, from very elementary to advanced ideas. The four issues described above with fractions and geometry appear in various combinations throughout.
Well, finding equivalence relations is a fun game for mathematics people, but so what? What can we do differently after finding them? I now do two things differently in my classes based on my thinking about equivalence:
To conclude, no matter what level you teach (or learn) at, keep looking for those hidden equivalence relations lurking throughout the mathematics curriculum. With a little awareness, we can help students avoid the traps and make more sense of this powerful tool.
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This two-part series is a summary of the longer paper, Textbook School Mathematics and the preparation of mathematics teachers.
TSM (Textbook School Mathematics) has dominated school mathematics curriculum and assessment for the past four decades, yet, in mathematics education, TSM is still the elephant in the room that everybody tries to ignore.
We will look at three examples of this phenomenon.
Example 1. The volume on The Mathematical Education of Teachers II ( MET2) rejects the temptation of teaching teachers only advanced mathematics and leaving them to find their own way in school mathematics. That is good. But its recommendation for the preparation of high school teachers, for example, is the completion of the equivalent of a math major and three courses with a primary focus on high school mathematics from an advanced standpoint. The suggested organizing principles for the three-semester course sequence are (MET2, p. 62):
Emphasize the inherent coherence of the mathematics of high school.
Develop a particular mathematical terrain in depth.
Develop mathematics that is useful in teachers’ professional lives.
There is no mention of teachers’ thirteen years of mis-education in TSM, much less what to do about it.
For example, we know how definitions are ignored or mangled in TSM: definitions are not important. A definition, teachers are told, is nothing more than “one more thing to memorize”. How then can they learn to start emphasizing the importance of definitions?
Take the concept of congruence: in K-8 TSM, students are taught that any two figures are congruent if they have the ‘same size and same shape,” but in high school “curvy” figures are forgotten and only polygons are considered: two polygons are congruent if corresponding sides and corresponding angles are equal. Now come the CCSSM which want congruence to have one definition in terms of reflections, translations, and rotations all through middle school and high school. Similar remarks hold for all other concepts, such as similarity, expression, graph of an equation, graph of an inequality, etc.
When we ask for such a sea change, would a genteel discussion in general and an in-depth investigation of a particular terrain be enough to bring it about?
I suggest that these three courses be used to give a systematic exposition of the high school mathematics curriculum at a level as close to the school classroom as possible, but in a way that is mathematically correct. Such an exposition will show teachers how definitions can be used productively in the school classroom as well as how school mathematics differs from TSM in terms of coherence, reasoning, precision, and purposefulness. If we want a sea change in teachers’ conception of mathematics, let us show them the way, from the ground up.
Few math departments have the resources to offer such courses, but one of them is at UC Berkeley: see the description of “Math 151–153″ in the Appendix of this article. Until we can provide teachers with a knowledge of correct school mathematics, the more esoteric recommendations in MET2—such as research experience for high school teachers—can wait.
Example 2. The CCSSM have made significant inroads in steering many topics away from TSM, but the CCSSM have also prefaced the content standards with eight Mathematical Practice Standards (MPS) for students. A confluence of unusual circumstances has created the misconception that equates the CCSSM with the MPS. The idea that, in order to implement the CCSSM, all it takes is to study the MPS has taken root. Let us take a reality check.
If teachers know correct mathematics, the substance of the MPS would be a natural side effect of this knowledge. Mentioning the MPS somewhere in the content stanards is definitely a good thing. Unfortunately, putting MPS front and center in the transition from TSM to the CCSSM puts the cart before the horse. Let us consider, for example, the second and third MPS that state:
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
How would these work out for the task of writing down the equation of the line passing through two given points, \( (1,2)\) and \((3,4)\)?
TSM only teaches how to do this by rote, because the slope of a line is incorrectly defined in TSM as the difference quotient of the coordinates of two points that are a priori given on the line. But the CCSSM want slope to be defined correctly so that any two points on the line can be used to compute its slope. Then it is self-evident that both of the following lead to an equation of the line: For any point \((x,y)\) on this line,
\[ \frac{y-2}{x-1}\ = \ \frac{4-2}{3-1} \quad \mbox{and} \quad \frac{y-4}{x-3}\ = \ \frac{4-2}{3-1}\]
Now the MPS exhort students to explain how the equations come about and to critique each other’s reasoning. Given that we have only provided teachers with a knowledge base of TSM and that students continue to get TSM from their textbooks, studying the MPS will help neither the students nor the teachers in this task. Instead of encouraging this fixation on the MPS, how about first helping teachers to replace their defective knowledge (TSM) with correct mathematics?
Example 3. A recent volume Principles to Actions (NCTM 2014) has the goal of describing “the conditions, structures, and policies that must exist for all students to learn.” We will refer to this volume as P-to-A.
P-to-A makes no mention of TSM or the need to help teachers overcome the damage done to their thinking by TSM. Nevertheless, it asks teachers to use “purposeful questions” to “help students make important mathematical connections, and support students in posing their own questions”, (P-to-A, pp. 35, 36). Given teachers’ immersion in TSM, what can they say when students ask for the purpose of learning the laws of (rational) exponents? In TSM, it could only be to ace standardized tests. P-to-A also says “effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding” (P-to-A, p. 42). Since TSM gets it all wrong even in something as mundane as solving a linear equation (see p. 22–25 of this article), teachers who know only TSM will be able to do nothing but transmit TSM’s pseudo-reasoning about the procedure of solving an equation to their students. There goes conceptual understanding out the window. And so on.
P-to-A enthusiastically recommends actions to realize these and other worthy learning goals seemingly without realizing that, given their damaged knowledge base, our teachers are not ready for these actions. On the issue of how to help teachers, all that P-to-A has to say is that they should be provided with all the necessary resources and professional development they need. Nothing about TSM. Since current professional development mainly recycles TSM, how can this possibly help?
TSM is the elephant in the room that everybody tries to ignore. This cannot go on.
Let us bring closure to this discussion. TSM comes from school textbooks, so why not just concentrate on getting rid of TSM by writing better textbooks? Two reasons: (1) The vicious circle syndrome: Staff writers in major publishers are themselves products of TSM. (2) The bottom-line mentality: In order to maximize the sales of their books, publishers do not publish anything teachers (products of TSM) don’t feel comfortable reading.
At the moment, the only hope of getting better school textbooks is for teachers to reject TSM-infested textbooks. Then, and only then, will publishers listen. Helping teachers to eradicate TSM is therefore not only imperative for improving their content knowledge, but it may also be the only way to get better school textbooks written.
The author is very grateful to Larry Francis for his many suggested improvements.
]]>This two-part series is a partial summary of the longer paper, Textbook School Mathematics and the preparation of mathematics teachers.
School mathematics education has been national news for at least two decades. The debate over the adoption of the Common Core State Standards for Mathematics (CCSSM) even became a hot-button issue in the midterm elections of 2014. This surge in the public’s interest in math education stems from one indisputable fact: school mathematics is in crisis.
From the vantage point of academia, two particular aspects of this crisis are of pressing concern: School textbooks are too often mathematically flawed, and in spite of the heroic efforts of many good teachers, the general level of math teaching in school classrooms is below acceptable.
Mathematicians like to attack problems head-on. To us, the solution is simple: Just write better school textbooks and design better teacher preparation programs. I will concentrate on the latter for now and will not return to the textbook problem until the end of Part 2.
An effective math teacher has to know the subject thoroughly (strong content knowledge) and be able to communicate with students (good pedagogy). However, my own involvement with the professional development of teachers in K–8 points strongly to the fact that, as of 2014, students’ nonlearning of mathematics has more to do with teachers’ content knowledge deficit than with deficiencies in pedagogy. If our short-term goal is to get schools out of this crisis as soon as possible, a focus on this content knowledge deficit must be our top priority.
Let me make clear at the outset that the blame for this deficit has to be laid squarely at the feet of the education establishment: the schools of education, statewide and district-wide administrators, and the mathematics community. Its systemic failure to provide teachers with the correct content knowledge they need for teaching mathematics is nothing short of scandalous. When teachers are themselves students in K-12, the mathematics they learn from their textbooks is the same fundamentally and seriously flawed mathematics they will in turn teach to their students. We call this body of knowledge Textbook School Mathematics (TSM). When they get to college as preservice teachers, they are (as of 2014) generally not made aware of the flaws in TSM, and are consequently not provided with any replacement for TSM. It therefore comes to pass that when it is their turn to teach students, they can only fall back on the TSM that they learned as K -12 students. This is how TSM gets recycled in schools, and this is how a body of unlearnable mathematical “knowledge” has come to rule the classrooms of our nation.
TSM has been around for at least four decades. Some may be taken aback by this statement, because didn’t the 1989 mathematics education reform correct all that was wrong with “traditional math”? Isn’t “reform math” at least different from “traditional math”? The short answer is that the two differ mostly in the packaging but not in their underlying mathematical substance: they both suffer from the same mathematical defects. How could this be otherwise when the people who brought us the reform were themselves victims of TSM?
Let us give a brief list of the defects of TSM. Note, however, that these defects are so deep and pervasive that they cannot be fully captured by a short list.
One can easily infer from this list that we consider it imperative for teachers to know the mathematics they have to teach in K–12 in a way that is consonant with the normal development of mathematics: the definitions leave no doubt about what is being discussed; the precise language minimizes misunderstanding; the reasoning and the coherence reduce learning by rote; and finally, knowing the mathematical purpose that a concept or skill serves gives students more motivation to learn it. This list is all about maximizing the learning outcome.
A typical illustration of TSM is the way the concept of percent is presented to students. The meaning TSM gives for percent is out of a hundred. Perhaps “out of a hundred” sounds clear in everyday language, but mathematically its lack of precision is unacceptable and unusable. Is percent perhaps a number? Because TSM gives no clear definition, percent has become one of the most feared topics in middle school. Without a definition for percent, students cannot reason their way to solutions of percent problems. Therefore, when they need to find the percent of a shaded area when 6 of 41 congruent squares are shaded, students are forced to resort to the rote skill of “setting up a proportion”: solve for \(x\) in \(6/41 = x/100\). In fact, rote memorization of procedures is a hallmark of what passes for learning in TSM.
Any hope of improving mathematics learning hinges on our ability to replace TSM in the school curriculum with correct mathematics. For a reason to be explained in Part 2, the starting point has to be helping our teachers shed their knowledge of TSM and learn how to do K–12 mathematics correctly. The mathematics community should take note that this cannot be accomplished simply by teaching future teachers good, advanced mathematics. University mathematics is fundamentally different from school mathematics. In short, the latter is an engineered version of the former. For example, while rational numbers can be dispatched in three lectures in a junior-level abstract algebra course, it would not do to teach ten-year-olds that a fraction is an equivalence class of ordered pairs of integers. Similarly, one cannot tell twelve-year-olds that constant speed means the distance function has a constant derivative. And so on. If we are serious about doing our share to resolve the education crisis, university campuses across the land will have to commit to teaching correct school mathematics to teachers until TSM is no more.
This will be a serious commitment. The absence of such a commitment thus far has frustrated many teachers who are hungry for correct content knowledge but have nowhere to turn for this knowledge. The education establishment has systematically let math teachers down. There should be no illusion, however, about the heavy responsibility that comes with this commitment: teachers do not shed the habits acquired over thirteen years of immersion in TSM without a protracted struggle and without a lot of help. The help they need translates into sustained hard work on our part. This is hardly glamorous work, but if mathematicians don’t do it, who will?
The author is very grateful to Larry Francis for his many suggested improvements.
]]>Introduction
The advanced proof-oriented courses for mathematics majors are typically taught in a lecture format, where much of the lecture is comprised of presenting definitions, theorems, and proofs. There is a general perception amongst mathematicians and mathematics educators that these lectures are not as effective as they could be. However, the issues of why these lectures are not effective and how they might be improved are not discussed often in the mathematics education literature. In my research, I have sought to address this issue. Through task-based interviews with students and discussions about pedagogy with mathematicians, as well as observations of lectures and students’ reactions to them, I have found that mathematics professors and mathematics majors have different expectations of lectures and these different expectations lead to barriers in communication. By expectations, I am referring to (i) what a student is supposed to learn from, or “get out of”, a lecture and (ii) how students should engage in the lecture to understand this content. As a consequence of these different expectations, students do not gain what mathematicians hoped they would from the lectures they attend. Below I describe four such differing expectations and how they might inhibit lecture comprehension, with the hope that discussing these differing expectations might help us improve the teaching of proof at the undergraduate level.
Expectation #1 – Students can learn a lot by filling in the logical details of the presented proofs
In a conversation with a mathematician about his teaching, he provided the following anecdote:
“I’m doing a reading course with a student on wallpaper groups and there is a very elegant, short proof on the classification of wallpaper groups written by an English mathematician. […] he’s deliberately not drawing pictures because he wants the reader to draw pictures. And so I’m constantly writing in the margin, and trying to get the student to adopt the same pattern. Each assertion in the proof basically requires writing in the margin, or doing an extra verification, especially when an assertion is made that is not so obviously a direct consequence of a previous assertion […] I write sub-proofs and check lots of examples” (Weber & Mejia-Ramos, 2011, p. 337).
Here, this professor is providing a common expectation shared by many mathematicians. When a student is reading a proof, the student is expected to seriously engage with each line of the proof, engagement that might consist of drawing pictures, considering examples, and writing sub-proofs. This is not only about checking the correctness of a proof; it is also an important means of building one’s understanding of mathematics. Indeed, mathematics professors feel that we may be robbing students of this opportunity if we did all of this work for them (cf., Lai, Weber, & Mejia-Ramos, 2012) and research has shown that encouraging students to meaningfully connect new statements within a proof with previous statements and their own knowledge base enhances proof comprehension (Hodds, Inglis, & Alcock, 2014).
Students see things differently. In an interview study, I asked 28 mathematics majors what makes a good mathematical proof. The following response was typical.
“For me, as a student, what else I would like to see is all the intermediate sorts of steps, things to help along, graphs, visual things. Things that recalled facts that perhaps I should know but you know, maybe not immediately at the tip of my tongue. That’s to me what makes a good mathematical argument” (Weber & Mejia-Ramos, 2014).
Sixteen of the 28 students gave a response like this, claiming all the intermediate steps of a proof should be provided. If students have to work at understanding the proof, then from their perspective, this was not a good proof. A survey with 175 upper-level mathematics majors and 83 mathematics professors verified the generality of these findings. Most mathematics majors (75%) agreed with a statement that a student should not have to spend time filling in logical gaps if a good proof was presented with them, while few mathematicians (27%) agreed with this statement (Weber & Mejia-Ramos, 2014).
It is easy to see how these different expectations can hinder communication in mathematics lectures. For a variety of reasons, mathematics professors leave some details out of the proofs that they present. Typically, mathematics majors will not view filling in these gaps as their responsibility or an opportunity to learn; rather they will simply see the incomplete proof as a low quality presentation by the lecturer.
Expectation #2 – Understanding a proof is more than just providing a justification for each individual step
While a complete understanding of a proof includes understanding how each step in the proof was justified, most mathematicians believe understanding a proof involves more than this. For instance, some proofs might demonstrate a technique that can be useful for solving other problems or proving other theorems, others might illustrate a new way of thinking about a concept (Mejia-Ramos & Weber, 2014; Weber, 2010a). When presenting some proofs to students, they hope that students learn these things (Weber, 2012). Most mathematics majors disagree with this. The large majority (75%) believe that if they understand how every step in a proof follows from previous work, they understand the proof completely (Weber & Mejia-Ramos, 2014). Consequently, mathematics majors tend not to invest the time trying to understand proofs holistically, in part because they are not aware that this is something they should do.
Expectation #3 – Students are expected to spend time studying proofs outside of class
Most mathematics professors believe that understanding a proof can be a lengthy and complicated process (Weber, 2012). When asked how long a mathematics major should ideally spend studying a proof outside of the classroom, their average response was between 30 and 37 minutes, with 81% giving a response of greater than 15 minutes. Again, mathematics majors see things differently. Their average response to the same question was 17 to 20 minutes, with only 41% giving a response of greater than 15 minutes. (Weber & Mejia-Ramos, 2014). In practice, mathematics majors probably spend much less time than that. In a study in which 28 mathematics majors were asked to determine the correctness of 10 purported proofs, no participant spent more than five minutes studying a single proof, even though participants acknowledged that they often did not fully understand the proofs that they were reading (Weber, 2010b).
Expectation #4 – What the professor is saying when he or she writes the proof is important
The job of a mathematics lecturer in advanced mathematics is hard. Two responsibilities of the lecturer are to help students (i) understand what constitutes a proof in advanced mathematics (i.e., what is an acceptable product) and (ii) understand how proofs can be written (i.e., what is the process by which that product is produced). These two things are sometimes in opposition to one another. Writing proofs effectively involves knowing how to choose a proving approach that is likely to be effective (e.g., Weber, 2001), but the proof itself focuses on the implementation of the approach. The proof itself is a verification, not a description of the problem-solving process (Selden & Selden, 2013).
One way that lecturers manage these competing responsibilities is by leaving a space on the blackboard designated for the “official proof”, while discussing how the proof was produced orally (Fukawa-Connelly et al, 2014; Weber, 2004). What the professor says aloud but does not write down is therefore a crucial part of his or her lecture.
We studied a lecture where a professor repeatedly emphasized the same important proving heuristic in a proof that he provided. After the lecture, we asked six students what they learned from this proof and none mentioned this heuristic. When we showed each student a video-recording of this part of the lecture again, we obtained the same result. Inspecting the notes that the students took offered an explanation for this phenomenon. Only one student wrote down anything the professor said aloud. The other five students’ notes were comprised entirely of what the professor wrote on the board (Fukawa-Connelly et al, 2014). Clearly if students focus on what the professor writes but not what he or she says, much of the insight in the professor’s oral comments will not be acknowledged by the student.
Discussion and recommendations
I have outlined four expectations that students hold about mathematics lectures which are at variance with mathematicians’ intentions and inhibit what they learn from these lectures. A point that I wish to emphasize is that I do not believe it is the students’ fault that they have these expectations. Lectures in advanced mathematics are a new experience for students. They need guidance about the nature of these lectures and what the role of this process should be. (How to study for a mathematics degree (Alcock, 2012) is a useful resource for students in this regard). Indeed, some lecturers hold the belief that the purpose of the lecture is to provide the students with the big ideas, intuitions, and motivations for a proof rather than focus on its logical details (Lai & Weber, 2014), but students do not view lectures in this way. Further, many of the students’ expectations are quite sensible given their previous experiences. For instance, in students’ previous exposure to proof, they were probably required to justify every claim that they made, even ones that seemed to be obvious. This is certainly the case– indeed, the motivation– for the two-column proofs in high school geometry. It is therefore understandable that they would expect their professors to provide a similar level of justification. We should not expect that students will naturally share mathematicians’ expectations; the professors will have to work to develop this shared understanding. Below are some recommendations as to how this can be done.
Explicitly communicate your expectations about lectures early in the semester. One reason for the discrepancy between mathematics professors and students’ expectations of lectures in mathematics is that these expectations are rarely the topic of explicit conversation. Stating what you hope to convey in your lectures, what you expect your students to attend to, and how they should study does not guarantee that they will do these things but it at least gives them the opportunity to appreciate their importance.
Assess what you want your students to know.
I have found that advanced mathematics lecturers strive to engage students with high-level or intuitive ways to understand the course content, but students’ assessments are predominantly on formal mathematics (i.e., stating definitions, writing proofs). This sends a message to students that the formal aspects of mathematics are what is important and the other aspects of mathematics are superfluous. I would recommend assessing students on high-level and informal aspects of mathematical proofs as well. Of course, this will motivate students to understand the material, but there are two additional benefits. First, the types of questions that students are asked can give them a better sense of how the material should be understood. Second, students’ responses to these questions will provide the instructor with a better sense of how the students are interpreting his or her lecture.
My colleagues and I have developed a template for asking questions designed to measure students’ understanding of a proof (Mejia-Ramos et al, 2012). These include questions regarding the holistic nature of a proof, such as asking students to apply the ideas of a proof to a specific example or diagram, having students use the technique in the proof to establish a different theorem, or presenting students with several summaries and asking which one captures the main idea of a proof that they read. These questions illustrate to students that understanding a proof does not solely consist of justifying each step within a proof and it provides incentive to study lecture proofs carefully outside of class.
Write down your key points. A typical lecturer speaks at a rate of over 100 words a minute while a typical student can transcribe less than 25 words a minute. Students cannot write down everything they hear; they must prioritize. Further, nearly everything that students do not write down is not recalled at a later time (Williams & Eggert, 2002). It is natural for students to focus on what is written on the blackboard. Written statements have a permanence that oral speech lacks and it is a traditional means by which a professor emphasizes importance. I would recommend that the key ideas that a professor wishes to emphasize in a proof should be written down on the blackboard or distribute to the students as lecture notes. If they are not, students will probably not include them in their notes and will consequently forget these lessons.
References
Alcock, L. (2012). How to Study for a Mathematics Degree. Oxford University Press.
Fukawa-Connelly, T., Lew, K., Mejia-Ramos, J.P. & Weber, K. (2014). Why lectures in advanced mathematics often fail. In Proceedings from the 38th Conference of the International Group for the Psychology of Mathematics Education, 3, 129-136. Vancouver, Canada.
Hodds, M., Alcock, L., & Inglis, M. (2014). Self-Explanation Training Improves Proof Comprehension. Journal for Research in Mathematics Education, 45(1), 62-101.
Lai, Y., & Weber, K. (2014). Factors mathematicians profess to consider when presenting pedagogical proofs. Educational Studies in Mathematics, 85(1), 93-108.
Lai, Y., Weber, K., & Mejía-Ramos, J. P. (2012). Mathematicians’ perspectives on features of a good pedagogical proof. Cognition and Instruction, 30(2), 146-169.
Mejia-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79(1), 3-18.
Mejia-Ramos, J. P., & Weber, K. (2014). Why and how mathematicians read proofs: further evidence from a survey study. Educational Studies in Mathematics, 85(2), 161-173.
Selden, A. & Selden, J. (2013). The genre of proof. Contribution to the chapter “Reflections on justification and proof”. In M. Fried & T. Dreyfus (Eds.) Mathematicians and mathematics educators: Searching for common ground. Springer. (pp. 237-260).
Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101-119.
Weber, K. (2004). Traditional instruction in advanced mathematics courses: A case study of one professor’s lectures and proofs in an introductory real analysis course. The Journal of Mathematical Behavior, 23(2), 115-133.
Weber, K. (2010a). Proofs that develop insight. For the Learning of Mathematics, 30(1), 32-36.
Weber, K. (2010b). Mathematics majors’ perceptions of conviction, validity, and proof. Mathematical Thinking and Learning, 12(4), 306-336.
Weber, K. (2012). Mathematicians’ perspectives on their pedagogical practice with respect to proof. International Journal of Mathematical Education in Science and Technology, 43(4), 463-482.
Weber, K., & Mejia-Ramos, J. P. (2011). Why and how mathematicians read proofs: An exploratory study. Educational Studies in Mathematics, 76(3), 329-344.
Weber, K. & Mejia-Ramos, J.P. (2014). Mathematics majors’ beliefs about proof reading. International Journal of Mathematics Education in Science and Technology, 45, 89-103.
Williams, R. L., & Eggert, A. C. (2002). Notetaking in college classes: Student patterns and instructional strategies. The Journal of General Education, 51(3), 173-199.
]]>Editor’s note: The editorial board believes that in our discussion of teaching and learning, it is important to include the authentic voices of undergraduate students reflecting on their experiences with mathematics. We thank Ms. Blackwell, Ms. Kaplan-Kelly, and Ms. Webster for contributing their essay. The American Mathematical Society maintains a list of summer mathematics programs for undergraduates and has published Proceedings of the Conference on Promoting Undergraduate Research in Mathematics as a resource for mathematicians interested in similar programs.
We were participants in the Summer Math Program for Women Undergraduates (SMP) at Carleton College, a program with the goal of encouraging and supporting women undergraduates in their study of mathematics during their first two years of college. For four weeks we took math classes, listened to math talks, went to problem sessions, and talked about math for fun. We had the opportunity to meet many mathematicians from across the country. The people we met did not fit into the mold of the solitary eccentric that popular culture would have us believe. We met mathematicians who defied negative stereotypes often attributed to people in STEM areas and especially to women who are interested in math. Learning about their projects and interests helped us to see ourselves as capable of becoming mathematicians as well. In talking to them, we started to see what our lives could be like if we pursued math as a career and learned that there was no single “correct” type of person we would need to become. SMP was also an opportunity to meet mathematicians who worked outside of academia, mainly in applied math, which was not an area many of us had been exposed to before. This expanded our view of what being a mathematician might be like and what we could achieve.
This program helped shape our view of mathematics in many different ways. SMP gave us an idea of what upper level math classes could look and feel like. We encountered abstract algebra in our Lie Theory course, and our Topology course presented an entirely new branch of math for most of us. For instance, in Topology, we covered topics such as how to play tic-tac-toe or chess on a torus and klein bottle, which was fun and not something we had seen before. In addition to experiencing upper level math classes, we attended math talks by visiting professors and current graduate students on different areas of math such as a lecture on Knot Theory, a presentation that included a demonstration with a Ruben’s tube, and a lecture given by a professor who studies the history of math. We saw how math was used in different ways and could overlap with other subjects. These talks expanded our mathematical knowledge and also showed us how we could fit into the math community by showing more of what was possible.
SMP provided us with a great introduction to the collaborative math world. In the courses we took, there was much more emphasis placed on collaboration than many of us had been used to in previous math classes. In our Topology class we were assigned to work in groups for homework problems. These homework groups became the first way many of us got to know fellow participants, and we bonded quickly over attempting new problems such as designing a digestive system for a two-dimensional square in Flatland. In Lie Theory, we turned in partner homework assignments and presented tough problems on the board in class. In both classes we also worked on final projects in the form of group presentations. Though the research was tough, we immensely enjoyed putting together the presentations with our group mates. Some groups showed their enthusiasm through added touches like origami demonstrations and knot models. Working in groups helped us improve our collaboration skills and our ability to communicate mathematically, but most importantly it showed us how math can be a social activity.
SMP gave us the chance to become part of a new close-knit math community. We were excited to meet other math students who were passionate about math for its own sake, rather than just as a means to do physics or computer science. At first, most interactions at SMP were facilitated by group homework, but these interactions quickly became friendships centered on a wonderful common interest. Since we all lived in the same dorm, we constantly had opportunities to build on these relationships. Even casual conversations over meals or in the lounge helped strengthen our community. One such conversation about gender in mathematics stretched long past the end of dinner and would have gone hours longer if we hadn’t needed to work on homework.
We were encouraged to meet women who were attending or had completed graduate school for math, and to learn from their experience. The people we met made a point of telling us all the “secrets” of being a mathematician – what courses to take as an undergraduate, how to find summer opportunities, how to apply to and succeed at graduate school, and much more. For example, we learned about how to apply and what to expect at REU programs from current REU participants at the University of Minnesota, and we attended information sessions given by math professors on declaring a math major and applying to graduate school. SMP also made it possible for us to meet former participants who are now professors. At the SMPosium, a two-day conference of former SMP participants who have since earned their PhDs, we heard talks from professional mathematicians working in a wide variety of fields. Many of the talks were in applied math, covering topics such as acoustics or using topology for data analysis. We also heard several talks in pure math, including talks about hyperbolic spaces and a complicated type of algebra. Hearing about the accomplishments of graduate students who were only a few years older than ourselves helped us think about what exciting discoveries we could make in the near future. In addition to learning math, SMPosium allowed us to expand our mathematical community. Through these connections we learned that working on mathematics does not have to be an isolating experience, despite common stereotypes to the contrary.
By combining classroom work, problem solving, and lectures with a community of people who are all invested and enthusiastic about math, SMP provided a holistic math experience. Many of us came into SMP viewing graduate school as an unreachable goal, and thoughts of a future in mathematics were unclear. Rather than leaving SMP burnt out, we left wanting to do more math. While we were at SMP, we had the opportunity to talk to many mathematicians and clarify our goals for the future. In particular, our professors were helpful in making plans to achieve our goals. Many of us met one-on-one with our professors to discuss our individual career options and future plans. We now know what is necessary to continue into graduate level work. We believe we are capable of succeeding in graduate school and we are starting to figure out what types of math interest us most. We left SMP with confidence in our ability to have a future in the mathematical community.
]]>Mathematics faculty and educational researchers are increasingly recognizing the value of the history of mathematics as a support to student learning. The expanding body of literature in this area includes recent special issues of Science & Education and Problems, Resources and Issues in Undergraduate Mathematics Education (PRIMUS), both of which include direct calls for the use of primary historical sources in teaching mathematics. Sessions on the use of primary historical sources in mathematics teaching at venues such as the Joint Mathematics Meetings regularly draw large audiences, and the History of Mathematics Special Interest Group of the Mathematical Association of America (HOMSIGMAA) is one of the largest of the Association’s twelve special interest groups. In this blog post, which is adapted from a recent grant proposal, we explore the rationale for implementing original sources into the teaching and learning of undergraduate mathematics, and then describe in detail one method by which faculty may do so, namely through the use of Primary Source Projects (PSPs).
Teaching from primary sources has long been common practice in the humanities and social sciences [11, 17]. Reading texts in which individuals first communicated their thinking offers an effective means of becoming mathematically educated in the broad sense of understanding both traditional and modern methods of the discipline [9, 18]. The use of original sources in the classroom promotes an enriched understanding of the subject and its genesis for instructors as well as students.
In contrast to many textbook expositions, which often present mathematical ideas in a distilled form far removed from the questions that motivated their development, original sources place these ideas in the context of the problem the author wished to solve and the setting in which the work occurred. Problems and the motivations for solving them are more apparent and natural in primary sources, and the works of these thinkers are more compelling than traditional textbook expositions. Exposing the original motivations behind the development of “esoteric” mathematical concepts may be especially critical for placing the subject “within the larger mathematical world,” thereby making it more accessible to students [22]. Further, primary texts seldom contain specialized vocabulary (which comes with later formalism), thereby promoting access to the ideas by students with varied backgrounds.
Precisely because they give students the opportunity to interpret results as they were originally presented and then reformulate them in modern terms, original source readings encourage robust understanding of mathematics. Engagement with the original problems from which concepts arose has been observed to invite questions of a different nature than students generally pose in more traditional approaches as it provides students with a basis for making their own reflections and developing their own judgment, thereby helping them to see how to develop ideas and reason with them on their own [12, 13]. Drawing on Sfard’s work on “mathematical discourse” and “commognition” [23], Kjeldsen and Blomhøj have further suggested that the reading and interpretation of original sources may even be essential for raising students’ awareness of what constitutes a proper definition, or deciding whether a solution is correct and complete [16], a key feature of success in the study of advanced mathematics. Jankvist further proposes that original source materials may help with the “transition problem” between educational levels [14].
Finally, the reading of original texts provides a means of responding to calls within the Science, Technology, Engineering, and Mathematics (STEM) education community to engage undergraduate students in authentic research at the frontiers of the field. While the importance of such engagement for socializing and retaining students within the disciplinary community has been noted by Project Kaleidoscope Director J. Narum [20] and others, the present research frontier in mathematics is generally too far removed from the undergraduate experience to make this possible. Engaging students with texts that represent the state of mathematical knowledge at an earlier stage of development can, however, provide undergraduates an opportunity to experience research at a frontier stage as it was practiced by some of the greatest mathematical minds throughout time.
Despite the benefits of primary source materials detailed above, and granting the wide availability of such materials via published collections and web resources [6, 8], there are significant challenges to incorporating primary sources directly into the classroom. Using secondary historical sources, such as [15], may suffice to reap some of the benefits of the original works; however, use of such sources carries its own difficulties, including the risk of placing too much emphasis on learning the history of mathematics per se, as opposed to using that history to support the learning of mathematics.
One approach to addressing these issues is through PSPs, which are curricular modules designed to teach core mathematical topics from primary historical sources rather than from standard textbooks. Each PSP is designed to cover its topic in about the same number of course days as classes would otherwise. With PSPs, rather than learning a set of ideas, definitions, and theorems from a modern textbook, students learn directly from mathematicians such as Leonhard Euler, Augustin-Louis Cauchy, or Georg Cantor. This distinction is crucial to PSPs: they are not designed to teach history; rather, they use history as a tool to better teach mathematics.
PSPs employ a selection of excerpts from primary historical sources that follows the discovery and evolution of the topic in question. Each PSP contains commentary about the historical author, the problem the author wished to solve, and information about how the subject has evolved over time. Exercises are woven throughout the project, requiring that students actively engage with the mathematics as they read and work through each excerpt. At appropriate junctures, students are also introduced to present-day notations and terminology and are asked to reflect on how modern definitions have evolved to capture key properties of solutions to problems posed in the past. Learning from the PSP via in-class activities and discussions replaces standard lectures and template blackboard calculations.
As an example, the PSP Networks and Spanning Trees [19], opens with Arthur Cayley’s discovery of a pattern for the number of (labeled) trees on n vertices. Cayley used the term “tree,” without any definition, to describe the logical branching when iterating the basic process of (partial) differentiation. Students are asked to follow in Cayley’s footsteps and arrive at the same observations, hinting at an algebraic pattern for the number of such trees. Students are then asked to find the gaps in Cayley’s 1889 “proof” [7], and to reflect on what would constitute a valid argument. This is followed by Heinz Prüfer’s rigorous counting of (labeled) trees [21], motivated by the problem of enumerating all possible railway networks on n (fixed) towns so that (i) the least number of railway segments is used, yet (ii) travel remains possible between any two towns. The module continues with Otakar Borůvka’s 1926 solution to finding the most economical way to connect n towns in a rural region to an electrical network. He devised an ingenious algorithm to solve this problem by connecting each town to its nearest neighbor, and then iterating the algorithm on connected components until a connected graph (tree) resulted [4, 5, 10]. After working through these specific applied problems of Prüfer and Borůvka, students are asked to reflect on how the modern definition of a tree captures the proprieties sought by both authors, and what lemmas or theorems have evolved from these historical sources.
Classroom implementation of this and other PSPs is extremely flexible, thereby enhancing their adaptability to a wide variety of institutional settings. Although they work best by utilizing a combination of in-class activities and out-of-class homework, PSPs can be completed individually or in small groups, or assigned as a one-to-two week assignment requiring a written paper addressing all the exercises in the PSP. Other instructors assign a PSP in parts over a four-to-five week time period, and ask students to submit written answers to each exercise in installments. The written portion of a PSP allows students to react to the historical sources, organize their thoughts through mathematical exposition, and rediscover groundbreaking ideas for themselves.
Faculty interested in finding a wide collection of PSPs are encouraged to consult [1, 2, 3].
References
[1] J. Barnett, G. Bezhanishvili, H. Leung, J. Lodder, D. Pengelley., I. Pivinka, and D. Ranjan, Learning Discrete Mathematics and Computer Science via Primary Historical Sources, http://www.cs.nmsu.edu/historical-projects, accessed on January 4, 2015.
[2] Barnett, G. Bezhanishvili, H. Leung, J. Lodder, D Pengelley and D. Ranjan, Teaching discrete mathematics via primary historical sources, http://www.math.nmsu.edu/hist_projects, accessed on January 4, 2015.
[3] J. Barnett, G. Bezhanishvili, H. Leung, J. Lodder, D. Pengelley, I. Pivkina, D. Ranjan, and M. Zack, Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science, Loci: Convergence (July 2013), DOI 10.4169/loci003984, http://www.maa.org/publications/periodicals/convergence/primary-historical-sources-in-the-classroom-discrete-mathematics-and-computer-science, accessed on January 18, 2015.
[4] O. Borůvka, O jistém problému minimálnim, (On a Certain Minimal Problem), Práce Moravské Přdovědecké Spolecnosti v Brně 3 (1926), 37–58.
[5] ______ , Přispěvek k řěsení oťazky ekonomické stavby elektrovodnich sítí (A Contribution to the Solution of a Problem on the Economical Construction of Power Networks), Elecktronický obzor 15 (1926), 153–154.
[6] R. Calinger, Classics of Mathematics, 2nd ed., Prentice-Hall, Engelwood Cliffs, New Jersey, 1995.
[7] A. Cayley, A Theorem on Trees, Quarterly Journal of Pure and Applied Mathematics 23 (1889), 376–378.
[8] L. Euler, The works of Leonhard Euler online, Available at http://eulerarchive.maa.org, accessed on January 4, 2015.
[9] M. Fried, Can Mathematics Education and History of Mathematics Coexist?, Science & Education 10 (2001), 391–408.
[10] R. L. Graham and P. Hell, On the History of the Minimum Spanning Tree Problem, Annals of the History of Computing 7 (1985), 43–57.
[11] M. de Guzmán, Enseñanza de las ciencias y la mathemática, Revista Iberoamericana de Education 043 (2007), 19–58.
[12] H. N. Jahnke, The use of original sources in the mathematics classroom, History in mathematics education: the ICMI study (Fauvel, J. and van Maanen, J., ed.), Kluwer Academic, Dordrecht, 2002, 291–328.
[13] U. T. Jankvist, The use of original sources and its possible relation to the recruitment problem, Proceedings of the eighth congress for the European society for research in mathematics education (B. Ubuz, Ç. Haser, and M. A. Mariotti, eds.), Middle East Technical University, Ankara, Turkey, 2013, 1900–1999.
[14] _______, On the Use of Primary Sources in the Teaching and Learning of Mathematics, International Handbook of Research in History, Philosophy and Science Teaching, Springer Verlag, New York, 2014, Matthews, M. (editor).
[15] V. Katz, A History of Mathematics: An Introduction, 2nd ed., Addison-Wesley, New York,1998.
[16] T. H. Kjeldsen and M. Blomhøj, Beyond motivation: history as a method for learning meta- discursive rules in mathematics, Educational Studies in Mathematics 10 (23 September 2011), DOI 10.1007/s 10649–011–9352–z.
[17] D. Klyve, L. Stemkowski, and E. Tou, Teaching and Research with Original Sources from the Euler Archive, Loci: Convergence (April 2011), DOI 10.4169/loci003672.
[18] R. Laubenbacher, D. Pengelley, and M. Siddoway, Recovering Motivation in Mathematics: Teaching with Original Sources, UME Trends 6, DOI 10.1007/s 11191–012–9470–8, Available at the website http://www.math.nmsu.edu/~history/ume.html, accessed January 4, 2015.
[19] J. Lodder, Networks and spanning trees, 32 page curricular module available at [1].
[20] J. Narum, Promising Practices in Undergraduate Science, Technology, Engineering, and Mathematics (STEM) Education, Evidence on Promising Practices in Undergraduate STEM Education – Commissioned Workshop Papers, National Academy of Sciences, June 30, 2008, Available at the website http://sites.nationalacademies.org/dbasse/bose/dbasse_080106, accessed January 4, 2015.
[21] H. Prüfer, Neuer Beweis eines Satzes über Permutationen, Archiv der Mathematik und Physik 3 (1918), no. 27, 142–144.
[22] N. Scoville, Georg Cantor at the Dawn of Point-Set Topology, Loci: Convergence (March 2012), DOI: 10.4169/loci003861.
[23] A. Sfard, Thinking as communicating: Human Development, the Growth of Discourse, and Mathematizing, Cambridge University Press, New York, 2008.
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